Evaluating Sensitivities ofBermudan Swaptions
Sebastian Schlenkrich
Christ Church College
University of Oxford
A thesis submitted in partial fulfillment of the MSc in
Mathematical Finance
March 20, 2011
Acknowledgements
I would like to thank my supervisor Prof. Mike Giles for helpful advices and dis-
cussions during the preparation of this thesis. Moreover I would like to express my
gratitute to d-fine GmbH for giving me the opportunity to attend the MSc in Mathe-
matical Finance programme. Furthermore I wish to thank Jan Riehme for supporting
me with ADTAGEO, many fruitful discussions, and helpful manuscript proofreading.
Above all I thank Sybille for her endless patience and support.
Contents
1 Scope and Setting 3
1.1 Notation and general setting . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Reference test problem . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Analytical Pricing Formulas for Swaptions and Bond Options 10
2.1 Black’76 formula for European swaptions . . . . . . . . . . . . . . . . 10
2.2 Analytical Formulas for the Hull White model . . . . . . . . . . . . . 12
2.2.1 Risk-neutral drift . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Bond and bond option formulas . . . . . . . . . . . . . . . . . 14
2.2.3 Specifying the model parameters . . . . . . . . . . . . . . . . 17
2.3 Calibration of the Hull White Model . . . . . . . . . . . . . . . . . . 19
2.3.1 Formulation of the optimization problem . . . . . . . . . . . . 21
2.3.2 Iterative solution of the non-linear problem . . . . . . . . . . . 22
3 Discretisation and Numerical Solution 25
3.1 Spatial Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.1 Discretisation Schemes . . . . . . . . . . . . . . . . . . . . . . 30
3.1.2 Determining the computational domain . . . . . . . . . . . . . 32
3.1.3 Using variable grids . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Sensitivity Evaluation by Automatic Differentiation 41
4.1 Differentiating the European swaption prices . . . . . . . . . . . . . . 45
4.2 Differentiating the Calibration . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Evaluating Derivatives of the PDE solution . . . . . . . . . . . . . . . 48
4.4 Linking the partial derivatives . . . . . . . . . . . . . . . . . . . . . . 51
5 Conclusions 53
1
A Derivative Approximation for Normally Distributed Grids 55
B Fundamentals of Automatic Differentiation 59
B.1 Evaluation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
B.2 Forward mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
B.3 Reverse mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
B.4 General Graph Reduction . . . . . . . . . . . . . . . . . . . . . . . . 66
C Option Pricing via Integration 70
C.1 Risk Neutral Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 71
C.2 Forward Neutral Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 73
C.3 Solving the Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Bibliography 76
2
Chapter 1
Scope and Setting
The Hull White interest rate model is one of the classical interest rate models in
finance. It was proposed in [HW90] as an extension of the Vasicek model. The model
yields analytical formulas for bonds and European bond options. With time inhomo-
geneous model parameters it can be fitted to an observed term structure of interest
rates and a term structure of volatilities. The resulting calibrated model can then
be used to price more exotic interest rate derivatives. Particular financial deriva-
tives priced by the Hull White model are Bermudan bond options and Bermudan
swaptions.
The evaluation of sensitivities in the Hull White model with respect to changes
in the yield curve (i.e. Deltas and Gammas) are discussed, e.g. in [Hen04]. Risk
sensitivities of Bermudan swaptions (also with a focus on changes in the yield curve)
are elaborated in [Pit04].
Key risk factors for Bermudan swaptions are market observed Black’76 volatilities
of European swaptions. Hence the sensitivity of the price with respect to changes
in the volatility is of particular interest. A market standard method for sensitivity
evaluation is bumping the input risk factors, re-evaluate the derivative price and
compute a finite difference approximation of the sensitivity. This approach may not
work appropriately for Bermudan swaption since the pricing involves an iterative
calibration procedure and a numerical solution on a PDE grid (or a tree) which both
introduce numerical errors.
We propose the application of methods of Automatic Differentiation to the pricing
procedure for Bermudan swaptions. This approach yields derivatives of the numerical
scheme within machine precision. It incorporates differentiating the calibration and
the parabolic PDE integration. Our numerical results confirm the applicability and
accuracy of this approach.
3
The thesis is organized as follows: In this chapter we define the general notation
and setting. Moreover we describe a reference test problem and market data used
for the evaluation of the numerical results. Chapter 2 presents analytical formulas
for European swaptions and bond options in the Black’76 and Hull White model.
These formulas are used to define the objective function for the calibration. The
numerical solution of Bermudan bond options on a PDE grid is elaborated in Chapter
3. We study several aspects of the efficient discretisation and parametrization of the
numerical scheme. In Chapter 4 we elaborate the sensitivity evaluation in our Hull
White model. Finally we summarize our conclusions in Chapter 5.
1.1 Notation and general setting
We aim at pricing Bermudan swaptions. A Bermudan swaption gives the option
holder the right to enter an interest rate swap at predefined dates. The underlying
swap is assumed to exchange a fixed simple compounding rate R against a floating
rate Li. Typical floating rate indices are interbank offered rates (Ibor) such as Euribor
or Libor. Forthcoming we use the general term Ibor rate. The subscript i indicates
the dependence of the floating rate on the accrual period.
Fixed leg coupon payment dates are denoted by S1, . . . , SM and S0 is the start
date of the first coupon period. Year fractions associated with the fixed leg coupon
periods are τ1, . . . , τM . Moreover we assume that fixed leg payment dates are a subset
of the floating rate payment dates. In the Euro market fixed payments are usually
annual and the fixed leg day count convention is 30/360.
Floating leg payment dates are given by S1, . . . , SM and S0 is again the start date
of the first coupon period. Corresponding year fractions are η1, . . . , ηM . The fixed
and float leg must belong to the same time period, i.e. S0 = S0 and SM = SM .
The price of a risk free zero coupon bond at observation time t = 0 with maturity
T (t ≤ T ) is given by P (t, T ). The mapping T 7→ P (t, T ) represents the yield curve
at observation time t. The yield curve may be inferred from deposit, forward rate,
and swap rates quoted in the market. It is usually given as an interpolated set of zero
coupon bond prices (i.e. discount factors). We work in a single yield curve setting. For
example, if we price a swaption on a standard swap exchanging fixed against 6-month
Euribor the yield curve must be bootstrapped from quotes depending on the 6-month
Euribor, i.e. standard 6-month Euribor swap quotes. By this single curve setting we
assume that we can fund and invest at rates given by the yield curve. This assumption
is important to bridge the gap from the pricing of swaptions to the pricing of bond
4
options. However in recent market situations we face significant liquidity, basis spread
and credit risks which are not covered by this setting. Although to our knowledge
this setting is market standard for the pricing of Bermudan swaptions it should be
used with care.
Ibor floating rates are reference rates for short term deposits. The discount factor
P (t, T ) inferred from an Ibor rate Lfix fixed for the time period [t, T ] is given by
P (t, T ) =1
1 + Lfix η(t, T ).
Here η(t, T ) denotes the year fraction between observation time t and maturity time
T . Correspondingly, the future Ibor rate Lfixj as observed at Sj−1 for the accrual
period [Sj−1, Sj] becomes
Lfixj =1− P (Sj−1, Sj)
η(Sj−1, Sj) P (Sj−1, Sj).
The corresponding forward Ibor rate Lj(t) as seen at observation time t is defined as
Lj(t) =
P (t, Sj−1)− P (t, Sj)
η(Sj−1, Sj) P (t, Sj)for t ≤ Sj−1
Lfixj for t > Sj−1
Using risk neutral valuation [BM07] we find that
Lj(t) = ESj
[Lfixj | F(t)
].
The expectation ESj is taken in the Sj-forward measure conditional on the information
at observation time t (characterized by the filtration F(t)). As a consequence the time
t (t < Sj−1) price of a cash flow Lfixj η(Sj−1, Sj) paid at Sj becomes
P (t, Sj) ESj
[Lfixj η(Sj−1, Sj) | F(t)
]= P (t, Sj) Lj(t) η(Sj−1, Sj)
= P (t, Sj−1)− P (t, Sj).
The price of the underlying swap at time t ≤ S0 is determined by discounting the
fixed and forward floating leg cash flows. For a (fixed) receiver swap it becomes
Swap(t) = RM∑i=1
τiP (t, Si)︸ ︷︷ ︸FixedLeg(t)
−M∑j=1
Lj(t)ηjP (t, Sj)︸ ︷︷ ︸FloatLeg(t)
.
5
We require that the swaps float leg day count convention coincides with the day
count convention of the underlying Ibor rate. That is ηj = η(Sj−1, Sj). Moreover,
the accrual periods of the Ibor rates Lfixj must start and end at the payment dates
exactly. Since we are working in our single curve environment the time t price of the
floating leg can be simplified to
FloatLeg(t) =M∑j=1
P (t, Sj−1)− P (t, Sj)
= P (t, S0)− P (t, SM)
= P (t, S0)− P (t, SM).
Thus we obtain the swap pricing formula
Swap(t) = R
M∑i=1
τiP (t, Si) − [P (t, S0)− P (t, SM)]
which only depends on the fixed rate R, the fixed leg schedule S0, . . . , SM , and the
yield curve T 7→ P (t, T ). Rearranging terms yields that the swap can also be inter-
preted as the time t price of a risk free (forward) bond contract with unit bond price
paid at S0, fixed coupons Rτi paid at Si for i = 1, . . . ,M and unit notional payment
at SM . That is
Bond(t) = −P (t, S0)︸ ︷︷ ︸bond price
+M∑i=1
RτiP (t, Si) + P (t, SM)︸ ︷︷ ︸coupons and notional
and
Swap(t) = Bond(t).
In practice swaps are usually priced incorporating the basis spread between differ-
ent forward Ibor indices, e.g. 6-month Euribor against 3-month Euribor. This means
that the forward Ibor rates Lj(t) may be evaluated from a yield curve different to the
discount curve given by the discount factors P (t, T ). Additionally, rolling and busi-
ness day conventions of the swap may yield payment dates Sj that have a small lag
(of one or two days) to the end dates of the accrual periods of the floating rates Lfixj .
In such situations the above equivalence between swaps and (risk free) bonds only
holds approximately. Nevertheless, for the pricing of swaptions as bond options (or
vice versa) the equivalence between swaps and bonds is usually assumed in practice.
A swaption gives its holder the right to enter a swap at a given strike rate R.
The swaption is considered to be of European style if the right may be exercised
6
at a single predefined date. Bermudan swaptions bare the right to enter a fixed
maturity swap at several predefined exercise dates. At an exercise date T with Si0 =
mini=1,...,M {Si|Si ≥ T} the payoff of the swaption is given by
Swaption(T ) = [ω Swap(T )]+
=
[ω
{R
M∑i=i0+1
τiP (T, Si)− [P (T, Si0)− P (T, SM)]
}]+
.
Here ω ∈ {−1,+1} distinguishes between a payer (−1) and receiver (+1) swaption
and [ · ]+ abbreviates max{ · , 0}.Exploiting the equivalence between swaps and bonds yields that the swaption may
be interpreted as a coupon bond option with payoff
CBO(T ) =
ωR
M∑i=i0+1
τiP (T, Si)︸ ︷︷ ︸coupons
+P (T, SM)︸ ︷︷ ︸notional
− P (T, Si0)︸ ︷︷ ︸strike
+
.
The underlying bond coupons equal the fixed leg payments of the swap. At maturity
the unit notional is paid and the option strike equals the unit notional. In order
to obtain analytical formulas for bond options in the Hull White model we have to
restrict the exercise dates to the settlement dates. That is we require that T = Si0 .
In cases where the option is not settled at the exercise date, i.e. the strike is not paid
at T , a corresponding approximated time-T strike could be determined by (forward)
discounting from T to Si0 . However, this approximation only holds, if the settlement
offset is small.
1.2 Reference test problem
We use a standard setting for a test problem to verify the applicability of the methods
elaborated in this thesis. If not stated otherwise we consider the pricing of a EUR
fixed maturity Bermudan Swaption. The underlying swap receives 4% fixed on an
annual 30/360 day count basis against an Ibor floating rate. The swap matures in
31 years. It may be entered annually with the first exercise date being in one year.
Hence we have 30 exercise dates in 1Y, 2Y, to 30Y. We use reference market data as
of February 2010. The yield curve and swaption volatility surface are illustrated in
Figure 1.1 and Figure 1.2.
7
Figure 1.1: EUR yield curve for reference test problem.
time to maturity in years
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 5 10 15 20 25 30 35
zero
rate
(cont.
com
p.,
act/
365
)
0
Figure 1.2: Implicit EUR swaption volatility surface for a strike of 4%.
15
20
25
510
1520
2530
10
Black’76
15
20
11121314151617181920
swap tenor in years
time to excercise in years
5
10
volatilities
8
The Hull White model is calibrated to replicate the prices of reference European
swaptions. In our test setting the 30 reference European swaptions correspond to each
of the Bermudan exercise dates. Denote Swaptioni the reference European swaption
in i years (i = 1, . . . , 30). Its payoff is equivalent to a coupon bond call option CBOi
with 31 − i coupon payments in i + 1 to 31 years plus a unit notional payment in
31 years. The reference European swaptions are particularly used to estimate the
accuracy of the numerical PDE methods.
9
Chapter 2
Analytical Pricing Formulas forSwaptions and Bond Options
In this chapter we elaborate analytical pricing formulas for swaptions and bond op-
tions. The formulas are used to calibrate the Hull White model to market observable
data. Moreover we use the analytical pricing formulas of the Hull White model to
verify the numerical methods.
2.1 Black’76 formula for European swaptions
European swaptions may be valued in the Black’76 framework. For a derivation of
the formulas see for example [BM07]. The payoff of the swaption is rewritten as
Swaption(T ) =
(M∑
i=i0+1
τiP (T, Si)
)[ω
(R− P (T, Si0)− P (T, SM)∑M
i=i0+1 τiP (T, Si)
)]+
.
In this representation the annuity is given by
Annuity(T ) =M∑
i=i0+1
τiP (T, Si)
and the (forward) par swap rate is denoted by
Y (T ) =P (T, Si0)− P (T, SM)∑M
i=i0+1 τiP (T, Si).
With this notation the swaption payoff becomes
Swaption(T ) = Annuity(T ) [−ω (Y (T )−R)]+ .
Thus an European receiver (payer) swaption is equivalent to an European put (call)
on the forward par swap rate Y (T ) with strike R.
10
The Black’76 model assumes that conditional on the information available at
observation time t the forward par swap rate Y (T ) is log-normally distributed with
mean Y (t) and variance σ2B76δ(t, T ). The function δ( · , · ) denotes the year fraction
function applied to scale the Black’76 volatility σB76. The resulting price of the
European swaption at observation time t becomes
Swaption(t) = Annuity(t) · Black76(Y (t), R, σB76, δ(t, T ),−ω). (2.1)
The Black’76 formula for European puts (ω = −1) and calls (ω = +1) with forward
price F , strike K, volatility σ, and time to maturity τ is
Black76(F,K, σ, τ, ω) = ω [FΦ(ωd1)−KΦ(ωd2)] ,
d1,2 =log (F/K)
σ√τ
± σ√τ
2.
Market prices of European swaptions are quoted in terms of implicit Black’76
volatilities. These quotes are given for several times to maturity δ(t, T ) and remaining
swap tenors δ(Si0 , SM). Moreover the prices and implicit volatilities depend on the
moneyness of the swaption strike rate R. The moneyness for swaptions is measured by
the absolute difference R−Y (t). As a result a swaption volatility cube is spanned by
the time to exercise, the remaining swap tenor, and the moneyness. To evaluate prices
of specific swaptions for which no direct volatility quote is available interpolation
schemes are used.
