A Bond Option Pricing Formula in the Extended Cox-Ingersoll-Ross Model · 2016-08-24 · A Bond...

A Bond Option Pricing Formula in the ExtendedCox-Ingersoll-Ross Model

Joint work with Zheng Liu, Henry Schellhorn

Presented by Qidi Peng

Institute of Mathematical Sciences, Claremont Graduate University, U.S.

Conference on Stochastic Asymptotics & Applications, Sixth Western

Conference on Mathematical Finance

Sept. 25, 2014

Q.Peng (CGU) Extended CIR Model Sept. 25, 2014 1 / 21

1 The extended CIR (ECIR) modelFrom Vasicek to ECIR modelMotivation

2 First Main Result: Distribution of r(t)

3 Second Main Result: Bond Option Pricing in ECIR

4 Future Work


The extended CIR (ECIR) model

Plan




4 Future Work


The extended CIR (ECIR) model From Vasicek to ECIR model

Vasicek model, start of our travel...

In finance, the Vasicek model (1977) is a mathematical model whichdescribes the evolution of interest rates.

{dr(t) = b(a− r(t))dt + σ dW (t)r(0) = r0 > 0.

Advantages:

- The first one to capture the ”mean reversion” property.- a > 0 presents the long run equilibrium value. b > 0denotes the speed of ”mean reversion”.

Inconvenience:

- The interest rate might be negative.



This inconvenience can be fixed by Cox-Ingersoll-Ross model,exponential Vasicek model, Black-Derman-Toy model andBlack-Karasinski model, etc.

We study CIR model, because it is difficult and challenging.



CIR Model, ”better” than Vasicek....

The CIR model was first introduced by Cox, Ingersoll and Ross (1985).

{dr(t) = b(a− r(t))dt + σ

√

r(t)dW (t)r(0) = r0 > 0.

Advantages:

− When the dimension (integer) 4abσ2 ≥ 2, the solution is strictly

positive.− The CIR model is not a generalization of Vasicek model, since it isa sum of independent squared Ornstein-Uhlenbeck process (providedthat 4ab

σ2 is integer), while the Vasicek model is anOrnstein-Uhlenbeck process.

Inconveniences:

− The volatility is constant (=⇒ time-dependent model needed).− The assumption that 4ab

σ2 is integer generally doesn’t fit real data(=⇒ extension of noncentral chi-square distribution).



ECIR model, more generalized than CIR...

To remedy the inconveniences of CIR model, we consider an extended CIRmodel (Hull and White, 1990)

{dr(t) =

(θ(t)− b(t)r(t)

)dt + σ(t)

√

r(t)dW (t)r(0) = r0 > 0.

Some known results:

(Maghsoodi, 1996) If4θ(t)σ2(t) ≡ d ∈ N/{0}, then

(Σ(0, t)

)−1r(t) ∼ χ2

(

d ,r0e

−∫ t

0 b(u)du

Σ(0, t)

)

,

where Σ(0, t) := 14

∫ t

0 e−∫ t

vb(u)duσ2(v )dv and χ2(a,λ) denotes a

noncentral chi-square distribution with d degrees of freedom andnoncentrality parameter λ.

(Maghsoodi, 1996) When d = 1, r(t) visits 0 a finite number oftimes over any compact time interval; when d > 1, r(t) > 0.


The extended CIR (ECIR) model Motivation

Begin to participate in the project :)

Question 1: The distribution of (Σ(0, t))−1r(t), if 4θ(t)σ2(t)

≡ d ∈ R+?

Question 2: The distribution of (Σ(0, t))−1r(t), if4θ(t)σ2(t) = d(t) ∈ R

+?

Answer to Question 1:

(Σ(0, t)

)−1r(t) ∼ χ2

(

d ,r0e

−∫ t

0b(u)du

Σ(0, t)

)

,

where χ2(a,λ) denotes a noncentral chi-square distribution with d

(real-valued) degrees of freedom and noncentrality parameter λ, i.e.,

E(e iωχ2(a,λ)) =exp( iλω

1−2iω )

(1− 2iω)a/2.

Remark: r(t) ≥ 0, if r0 > 0 and d > 0.

The answer to Question 2 is our first main result.


First Main Result: Distribution of r(t)

Plan




4 Future Work



”It is difficult to derive the probability distribution of the squared Besselprocesses with time-varying dimensions explicitly.” ——By Shirakawa(2002).



Only have characteristic function :(

Theorem 1: Assume θ(·), σ(·) are continuous positive-valued and d(·) iscontinuously differentiable and positive-valued. The characteristic functionof the interest rate r(t) in the ECIR model is:

E(e iωr(t)) = exp

(

iω( r0e

−∫ t

0 b(u)du

1− 2iωΣ(0, t)−∫ t

0

θ(s)e−∫ t

sb(u)du

1− 2iωΣ(s, t)ds))

,

where:

Σ(s, t) :=1

4

∫ t

se−

∫ t

vb(u)duσ2(v )dv .

Remark: r(t) ≥ 0, if r0 > 0 and d(s) > 0 for all s ≥ 0.



