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MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePhase Transition
Phase Transition in a Log-normal Interest RateModel
Dan Pirjol1
1J. P. Morgan, New York
17 Oct. 2011
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
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MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePhase Transition
Outline
Introduction to interest rate modeling
Black-Derman-Toy model
Generalization with continuous state variable Binomial tree
BDT model with log-normal short rate in the terminal measure
Analytical solution Surprising large volatility behaviour
Phase transition
Summary and conclusions
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
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MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePhase Transition
[1] F. Black, E. Derman, W. ToyA One-Factor Model of Interest Rates
Fin. Analysts Journal, 24-32 (1990).[2] P. Hunt, J. Kennedy and A. Pelsser,Markov-functional interest rate modelsFinance and Stochastics, 4, 391-408 (2000)
[3] A. PelsserEfficient methods for valuing interest rate derivativesSpringer Verlag, 2000.
[4] D. Pirjol,Phase transition in a log-normal Markov functional model
J. Math. Phys 52, 013301 (2011).[5] D. Pirjol,Nonanalytic behaviour in a log-normal Markov functional model,arXiv:1104.0322, 2011.
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
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MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePhase Transition
Interest rates
Interest rates are a measure of the time value of money: what is thevalue today of $1 paid in the future?
0 5 10 15 20 25
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
USD Zero Curve
27Sep2011
ExampleZero curve R(t) for USD as of 27-Sep-2011. Gives the discount curve as
D(t) = exp(R(t)t).Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
M i i
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MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePhase Transition
Interest rates evolve in time
0 5 10 15 20 25
0.01
0.02
0.03
0.04
0.05
USD Zero Curve
22Jun to 27Sep
ExampleThe range of movement for the USD zero curve R(t) between22-Jun-2011 and 27-Sep-2011.
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
M ti ti
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MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePhase Transition
Yield curve inversion
0 5 10 15 20 25 30
4.25
4.5
4.75
5
5.25
5.5
5.75
6
USD Zero Rate
1Nov07
ExampleBetween 2006 and 2008 the USD yield curve was inverted - rates for 2years were lower than the rates for shorter maturities
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
Motivation
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MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePhase Transition
Interest rate modeling
The need to hedge against movements of the interest ratescontributed to the creation of interest rate derivatives.
Corporations, banks, hedge funds can now enter into many types ofcontracts aiming to mitigate or exploit/leverage the effects of theinterest rate movements
Well-developed market. Daily turnover for1:
interest rate swaps $295bn forward rate agreements $250bn interest rate options (caps/floors, swaptions) $70bn
This requires a very good understanding of the dynamics of theinterest rates markets: interest rate models
1The FX and IR Derivatives Markets: Turnover in the US, 2010, Federal Reserve
Bank of New YorkDan Pirjol Phase Transition in a Log-Normal Interest Rate Model
Motivation
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MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePhase Transition
Setup and definitions
Consider a model for interest rates defined on a set of discrete dates(tenor)
0 = t0 < t1 < t2 tn1 < tnAt each time point we have a yield curve.
Zero coupon bonds Pi,j: price of bond paying $1 at time tj, as of time ti.Pi,j is known only at time ti
Li = Libor rate set at time ti for the period (ti, ti+1)
Li =1
1
Pi,i+1 1
...0
Li
1 i i+1 n...
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
Motivation
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MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePhase Transition
Simplest interest rate derivatives: caplets and floorlets
Caplet on the Libor Li with strike K pays at time ti+1 the amount
Pay = max(Li(ti) K, 0)
Similar to a call option on the Libor Li
Caplet prices are parameterized in terms of caplet volatilities i via theBlack caplet formula
Caplet(K) = P0,i+1CBS(Lfwd
i , K, i, ti)
Analogous to the Black-Scholes formula.