Considering the smile. For the calibration of the Hull White model we require
swaption volatilities for several exercise dates and swap tenors. The fixed rates of
the swaptions in question are equal to the corresponding fixed rate of the Bermudan
swaption characterizing the calibration problem. Since the forward par swap rates
vary for the European swaptions also the moneynesses of the European swaptions
differ. Therefore we have to interpolate the smiles of the swaption volatilities.
As reported for example in [BM07] it is market practice to interpolate volatility
smiles by the approximated formulas of the SABR model [HKLW02]. We calibrate the
SABR model separately to available market quotes of swaption volatilities for a given
pair of exercise date and swap tenor. Using the resulting grid of SABR parameters
we build a volatility surface for the required strike R. Then the resulting volatility
surface is interpolated linearly and extrapolated constantly. A graph of the SABR
swaption volatility surface used for the numerical tests is given in Figure 1.2.
The modelling of the volatility smile is crucial for the pricing of swaptions. Both
the Black’76 and the Hull White model do not model the smile. In the Black’76 model
11
we use implicit volatilities which are determined from market prices of European
swaptions inverting the same Black’76 model before. The dependence of the Hull
White option prices to the strike is only taken into account by the calibration. Given
prices of reference European swaptions incorporate the smile. If the Hull White model
is calibrated to these European swaption it may replicate the smile of these specific
instruments. As the smile modelling is such an important issue it is subject to a
wide range of research activities. For further details see, for example [RMW09] and
references therein.
In our setting we assume the required Black’76 volatility surface given for the
strike rate in question. As mentioned before this surface may be evaluated using a
SABR model. In a simplified setting one could also neglect the smile and use the
quoted at-the-money swaption volatility surface.
2.2 Analytical Formulas for the Hull White model
The Hull White model [HW90] specifies a stochastic process for the short rate r(t).
The model is given by
dr(t) = [θ(t)− ar(t)] dt+ σ(t)dW (t).
Here θ(t) denotes the risk neutral drift, a the constant mean reversion parameter,
and σ(t) the volatility of the short rate. W (t) is a Brownian motion under the risk
neutral measure with the bank account as numeraire.
2.2.1 Risk-neutral drift
The stochastic differential equation of the short rate may be solved for r. We find
that for t > T
r(t) = e−a(t−T )
[r(T ) +
∫ t
T
ea(u−T ) (θ(u)du+ σ(u)dW (u))
].
Given the information at time T the price of a zero coupon bond with maturity S > T
becomes
ZCB(T, S) = EQ
e−∫ S
T
r(t)dt| F(T )
.It follows that∫ S
T
r(t)dt =
∫ S
T
e−a(t−T )
[r(T ) +
∫ t
T
ea(u−T ) (θ(u)du+ σ(u)dW (u))
]dt. (2.2)
12
The integral may be decomposed as B(T, S)r(T ) +X(T, S) + Y (T, S) with
B(T, S) =
∫ S
T
e−a(t−T )dt =1− e−a(S−T )
a,
X(T, S) =
∫ S
T
e−a(t−T )
∫ t
T
ea(u−T )θ(u)du dt =
∫ S
T
θ(u)B(u, S)du,
Y (T, S) =
∫ S
T
e−a(t−T )
∫ t
T
ea(u−T )σ(u)dW (u) dt =
∫ S
T
σ(u)B(u, S)dW (u).
The term for X(T, S) may be derived by changing the order of integration∫ S
T
∫ t
T
e−a(t−T )ea(u−T )θ(u) du dt =
∫ S
T
∫ S
u
e−a(t−T )ea(u−T )θ(u) dt du
=
∫ S
T
θ(u)
(∫ S
u
e−a(t−u) dt
)du
=
∫ S
T
θ(u)B(u, S) du.
Y (T, S) follows analogously. Moreover, we have for B(T, S) that B(S, S) = 0,
BS(T, S) = e−a(S−T ), BS(S, S) = 1, and BSS = −aBS(T, S).
The only stochastic term in (2.2) is Y (T, S). Since Y (T, S) is an Ito integral with
deterministic integrand we find that
Y (T, S) ∼ N(
0,
∫ S
T
σ2(u)B2(u, S)du
).
For reference see, for example [Shr04, Th. 4.4.9]. Thus, given the information at time
T , we find that e−∫ ST r(t)dt is log-normally distributed and
ZCB(T, S) = exp
{−(B(T, S)r(T ) +
∫ S
T
θ(u)B(u, S)du− 1
2
∫ S
T
σ2(u)B2(u, S)du
)}.
The Hull White model should reproduce the initial market observed yield curve.
Therefore we must have that at observation time t the formula for the zero coupon
bond prices ZCB(t, S) equals the discount factors P (t, S) for all S ≥ t. For that
purpose we define the continuously compounded forward rates f(t, S) as
f(t, S) = −∂ log(P (t, S)
∂S.
We assume that the yield curve is modelled such that f(t, S) is continuous in S and
at least piecewise continuous differentiable. Then we can write
f(t, S) = r(t) +
∫ S
t
fS(t, u)du.
13
and
P (t, S) = e−∫ St f(t,v)dv = e−
∫ St (r(t)+
∫ vt fS(t,u)du)dv.
From P (t, S) = ZCB(t, S) and the properties of B(t, S) follows that
f(t, S) = BS(t, S)r(t) +
∫ S
t
θ(u)BS(u, S)du−∫ S
t
σ2(u)B(u, S)BS(u, S)du,
fS(t, S) = θ(S)− af(t, S)−∫ S
t
σ2(u)B2S(u, S)du.
This yields the risk neutral drift of the Hull White model as
θ(S) = fS(t, S) + af(t, S) +
∫ S
t
σ2(u)B2S(u, S)du
A Hull White model with risk neutral drift is arbitrage free in an economy with all
zero coupon bonds as tradeable assets.
2.2.2 Bond and bond option formulas
We use the Hull White model with risk neutral drift calibrated to the observed yield
curve to determine prices of bonds and bond options in the Hull White model. The
formulas are applied to calibrate the short rate volatility as described in the forth-
coming section.
Zero coupon bonds. We derive an expression of the zero coupon bond ZCB(T, S)
given the information about the short rate r(T ) at time T in a model that replicates
the yield curve at observation time t ≤ T . For that purpose we consider again the
prices of the discount factors P (t, T ) and P (t, S) in the Hull White model. We have
P (t, S) = e−(B(t,S)r(t)+∫ St θ(u)B(u,S)du− 1
2
∫ St σ2(u)B2(u,S)du).
For the price of the discount factor P (t, T ) we substitute the equivalence B(u, T ) =
B(u, S)−BS(u, T )B(T, S) and get
P (t, T ) = e−(B(t,T )r(t)+∫ Tt θ(u)[B(u,S)−BS(u,T )B(T,S)]du− 1
2
∫ Tt σ2(u)[B(u,S)−BS(u,T )B(T,S)]2du).
14
It follows that
log
(P (t, T )
P (t, S)
)= [B(t, S)−B(t, T )] r(t) +
∫ S
T
θ(u)B(u, S)du
+B(T, S)
∫ T
t
θ(u)BS(u, S)du− 1
2
∫ S
T
σ2(u)B2(u, S)du
−B(T, S)
∫ T
t
σ2(u)B(u, S)BS(u, T )du
+1
2B2(T, S)
∫ T
t
σ2(u)B2S(u, T )du.
Rearranging terms and further substitutions yield
log
(P (t, T )
P (t, S)
)=
∫ S
T
θ(u)B(u, S)du− 1
2
∫ S
T
σ2(u)B2(u, S)du+B(T, S)f(t, T )
−B(T, S)
∫ T
t
σ2(u) [B(u, S)−B(u, T )]BS(u, T )du
+1
2B2(T, S)
∫ T
t
σ2(u)B2S(u, T )du
=
∫ S
T
θ(u)B(u, S)du− 1
2
∫ S
T
σ2(u)B2(u, S)du+B(T, S)f(t, T )
− 1
2B2(T, S)
∫ T
t
σ2(u)B2S(u, T )du.
We end up with ∫ S
T
θ(u)B(u, S)du− 1
2
∫ S
T
σ2(u)B2(u, S)du =
log
(P (t, T )
P (t, S)
)−B(T, S)f(t, T ) +
1
2B2(T, S)
∫ T
t
σ2(u)B2S(u, T )du
Given that the Hull White model is calibrated to data at time t the price of a zero
coupon bond at time T , with maturity S, and realized short rate r becomes
ZCB(t;T, S, r) = A(t;T, S) e−B(T,S) r
with
A(t;T, S) =P (t, S)
P (t, T )exp
{B(T, S)f(t, T )− B(T, S)2
2
∫ T
t
σ2(u)B2S(u, T )du,
}
15
Zero coupon bond options. Consequently, we can derive the price of a zero
coupon bond option at observation time t. We have from (2.2) that for T > t
V ar [r(T ) | F(t)] =
∫ T
t
σ2(u)B2S(u, T )du.
Hence the time t price of a zero coupon bond is log-normally distributed with variance
σ2P = B2(T, S)
∫ T
t
σ2(u)B2S(u, T ).
Moreover, we have in the T -forward measure that
ET [ZCB(t;T, S, r(T )) | F(t)] =ZCB(t; t, S, r(t))
P (t, T )=
P (t, S)
P (t, T ).
As a result we find that in the T -forward measure the price of a zero coupon bond
is log-normally distributed with mean P (t, S)/P (t, T ) and variance σP . Hence for a
call or put option on a zero coupon bond we can use the Black’76 formula. The time
t price of an option on a zero coupon bond with exercise date T , bond maturity S,
and strike price K paid at T is given by
ZCO(t;T, S,K, ω) = P (t, T ) Black76(P (t, S)/P (t, T ), K, σP , 1, ω).
Coupon bonds and coupon bond options. A coupon bond with cash flows ci
at coupon payment dates Si is determined by the sum of the scaled zero coupon bond
prices, i.e.
CB(t;T, S1, . . . , SM , r) =∑Si≥T
ci ZCB(t;T, Si, r).
An option on a coupon bond with cash flows ci at coupon payment dates Si, exercise
date T , and strike price K may be valued using Jamshidian’s decomposition [Jam89].
This approach requires to solve the equation
CB(t;T, S1, . . . , SM , r?) = K
for the short rate r?. Using the resulting short rate r? we can evaluate corresponding
individual strikes Ki by
Ki = ZCB(t;T, Si, r?).
With these individual strikes the coupon bond option can be prices as a sum of zero
coupon bond options. We have that
CBO(t, T, S1, . . . , SM , ω) =∑Si≥T
ci ZCO(t, T, Si, Ki, ω). (2.3)
16
2.2.3 Specifying the model parameters
So far we did not specify how we model the functional parameters in the Hull White
model. The derived formulas in the Hull White model require the current yield curve
and the short rate volatility function.
Piecewise constant volatility. It is common to assume that the volatility is piece-
wise constant between two exercise dates of a Bermudan swaption. Let T0 = t and
denote the Bermudan exercise dates with T1, . . . , TN then we have that
σ(t) = σj for t ∈ (Tj−1, Tj], j = 1, . . . , N.
In this setting we can evaluate C(T, S) =∫ Ttσ2(u)B2
S(u, T )du. Additionally, we
incorporate the year fraction function δ(·, ·) which measures the time in years in the
model between two dates. Usually the year fraction is determined on an act/365 day
count basis. The zero coupon bond at an exercise date Tj becomes
ZCB(t;Tj, S, r) = A(t;Tj, S) e−B(Tj ,S)r
with
A(t;Tj, S) =P (t, S)
P (t, Tj)exp
{B(Tj, S)f(t, Tj)−
B(Tj, S)2
2C(T, S)
},
B(Tj, S) =1− e−aδ(Tj ,S)
a,
C(Tj, S) =
j∑k=1
σ2k
2a
[e−2aδ(Tk,S) − e−2aδ(Tk−1,S)
].
Moreover, the variance of the bond price can be rewritten as
σ2P = B2(Tj, S)C(Tj, S).
For the calibration procedure it is important to note that in this setting prices of
a zero and coupon bonds as well as zero and coupon bond options with exercise date
Tj depends only on short rate volatilities σ1 to σj. The price is independent from
volatilities corresponding to times larger than Tj.
Continuous forward rates A key feature of the implementation is the specifi-
cation of the initial yield curve. As pointed out in Section 1.1 the yield curve is
determined in terms of a set of discount factors 1 = P (t, T0), . . . , P (t, TN) for given
maturities t = T0 to TN . The grid of discount factors is in general independent
17
(and different) from the grid of Bermudan exercise and cash flow dates. Between the
dates of the yield curve grid we have to interpolate the discount factors. The type of
interpolation specifies the functional relation of the yield curve.
The calibration of the Hull White model to the initial yield curve is based on
the assumption that discount factors P (t, S) can be represented in terms of first
derivatives fS(t, S) of the forward rates, i.e.
P (t, S) = exp
{−∫ S
t
(r(t) +
∫ v
t
fS(t, u)du
)dv
}.
This assumption is equivalent to the requirement that the forward rates
f(t, S) = −∂ log(P (t, S))
∂S
are continuous and (at least) piecewise continuous differentiable. Such a strong
smoothness condition is not obvious because the zero coupon bond option prices
depend only on discount factors and bond prices depend only on forward rates. How-
ever, this condition becomes observable if one compares analytic and numeric prices
in the Hull White model.
For example, log-linear interpolation of discount factors is often used to model the
initial yield curve. Unfortunately such a model implies piecewise constant forward
rates which are in general not continuous. A similar situation occurs if zero rates are
interpolated linearly to obtain discount factors. The resulting forward rates are not
continuous. Some results related to that issue are elaborated in [HW06].
In our setting we model the initial yield curve by cubic C2-spline interpolation
applied to the set of points
{(T0, logP (t, T0)) , . . . , (TN , logP (t, TN))} .
Cubic C2-splines are twice continuous differentiable and thus the forward rates are
continuous differentiable. The drawback of cubic C2-spline interpolation is, however,
that it does not necessarily preserve monotonicity. This could in principle yield nega-
tive forward rates. In such situations a monotonicity preserving c-spline interpolation
can be used instead.
We study the practical influence of the yield curve interpolation to our Hull White
PDE model by a simple test problem. For that purpose we use a yield curve given
by the following three discount factors
{(0, 1), (1Y, 0.988003933), (2Y, 0.969893541)} , (2.4)
18
a constant short rate volatility of 5% and mean reversion of 5%. The price of a Euro-
pean zero coupon bond option with exercise in 1Y and maturity in 2Y is evaluated by
the analytical formula and the PDE solver of the Hull White model. We compare the
resulting prices using log-linear and log-cubic C2-spline interpolation of the discount
factors in the upper part (2.4) of Table 2.1. The computations are repeated for an
alternative yield curve given by
{(0, 1), (6M, 0.995578534), (1Y, 0.988003933), (2Y, 0.969893541)} (2.5)
which includes an additional 6M discount factor. The corresponding numerical results
are given in the lower part (2.5) of Table 2.1.
Table 2.1: Approximation of the numerical solution for log-linear and log-cubic inter-polation.
Yield Curve Interpolation Analytic Price Numerical Price Relative Error(2.4) log-linear 1.091061e-02 1.091060e-02 6.1e-07
log-cubic 1.091061e-02 1.091059e-02 2.0e-06(2.5) log-linear 1.091061e-02 1.314207e-02 2.0e-01
log-cubic 1.091061e-02 1.091052e-02 8.2e-06
As expected the analytical price of the zero coupon bond option is independent
of the yield curve and interpolation method applied. This is because the required
discount factors at 1Y and 2Y are given exactly and do not need to be interpolated.