Our result extends Maghsoodi (1996)’s

An equivalent version of E(e iωr(t)):

E(e iωr(t)) =

exp

(

iωr0e−∫ t0 b(u)du

1−2iωΣ(0,t)− 1

2

∫ t

0d ′(s) log

(

1− 2iωΣ(s, t)ds))

(1− 2iωΣ(0, t))d(0)/2.

(1)

When d(t) ≡ d ∈ N/{0},

E(e iωr(t)) =

exp

(

iωr0e−∫ t0 b(u)du

1−2iωΣ(0,t)

)

(1− 2iωΣ(0, t))d(0)/2⇐⇒

(Σ(0, t)

)−1r(t) ∼ χ2

(

d(0),r0e

−∫ t

0b(u)du

Σ(0, t)

)

,

which generalizes Maghsoodi (1996)’s result.

When d(t) ≡ d ∈ R+, (1) generalizes the answer to Question 1.



Our result fits Fokker-Planck equation

Fokker-Planck equation for the density f (x , t) of r(t):

∂f (x , t)

∂t= b(t)f (x , t)+

(σ2(t)+ b(t)x− θ(t)

)∂f (x , t)

∂x+

xσ2(t)

2

∂2f (x , t)

∂x2.

-”Don’t know f (x , t)!”-”Try with Fourier transform of Fokker-Planck.”

The Fourier transform of Fokker-Planck equation with respect to x is

∂f (ω, t)

∂t= −θ(t)iωf (ω, t)−

(b(t)ω +

iω2σ2(t)

2

)∂f (ω, t)

∂ω. (2)

E(e−iωr(t)) =√2π f (ω, t) fits (2).


Second Main Result: Bond Option Pricing in ECIR

Plan




4 Future Work



”Although these researches (Maghsoodi, 2006; Roger, 1995) enable us tostudy the special case of extended CIR model, they do not permit thestudy of the general case.” —–By Shirakawa (2002).



Bond option pricing in ECIR

Let P(t,T ) be the price of a zero-coupon bond at time t, withmaturity T .A classical result (Shreve, 2004):

P(t,T ) = exp(

− r(t)C (t,T ) + A(t,T ))

,

where C (t,T ) satisfies the Riccati equation:{

∂C (t,T )∂t

− b(t)C (t,T )− 12σ2(t)C 2(t,T ) + 1 = 0;

C (T ,T ) = 0.

A(t,T ) = −∫ T

tθ(u)C (u,T )du.

Let K be the strike price of the option. The price C (0) at time 0 of aEuropean option is:

C (0) = E

[

exp(−∫ t

0r(u)du

)max

(P(t,T )−K , 0

)]

.



Change of measure and Laplace transform of density

Fix t as in P(t,T ). By a change of measure (t-forward measure Pt),

there exists a Brownian motion W t(s) such that

dr(s) =(

(−b(s) + σ2(s)C (s, t))︸︷︷︸

bt (s)

r(s)+ θ(s))

ds+ σ(s)√

r(s)dW t(s).

The Laplace transform of the density of the interest rate r(t) underthe t-forward measure P

t can be expressed as: for some suitable p,

Fr(t)(p) = exp

(

−p( r0e

−∫ t

0 bt (u)du

1+ 2pΣt(0, t)−∫ t

0

θ(s)e−∫ t

sbt (u)du

1+ 2pΣt(s, t)ds))

,

where

Σt(s, t) :=

1

4

∫ t

se−

∫ t

vbt (u)duσ2(v )dv .



Pricing C (0)

Theorem 2: The price C (0) is given by Laplace inversion:

C (0) =P(0, t)

2πilim

b→+∞

∫ a+ib

a−ibepr0 Ct(p)Fr(t)(p)dp,

where, Ct(p) is the Laplace transform of the payoff max(P(t,T )− K , 0):

Ct(p) =pK−p−C (t,T )eA(t,T ) − (p + C (t,T ))Kp+1

C (t,T )p − p2.

Proof: it is well-known (see, e.g., Lewis (2000)) that the price of anoption is given by the discounted inverse Laplace transform of the productof the Laplace transform of the payoff and the Laplace transform of thedistribution. �


Future Work

Plan




4 Future Work


Future Work

Next stop...

Conjecture: In ECIR model, r(t) > 0 for all t ≥ 0, providedd(0) > 1; r(t) visits 0 a finite number of times over any compacttime interval, if d(0) ∈ (0, 1].

Unsolved problem: The density of r(t)? Which should beautifullygeneralize the density of noncentral chi-square distribution, and willlargely simplify Theorem 2.


Future Work

References

Vasicek, Oldrich (1977). ”An Equilibrium Characterisation of theTerm Structure”. Journal of Financial Economics 5 (2): 177C188.

Cox, J.C., J.E. Ingersoll and S.A. Ross (1985). ”A Theory of theTerm Structure of Interest Rates”. Econometrica 53: 385C407.

Maghsoodi, Y. (1996). ”Solution of the Extended CIR Term Structureand Bond Option Valuation”. Mathematical Finance (6): 89C109.

Shirakawa, H. (2002). ”Squared Bessel Processes and TheirApplications to the Square Root Interest Rate Model”. Asia-PacificFinancial Markets 9 (3-4): 169-190.

Lewis, A. L. (2000). ”Option Valuation under Stochastic Volatility:with Mathematica Code.” Finance Press.


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