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
Motivation
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Black, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
Analogy with equities
Define the forward Libor Li(t) at time t, not necessarily equal to ti
Li(t) =1
Pt,iPt,i+1
1
This is a stochastic variable similar to a stock price, and its evolution canbe modeled in analogy with an equity
Assuming that Li(ti) is log-normally distributed one recovers the Blackcaplet pricing formula
Generally the caplet price is the convolution of the payoff with (L), theLibor probability distribution function
Caplet(K) =
0
dL(L)(L K)+
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
Motivation
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Black, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
Caplet volatility - term structure
t
ATM
Volatility hump at the short end
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
Motivation
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Black, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
ATM caplet volatility - term structure
0 2 4 6 8 10
20
40
60
80
100
1203m caplet vols
27Sep11
Actual 3m USD Libor caplet yield (log-normal) volatilities as of
27-Sep-2011Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBl k D T d l
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Black, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
Caplet volatility - smile shape
Fwd
ATM
(K)
Strike
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack De man To model
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Black, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
Modeling problem
Construct an interest rate model compatible with a given yield curve P0,iand caplet volatilities i(K)
1. Short rate models. Model the distribution of the short rates Li(ti) at
the setting time ti. Black, Derman, Toy model
Hull, White model - equivalent with the Linear Gaussian Model(LGM)
Markov functional model
2. Market models. Describe the evolution of individual forward LiborsLi(t)
Libor Market Model, or the BGM model.
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack Derman Toy model
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Black, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
The natural (forward) measure
Each Libor Li has a different natural measure Pi+1
Numeraire = Pt,i+1, the zero coupon bond maturing at time ti+1The forward Libor Li(t)
Li(t) =1
Pt,iPt,i+1
1
is a martingale in the Pi+1 measure
Li(0) = Lfwdi = E[Li(ti)]
This is the analog for interest rates of the risk-neutral measure for equities
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack Derman Toy model
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Black, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
Simple Libor market model
Simplest model for the forward Libor Li(t) which is compatible with agiven yield curve P0,i and log-normal caplet volatilities iLog-normal diffusion for the forward Libor Li(t): each Li(t) driven by itsown separate Brownian motion Wi(t)
dLi(t) = Li(t)idWi(t)
with initial condition Li(0) = Lfwdi , and Wi(t) is a Brownian motion inthe measure Pi+1
Problem: each Libor Li(t) is described in a different measure.
We would like to describe the joint dynamics of all rates in a commonmeasure.This line of argument leads to the Libor Market Model. Simplerapproach: short rate models
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
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Black, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
The Black, Derman, Toy model
Describe the joint distribution of the Libors Li(ti) at their setting times ti
n1
0
Li
1 i i+1 n... ...
L L L L0 1 n2
Libors (short rate) Li(ti) are log-normally distributed
Li(ti) = Lieix(ti) 12
2i ti
where Li are constants to be determined such that the initial yield curveis correctly reproduced (calibration)
x(t) is a Brownian motion. A given path for x(t) describes a particularrealization of the Libors Li(ti)
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
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, , yModel with log-normal rates in the terminal measure
Phase Transition
Black, Derman, Toy model
The model is formulated in the spot measure, where the numeraireB(t) is the discrete version of the money market account.
B(t0) = 1
B(t1) = 1 + L0
B(t2) = (1 + L0)(1 + L1)
Model parameters:
Volatility of Libor Li is i Coefficients Li
The volatilities i are calibrated to the caplet volatilities (e.g. ATM vols),and the Li are calibrated such that the yield curve is correctly reproduced.
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
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Model with log-normal rates in the terminal measurePhase Transition
Black, Derman, Toy model - calibration
Price of a zero coupon bond paying $1 at time ti is given by anexpectation value in the spot measure
P0,i = E 1
B(ti)
= E
1(1 + L0)(1 + L1(x1))
(1 + Li1(xi1))
The coefficients Lj can be determined by a forward induction:
P0,1 =1
1 + L0 L0
P0,2 = P0,1E
11 + L1(x1)
= P0,1+
dx2t1
e1
2t1 x2
11 + L1e1x
12
21t1
L1and so on... Requires solving a nonlinear equation at each time step
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
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Model with log-normal rates in the terminal measurePhase Transition
BDT tree
The model was originally formulated on a tree.Discretize the Markovian driver x(t), such that from each x(t) it can
jump only to two values at t = t +
x(t)
xup(t + )
xdown(t + )
d
dd
Mean and variance
E[x(t + ) x(t)] = 12
1
2
= 0
E[(x(t + )
x(t))2] =
1
2
+1
2
=
Choose xup = x +
, xdown = x
with equal probabilities such thatthe mean and variance of a Brownian motion are correctly reproduced
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
M d l i h l l i h i l
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Model with log-normal rates in the terminal measurePhase Transition
BDT tree - Markov driver x(t)
0
+1
1
+2
0
2
E
0 1 2
t
Inputs:
1. Zero coupon bonds P0,iEquivalent with zero rates Ri
P0,i =1
(1 + Ri)i
2. Caplet volatilities i
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
M d l ith l l t i th t i l
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Model with log-normal rates in the terminal measurePhase Transition
BDT tree - the short rate r(t)
r0
r1e1 12
21
r1e1 12
21
r2
e22
222
r2e222
r2e22222
dd
d
dd
d
dd
d
E
0 1 2
t
Calibration: Determiner0, r1, r2, such that thezero coupon prices arecorrectly reproduced
Zero coupon bonds P0,iprices
P0,i = E 1
Bi
Money market account B(t)- node and path dependent
B0 = 1
B1 = 1 + r(1)
B2 = (1 + r(1))(1 + r(2)) .Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
Model with log normal rates in the terminal measure
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Model with log-normal rates in the terminal measurePhase Transition
Calibration in detail
t = 1: no calibration needed
P0,1 =1
1 + r0
t = 2: solve a non-linear equation for r1
P0,2 =1
1 + r0
12
1
1 + r1e1 1221
+1
2
1
1 + r1e1 12 21
and so on for r2, .