For the three point yield curve we find accurate approximations of the numerical
scheme for both log-linear and log-cubic interpolation. In this situation log-linear
interpolation implies a constant and continuous forward rate curve between 0 and
1Y.
If we include an additional 6M discount factor log-linear interpolation implies a
piecewise constant forward rate curve which is (in general) not continuous. Thus it
can not be modelled by the Hull White model. As a result we see that the numerical
PDE solution of the zero coupon bond option deteriorates. In contrast to the log-
linear discount factor interpolation log-cubic discount factor interpolation yields an
accurate approximation of the numerical solution.
2.3 Calibration of the Hull White Model
In Section 1.1 we illustrate that a forward receiver swap contract is equivalent to a
contract to buy a coupon bond. Consequently, a European receiver swaption with
19
strike rate R and fixed leg year fractions τ1, . . . , τM is equivalent to a call option on
a coupon bond with coupons Rτi and unit strike price.
Calibrating the Hull White model means choosing the model parameters such that
the model prices for European coupon bond options given by Equation (2.3) coincide
in a well defined way with market prices of European swaptions determined from
quoted Black’76 swaption volatilities and Equation (2.1). In our setting the model
parameters are the piecewise constant Hull White volatility values σj and the mean
reversion parameter a.
In a test case we analyse the calibration of the mean reversion a and a constant
short rate volatility σ to 10 EUR denominated at-the-money European swaptions
with exercises ranging from 10 to 19 years and fixed maturity in 20 years. We use
market data as of September 2009. The resulting objective function aimed to be
minimized and the convergence history of an optimization run with a constrained
Newton’s method are illustrated in Figure 2.1.
Figure 2.1: Convergence history for the simultaneous calibration of volatility andmean reversion.
Zoom to the last seven iteration steps
Convergence history
f(x) =∑10
i=1
(PV HW
i − PV B76i
)2
mean reversion short rate volatility
0.0001
0.10.08
0.060.04
0.02
6e-05
00.02
0.040.06
0.080.1
0
0
0.1
0.2
0.3
2e-05
The numerical example demonstrates that there is only very limited progress w.r.t.
the objective function in the last iterations. Moreover there is a steep descent towards
a convergence valley. This observation shows that calibrating the mean reversion to
European swaptions is a rather ill-posed problem. As a result we assume the mean
reversion parameter a to be predefined. However, Bermudan prices do depend on the
choice of the mean reversion. Hence, if market prices for Bermudan swaptions are
available then the mean reversion parameter can be chosen to replicate these prices.
This coincides with procedures proposed for example in [Hag].
20
2.3.1 Formulation of the optimization problem
For the calibration of the Hull White model we specify a mean reversion parameter a.
Then we consider the difference between the Hull White model price and the market
price of a set of European swaptions. For each exercise date j = 1, . . . , N of the
underlying Bermudan swaption a European swaption with swap maturity equal to
the Bermudan swap maturity is chosen. That is, in terms of the Equations (2.3) and
(2.1) we consider the differences
CBO(t, Tj, S1, . . . , SM , ω)− Swaptionj(t) for j = 1, . . . , N.
Since we use a piecewise constant short rate volatility we may write the Hull White
model price in terms of the relevant short rate volatility values. Thus we have
CBO(σ1, . . . , σj; t, . . .)− Swaptionj(t) for j = 1, . . . , N.
The resulting objective function is formulated by reordering independent vari-
ables (i.e. short rate volatilities) and dependent variables (i.e. residuals of reference
swaption prices). We define
x = (x1, . . . , xN)> = (σN , . . . , σ1)>,
gi(x) = CBO(x;TN−i, S1, . . . , SM , ω)− Swaption(t) for i = 0, . . . , N − 1,
and F : RN → RN , F (x) = (g1(x), . . . , gN(x))> .
The reordering of the volatilities and the reference instruments has no influence on
the modelling of the calibration problem. However it yields, that each individual
function gi(x) only depends non-trivially on elements xi to xN of x. This gives the
advantageous property that the Jacobian of F is an upper triangular matrix.
In principal one can aim at solving the non-linear system of equations F (x) = 0.
Since the Jacobian of F has an upper triangular form this problem could be split into
a sequence of one dimensional non-linear problems. One could start to solve for gN ,
then gN−1 up to g1. Unfortunately this approach fails if at least one of the market
prices could not be replicated by the model exactly. However we want our calibration
to be robust enough to cope with such market situations.
Instead of solving the set of non-linear equations F (x) = 0 directly we consider
the non-linear least squares formulation
minx
1
2F (x)>F (x).
21
Moreover we want to require the volatilities to lie within some reasonable bounds. In
particular, we want to ensure that the volatilities are positive. Therefore we consider
some scalar boundaries a and b and the component wise box constraint a ≤ x ≤ b.
Hence we aim at solving
mina≤x≤b
1
2F (x)>F (x). (2.6)
The formulation as non-linear least squares problem also allows us in principle to
choose a coarser time discretisation of the short rate volatilities. In market situations
with few liquid volatility quotes this yields more stable calibration results.
2.3.2 Iterative solution of the non-linear problem
The non-linear least squares problem in (2.6) may be solved iteratively by a Gauss-
Newton method. For further reference see for example [NW06]. In each Gauss Newton
iteration step with given iterate x a step direction s is evaluated by solving the linear
system
F ′(x)>F ′(x)s = −F ′(x)>F (x).
In general the linear system is solved by computing a QR factorization of the Jacobian
F ′(x). This requires O(N3) operations. Since the Jacobian F ′(x) of the Hull White
calibration problem is already an upper triangular matrix its QR factorization is
trivial. Thus the linear system may be solved within O(N2) operations.
The Gauss Newton iteration is stabilized by a line search in the range of the
objective function F . For given function values F (x) and F (x + s) a step multiplier
λ ∈ R is determined by
λ =F (x)> [F (x)− F (x+ s)]
[F (x)− F (x+ s)]> [F (x)− F (x+ s)].
The state x + s is accepted as the next iterate if λ ≥ 12
+ ε for a suitable choice of
ε ∈ (0, 1/6). If x+ s is not accepted the Gauss Newton step s is replaced by λ · s with
λ =
λ, for ε < |λ| < 1/2 + ε
ε, for 0 ≤ λ ≤ ε
−ε, for − ε ≤ λ < 0
−(1/2 + ε) for λ < −(1/2 + ε)
.
The line search procedure is repeated with the function values F (x) and F (x+ λ · s)until acceptance. Further references and convergence results concerning this line
search approach can be found in [Gri86].
22
If we find that a Gauss-Newton step x = x+ s would result in an iterate violating
the box constraints a ≤ x ≤ b then the components of x are projected to the bounds.
That is, for x = (x1, . . . , xN)> we set x = (x1, . . . , xN)> with
xi =
a, for xi < a
x, for a ≤ xi ≤ b
b, for xi > b
, i = 1, . . . , N.
In such a case we get a new step direction s = x−x. The line search is then performed
along the altered step direction s.
The Gauss Newton iteration may be summarized as follows: Starting with an
initial guess x(0) we evaluate for i = 0, 1, 2, . . .
1. solve F ′(x(i))>F ′(x(i))s(i) = −F ′(x(i))>F (x(i)),
2. determine a suitable step multiplier λ(i) ∈ R (by considering the box con-
straints),
3. evaluate the next iterate x(i+1) = x(i) + λ(i) · s(i).
Numerical results The Hull White model is parametrized with a mean reversion
of 5%. The upper bound of the short rate volatility is given by 50%. As lower bound of
the short rate volatility we choose 0.1%. The iterations are initialized with a constant
volatility of 10%. Figure 2.2 illustrates the convergence history of the Gauss Newton
method applied to our reference test problem. The resulting piecewise constant short
rate volatility function is shown in Figure 2.3. This short rate volatility function is
applied in the PDE methods described in the forthcoming chapters.
23
Figure 2.2: Convergence history for the calibration of the short rate volatility byGauss Newton method.
iteration count i
residual ‖F (x(i))‖2
5 61e-18
1e-06
2
1e-12
1
0 1 3 4
Figure 2.3: Short rate volatility calibrated to coterminal swaptions.
time in year fraction15 25 300 5 10
1.6%
1.4%
1.2%
1.0%
0.8%
0.6%
20
24
Chapter 3
Discretisation and NumericalSolution
In Section 2.2 we elaborate (auxiliary) analytical pricing formulas for European bond
options in the Hull White model. The aim of this thesis is the pricing and sensitivity
evaluation of Bermudan swaptions which are replicated as Bermudan bond options.
However there are no analytical pricing formulas for Bermudan bond options in the
Hull White model. Hence we have to apply numerical schemes for the pricing of
Bermudan bond options.
The pricing of exotic derivatives (in particular Bermudan bond options) in the
Hull White model may be realized by discretising the stochastic process of the short
rate r(t) in a recombining tree. For details see for example [BM07]. Under certain
constraints the trinomial tree method is equivalent to an explicit Euler finite differ-
ence scheme ([Duf06]). The explicit Euler scheme has limited stability properties
and may yield artificial oscillations in the numerical solution. Therefore we use the
alternative approach of formulating and solving the Hull White partial differential
equation (PDE).
In this chapter we describe the numerical methods for the pricing of Bermudan
swaptions in the Hull White model. The methods elaborated could easily be adapted
to other interest rate payoffs. The time t price V (t, r(t)) of a contingent claim with
payoff V (T, r(T )) = p(r(T )) (T > t) is given by
V (t, r(t)) = EQ[e∫ Tt r(s)dsp(r(T )) | F(t)
].
The process of the short rate r(t) in the Hull White model is
dr(t) = [θ(t)− ar(t)] dt+ σ(t)dW (t)
25
with W (t) a Brownian motion under the risk neutral measure Q. Applying Feynman-
Kac’s theorem (see for example [Shr04]) yields that the price V (t, r) for t < T satisfies
the partial differential equation
Vt +σ2(t)
2Vrr + [θ(t)− ar]Vr − rV = 0
with terminal condition V (T, r) = p(r).(3.1)
The payoff structure of a Bermudan option may be considered as a sequence of
terminal conditions. Let T1 to TN be the Bermudan exercise dates. If TN−1 < t < TN
and the option is not exercised then the time t price of the Bermudan option is
equivalent to a European option with exercise date TN . Denote pj(r) the payoff of
the Bermudan option at exercise date Tj. Then the price of the Bermudan at Tj is
V (Tj, r) = max
{pj(r), lim
t↓TjV (t, r)
}.
between the exercise dates and before the first exercise date the Hull White PDE
holds. Hence we may solve the Bermudan option by successively integrating back-
wards from TN to observation time t = 0.
Figure 3.1 illustrates the numerical solution V (t, r) of the Bermudan swaption
reference test problem. Moreover, we mark the solution V (t, f(0, t)) along the forward
rate observed at observation time t = 0. The desired time t = 0 price of the Bermudan
swaption becomes V (0, f(0, 0)).
In the forthcoming sections we describe the spatial and temporal discretisation of
the terminal value problem (3.1). We discuss different approaches for the numerical
solution and address the issues of approximating the boundary conditions.
3.1 Spatial Discretisation
For the discretisation of the Hull White terminal value problem we use the method of
lines [GR94]. That is we first discretise the spatial direction which is the short rate r.
This yields a system of ordinary differential equations with a terminal condition. The
system of ordinary differential equations is subsequently solved in a general setting.
The terminal value problem (3.1) is rewritten using the linear operator L as
Vt(t, r) = −L[V ](t, r) with L[V ] =σ2(t)
2Vrr + [θ(t)− ar]Vr − rV.
For the spatial discretisation we define a grid of n+ 1 short rate values
r0 < r1 < . . . < rn.
26
Figure 3.1: Numerical solution of the Bermudan swaption reference test problem.
25
35 0
0.02 0.04
0.06 0.08
0.1
0
0.05
0.1
0 5
10 15
20 25
30 35
0 0.02
0.04 0.06
0.08 0.1
0
0.05
0.1
0.15
0.2
0.15
0.2
10 15
0 5
20
30
Short Rate r
Numerical Solution V (t, r) of Bermudan Swaption
Time to Maturity t
Time to Maturity tShort Rate r
Forward Rate Solution V (t, f(t))
27
The discretisation width is given by
δ = maxi=1,...,n
{ri − ri−1} .
Each short rate grid point ri is associated with a component function V δi (t) ≈ V (t, ri)
for i = 0, . . . , n. The component functions V δi are combined in the vector valued
function V δ, i.e.
V δ(t) =(V δ
0 (t), . . . , V δn (t)
)>.
The partial derivatives at the interior short rate grid points r1, . . . , rn−1 are ap-
proximated by centred finite differences which are defined as
V δr,i(t) =
V δi+1(t)− V δ
i−1(t)
ri+1 − ri−1
,
V δrr,i(t) = 2
[V δi+1(t)− V δ
i (t)]
(ri − ri−1)−[(V δ
i (t)− V δi−1(t)
](ri+1 − ri)
(ri − ri−1)(ri+1 − ri−1)(ri+1 − ri).
If the solution V of the Hull White terminal value problem is sufficiently smooth (4th
derivatives must be continuous) and the spatial discretisation is exact, i.e. Vi(t) =
V (ri, t) for i = 0, . . . , n then we get from Taylor expansion that
V δr,i(t) =
∂V
∂r(ri, t) +
1
2
∂2V
∂r2(ri, t) [(ri+1 − ri)− (ri − ri−1)] +O(δ2), (3.2)
V δrr,i(t) =
∂2V
∂r2(ri, t) +
1
3
∂3V
∂r3(ri, t) [(ri+1 − ri)− (ri − ri−1)] +O(δ2). (3.3)
Hence for general grids the approximation is first order accurate and for equidistant
grids with δ = ri+1− ri = ri− ri−1 for i = 1, . . . , n− 1 the approximation is of second
order. If there is a smooth transformation from a uniform grid to the general short
rate grid we may also get a second order approximation. For further details see, e.g.
[HV03].
The Hull White terminal value problem (3.1) is defined for r ∈ R. For the numer-
ical treatment we have to truncate the domain for the short rate r. At the boundaries
some reasonable conditions have to be imposed. As there are no natural choices for
the boundary conditions it is market standard to use linear boundary conditions.
That is, we impose that
∂2V
∂r2(t, r0) =
∂2V
∂r2(t, rn) = 0.
The stability of the linear boundary condition in the related context of solving the
Black Scholes PDE is analysed, for example in [WFV04]. With this condition the
28
numerical approximation of the second derivative V δrr also vanishes for r0 and rn.
Using some ghost grid points at the boundaries and the linear boundary condition
yield that the numerical approximation of the first derivative V δr become a one-sided
finite differences
V δr,0(t) =
V δ1 (t)− V δ
0 (t)
r1 − r0
, V δr,n(t) =
V δn (t)− V δ
n−1(t)
rn − rn−1
.
The spatial discretisation of the partial derivatives of V allows us to define the
discretised linear operator Lδ of the Hull White PDE. We have
Lδ[V δ]
(t) =
(σ2(t)
2V δrr,i(t) + [θ(t)− ari]V δ
r,i(t)− riV δi (t)
)>i=0,...,n
.
Substituting the definitions for V δr,i and V δ
rr,i yields the matrix vector representation
of Lδ as
Lδ[V δ]
(t) = Lδ(t)V δ(t) with Lδ(t) ∈ R(n+1)×(n+1).