Once ri are known, the tree can be populated with values for the shortrate r(t) at each time t
Products (bond options, swaptions, caps/floors) can be priced by working
backwards through the tree from the payoff time to the presentDan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measure
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Model with log-normal rates in the terminal measurePhase Transition
Pricing a zero coupon bond maturing at t = 2
P0,2
Pup1,2 =
11+rup1
Pdown1,2 =1
1+rdown1
1
1
1
dd
d
dd
d
ddd
We know rup1 = r1e1 12
21 and rdown1 = r1e
1 12 21 can find the bond
prices Pt,2 for all t
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measure
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Model with log normal rates in the terminal measurePhase Transition
BDT model in the terminal measure
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measure
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gPhase Transition
BDT model in the terminal measure
Keep the same log-normal distribution of the short rate Li as in the BDTmodel, but work in the terminal measure
Li(ti) = Lieix(ti) 12
2i ti
Li(ti) = Libor rate set at time ti for the period (ti, ti+1)
Numeraire in the terminal measure: Pt,n, the zero coupon bond maturingat the last time tn
...0
L i
1 i i+1 n...
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measure
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gPhase Transition
Why the terminal measure?
Why formulate the Libor distribution in the terminal measure?
Numerical convenience. The calibration of the model is simpler thanin the spot measure: no need to solve a nonlinear equation at eachtime step
The model is a particular parametric realization of the so-calledMarket functional model (MFM), which is a short rate model aimingto reproduce exactly the caplet smile. MFM usually formulated inthe terminal measure.
More general functional distributions can be considered in theMarkov functional model Li(ti) = Lif(xi), parameterized by anarbitrary function f(x). This allows more general Libor distributions.
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measure
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Phase Transition
Preview of results
1. The BDT model in the terminal measure can be solved analytically forthe case of uniform Libor volatilities i = . Solution possible (inprinciple) also for arbitrary i, but messy results.
2. The analytical solution has a surprising behaviour at large volatility:
The convexity adjustment explodes at a critical volatility, such thatthe average Libors in the terminal measure (convexity-adjustedLibors) Li become tiny (below machine precision)
This is very unusual, as the convexity adjustments are supposed tobe well-behaved (increasing) functions of volatility
The Libor distribution function in the natural measure collapses tovery small values (plus a long tail) above the critical volatility
Caplet volatility has a jump at the critical volatility, after which itdecreases slightly
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePh T i i
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Phase Transition
What do we expect to find?2
Convexity adjusted Libor Li = related to the price of an instrumentpaying Li(ti, ti+1) set at time ti and paid at time tn
Price = P0,nEn[Li] = P0,nLi
Floating payment with delay: the convexity adjustment depends on thecorrelation between Li and the delay payment rate L(ti+1, tn)
...0
Li
1 i i+1 n...
2
Argument due to Radu Constantinescu.Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePh T iti
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Phase Transition
Convexity adjustment
Consider an instrument paying the rate Lab at time c. The price isproportional to the average of Lab in the c-forward measure
Price = P0,cEc[Lab]
ab
0
L
a b
ab
c
L bcL
Can be computed approximatively by assuming log-normally distributedLab and Lbc in the b-forward measure, with correlation
Ec[Lab] Lfwdab
1 Lfwdbc (c b)(eabbc
ab 1) + O((Lfwdbc (c b))2)
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePhase Transition
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Phase Transition
Convexity adjustment
The convexity adjustment is negative if the correlation between Lab andLbc is positive
Ec[Lab] Lfwdab
1 Lfwdbc (c b)(eabbc
ab 1) + O((Lfwdbc (c b))2)
0 2 4 6 8
0.2
0.4
0.6
0.8
1
1.2
1.4
a
cbexp1 Convexity-adjusted Libors Lifor 3m Libors (b = a + 0.25 )
Parameters:
ab = bc = 40% = 20%, c = 10 years
Naive expectation: The convexity adjustment is largest in the middle ofthe time simulation interval, and vanishes near the beginning and the end.
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
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Phase Transition
Convexity adjusted Libors - analytical solution
The solution for the convexity adjusted LiborsLi for several values of thevolatility
Simulation parameters: Lfwdi = 5%, n = 40, = 0.25
5 10 15 20 25 30 35 40
1
2
3
4
5
=0.2
=0.3
=0.4
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
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Phase Transition
Surprising results
For sufficiently small volatility , the convexity-adjusted Libors Liagree with expectations from the general convexity adjustmentformula
For volatility larger than a critical value cr, the convexityadjustment grows much faster.
The model has two regimes, of low and large volatility, separated bya sharp transition
Practical implication: the convexity-adjusted Libors Li become very
small, below machine precision, and the simulation truncates themto zero
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
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Explanation
The size of the convexity adjustment is given by the expectation value
Ni = E[Pi,i+1ex 12
2ti]
Recall that the convexity-adjusted Libors are Li = Lfwdi /Ni
0 0.1 0.2 0.3 0.4 0.5
2
4
6
8
10
Plot of log Ni vs the volatility
Simulation with n = 40 quarterlytime steps
i = 30, t = 7.5, r0 = 5%
Note the sharp increase after a critical volatility cr 0.33
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
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Explanation
The expectation value as integral
Ni = E[Pi,i+1ex 12
2ti] =
+
dx2ti
e 12ti x
2
Pi,i+1(x)ex 12
2ti
5 10 15 17.50
0.5
1
1.5
2
x
The integrand
Simulation with n = 20 quarterlytime steps i = 10, ti = 2.5
=
0.4 (solid)0.5 (dashed)
0.52 (dotted)
Note the secondary maximum which appears for super-critical volatilityat x
10
ti. This will be missed in usual simulations of the model.
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
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More surprises: Libor probability distribution
Above the critical volatility > cr, the Libor distribution in the naturalmeasure collapses to very small values, and develops a long tail
0 0.02 0.04 0.06 0.08 0.1
10
20
30
40
50
60
70
0.050.35
0.1
0.2
= =
=
=
L
Simulation with n = 40 quarterly time steps = 0.25. The plot refers tothe Libor L30 set at time ti = 7.5.
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
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Caplet Black volatility
0 0.1 0.2 0.3 0.4
0
0.25
0.5
0.75
1
1.25
1.5Black curve: ATM caplet volatility
Red curve: equivalent log-normalLibor volatility
2LNti = logE[L21]
E[L1]2
Simulation with n = 40 quarterly time steps = 0.25. The plot refers tothe Libor L30 set at time ti = 7.5
For small volatilities, the ATM caplet vol is equal with the Libor vol i.Above the critical volatility, the ATM caplet vol increases suddenly
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
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Caplet smile
Above the critical volatility the caplet implied volatility develops a smile
0.01 0.02 0.03 0.04 0.05 0.06
0.2
0.4
0.6
0.8
1
1.2
1.4
0.10.20.3
=0.34
0.35=
=
==
Simulation with n = 40 quarterly time steps = 0.25. The plot refers tothe Libor L30 set at time ti = 7.5.
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
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Conclusions
For sufficiently small volatility, the model with log-normallydistributed Libors in the terminal measure produces a log-normalcaplet smile
The probability distribution for the Libors in the natural (forward)measure is log-normal
Above a critical volatility cr the Libor probability distributioncollapses at very small values, and develops a fat tail
The caplet Black volatility increases suddenly above the criticalvolatility, and develops a smile
These effects are due to a coherent superposition of convexityadjustments
In practice we would like to use the model only in the sub-critical regime.Under what conditions does this transition appear, and how can we findthe critical volatility?