The (n+ 1)-dimensional matrix Lδ(t) is of the form
Lδ(t) =
d0 u0
l1. . . . . .. . . . . . un−1
ln dn
.
For the matrix elements we get the expressions
li =σ2(t)
(ri − ri−1)(ri+1 − ri−1)− θ(t)− ariri+1 − ri−1
,
di = − σ2(t)
(ri+1 − ri)(ri − ri−1)− ri,
ui =σ2(t)
(ri+1 − ri)(ri+1 − ri−1)+
θ(t)− ariri+1 − ri−1
for i = 1, . . . , n − 1. The first and last row of the matrix Lδ(t) is determined by the
boundary conditions of the discretised Hull White PDE. We have
d0 = −θ(t)− ar0
r1 − r0
− r0, u0 =θ(t)− ar0
r1 − r0
,
ln = −θ(t)− arnrn − rn−1
, dn =θ(t)− arnrn − rn−1
− rn.
As a result we get the linear time-dependent system of ordinary differential equa-
tionsdV δ
dt(t) = −Lδ(t)V δ(t)
with terminal condition
V δ(T ) = (p(r0), . . . , p(rn))> .
29
3.1.1 Discretisation Schemes
In this section we describe two approaches for the definition of the short rate grid
r0 < r1 < . . . < rn.
Uniform discretisation scheme. As a standard short rate grid we choose an odd
number of grid points r0 to rn. The center of the short rate grid rn/2 equals the short
rate at observation time t = 0, i.e.
rn/2 = r(0) = f(0, 0).
Thus the time t = 0 option value is directly given by Vn/2(0) and does not need to
be interpolated. Moreover, we specify a maximum deviation s from the current short
rate and set r0 = f(0, 0) − s and rn = f(0, 0) − s. The remaining grid points are
placed uniformly between r0 and rn. The discretisation width becomes
δ =2s
n.
This discretisation scheme yields a second order accurate spatial discretisation.
The (numerical) order of convergence is verified and illustrated in Figure 3.2. We
show the relative error
erri =
∣∣∣∣∣CBOPDEi − CBOanalytical
i
CBOanalyticali
∣∣∣∣∣between the analytical and numerical price of the ith European reference bond option
(i ∈ {1, 10, 20, 30}) of our reference test problem. The deviation s is chosen as 20%
and the relative error in the time integration is limited to 1.0e− 8.
Normally distributed discretisation scheme. For the pricing of an option in
the Hull White model we are usually interested in the time t option value at the
current realised short rate r(t) = f(t, t). Thus it appears reasonable to improve the
approximation at the current short rate by choosing a smaller grid size near r(t) than
at the boundaries of the numerical domain. Moreover, the short rate in the Hull
White model is normally distributed.
Therefore we construct a short rate grid which mimics a normal distribution of the
grid points. We choose again an odd number of grid points r0 to rn and a maximum
deviation s. The short rate grid is defined as
ri =Φ−1
(i+1n+2
)Φ−1
(n+1n+2
) s+ r(t) for i = 0, . . . , n.
30
Figure 3.2: Asymptotic approximation of equidistantly discretised PDE solution ofreference European swaptions.
Excercise Date 10
Reference 10/n2
Excercise Date 1
Excercise Date 20 Excercise Date 30
Number of spatial grid points nNumber of spatial grid points n
Relative Error of Numerical Solution
0.01
1
11 21 51 101 201 501 1001
0.0001
11 21 51 101 201 501 1001
0.0001
0.01
1
31
The function Φ−1 is the inverse cumulative normal distribution function. As for the
uniform grid we get rn/2 = r(t), r0 = r(t) − s, and rn = r(t) + s. The discretisation
width becomes
δ =
[1−
Φ−1(
nn+2
)Φ−1
(n+1n+2
)] s.We mentioned before that the spatial approximation with central differences in
(3.2) and (3.3) is only first order accurate for general grids. However, as elaborated
in Appendix A, we get for the normally distributed grid that
(ri+1 − ri)− (ri − ri−1) ≤ Cδ2.
The positive constant C is essentially independent of the discretisation. Hence we
get that in practice this discretisation scheme is also second order accurate.
Figure 3.3 demonstrates the numerical order of convergence for the normally dis-
tributed grid applied to our reference test problem. The setting is equal to the
numerical tests for the uniform short rate grid.
Comparing the numerical results in Figure 3.2 and Figure 3.3 shows that the
approximation with normally distributed short rate grid yields more accurate results
than the approximation with uniformly distributed short rate grid.
3.1.2 Determining the computational domain
The truncation of the domain in which we numerically solve the Hull White PDE
and the linear boundary condition impose an error on the numerical solution. Given
a fixed number of n+ 1 short rate grid points it is a priori not clear where to place r0
and rn. On the one hand it is advantageous to place them far away of the initial short
rate. This reduces the error of the wrong boundary condition. On the other hand a
large computational domain requires a large discretisation width. This increases the
error of the approximation of the derivatives by central differences.
We investigate the influence of the choice of the computational domain. For that
purpose we solve our reference test problem with uniform and normally distributed
grids, 1001 short rate grid points, and varying short rate deviation s. The relative
errors erri between the analytical and numerical price of the ith European reference
bond option (i ∈ {1, 10, 20, 30}) of our reference test problem are compared in Figure
3.4.
The results in Figure 3.4 demonstrate that the overall approximation error for
small deviation s usually first declines rapidly. For small values of s the distance
between the boundaries of the domain and point of interest rn/2 (i.e. the midpoint
32
Figure 3.3: Asymptotic approximation of normally distributed short rate grid PDEsolution of reference European swaptions.
Excercise Date 1
Relative Error of Numerical Solution
Number of spatial grid points nNumber of spatial grid points n
Excercise Date 30
Excercise Date 10
Excercise Date 20
Reference 10/n2
2121 51 101 201 501 1001 51 101 201 501 1001
0.0001
1
0.01
0.0001
0.01
1
1111
33
Figure 3.4: Approximation of numerical solution of reference European swaptions forvarying sizes s of the computational domain.
Uniform Short Rate GridNormally Distributed Short Rate Grid
Excercise Date 1
Deviation s of Short Rate Grid Deviation s of Short Rate Grid
Rela
tive
Err
or
of
Num
eri
cal
Solu
tion
Rela
tive
Err
or
of
Num
eri
cal
Solu
tion
Excercise Date 10
Excercise Date 30Excercise Date 20
0.4
0.001
0.0001
0.001
0.01
0.1
10
0.01
0.1 0.2 0.3 0.4 0.5 0.5
0.1
1
0 0.1 0.2 0.3
0.0001
34
of the domain) is small. By the linear boundary condition of the Hull White PDE
we introduce a modelling error rather close to the final point of interest. In contrast
the spatial discretisation width is small and thus the numerical approximation error
is small. Hence for small s the error of the linear boundary condition dominates the
approximation error of the finite difference approximation. For larger s the overall
approximation error increases moderately. Now the approximation error of the finite
difference approximation dominates the error of the linear boundary condition.
Unfortunately the optimal deviation which minimizes the overall approximation
error is not obviously determined. It depends on the spatial discretisation and the
integration time. Since the error of the finite difference approximation growths sub-
stantially smaller than the error of the linear boundary condition decreases it appears
reasonable to overestimate the computational domain. For our computations we use
values of s larger than or equal to 0.2.
Additionally, the results in Figure 3.4 demonstrate that the normally distributed
short rate grid usually yields a more accurate approximation of the numerical solution.
3.1.3 Using variable grids
So far we considered placing the center of the short rate grid at the value of the
current observable short rate. As an alternative approach we investigate the strategy
to place the center of the short rate grid at the effective strike rate at the exercise
dates Tj of the Bermudan. This method is motivated by the fact that the payoff is
in general not differential at the strike rate and that this region influences the option
price significantly.
Suppose t ∈ (Tj−1, Tj] and pj(r) is the payoff of the Bermudan option if exercised
at Tj. We assume that pj(r) is monotone in r. This holds in particular for call and
put options on bonds where we have that
pj(r) = ω (CBj(t, Tj, . . . , r)−Kj) .
Here CBj(t, Tj, . . . , r) is the coupon bond, Kj the strike price at Tj, and ω ∈ {−1,+1}describes call and put options. The price of the coupon bond is monotone in r. Hence
pj is also monotone in r. We determine r?j such that
pj(r?j ) = lim
t↓TjV δ(Tj, r
?j ). (3.4)
The function V δ(t, r) ≈ V (t, r) describes the available numerical solution of the option
price function. In our implementation it is obtained by C2-spline interpolation of the
35
components of V δ(t). For t > TN we set V (t, r) = 0. Equation (3.4) is solved by
Newton’s method. This approach ensures that for the Bermudan exercise condition
V δ(Tj, r) = max
{pj(r), lim
t↓TjV δ(t, r)
}there is a short rate grid point at the in general not differentiable short rate r?j .
The accuracy of the varying grid approach is analysed by means of our reference
test problem. We repeat the numerical computations of the uniform and normally
distributed grids for varying deviations s. However, at each exercise date we re-center
the short rate grid as described above. At observation time t = 0 the final option
value is also determined by C2-spline interpolation of the available numerical solution.
The results are illustrated in Figure 3.5.
Figure 3.5: Comparison of fixed and variable grid approximation of numerical solutionof reference European swaptions for varying sizes s of the computational domain.
Variable Normally Distributed Short Rate GridFixed Normally Distributed Short Rate Grid
Excercise Date 30
Rela
tive
Err
or
of
Num
eri
cal
Solu
tion
Rela
tive
Err
or
of
Num
eri
cal
Solu
tion
Deviation s of Short Rate Grid Deviation s of Short Rate Grid
Excercise Date 10Excercise Date 1
Excercise Date 20
0.2
0.001
0 0.1 0.2 0.3 0.4 0.5 0.3
0.01
0.4 0.5
0.0001
0.001
0.01
0.1
1
0.1
1
0 0.1
0.0001
The results in Figure 3.5 indicate that the variable grid approach does not improve
the approximation of the numerical solutions substantially. Therefore we do not use
it for further computations.
36
3.2 Time Integration
The spatial discretisation yields a system of ordinary differential equations
V δ(t) = −Lδ(t)V δ(t), V δ(t) ∈ Rn+1, Lδ(t) ∈ R(n+1)×(n+1)
with terminal conditions
V δ(Tj) = V δ,j for j = 1, . . . , N.
We solve the sequence of terminal value problems by Crank-Nicolson’s method. Let
V δt ≈ V δ(t) be the numerical solution of the terminal value problem at time t. The
numerical solution V δt−h at a previous time step t− h for h > 0 is determined by the
equationV δt − V δ
t−h
h= −
Lδ(t)V δt + Lδ(t− h)V δ
t−h
2≈ V δ(t− h/2).
Solving for V δt−h yields
V δt−h =
[I − h
2Lδ(t− h)
]−1 [I +
h
2Lδ(t)
]V δt . (3.5)
Since Lδ is of tridiagonal form the matrix vector multiplication U δt =
[I + h/2Lδ(t)
]V δt
may be evaluated within O (n+ 1) operations. The linear system[I − h
2Lδ(t− h)
]V δt−h = U δ
t
is solved directly by an LU decomposition. Again since Lδ is of tridiagonal form the
LU decomposition and the system solution requires O (n+ 1) operations.
The Crank-Nicolson method is consistent of order 2. That is if the solution is
sufficiently smooth and V δt = V δ(t) then∥∥V δ
t−h − V δ(t− h)∥∥ ≤ C h2
for a positive constant C independent of h. Moreover, the method is A-stable. For a
proof see, for example [HNW93] and [HW91].
The solution of the Bermudan option is evaluated in the following fashion. Starting
at the last exercise date TN we set the terminal value V δTN
equal to the payoff, i.e.
V δTN
= V δ,N =
max {ω [CBN(t, TN , . . . , r0)−KN ] , 0}...
max {ω [CBN(t, TN , . . . , rn)−KN ] , 0}
.
37
Then we choose a sequence of time steps tj such that TN−1 = t0 < t1 < . . . < tm = TN
with step sizes hj = tj − tj−1 for j = 1, . . . ,m. We evaluate successively
V δtj−1
=
[I − hj
2Lδ(tj−1)
]−1 [I +
hj2Lδ(tj)
]V δtj
for j = m, . . . , 1.
The payoff condition at TN−1 is imposed by
V δ,N−1 =
max{ω [CBN(t, TN−1, . . . , r0)−KN−1] , V δ
t0,0
}...
max{ω [CBN(t, TN−1, . . . , rn)−KN−1] , V δ
t0,n
} .
Here V δt0,i
(i = 0, . . . , n) denotes the ith component of the vector
V δt0
=(V δt0,0, . . . , V δ
t0,n
)>.
The numerical integration and the evaluation of the Bermudan payoff condition are
repeated analogously to the first exercise date T1. Finally the solution at observation
time t = 0 < T1 is evaluated by numerical integration backwards from T1 to 0.
Numerical stability of Crank-Nicolson’s scheme. Although Crank-Nicolson’s
scheme is A-stable it may yield poor numerical results if the initial data is not contin-
uous. This drawback can be overcome by the use of a Rannacher’s startup procedure.
For details see, e.g. [GC06]. This approach uses initial Backward Euler steps and
then proceeds with Crank-Nicolson steps.
For our setting and test problem we also tried Backward Euler steps and compared
the accuracy of the numerical solution to the available analytical solution of the
reference European swaptions. However, for the prices as well as for the Vegas with
Backward Euler we did not find significant differences in the approximation compared
to Crank-Nicolson’s scheme (despite the property that Backward Euler is only first
order consistent).
Adaptive step size control. The overall error of the numerical solution of the Hull
White terminal value problem consists of the modelling error of the linear boundary
condition, the error by spatial discretisation, and the error by numerical integration.
The impact of the modelling and spatial discretisation error are analysed in Section
3.1. To control the overall integration error we aim at estimating and limiting the
local numerical error of the time integration steps.
In each integration step we evaluate two numerical solutions V δt and V δ
t . The
solution V δt is evaluated by Crank-Nicholson’s method with step size h. Thus the
38
local error of V δt is of order h2. V δ
t is a more accurate numerical solution with a
local error of order h3. The more accurate solution acts as a local proxy of the exact
solution of the ODE terminal value problem. The local relative error is determined
by
errrel = maxi=0,...,n
∣∣∣V δt,i − V δ
t,i
∣∣∣max{|V δ
t,i|, ε}
.
The parameter ε is chosen with 1.0e − 8 to prevent floating point overflow in the
implementation.
The higher order numerical solution V δt is evaluated by Richardson extrapolation.
For reference see for example [HNW93]. Since V δt is evaluated by Crank-Nicholson’s
method with step size h we have that
V δt = V δ(t) + Ch2 +O(h3).
Richardson extrapolation is based on evaluating a second numerical solution V δt by
two Crank-Nicholson steps with half the step size. This yields that
V δt = V δ(t) + C
(h
2
)2
+O(h3).
The higher order solution is given by
V δt =
4V δt − V δ
t
3= V δ(t) +O(h3).
Using Richardson extrapolation we take errrel as the local relative error. Given
that errrel ≤ tol for a globally defined relative tolerance tol the integration step from
t+h to t is accepted. If errrel > tol the integration step is rejected and the integration
step size h is reduced. As adaptive step selection strategy we use the simple approach
of evaluating a preliminary step multiplier
λ =
√tol
errrel.
The step multiplier λ applied to the current step size h is determined considering
some technical restrictions
λ =
λmin for λ < λmin
λ for λmin ≤ λ < λfloor
λfloor for λfloor ≤ λ < 1
1 for 1 ≤ λ < λcap
λ for λcap ≤ λ < λmax
λmax for λmax ≤ λ
.
39
The mapping λ 7→ λ is also illustrated in Figure 3.6. The restricting parameters are
chosen rather conservative by λmin = 0.5, λfloor = 0.8, λcap = 1.5, and λmax = 2.0.
These choices should reduce the number of rejected steps as much as possible. the
new step size for both accepted and rejected integration steps is then given by
hnew = λh.
Figure 3.6: Step size restriction function for adaptive time stepping
λfloor
�������
λmin λcap
�������
1
1
λmax
Moreover we impose a lower boundary of 1.0e − 8 on the resulting new step
size hnew. This restriction should prevent the integration algorithm of getting stuck.
Additionally, we restrict the step size if it would integrate over the next exercise date
or the observation time.
40
Chapter 4
Sensitivity Evaluation byAutomatic Differentiation
The key issue of this thesis is the efficient evaluation of sensitivities for Bermudan
Swaptions in the Hull White interest rate model. A key input parameter of the pricing
problem of Bermudan swaptions is the Black’76 European swaption volatility surface.
In this chapter the focus lies on the evaluation of Vegas, i.e. the sensitivities of the
price with respect to the volatilities.
We elaborate how these sensitivities may be evaluated by applying methods of
Automatic Differentiation to the numerical PDE solver. Moreover we propagate the
PDE sensitivities backwards differentiating the calibration procedure. Finally we may
obtain the sensitivity of the Bermudan price with respect to the Black’76 volatilities
in the volatility surface.
Methods of Automatic differentiation apply the chain rule of differentiation to
the elementary operations of a computer program. For a comprehensive treatment of
these methods we refer to the Literature, e.g. [GW08]. Appendix B illustrates the
key concepts by means of an example. In this chapter we mention the Automatic
Differentiation (AD) tools and methods applied but do not explain them in detail.
The numerical results of this thesis are evaluated by programs written in C++. All
routines relevant for the sensitivity evaluation are defined as function templates. For
the Bermudan price evaluation the function templates are instantiated using intrinsic
floating point data types (i.e. doubles). For the sensitivity evaluation we use AD
tools based on operator overloading. These tools define an active data type which
facilitates the automatic sensitivity evaluation. The pricing routines are instantiated
with this active data type if sensitivities should be evaluated.
The combination of template based programming and operator overloading AD
tools yields an efficient and flexible approach to incorporate sensitivity evaluation
41
to pricing procedures. Usually, only minor changes to the original (double-based)
program are necessary to allow for the sensitivity evaluations. A key advantage is
that the template based programming requires maintaining only one source code.
That is the same template functions are used for only price evaluation and sensitivity
evaluation (even with possibly different AD tools).
Comparison of Automatic Differentiation to finite difference approxima-
tions. In practice sensitivities are often evaluated as finite difference approxima-
tions. This is the input parameters are bumped by a small value and the objective
price is re-evaluated. This approach is easy to implement in an existing pricing or
risk management system and hence widely applied.
It is well known that the finite difference approach may be less accurate and not
efficient, particularly for many input parameters. For our reference test problem we
evaluate the average Vega as the sensitivity of the Bermudan price with respect to
a small parallel shift of the Black’76 volatility surface. We compare the results for
several spatial discretisations obtained by Automatic Differentiation and central finite
differences with varying finite difference step size h = 10−2, . . . , 10−8. The results are
given in Figure 4.1.
We find from the results in Figure 4.1 that particularly for smaller problem dis-
cretisations finite differences yield no acceptable sensitivities. Even for larger discreti-
sations the Vega determined by finite differences deteriorates if the step size is chosen
a little too small. However, for larger finite difference step sizes the approximation
error may become significant. In contrast to the finite difference approach Automatic
Differentiation yields stable and accurate approximations of the Bermudan Vega.
In the numerical experiment corresponding to the results of Figure 4.1 we did not
fix the time discretisation of the numerical PDE solver. Thus the bumping of the
input parameters (implicitly) also changes the adaptive time stepping. This may be
a significant reason of the poor behaviour of the finite difference approach.
Sensitivities with respect to the interest rate curve. For hedging and risk
management sensitivities of the price with respect to the yield curve (i.e. Deltas)
are also relevant. In principle these sensitivities can also be evaluated by methods
Automatic Differentiation. In this thesis we do not cover the evaluation of these
sensitivities. The evaluation of risk sensitivities (in particular Deltas) for Bermudan
swaptions is elaborated, e.g. in [Pit04]. Sensitivities of European swaptions in the
Hull White model are discussed, e.g. in [Hen04].
42
Figure 4.1: Comparison of average Vega evaluated by central finite differences andAutomatic Differentiation.
average vega by finite differences
101 grid points
finite difference step size
1001 grid points501 grid points
finite difference step size
average vega by Automatic Differentiation
201 grid points
0.8
1.2
1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01
0.6
0.8
1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01
0.4
0.6
0.4
1.2
1
1
43
Evaluating Vegas in the Hull White PDE model. In this chapter we describe
the sensitivity evaluation of the Hull White Bermudan swaption price PV HW with
respect to the Black’76 swaption volatilities σB76 of the European reference swaptions
and the swaption volatility surface ΣB76. The pricing of Bermudans in the Hull White
model may be split into the following steps:
1. Interpolating the Black’76 volatility surface and evaluating prices for reference
European swaptions (see Section 2.1).
2. Evaluating a piecewise constant short rate volatility function by calibrating the
Hull White model to the prices of reference European swaptions (see Section
2.3).
3. Solving the Hull White PDE model for the Bermudan swaption price (see Chap-
ter 3).
Hence we get the following mapping
ΣB76Surface
Interpolation7−→ σB76 Swaption7−→ PV B76 Calibration7−→ σHWBermudanHW7−→ PV HW .
(4.1)
Here
ΣB76 =
σB761,1 . . . σB76
1,L...
...σB76K,1 . . . σB76
K,L
represents the European swaption Black’76 volatility surface and
σB76 =(σB76
1 , . . . , σB76N
)>,
PV B76 =(PV B76
1 , . . . , PV B76N
)>describe the interpolated Black’76 volatilities and corresponding prices for the refer-
ence European swaptions. The calibrated values of the piecewise constant short rate
volatility function are denoted by
σHW =(σHW1 , . . . , σHWN
)>and the resulting price of the Bermudan swaption is PV HW . The sensitivity of the
Bermudan swaptions price PV HW with respect to the Black’76 volatility surface ΣB76
is determined by differentiating each mapping in the evaluation procedure (4.1). The
resulting sensitivity is determined as the product of the individual derivatives by
applying the chain rule of differentiation.
44
For the sensitivity of the Bermudan swaption price in the Hull White model PV HW
with respect to the Black’76 swaption volatility surface ΣB76 we get
dPV HW
dΣB76=
dPV HW
dσHW· dσHW
dPV B76· dPV
B76
dσB76· dσ
B76
dΣB76.
The first derivative dPV HW/dσHW can also be considered the gradient of the PDE
solution w.r.t. the short rate volatility values. Its evaluation is described in Section
4.3. The second term dσHW/dPV B76 corresponds to the differentiation of the Hull
White model calibration. This is elaborated in Section 4.2. The Black’76 European
swaption Vegas and the derivatives of the Black’76 volatility surface interpolation are
given by dσB76/dΣB76 and dσB76/dΣB76. This is described in Section 4.1.
4.1 Differentiating the European swaption prices
The Black’76 swaption volatility surface ΣB76 is interpolated bilinearly to obtain the
the reference Black’76 volatilities σB76i for i = 1, . . . , N . Suppose Ti is the exercise
date of the ith European reference Swaption and Si = SM − Ti its underlying swap
tenor (in years). Let TB761 , . . . , TB76
K be the exercise dates and S1, . . . , SL be the swap
tenors of the volatility surface ΣB76. We choose TB76k and SB76
l such that
TB76k < Ti ≤ TB76
k+1 and SB76l < (SM − Ti) ≤ SB76
l+1 .
The interpolated Black’76 volatility becomes
σB76i =
(TB76k+1 − Ti)(SB76
l+1 − Si)(TB76
k+1 − TB76k )(SB76
l+1 − SB76l )
σB76k,l +
(Ti − TB76k )(SB76
l+1 − Si)(TB76
k+1 − TB76k )(SB76
l+1 − SB76l )
σB76k+1,l+
(TB76k+1 − Ti)(Si − SB76
l )
(TB76k+1 − TB76
k )(SB76l+1 − SB76
l )σB76k,l+1+
(Ti − TB76k )(Si − SB76
l )
(TB76k+1 − TB76
k )(SB76l+1 − SB76
l )σB76k+1,l+1.
Consequently the sensitivity of the reference volatility σB76i with respect to the volatil-
ity quote σB76k,l in the surface is
dσB76i
dσB76k,l
=(TB76
k+1 − Ti)(SB76l+1 − Si)
(TB76k+1 − TB76
k )(SB76l+1 − SB76
l ).
The sensitivity of the reference volatility σB76i with respect to the other volatility
quotes used for the interpolation follows analogously.
45
The European swaption price in the Black’76 formula (2.1) is given by
Swaption(t) = Annuity(t) · (−ω) [Y (t)Φ(−ωd1)−RΦ(−ωd2)] ,
d1,2 =log (Y (t)/R)
σB76√τ± σB76
√τ
2.
The derivative of the swaption price with respect to the input volatility is given by
dSwaption(t)
dσB76= Annuity(t)Y (t)
e−d21/2
√2π
√τ .
Hence we can evaluate the sensitivity of the Black’76 prices of the reference European
swaptions with respect to the volatility surface by analytical formulas.
The implementation of the analytical formulas for the interpolation and the swap-
tion formula in a computer program would require a careful tracking of which surface
volatility σB76k,l contributes to a reference swaption volatility σB76
i . We circumvent this
effort by applying the AD tool ADTAGEO [RG09] to the interpolation and Swaption
price evaluation.
ADTAGEO builds up the internal computational graph between the input volatil-
ities of the volatility surface and the output swaption prices. It evaluates the Jacobian
of all independent and dependent variables. That is it allows the evaluation of the
derivatives of each swaption price with respect to each volatility input. This ap-
proach may not be optimal considering the computational effort. An AD reverse
mode implementation could be more efficient.
However the overall computational effort of the evaluation of Black’76 prices in the
Bermudan pricing procedure is negligible compared to the calibration and the PDE
solution. We prefer to exploit the easy to use programming interface of ADTAGEO
compared to a slightly more sophisticated incorporation of the pure AD reverse mode.
4.2 Differentiating the Calibration
In the calibration procedure of the Hull White model we aim at solving the minimi-
sation problem
mina≤σHW≤b
1
2F (σHW )>F (σHW ),
see (2.6). The objective function F : RN → RN may be formulated using the notation
in (4.1) as
F (σHW ) = CBOHW (σHW )− PV B76.
46
The function CBOHW (σHW ) summarizes the evaluation of the swaption equivalent
bond options by formula (2.3) depending on the short rate volatility values in σHW .
We point out that the structure of the short rate volatility σHW ∈ RN and the
number of reference European options prices PV B761 , . . . , PV B76
N is chosen such that
the function F maps from RN to RN .
Usual market situations allow to find a solution σHW of the calibration problem
such that
F (σHW ) = 0 ∈ RN , i.e. CBOHW (σHW ) = PV B76.
Moreover it appears reasonable to assume that the Jacobian F ′(σHW ) = CBOHW ′(σHW )
at the solution is non-singular. We would like to point out that these assumptions
are local properties close to the optimal value σHW . The more general formulation as
minimisation problem in Section 2.3.1 is intended to avoid iteration steps to regions
where the objective function may not be evaluated and the algorithm breaks down.
Since we assume F ′(σHW ) non-singular and F (σHW ) = 0 we may apply the im-
plicit function theorem. This yields that there is a vicinity U ⊂ Rn around PV B76
and a differentiable function CBOHW−1: U → RN with
σHW = CBOHW−1(PV B76).
We get for the Jacobian of CBOHW−1that
dCBOHW−1
dPV(PV B76) =
[F ′(σHW )
]−1=
[dCBOHW
dσ(σHW )
]−1
.
That is the sensitivity of the calibrated short rate volatility σHW with respect to the
input Black’76 prices of reference swaptions PV B76 is given by
dσHW
dPV B76=
[dCBOHW
dσ(σHW )
]−1
.
If the calibration problem is solved by a Gauss-Newton method as described in
Section 2.3.2 the Jacobian F ′(σHW ) = dCBOHW
dσ(σHW ) is already available from the
last iteration. Moreover the Jacobian is already decomposed into a QR factorisa-
tion which allows an efficient evaluation of inverse Jacobian vector products. In our
situation we are particularly interested in evaluating
dPV HW
dσHW· dσHW
dPV B76= ∇PV HW> ·
[dCBOHW
dσ(σHW )
]−1
.
The derivative evaluation of the coupon bond option formula in the Hull White model
is implemented using the AD tool ADTAGEO again.
47
If the Jacobian of the objective function is not non-singular at the solution or
if the calibration problem is not formulated as a system of (non-linear) equations
then we can not use the approach above. In such situations a general fixed point
iteration may be used to calibrate the model. The derivative of the calibrated short
rate volatility with respect to input reference prices may then be evaluated by reverse
accumulation as described in [Chr94].
4.3 Evaluating Derivatives of the PDE solution
The price of the Bermudan swaption is evaluated as the solution V (t, f(t, t)) of the
Hull White PDE
Vt +σ2(t)
2Vrr + [θ(t)− ar]Vr − rV = 0
with Bermudan exercise conditions
V (Ti, r) = max
{limt↓Ti
V (t, r), [ω (CBi(Ti, r, σ(·), . . .)−Ki)]+
}, i = 1, . . . , N.
The volatility is piecewise constant between the exercise dates T1, . . . , TN (T0 = t),
i.e.
σ(t) = σHWi for Ti−1 < t ≤ Ti.
For details see Section 3.
The numerical Bermudan swaption price PV HW = V δ(t, f(t, t)) can be considered
a function of the short rate volatility values σHW1 , . . . , σHWN . The volatilities enter the
evaluation procedure of PV HW in the (boundary) exercise conditions and in the
linear operator of the differential equation. Thus we can evaluate the sensitivity
of the Bermudan price PV HW with respect to the short rate volatilities σHW =
(σHW1 , . . . , σHWN )> by differentiating all the elementary operations involved in the
evaluation.
In our implementation we use the tapeless vector forward of the AD tool ADOL-
C [GJU96] to differentiate the Hull White PDE solver. This approach is based on
operator overloading. For each elementary operation the corresponding derivatives are
directly evaluated and propagated along with function values. Since we are interested
in the derivative of the Bermudan price with respect to each of the N short rate
volatilities we apply the vector forward mode with N tangential directions.
The Bermudan price as a function of the short rate volatilities can be considered
a mapping of N independent variables to one dependent variable. The derivative
48
dPV HW/dσHW is essentially a gradient. For such problems it is usually computation-
ally more efficient to use the reverse mode of AD. However, the reverse mode requires
the recording (taping) of all intermediate variables of the evaluation procedure. For
applications involving a time stepping the memory effort may be tremendous. This is
particularly the case for small discretisations in space and time. The memory effort
may be reduced by checkpointing strategies [SW10] which successively re-evaluate
and store sequences of the evaluation procedure.
Bermudan swaptions usually involve a rather limited number of N exercise dates.