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
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Phase Transition
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePhase Transition
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The generating function
We would like to investigate the nature of the discontinuous behaviourobserved at the critical volatility, and to calculate its value
Define a generating function for the coefficients c(i)j giving the one-step
zero coupon bond
f(i)(x) =ni1j=0
c(i)j x
j
This was motivated by a simpler solution for the recursion relation
The expectation value which displays the discontinuity is simply
Ni =
ni1j=0
c(i)j e
j2ti = f(i)(e2ti)
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePhase Transition
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Properties of the generating function
f(i)(x) is a polynomial in x of degree n i 1
f(i)(x) = 1 + c(i)1 x + c
(i)2 x
2 + + c(i)ni1xni1
where the coefficients are all positive and decrease with j
Can be found in closed form in the zero and infinite volatility limits 0,
It has no real positive zeros, but has n i 1 complex zeros. They arelocated on a curve surrounding the origin.
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePhase Transition
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Complex zeros of the generating function
Example: simulation with n = 40 quarterly time steps = 0.25, flatforward short rate r0 = 5%
-4 -2 0 2 4
-4
-2
0
2
4
i30, psi0.3
The zeros of the generating functionat i = 30, corresponding to the
Libor set at ti = 7.5 years
Number of zeros = n i 1 = 9
Volatility = 30%
Key mathematical result: The generating function f(i)(x) is continuousbut its derivative has a jump at the point where the zeros pinch the realpositive axis
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePhase Transition
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Complex zeros - volatility dependence
-4 -2 0 2 4
-4
-2
0
2
4
i30, psi0.3
-4 -2 0 2 4
-4
-2
0
2
4
i30, psi0.31
-4 -2 0 2 4
-4
-2
0
2
4
i30, psi0.32
-4 -2 0 2 4
-4
-2
0
2
4
i30, psi0.33
0 0.1 0.2 0.3 0.4 0.5
2
4
6
8
10
Solid curve: plot off(i)
(e
2ti
)
Criterion for determining the critical volatility: The critical point at whichthe convexity adjustment increases coincides with the volatility where thecomplex zeros cross the circle of radius R1 = e
2ti
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePhase Transition
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Complex zeros - effect on caplet volatility
-4 -2 0 2 4
-4
-2
0
2
4
i30, psi0.3
-4 -2 0 2 4
-4
-2
0
2
4
i30, psi0.31
-4 -2 0 2 4
-4
-2
0
2
4
i30, psi0.32
-4 -2 0 2 4
-4
-2
0
2
4
i30, psi0.33
0 0.1 0.2 0.3 0.40
0.25
0.5
0.75
1
1.25
1.5
Black curve: ATM caplet volatility
vs
Simulation with n = 40 quarterly time steps, at i = 30The turning point in ATM coincides with the volatility where thecomplex zeros cross the circle of radius R1 = e
2ti
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePhase Transition
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Phase transition
The model has discontinuous behaviour at a critical volatility cr
The critical volatility at time ti is given by that value of the modelvolatility for which the complex zeros of the generating functionf(i)(z) cross the circle of radius e
2ti
The position of the zeros and thus the critical volatility depend on
the shape of the initial yield curve P0,i For a flat forward short rate P0,i = e
r0ti the zeros are zk = er0xk,
where xk are the complex zeros of the simple polynomial
Pn(x) =1
1
e
r0+ x + x2 + + xni1
Approximative solution for the critical volatility at time ti
2cr 1
i(n i 1) log 1
r0
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
MotivationBlack, Derman, Toy model
Model with log-normal rates in the terminal measurePhase Transition
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Phase transition - qualitative features
The critical volatility decreases as the size of the time step isreduced, approaching a very small value in the continuum limit
The critical volatility increases as the forward short rate r0 isreduced, approaching a very large value as the rate r0 becomes very
small the applicability range of the model is wider in the smallrates regime
The phenomenon is very similar with a phase transition in condensedmatter physics, e.g. steam-liquid water condensation/evaporation, orwater freezing
The Lee-Yang theory of phase transitions relates such phenomena to theproperties of the complex zeros of the grand canonical partition function.
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
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Motivation
Black, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
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Conclusions
The Black, Derman, Toy model with log-normally distributed Liborin the terminal measure can be solved exactly in the limit ofconstant and uniform rate volatility
The analytical solution shows that the model has two regimes atlow- and high-volatility, with very different qualitative properties
The solution displays discontinuous behaviour at a certain criticalvolatility cr
Low volatility regime
Log-normal caplet smile Well-behaved Libor distributions
High volatility regime
The convexity adjustment explodes Libor pdf is concentrated at very small values, and has a long tail A non-trivial caplet smile is generated
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
Motivation
Black, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
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Comments and questions
A similar behaviour is expected also in a model with non-uniformLibor volatilities (time dependent), but the form of the analyticalsolution is more complicated
Is this phenomenon generic for models with log-normally distributedshort rates, e.g. the BDT model, or is it rather a consequence ofworking in the terminal measure?