Hence the number of short rate volatility values is also not that much. In our (rather
extreme) reference test problem we incorporate 30 exercise dates. Consequently the
additional computational effort of the forward mode compared to the reverse mode
is still acceptable.
The more important point for the application of the AD forward mode in the PDE
solver is that it allows us to solve for several solutions of the Hull White PDE and its
derivatives in parallel. This is particularly important if we want to estimate the accu-
racy of the solution and the gradient of the numerical scheme. In our implementation
we solve the Hull White PDE for the exercise conditions of the Bermudan and in
addition for the boundary conditions of each of the N European reference swaptions.
The numerical results for prices and gradients are compared to the (semi-) analytical
prices and gradients available from the calibration procedure.
We illustrate the accuracy of the differentiation of the PDE code for our reference
test problem by evaluating the sensitivity of the numerical prices of the ith reference
swaptions PV HWi with respect to a parallel shift in the short rate volatility curve.
That is we evaluate
˙PViHW
=N∑j=1
dPV HWi
dσHWj.
This quantity is compared to the (average) derivative of the corresponding analytical
bond option price
˙CBOi(t, . . .) =N∑j=1
dCBOi(t, . . .)
dσHWj.
The relative error of the derivative of the numerical scheme is evaluated as
erri =
∣∣∣∣∣ ˙PViHW
˙CBOi(t, . . .)− 1
∣∣∣∣∣ .Figure 4.2 illustrates the results of the accuracy of the derivative evaluation for ref-
erence swaptions of our standard test problem.
49
Figure 4.2: Asymptotic approximation of average sensitivities by Automatic differen-tiation of reference European swaptions
number of spatial grid points n
Relative Error of Sensitivity
number of spatial grid points n
Excercise Date 30Excercise Date 20
Excercise Date 1Excercise Date 1
Reference 10/n2
501
0.001
0.01
201101 1001
0.0001
0.001
0.01
101 201 501 1001
0.0001
50
The results in Figure 4.2 show that the derivatives evaluated by differentiating the
numerical scheme approximate the analytical solutions. However, the convergence of
the numerical derivative to the analytical derivative is considerably slower than the
convergence of the numerical PDE solution to the analytical solution. This is related
to observations reported, for example in [EB99] and [AS09]
4.4 Linking the partial derivatives
According to the description in the Sections 4.1, 4.2, and 4.3 we can evaluate par-
tial derivatives of the Bermudan swaption pricing procedure. The individual partial
derivatives are combined in reverse order of the evaluation procedure. That is we first
evaluate the N -dimensional transposed-vector-matrix multiplication
dPV HW
dPV B76=
dPV HW
dσHW· dσHW
dPV B76.
Here N is the number of exercise dates. This results in a (row) vector of dimension
N .
In the next step we evaluate
dPV HW
dσB76=
dPV HW
dPV B76· dPV
B76
dσB76.
The derivative dPV B76/dσB76 is of diagonal form. Hence this requires only N mul-
tiplications. As a result we get the sensitivity of the Bermudan price with respect
to the interpolated Black’76 volatilities. Figure 4.3 shows the resulting sensitivities
dPV B76/dσB76 for our reference test problem.
If we are also interested in the sensitivity of the Bermudan price with respect to the
quoted Black’76 volatilities we may apply the derivatives of the surface interpolation.
That is we evaluatedPV HW
dΣB76=
dPV HW
dσB76· σB76
dΣB76.
The derivative σB76/dΣB76 is a N -by-K ·L matrix where K ·L is the number of quotes
in the volatility surface. This matrix can be rather big. However it is also sparse. For
our reference test problem the final sensitivities of the Bermudan price with respect
to the quoted Black’76 volatilities are given in Table 4.1.
51
Figure 4.3: Sensitivity of Bermudan swaption price with respect to interpolatedBlack’76 volatilities of reference European swaptions.
Excercises i
sen
siti
vit
ydPV
HW/dσB
76
i
0.04
7 9 11 13 15 17 19 21 23 25 27 29
0.05
0.06
310
0.01
0.02
0.03
5
Table 4.1: Sensitivity of Bermudan swaption price with respect to Black’76 volatilitysurface.
Exercise Swap tenors (years)(years) 1 5 10 20 30
2 0 0 0 5.9E-04 9.6E-033 0 0 0 6.3E-03 2.5E-024 0 0 0 1.7E-02 3.9E-025 0 0 0 2.2E-02 3.3E-026 0 0 0 2.6E-02 2.6E-027 0 0 0 2.7E-02 1.8E-028 0 0 0 2.5E-02 1.1E-029 0 0 0 3.3E-02 8.2E-0310 0 0 5.2E-03 7.8E-02 3.0E-0315 0 0 4.2E-02 7.1E-02 020 0 5.0E-03 6.9E-02 1.4E-02 025 2.4E-03 2.8E-02 1.8E-02 0 030 7.0E-03 5.5E-03 0 0 0
52
Chapter 5
Conclusions
In this thesis we investigate the sensitivity evaluation of Bermudan swaption prices
with respect to the market observable Black’76 volatilities. For the pricing of the
Bermudan swaption we apply a Hull White model with piecewise constant short
rate volatility function. The Bermudan swaption pricing involves the evaluation of
reference European swaption prices, the calibration of the short rate volatilities of the
Hull White model, and the numerical evaluation of the Bermudan price. Sensitivities
of all the valuation steps are evaluated by methods of Automatic Differentiation (AD).
Numerical results are reported for a reference test problem.
A key result of this thesis is that Automatic Differentiation yields accurate and
efficient sensitivities of Bermudan swaptions (see Chapter 4). It circumvents the
disadvantages of finite difference approximations. We find that template based pro-
gramming and AD tools based on operator overloading provide an efficient and flexible
approach to incorporate sensitivity evaluation in complex pricing procedures.
Besides the sensitivity evaluation we analysed the implementation of the Hull
White model on a PDE grid. It became visible in Subsection 2.2.3 that the numerical
solution of the Hull White model on a PDE grid requires an initial yield curve that
provides continuous forward rates. The PDE discretisation requires to bound the
computational domain and approximate the boundary condition. We find in Sub-
section 3.1.2 that the boundary discretisation error is small if the domain is chosen
sufficiently large. Our numerical results indicate that a short rate radius of 20% to
30% around the initial short rate value appear to be a reasonable choice. Moreover,
we find that the accuracy of the numerical solution may be increased substantially if
we apply the normally distributed short rate grid as elaborated in Subsection 3.1.1.
For the time integration we get in that adaptive step size control by Richardson’s
extrapolation (see 3.2) is crucial for the efficiency of the overall procedure.
53
Additionally, the calibration procedure of the Hull White model may be differ-
entiated with only minor additional computational effort. If the short rate volatility
calibration problem is formulated as in Subsection 4.2 the required derivatives are
already available in a computationally desirable form from the calibration procedure
itself.
Consequently, we can consider the results in this thesis to be another proof of
concepts that Automatic Differentiation may improve the sensitivity evaluation in a
wide range of applications. Further research might concern the sensitivity evaluation
in other financial models and with respect to further market and model parameters
like interest rates, mean reversion speeds, or correlations. The results may yield a
better understanding of the risk profile of exotic products. The adoption to other
models could be realized, for example, by extending existing pricing libraries to allow
the incorporation of Automatic Differentiation tools.
54
Appendix A
Derivative Approximation forNormally Distributed Grids
In Section 3.1 we elaborate that the discretisation error of the finite difference ap-
proximation V δr and V δ
rr to the first and second spatial derivative ∂V∂r
and ∂2V∂r2
becomes
V δr (t, ri)−
∂V
∂r(ri, t) =
1
2
∂2V
∂r2(ri, t) [(ri+1 − ri)− (ri − ri−1)] +O(δ2),
V δrr(t, ri)−
∂2V
∂r2(ri, t) =
1
3
∂3V
∂r3(ri, t) [(ri+1 − ri)− (ri − ri−1)] +O(δ2),
for i = 1, . . . , n− 1. The spatial discretisation width δ is
δ = maxi=1,...,n
{ri − ri−1}.
For uniform grids with (ri+1 − ri) − (ri − ri−1 = 0 we get immediately that the
discretisation error is of order O(δ2). This yields quadratic order of convergence of
the numerical scheme. If we have a non-uniform grid the derivative approximation
is in the general case only of order O(δ) which reduces the convergence order of the
numerical scheme accordingly.
For the normally distributed short rate grid we choose the grid points ri according
to
ri =Φ−1
(i+1n+2
)Φ−1
(n+1n+2
) s+ r(t) for i = 0, . . . , n.
Denote ui = i+1n+2
and hn = 1n+2
. Then
(ri+1−ri)−(ri−ri−1) =s
Φ−1(un)
[(Φ−1(ui + hn)− Φ−1(ui)
)−(Φ−1(ui)− Φ−1(ui − hn)
)]︸ ︷︷ ︸d2Φ−1(ui)
55
for i = 1, . . . , n − 1. Since Φ−1 is continuously differentiable there are constants
ρ, η ∈ [0, 1] such that
d2Φ−1(ui) = Φ−1′(ui + ρhn)hn − Φ−1′(ui − ηhn)hn.
Moreover since Φ−1′ is also continuously differentiable there is a constant λ ∈ [−1, 1]
such that
d2Φ−1(ui) = Φ−1′′(ui + λhn)(ρ+ η)h2n
=Φ−1′′(ui + λhn)
Φ−1′(ui + ρhn)2 (ρ+ η)
(Φ−1(ui + hn)− Φ−1(ui)
)2.
SinceΦ−1′′(ui + λhn)
Φ−1′′(ui)→ 1 and
Φ−1′(ui)
Φ−1′(ui + ρhn)→ 1 for hn → 0
we get
|(ri+1 − ri)− (ri − ri−1)| ≤ CΦ−1(un)
s
∣∣∣∣∣Φ−1′′(ui)
Φ−1′(ui)2
∣∣∣∣∣ |ri+1 − ri|2
for a positive constant C <∞ (usually in the order of one). Moreover it follows that
Φ−1′(ui) =1
Φ′ (Φ−1(ui))and Φ−1′′(ui) = −Φ′′ (Φ−1(ui))
Φ′ (Φ−1(ui))3 .
Exploiting that
Φ′(x) =1√2π
exp
(−x
2
2
)and thus Φ′′(x) = −xΦ′(x)
yields
|(ri+1 − ri)− (ri − ri−1)| ≤ C
sΦ−1(un) |Φ−1(ui)| |ri+1 − ri|2 . (A.1)
Since |Φ−1(ui)| < Φ−1(un) follows
|(ri+1 − ri)− (ri − ri−1)| < C
s
[Φ−1(un)
]2 |ri+1 − ri|2 .
Unfortunately Φ−1(un) = −Φ−1(
1n+2
)is not bounded for n→∞. However, it grows
only very moderately. For (computationally) reasonable numbers of discretisation
grid points n we may consider it bounded above. Figure A.1 illustrates the growth
of the factor [Φ−1(un)]2.
Moreover, we have in Equation (A.1) that Φ−1(ui) = 0 for ui = 12. Hence for ui
in a vicinity of 12
we may expect that the additional error by the non-uniform grid
discretisation almost vanishes. For s = 1 and n = 1001 we illustrate the factor
(ri+1 − ri)− (ri − ri−1)
max{
(ri+1 − ri)2 , (ri − ri−1)2}56
Figure A.1: NormGridFactor
[Φ−1 (un)
]2
number of grid points n
����""""""""""""��������������������
0
30
40
10310 105 107 109
10
20
Figure A.2: RFactor
(ri+1−ri)−(ri−ri−1)
max{(ri+1−ri)2,(ri−ri−1)2}
spatial dimention r
-10
5
10
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-5
0
57
for i = 1, . . . , n− 1 in Figure A.2.
We may conclude that the discretisation error of the finite difference approxima-
tion for the first and second derivatives of V with normally distributed grid points
becomes ∣∣∣∣V δr (t, ri)−
∂V
∂r(ri, t)
∣∣∣∣ =
[Φ−1
(1
n+ 2
)]2
O(δ2) +O(δ2),
∣∣∣∣V δrr(t, ri)−
∂2V
∂r2(ri, t)
∣∣∣∣ =
[Φ−1
(1
n+ 2
)]2
O(δ2) +O(δ2),
for i = 1, . . . , n − 1. Hence for the normally distributed short rate grid we get
essentially a second order approximation. This confirms the quadratic convergence
observed in the results of Figure 3.3.
58
Appendix B
Fundamentals of AutomaticDifferentiation
Automatic Differentiation (AD) covers principles and techniques of derivative evalua-
tions that make use of the computational representation of a function. In applications
a mathematical function is usually given as a computer program (in some high level
programming language) with input and output variables. Methods of AD allow the
efficient evaluation of the derivatives of selected outputs with respect to selected in-
puts. The concepts of AD are implemented by tools that use the computational rep-
resentation of a function given in a programming language like C/C++, Fortran,
or Matlab. The following details are mainly based on the book by A. Griewank
[GW08]. It is also referred to in the Literature for a more detailed discussion.
The main ideas of AD are the representation of a functional relation as a se-
quence of elemental differentiable operations, and the application of the chain-rule of
differentiation in forward or reverse order to this sequence.
Besides the elemental concepts of the forward and reverse mode, there are also
methods for the evaluation of higher order derivatives and the efficient evaluation of
derivatives in a number of directions. For more detailed discussion, it is referred to
the references at www.autodiff.org.
B.1 Evaluation procedure
To illustrate the principles of AD, consider the function described by the Black’76
formula. The Black’76 formula for European puts (ω = −1) and calls (ω = +1) with
59
forward price F , strike K, volatility σ, and time to maturity τ is
Black76 : (0,∞)4 × {−1,+1} → (0,∞),
Black76(F,K, σ, τ, ω) = ω [FΦ(ωd1)−KΦ(ωd2)] ,
d1,2 =log (F/K)
σ√τ
± σ√τ
2.
By splitting the evaluation of Black76() into a sequence of unary and binary ele-
mental operations, one may get a list of operations as shown in Figure B.1, where
vi denote intermediate variables and fi(·) stands for the elemental operations. This
Figure B.1: Evaluation procedure of Black’76 formula
v−3 = x1 = Fv−2 = x2 = Kv−1 = x3 = σv0 = x4 = τv1 = v−3/v−2 ≡ f1(v−3, v−2)v2 = log(v1) ≡ f2(v1)v3 =
√v0 ≡ f3(v0)
v4 = v−1 · v3 ≡ f4(v−1, v3)v5 = v2/v4 ≡ f5(v2, v4)v6 = 0.5 · v4 ≡ f6(v4)v7 = v5 + v6 ≡ f7(v5, v6)v8 = v7 − v4 ≡ f8(v7, v4)v9 = ω · v7 ≡ f9(v7)v10 = ω · v8 ≡ f10(v8)v11 = Φ(v9) ≡ f11(v9)v12 = Φ(v10) ≡ f12(v10)v13 = v−3 · v11 ≡ f13(v−3, v11)v14 = v−2 · v12 ≡ f14(v−2, v12)v15 = v13 − v14 ≡ f15(v13, v14)v16 = ω · v15 ≡ f16(v15)y1 = v16
representation of Black76() is considered as an evaluation procedure. Note that ele-
mentary functions may involve constants and parameters which are not considered to
be differentiated. For example, in Figure B.1 f6(v4) = 0.5·v4 or f9(v7) = ω ·v7. involve
factors 0.5 or ω which act as constants. The evaluation procedure of a function can be
represented as a directed acyclic computational graph connecting independent input
variables with dependent output variables. A computational graph of the Black76()
60
function based on the elementary operation sequence in Figure B.1 is given in Figure
B.2.