Are there other interest rate models displaying similar behaviour?
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
Motivation
Black, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
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BDT model in the terminal measure - calibration
and exact solution
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
Motivation
Black, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
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Calibrating the model by backward recursion
The non-arbitrage condition in the terminal measure tells us that the zerocoupon bonds divided by the numeraire should be martingales
Pi,j
Pi,n= E[
1
Pj,n|Fi] , for all pairs (i,j)
Denote the numeraire-rebased zero coupon bonds as
Pi,j =Pi,j
Pi,n
Choose the two cases (i,j) = (i, i + 1)
Pi,i+1(xi) = E[Pi+1,i+2(1 + Li+1)|Fi] (i,j) = (0, i)
P0,i = E[Pi,i+1(1 + Li)]
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
Motivation
Black, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
R f P ( ) L
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Recursion for Pi,i+1(xi), Li
The two non-arbitrage relations can be used to construct recursivelyPi,i+1(xi) and Li working backwards from the initial conditions
Pn1,n(x) = 1 , Ln1 = Lfwdn1
No root finding is required at any step.
The calculation of the expectation values requires an integration overxi+1 at each step. Usual implementation methods:
1. Tree. Construct a discretization for the Brownian motion x(t)
2. SALI tree. Interpolate the function Pi,i+1(xi) between nodes andperform the integrations numerically
3. Monte Carlo implementation
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
Motivation
Black, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
A l i l l i f if
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Analytical solution for uniform
Consider the limit of uniform Libor volatilities i = The model can be solved in closed form starting with the ansatz
Pi,i+1(x) =
ni1j=0
c(i)j e
jx 12j22ti
Matrix of coefficients c(i)j is triangular (e.g. for n = 5)
c =
c(n1)0 0 0 0 0
c(n2)0 c
(n2)1 0 0 0
c(n
3)0 c
(n
3)1 c
(n
3)2 0 0
c(1)0 c
(1)1 c
(1)2 c
(1)3 0
c(0)0 c
(0)1 c
(0)2 c
(0)3 c
(0)4
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
Motivation
Black, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
R i l i f h ffi i(i )
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Recursion relation for the coefficients c(i)
j
The coefficients c(i)j and convexity adjusted Libors Li can be determined
by a recursion
c(i)j = c
(i+1)j + Li+1c
(i+1)j
1 e(j1)2ti+1
Li =P0,i P0,i+1
ni1j=0 c(i)j e
j2ti= Lfwdi
P0,i+1
ni1j=0 c(i)j e
j2ti
starting with the initial condition
Ln1 = P0,n1 1 , Pn1,n(x) = 1
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
Motivation
Black, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
R i f th ffi i t(i )
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Recursion for the coefficients c(i)
j
Linear recursion for the coefficients c(i)j . They can be determined
backwards from the last time point starting with c(n1)0 = 1
c(i)j = c
(i+1)j + Li+1c
(i+1)j1 e
(j1)2ti+1
i = n 1 :
i = n 2 :
i = n 3 :
c(n1)0
c
(n
2)
0
c(n3)0
c
(n
2)
1
c(n3)1 c
(n3)2
T
dd
ds
T T
dd
ds
dd
ds
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
Motivation
Black, Derman, Toy modelModel with log-normal rates in the terminal measure
Phase Transition
S l ti f th d l
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Solution of the model
Knowing Pi,i+1
(x) and Li
one can find all the zero coupon bond prices
Pi,j(x) =Pi,j(x)
Pi,i+1(x)(1 + Liiex 12 2ti)
where
Pi,j(x) = E 1
Pj,n|Fi
= E[Pj,j+1(1 + Ljjexj 12
2tj))|Fi]
=
nj1k=0
c(j)k e
kx 12 (k)2ti
+Ljj
nj1k=0
c(j)k e
(k+1)x 12 (k2+1)2ti+k
2(tjti)
All Libor and swap rates can be computed along any path x(t)
Dan Pirjol Phase Transition in a Log-Normal Interest Rate Model
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