Figure B.2: Computational Graph of Black’76 Formula
v−1
v4
v6
v7
v8
v10
v−3
v12
v14v13
v11
v1
v2
v5
v9
v15
v16
v0
v3
v−2
Generally, be F : Rn → Rm and fi : Rni → Rmi . The relation j ≺ i denotes that
vi ∈ R depends directly on vj ∈ R. If for all x ∈ Rn and y ∈ Rm with y = F (x) holds
thatvi−n = xi i = 1, . . . , nvi = fi(vj)j≺i i = 1, . . . , l
ym−i = vl−i i = m− 1, . . . , 0,(B.1)
61
then (B.1) is called an evaluation procedure of F with elementary operations fi.
We assume differentiability of all elementary operations fi (i = 1, . . . , l). Then the
resulting function F is also differentiable.
B.2 Forward mode
By differentiating the elementary operations fi in the evaluation procedure (B.1)
totally and applying the chain-rule of differentiation directional derivatives can be
evaluated. Denote ui = (vj)j≺i ∈ Rni . Then ui collects the arguments of the operation
fi. In addition to each elementary function evaluation vi = fi(ui) we evaluate the
derivatives vi by
vi =∑j≺i
∂
∂vjfi(ui) · vj.
Abbreviating ui = (vj)j≺i and fi(ui, ui) = f ′i(ui) · ui, leads to the forward mode of AD
by[vi−n, vi−n] = [xi, xi] i = 1, . . . , n
[vi, vi] =[fi(ui), fi(ui, ui)
]i = 1, . . . , l
[ym−i, ym−i] = [vl−i, vl−i] i = m− 1, . . . , 0.
(B.2)
Here, the initializing derivative-values xi−n for i = 1 . . . n are given and determine the
direction of the tangent. This representation can also be interpreted as an evaluation
procedure. Hence with x = (xi) ∈ Rn and y = (yi) ∈ Rm, the forward mode of AD
evaluates
y = F ′(x)x.
For a proof of this statement, it is referred to [GW08]. The forward differentiation of
the Black’76 example is given in Figure B.3.
The additional computational effort of the forward differentiation consists of the
differentiation of the elemental operation fi(ui), the multiplication with the corre-
sponding values in ui, and the summation. A distinguished count of memory moves,
additions, multiplications, and non-linear operations together with a weighted mea-
sure of its computational effort yields, according to [GW08], that
OPS (F ′(x)x) ≤ cf OPS (F (x)) , where cf ∈[
2,5
2
].
OPS(·) is a measure of the computational effort and the multiplier cf depends on the
weighting of the effort of moves, additions, multiplications and non-linear operations.
With this estimation, one may state that the computational effort of the evaluation
62
Figure B.3: Forward Differentiation of Black’76 Formula
v−3 = x1 = F v−3 = 0v−2 = x2 = K v−2 = 0v−1 = x3 = σ v−1 = 1v0 = x4 = τ v0 = 0v1 = v−3/v−2 v1 = v−3/v−2 − v1 · v−2/v−2
v2 = log(v1) v2 = v1/v1
v3 =√v0 v3 = 0.5 · v0/v3
v4 = v−1 · v3 v4 = v−1 · v3 + v−1 · v3
v5 = v2/v4 v5 = v2/v4 − v5 · v4/v4
v6 = 0.5 · v4 v6 = 0.5 · v4
v7 = v5 + v6 v7 = v5 + v6
v8 = v7 − v4 v8 = v7 − v4
v9 = ω · v7 v9 = ω · v7
v10 = ω · v8 v10 = ω · v8
v11 = Φ(v9) v11 = φ(v9) · v9
v12 = Φ(v10) v12 = φ(v10) · v10
v13 = v−3 · v11 v13 = v−3 · v11 + v−3 · v11
v14 = v−2 · v12 v14 = v−2 · v12 + v−2 · v12
v15 = v13 − v14 v15 = v13 − v14
v16 = ω · v15 v16 = ω · v15
y1 = v16 y1 = v16
63
of the forward mode of AD is bounded by the computational effort of 52
function
evaluations. It is to note that a forward mode includes a function evaluation, because
the evaluation of f ′i requires also ui for all i.
In practice, a concrete run-time ratio of the derivative evaluation compared to the
function evaluation may differ from the theoretical result. It depends on the specific
program of the function, the AD tool applied, and the hardware used.
B.3 Reverse mode
In the forward mode, the chain-rule is applied in the same order as the function
evaluation itself, starting at the independent variables and evaluating the dependent
variables. On the contrary, the chain-rule can be applied in the reverse order of the
function evaluation. This means, the sensitivities of the intermediate variables vi
with respect to the dependent variables yi are computed successively and in reverse
order. Note that this requires a preceding function evaluation where all overwritten
program variables are stored for restoration in the reverse sweep.
The reverse mode evaluation requires auxiliary derivative values vj. These inter-
mediate variables are initialized to zero before the reverse mode evaluation. For each
elementary operation fi and all intermediate variables vj with j ≺ i,
vj + = vi ·∂
∂vjfi(ui)
is evaluated. In other words, for each arguments of fi the partial derivative is derived.
Denoting ui = (vj)j≺i ∈ Rni and fi(ui, vi) = vi · f ′i(ui), the incremental reverse
mode of AD is given by the evaluation procedure
vi−n = xi i = 1, . . . , nvi = fi(vj)j≺i i = 1, . . . , l
ym−i = vl−i i = m− 1, . . . , 0vi = yi i = 0, . . . ,m− 1ui + = fi(ui, vi) i = l, . . . , 1xi = vi i = n, . . . , 1.
(B.3)
In this representation all intermediate variables vi are assigned only once. Here the
initializing values yi are given. They represent a weighting of the dependent variables
yi. The vector y = (yi) can also be interpreted as normal vector of a hyperplane in
the range of F . With y = (yi) and x = (xi), the reverse mode of AD yields
xT = ∇[yTF (x)
]= yTF ′(x).
64
Figure B.4: Reverse Mode Evaluation of Black’76 formula
v−3 = x1 = Fv−2 = x2 = Kv−1 = x3 = σv0 = x4 = τv1 = v−3/v−2
v2 = log(v1)v3 =
√v0
v4 = v−1 · v3
v5 = v2/v4
v6 = 0.5 · v4
v7 = v5 + v6
v8 = v7 − v4
v9 = ω · v7
v10 = ω · v8
v11 = Φ(v9)v12 = Φ(v10)v13 = v−3 · v11
v14 = v−2 · v12
v15 = v13 − v14
v16 = ω · v15
y1 = v16
v16 = y1 = 1v15 += ω · v16
v13 += v15; v14 += (−1) · v15
v−2 += v12 · v14; v12 += v−2 · v14
v−3 += v11 · v13: v11 += v−3 · v13
v10 += φ(v10) · v12
v9 += φ(v9) · v11
v8 += ω · v10
v7 += ω · v9
v7 += v8; v4 += (−1) · v8
v5 += v7; v6 += v7
v4 += 0.5 · v6
v2 += v5/v4; v4 += (−1) · v5 · v5/v4
v−1 += v3 · v4; v3 += v−1 · v4
v0 += 0.5 · v3/v3
v1 += v2/v1
v−3 += v1/v−2; v−2 += (−1) · v1 · v1/v−2
τ = x4 = v0
σ = x3 = v−1
K = x2 = v−2
F = x1 = v−3
65
For a proof of this statement, see [GW08]. An example code list of the reverse mode
differentiated Black’76 formula is given in Figure B.4.
In analogy to the forward mode, the additional computational effort of the reverse
mode can be estimated by considering memory moves, additions, multiplications,
and non-linear operations. This includes again a function evaluation. According to
[GW08] follows
OPS(yTF ′(x)
)≤ cr OPS ((F (x)) , where cr ∈ [3, 4].
It is of particular importance that the reverse mode is independent of the dimen-
sions n of the input variables. In the special case of scalar functions F : Rn → R the
gradient ∇F (x) can be evaluated at an effort of four function evaluations independent
of the number n of input variables.
B.4 General Graph Reduction
The computational graph of a function can be exploited to evaluate derivatives. If the
evaluation procedure of a function with given input variables and parameters is known
we can in principle build up the computational graph. Each vertex in the graph is
associated with an intermediate variable with known value. The edge connecting an
independent variable vj with a dependent variable vi is associated with the derivative
dvi/dvj. As the immediate functional relation between vj and vi is known by the
elementary operation for vi and the intermediate variable values are known we can
evaluate all derivatives between intermediate variables.
Derivatives of dependent output variables with respect to independent input vari-
ables may be evaluated by the repeated elimination of the vertices of intermediate
variables. This can be done by applying the chain rule of differentiation. For a rigor-
ous description of that procedure we refer to [GW08]. Forthcoming we illustrate the
principles of the elimination by means of an example.
A subgraph of the evaluation procedure for the Black’76 formula is given in Figure
B.5. It describes the evaluation of
v8 =v2
v4
− v4
2and v9 = ω
(v2
v4
− v4
2
).
The total derivatives are given by
dv8
dv2
=1
v4
,dv8
dv4
= −v5
v4
− 1
2,
dv9
dv2
=ω
v4
,dv8
dv4
= ω
(−v5
v4
+1
2
).
66
Figure B.5: Subgraph of Black’76 Formula
dv9
dv7= ω
v6 = v42
v5 = v2/v4 v9 = ωv7
v8 = v7 − v4
v7 = v5 + v6
v2
v4
dv5
dv2= 1
v4
dv7dv5
= 1
dv7
dv6= 1
dv5
dv4= −v5
v4
dv6dv4
= 12
dv8dv4
= −1
dv8dv7
= 1
The derivatives of the dependent variables v8 and v9 with respect to the inde-
pendent variables v2 and v4 may be determined by successively eliminating vertices
in the graph of Figure B.5. As a first step we may eliminate the vertex of v6.
The chain rule of differentiation yields that the derivative of v7 w.r.t. v4 becomes
dv7/dv4 = dv7/dv6 · dv6/dv4. The resulting graph is illustrated in Figure B.6.
Figure B.6: Subgraph of Black’76 Formula after Elimination of v6
dv7dv4
= 12 · 1
v5 v9
v8
v7
v2
v4
dv5
dv2= 1
v4
dv7dv5
= 1dv5
dv4= −v5
v4
dv8dv4
= −1
dv8dv7
= 1
dv9
dv7= ω
In the next step we may eliminate the vertex of v5. This introduces a new direct
edge between v2 and v7. The corresponding derivative is dv7/dv2 = dv7/dv5 · dv5/dv2.
Moreover, besides the existing edge between v4 and v8 the elimination of v5 introduces
another connection of v4 to v8 with derivative dv8/dv5 · dv5/dv4. The value of that
derivative is added to the existing derivative dv5/dv4. The resulting graph is shown
in Figure B.7.
67
Figure B.7: Subgraph of Black’76 Formula after Elimination of v6 and v5
dv7
dv4= 1
2 −v5v4
v2
v4
v9
v8
v7
dv8
dv4= −1
dv8dv7
= 1
dv9
dv7= ω
dv7dv2
= 1v4
The last step requires the elimination of the vertex of v7. The derivatives are eval-
uated analogously to the preceding steps. This yields the graph illustrated in Figure
B.8. All dependent variables are directly connected to all independent variables. The
derivatives at the edges describe the complete Jacobian of the mapping(v2
v4
)7→(v8
v9
).
Figure B.8: Subgraph of Black’76 Formula after Elimination of v6, v5, and v7
dv8
dv2= 1
v4
v2
v4
v9
v8
dv9
dv2= 1
v4· ω
dv8dv4
= − 12 −
v5v4
dv9dv4
=(
12 −
v5v4
)· ω
Both the forward and reverse mode of AD may be described in the context of
graph elimination. The order of the vertex elimination in these cases is determined
by the selected AD mode. The additional computational effort for the derivative
evaluation depends on the order of the vertex elimination. The problem of finding
68
the optimal order which causes the minimum additional computations, i.e. Optimal
Jacobian Accumulation Problem, is NP-complete [Nau08].
For this thesis we apply the AD tools ADOL-C [GJU96] and ADTAGEO [RG09].
ADOL-C is based on operator overloading in C++. It allows the evaluation of first
and higher order derivatives with forward and reverse mode. We use particularly its
tapeless vector forward mode.
The AD tool ADTAGEO is based on graph elimination. It builds up a computa-
tional graph of independent and dependent variables. However, it does not construct
the entire computational graph. Whenever a variable is deallocated or reassigned
its vertex in the graph is instantaneously eliminated. Graph elimination AD tolls
often require more computational effort than tools based on operator overloading or
source transformation. However, ADTAGEO’s programming interface allows easy
and flexible derivative evaluation.
69
Appendix C
Option Pricing via Integration
In this chapter we elaborate an alternative approach for the pricing of Bermudan
bond options in the Hull White model. The method is based on the reformulation of
the Hull White model in the time-T neutral measure. For references, see for example
[BM07]. The fundamental theorem of asset pricing yields for the price V (t, r(t)) of a
security depending on the time t and a (also time-dependent) risk factor r(t) that
V (t, r(t)) = ZCB(0; t, T, r(t)) · ET [V (T, r(T )) | F(t)] for t < T.
In this representation ZCB(0; t, T, r(t)) is the time-t price of a zero coupon bond
maturing at time T . The zero coupon price depends on the model calibration at
time-0 and the time-t state of the risk factor r(t). For the Hull White model the risk
factor is the short rate. The analytical formula for ZCB() is elaborated in Section
2.2.2.
The expectation ET conditional on the information at time t is evaluated in the
time-T neutral measure. That is the numeraire applied is the zero coupon bond
maturing at time T . In our setting the price of the numeraire is given by the ZCB()
formula. For the pricing of the option in this setting we require the dynamics of the
short rate r(t) in the time-T neutral measure.
Provided we can evaluate ET [V (T, r(T )) | F(t)] for a given option price or payoff
at time T then we can also price Bermudan bond options. We discretise the short rate
by a grid r0, . . . , rn. Analogously to the PDE approach we start at the last exercise
date TN and work backwards in time. We evaluate the auxilliary option price
V (TN−1, rj) = ZCB(0;TN−1, TN , rj) · ETN [pN(r(TN)) | F(TN−1)]
for j = 0, . . . , n. Here pN(r) is the Nth payoff function of the Bermudan. The option
price at TN−1 then becomes
V (TN−1, rj) = max{V (TN−1, rj), pN−1(rj)
}for j = 0, . . . , n.
70
The resulting discrete points V (TN−1, r0), . . . , V (TN−1, rn) are interpolated to model
the option price function V (TN−1, r) at time TN−1 and intermediate short rate points
r. We may proceed evaluating
V (TN−2, rj) = ZCB(0;TN−2, TN − 1, rj) · ETN−1 [V (TN−1, r(TN−1)) | F(TN−2)] ,
V (TN−2, rj) = max{V (TN−2, rj), pN−2(rj)
}for j = 0, . . . , n. These steps are repeated until V (T1, r) is available. The desired
price of the Bermudan option is finally determined as
V (0, r(0)) = P (0, T1) · ET1 [V (T1, r(T1)) | F(0)] .
In the following sections we elaborate the dynamics of the short rate in the Hull
White model. We start by describing the risk neutral dynamics since the model is
initially formulated in this setting. By changing the measure we continue elaborating
the dynamics in the time-T neutral measure. Finally we describe the numerical
evaluation of the expectation.
C.1 Risk Neutral Dynamics
The Hull white model in Section 2.2 yields the short rate process
r(T ) = e−a(T−t)[r(t) +
∫ T
t
ea(u−t) (θ(u)du+ σ(u)dW (u))
]for t < T . Here W (t) is a Brownian motion under the risk neutral measure. The
numeraire under the risk neutral measure is the generic bank account which accruals
with the short rate. The price of the bank account Q(t) is given by
Q(t) = exp
{∫ t
0
r(u)du
}.
We find immediately that r(T ) conditional on the information at time t is normally
distributed. The variance is given by
V ar [r(T ) | F(t)] =
∫ T
t
e−2a(T−u)σ2(u)du.
For piecewise constant volatility functions σ(t) the variance can be evaluated analyt-
ically. The expectation becomes
EQ [r(T ) | F(t)] = e−a(T−t)r(t) +
∫ T
t
e−a(T−u)θ(u)du.
71
Our model is calibrated to the observed yield curve at time 0. Hence the drift θ(u) is
θ(u) =∂f(0, u)
∂u+ af(0, u) +
∫ u
0
e−2a(u−s)σ2(s)ds.
Consequently, we have to evaluate∫ T
t
e−a(T−u)
[∂f(0, u)
∂u+ af(0, u)
]du︸ ︷︷ ︸
I1(t,T )
+
∫ T
t
e−a(T−u)
∫ u
0
e−2a(u−s)σ2(s)ds du︸ ︷︷ ︸I2(t,T )
.
The first integral I1(t, T ) may be easily solved as
I1(t, T ) =[e−a(T−u)f(0, u)
]Tu=t
= f(0, T )− e−a(T−t)f(0, t).
The second interval is split again as
I2(t, T ) =
∫ T
t
e−a(T−u)
∫ t
0
e−2a(u−s)σ2(s)ds du︸ ︷︷ ︸I3(t,T )
+
∫ T
t
e−a(T−u)
∫ u
t
e−2a(u−s)σ2(s)ds du︸ ︷︷ ︸I4(t,T )
Now we can change the order of integration. For the integral I3(t, T ) we get
I3(t, T ) =
∫ t
0
∫ T
t
e−a(T−u)e−2a(u−s)du ds
=
∫ t
0
σ2(s)
∫ T
t
e−a(T−u)−2a(u−s)du ds
=
∫ t
0
σ2(s)
[e−a(T−u)−2a(u−s)
−a
]Tu=t
ds
=
∫ t
0
σ2(s)
a
[e−a(T−t)e−2a(t−s) − e−2a(T−s)] ds.
The integral I4(t, T ) becomes
I4(t, T ) =
∫ T
t
∫ T
s
e−a(T−u)e−2a(u−s)σ2(s)du ds
=
∫ T
t
σ2(s)
∫ T
s
e−a(T−u)−2a(u−s)du ds
=
∫ T
t
σ2(s)
[e−a(T−u)−2a(u−s)
−a
]Tu=s
ds
=
∫ T
t
σ2(s)
a
[e−a(T−s) − e−2a(T−s)] ds.72
As a result we end up with
EQ [r(T ) | F(t)] = f(0, T ) + e−a(T−t) [r(t)− f(0, t)] +∫ t
0
σ2(s)
a
[e−a(T−t)e−2a(t−s) − e−2a(T−s)] ds+∫ T
t
σ2(s)
a
[e−a(T−s) − e−2a(T−s)] ds.
The expectation EQ [r(T ) | F(t)] and the variance V ar [r(T ) | F(t)] completely de-
scribe the normally distributed short rate r(T ).
C.2 Forward Neutral Dynamics
The dynamics of the short in the time-T neutral measure are evaluated by comparing
the dynamics of the bank account compared to the time-T maturing zero coupon
bond. The dynamics of the bank account are given by
dQ(t) = r(t)Q(t)dt.
In the Hull White model the price of a zero coupon bond is given by
ZCB(t, T, r(t)) = exp
{−(B(t, T )r(t) +
∫ T
t
θ(u)B(u, T )du− 1
2
∫ T
t
σ2(u)B2(u, T )du
)}with B(t, T ) =
[1− e−a(T−t)] /a. Applying Ito’s lemma yields the zero coupon bond
dynamics
dZCB(t, T, r(t))
ZCB(t, T, r(t))=
[−Bt(t, T )r(t) + θ(t)B(t, T )− 1
2σ2(t)B2(t, T )
]dt
−B(t, T )dr(t) +1
2B2(t, T ) [dr(t)]2
With [dr(t)]2 = σ2(t)dt and Bt(t, T ) = aB(t, T )− 1 follows
dZCB(t, T, r(t))
ZCB(t, T, r(t))= [−Bt(t, T )r(t) + θ(t)B(t, T )] dt−B(t, T )dr(t)
= [(1− aB(t, T ))r(t) + θ(t)B(t, T )] dt−B(t, T )dr(t).
We choose ZCB(t, T, r(t)) as the numeraire. The fundamental theorem of asset
pricing (???) yields that the price process of any asset in our economy discounted with
that numeraire is a martingale. In particular we must have that Q(t)/ZCB(t, T, r(t))
is a martingale. We get
d
(Q(t)
ZCB(t, ·)
)=
Q(t)
ZCB(t, ·)
[dQ(t)
Q(t)− dZCB(t, ·)
ZCB(t, ·)+
(dZCB(t, ·)ZCB(t, ·)
)2].
73
By substituting the processes for Q(t) and ZCB(t, T, r(t)) follows
d
(Q(t)
ZCB(t, ·)
)=
Q(t)
ZCB(t, ·)
[r(t)dt− (1− aB(t, T ))r(t)dt
− θ(t)B(t, T )dt+B(t, T )dr(t) +B(t, T )2σ2(t)dt]
=Q(t)
ZCB(t, ·)B(t, T )
[dr(t)−
(θ(t)−B(t, T )σ2(t)− ar(t)
)dt].
The martingale condition of Q(t)/ZCB(t, T, r(t)) and the fact that the variance of
r(t) is invariant with respect to a change of measure yield that
dr(t)−[θ(t)−B(t, T )σ2(t)− ar(t)
]dt = σ(t)dW T (t).
The processW T (t) is a Brownian motion in the time-T forward measure with ZCB(t, T, r(t))
as numeraire. Consequently, the dynamics of the Hull White model in that forward
measure are
dr(t) =[θ(t)−B(t, T )σ2(t)− ar(t)
]dt+ σ(t)dW T (t).
Thus the change of measure yields the additional drift term −B(t, T )σ2(t)dt in the
time-T forward measure compared to the risk neutral measure.
Using the results in Section C.1 for for the expectation of r(T ) in the risk neutral
measure we get that the expectation ET in the forward neutral measure becomes
ET [r(T ) | F(t)] = EQ [r(T ) | F(t)]−∫ T
t
σ2(u)e−a(T−u)B(u, T )du.
Moreover, we find that∫ T
t
σ2(u)e−a(T−u)B(u, T )du =
∫ T
t
σ2(u)
a
[e−a(T−u) − e−2a(T−u)
]du = I4(t, T ).
The integral term I4(t, T ) already appeared in the risk neutral expectation in Section
C.1. As a result we get the forward neutral expectation as
ET [r(T ) | F(t)] = f(0, T ) + e−a(T−t) [r(t)− f(0, t)] +∫ t
0
σ2(s)
a
[e−a(T−t)e−2a(t−s) − e−2a(T−s)] ds.
In particular, we see that the expectation depends on the volatility only from time 0
to time t. The volatility between t and T only occurs in the variance term.
74
C.3 Solving the Integral
At the beginning of this chapter we elaborate that an option price in the Hull White
model can be evaluated by
V (t, r(t)) = ZCB(0; t, T, r(t)) · ET [V (T, r(T )) | F(t)] for t < T.
In this Section we focus on the numerical evaluation of ET [V (T, r(T )) | F(t)]. The
risk factor (i.e. the short rate) is normally distributed with mean µ = ET [r(T ) | F(t)]
and variance σ2 = V ar [r(T ) | F(t)]. The expressions for µ and σ2 are evaluated in
the preceding sections.
The distributional properties of r(T ) yield that
ET [V (T, r(T )) | F(t)] =1√
2πσ2
∫ +∞
−∞V (T, u) exp
{−(u− µ)2
2σ2
}︸ ︷︷ ︸
q(u)
du.
The integration is split into two parts as
ET [V (T, r(T )) | F(t)] =1√
2πσ2
[∫ µ
−∞q(u)du +
∫ +∞
µ
q(u)du
].
For the numerical integration of∫ +∞µ
q(u)du we specify a tolerance tol and use an
adaptive discretisation scheme. We set u0 = µ and guess an initial discretisation
width h0. In each step i = 0, 1, . . . the numerical integral from ui to ui + hi is
evaluated by the trapezoidal rule and Simpson’s rule, i.e.
ITrapi =hi2
(q(ui) + q(ui + hi))
ISimpi =hi6
(q(ui) + 4q(ui + hi/2) + q(ui + hi))
Provided q(u) is sufficiently smooth for u ∈ [ui, ui + hi] we get the approximations∫ ui+hi
ui
q(u)du = ITrapi +h3i
12f ′′(ξ) = ISimpi +
h5i
2880f (4)(η)
for ξ, η ∈ [ui, ui + h]. The error of the numerical integration erri is estimated by
erri =∣∣∣ITrapi − ISimpi
∣∣∣ .The step i is accepted if
erri ≤ hi · tol.
75
If the step is accepted we set ui+1 = ui + hi. The new step size hi+1 is determined by
hi+1 = hi ·min
{√tol
erri + ε, 2
}.
The constant ε = 10−32 is incorporated to avoid division by zero. If the integration
step is not accepted, i.e. erri > hi · tol, then we set hi ← hi/2 and repeat step i until
acceptance.
The numerical integration is repeated for i = 0, 1, . . . , n until the residual part∫ +∞un+1
q(u)du is sufficiently small. Unfortunately for general payoffs this can not be
evaluated exactly. We use a heuristic argument and integrate until
max
{[1− Φ−1
(un+1 − µ
σ
)], V (un+1)
[1− Φ−1
(un+1 − µ
σ
)]}≤ tol2.
Finally the numerical integral is summed up as∫ +∞
µ
q(u)du ≈n∑i=0
ISimpi .
The second integral∫ µ−∞ q(u)du is treated analogously. As a result we end up with
an approximation of ET [V (T, r(T )) | F(t)] which is considered to be bounded by tol.
76
Figure C.1: Asymptotic approximation of PDE solution of reference European swap-tions.
Excercise Date 20
Excercise Date 1
Relative Error of Numerical Solution
Number of spatial grid points n Number of spatial grid points n
Reference 10/n2
Excercise Date 30
Excercise Date 10
0.0001
101
1e-07
1e-06
1e-05
201 501 1001 201
0.0001
0.001
101
1e-07
501 1001
1e-06
1e-05
0.001
77
Figure C.2: Asymptotic approximation of average sensitivities of reference Europeanswaptions
number of spatial grid points n
Excercise Date 10Excercise Date 1
Excercise Date 30
Reference 10/n2Relative Error of Sensitivity
number of spatial grid points n
Excercise Date 20
0.0001
501 1001
0.001
101 201 501 1001 101
0.01
0.001
0.0001
1e-07
1e-06
1e-05
201
78
References
[AS09] M. Alexe and A. Sandu. On the discrete adjoints of adaptive time step-
ping algorithms. Journal of Computational and Applied Mathematics,
233:1005–1020, 2009.
[BM07] D. Brigo and F. Mercurio. Interest Rate Models - Theory and Practice.
Springer-Verlag, 2007.
[Chr94] B. Christianson. Reverse accumulation and attractive fixed points. Opti-
mization Methods and Software, 3:311–326, 1994.
[Duf06] D.J. Duffy. Finite difference methods in financial engineering. John Wiley
& Sons Ltd, 2006.
[EB99] P. Eberhard and C. Bischof. Automatic differentiation of numerical inte-
gration algorithms. Mathematics of Computation, 68:717–731, 1999.
[GC06] M.B. Giles and R. Carter. Convergence analysis of crank-nicolson and
rannacher time-marching. Journal of Computational Finance, 9:89–112,
2006.
[GJU96] A. Griewank, D. Juedes, and J. Utke. ADOL-C: A package for automatic
differentiation of algorithms written in C/C++. TOMS, 22:131–167, 1996.
[GR94] Chr. Großmann and H.-G. Roos. Numerik partieller Differentialgleichun-
gen. Teubner, 2 edition, 1994.
[Gri86] A. Griewank. The ”global” convergence of Broyden-like methods with a
suitable line search. J. Austral. Math. Soc. Ser. B, 28:75–92, 1986.
[GW08] A. Griewank and A Walther. Evaluating derivatives: principles and tech-
niques of algorithmic differentiation - 2nd ed. SIAM, 2008.
79
[Hag] P. S. Hagan. Evaluating and Hedging Exotic Swap Instruments via LGM.
Bloomberg Technical Report.
[Hen04] M. Henrard. Semi-explicit delta and gamma for european swaptions in
hull-white one-factor model. (Ewp-fin 0411036), 2004.
[HKLW02] P.S. Hagan, D. Kumar, A.S. Lesniewski, and D.E. Woodward. Managing
smile risk. Wilmott magazine, September:84–108, 2002.
[HNW93] E. Hairer, S. P. Norsett, and G. Wanner. Solving Ordinary Differential
Equations I. Springer, 1993.
[HV03] W. Hundsdorfer and J.G. Verwer. Numerical Solution ot Time-Dependent
Advection-Diffusion-Reaction Equations. Springer, 2003.
[HW90] J.C Hull and A. White. Pricing interest-rate-derivative securities. The
Review of Financial Studies, 3:573–592, 1990.
[HW91] E. Hairer and G. Wanner. Solving Ordinary Differential Equations II.
Springer, 1991.
[HW06] P.S. Hagan and G. West. Interpolation methods for curve construction.
Applied Mathematical Finance, 13:89–129, 2006.
[Jam89] F. Jamshidian. An exact bond option pricing formula. The Journal of
Finance, 44:205–209, 1989.
[Nau08] U. Naumann. Optimal jacobian accumulation is np-complete. Mathemat-
ical Programming, 112:427–441, April 2008.
[NW06] J. Nocedal and S. J. Wright. Numerical Optimization. Springer-Verlag,
2006.
[Pit04] V.V. Piterbarg. Risk sensitivities of bermudan swaptions. International
Journal of Theoretical and Applied Finance, 7:465–510, 2004.
[RG09] J. Riehme and A. Griewank. Algorithmic differentiation through au-
tomatic graph elimination ordering (ADTAGEO). In U. Naumann,
O. Schenk, H.D. Simon, and S. Toledo, editors, Combinatorial Scien-
tific Computing, number 09061 in Dagstuhl Seminar Proceedings. Schloss
Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany, 2009.
80
[RMW09] R. Rebonato, K. McKay, and R. White. The SABR/LIBOR Mar-
ket Model: Pricing, Calibration and Hedging for Complex Interest-Rate
Derivatives. John Wiley & Sons Ltd, 2009.
[Shr04] S. Shreve. Stochastic Calculus for Finance II - Continuous-Time Models.
Springer-Verlag, 2004.
[SW10] P. Stumm and A. Walther. New algorithms for optimal online checkpoint-
ing. SIAM Journal on Scientific Computing, 32:836–854, 2010.
[WFV04] H. Windcliff, P.A. Forsyth, and K.R. Vetzal. Analysis of the stability of
the linear boundary condition for the black-scholes equation. J. Compu-
tational Finance, 8:65–92, 2004.
81
