Presence of Jumps
Nicolas Krampe
Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam
Faculty of Economics and Business Amsterdam School of Economics
Author: Nicolas Krampe
Student nr: 11374942
Email: nicolas.krampe@student.uva.nl
Date: July 11, 2017
Supervisor: Prof. Dr. Ir. Michel Vellekoop Second reader: Prof. Dr. R.J.A. Laeven
This document is written by Nicolas Krampe who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Nicolas Krampe, July 11, 2017
Abstract
In this thesis we gain a thorough understanding on why and what ef-fect allowing for jumps in the stock price process has on the American put option value as well as on its early exercise boundary. In this con-text we discuss rebalancing costs which are the main driver of these effects.
The approach proposed in this thesis closely follows the one of
Chiarella and Ziogas (2009). We derive formulas for the American put option value as well as of its early exercise boundary by rewriting the homogeneous integro-partial-differential-equation (IPDE) defined on an unknown interval into an inhomogeneous IPDE defined on a known interval. By using the Fourier transform we are able to trans-from this inhomogeneous IPDE into an inhomogeneous ordinary differ-ential equation (ODE). The solution to this inhomogenous ODE can be readily found. The resulting solution for the American put option value and its early exercise boundary is a set of two dependent inte-gral equation which can be solved by standard numerical techniques for Volterra integral equations.
After deriving these theoretical results which hold valid for all contin-uous jump size distributions, we discuss the numerical implementation for the specific case of the Merton jump diffusion model.
We find that jumps have a large impact on close at-the-money Amer-ican put option values. They do also affect the location and slope of the early exercise boundary.
Keywords Merton, Jump Diffusion Process, Early Exercise Boundary, Rebalancing Costs, Method of Lines, American Option, Put Option, Fourier Transformation, Integro Partial Differ-ential Equation
Preface v
1 Introduction 1
2 The Jump Diffusion Process and the Implied Rebalancing Costs 2
2.1 The Continuation Region and the Stopping Region . . . 2
3 Theoretical results on the IPDE of American Put Options 4 3.1 The Inhomogeneous IPDE of the American Put Option . . . 4
3.2 The Fourier Transform and the resulting inhomogeneous ODE . . . 5
3.3 The Pricing Formula for the American Put Option . . . 12
3.4 The Early Exercise Boundary at Expiration . . . 14
4 Investigation of Merton’s Jump Diffusion Model 16 4.1 The Delta of the American Put Option. . . 21
4.2 The Early Exercise Boundary at Expiration . . . 22
5 Numerical Implementation and Results 24 5.1 Numerical Implementation . . . 24
5.2 Numerical Results . . . 27
6 Conclusion 33
Appendix: Complimentary Figures and Tables 34
References 36
Having been of great support throughout the Master’s Thesis as su-pervisor I want to thank Prof. Michel Vellekoop for his interest in the topic and the help he provided. I also want to thank each of the authors I am referring to as without their outstanding prior research this thesis would have not been possible in this form. I am grateful to have gone through the process of writing this thesis as it has further expanded my knowledge on option pricing which was my initial motivation in the topic choice.
Introduction
The valuation of American options on equity has been discussed intensively in literature due to its early exercise feature. Until now no explicit pricing formula has been found, but numerous different approaches have been proposed on how to approximate the price of such an option. Since American options are very common derivatives traded in the market, the importance of accurate valuation of these products is obvious. Detemple
(2014) gives a detailed overview on the various pricing approaches.
The essence of every pricing model is the stock price process assumption. While it is still very common to assume pure diffusion processes, Ball and Torous(1985) found statistical evidence for a majority of NSYE listed common stocks exhibiting lognormally distributed jumps. Given these findings of leptokurtosis, which cannot be explained by a pure diffusion model, we will investigate the pricing of American put options given a jump diffusion stock price process.
Given a jump diffusion stock price process, the American option has an early exercise premium which cannot be derived independently of its option price. This turns out to be a major difficulty when trying to approximate American option values given such stock price processes. Chiarella and Ziogas (2009) derived two dependent integral equations for the early exercise boundary and option price of American call options given a jump diffusion stock price process. They solved this set by extending the approximation of Kim integral equations (Kim(1990)) proposed byKallast and Kivinukk (2003). In this thesis we will perform a similar approach to find an approximation for the early exercise boundary and option value of American put options given a jump diffusion stock price process.
The remainder of the thesis is structured as follows. In Chapter2, after giving a short introduction on the underlying stock price process, we provide intuition for why the early exercise premium differs between a pure diffusion and a jump diffusion process. After that, in Chapter3, we derive formulas for the early exercise boundary and the American put option price which, as in the the call case, will be a set of two dependent integral equations. In Chapter4 we focus on lognormally distributed jump sizes. We proceed in Section5.1with the numerical implementation of the pricing algorithm given Merton’s jump diffusion model. In Section5.2we discuss the resulting price differences as well as the differences in the early exercise boundaries between American put option when the underlying follows either a jump diffusion or a pure diffusion prices. In Chapter 6 we conclude.
The Jump Diffusion Process and
the Implied Rebalancing Costs
Before starting the discussion on how jump diffusion processes impact the value of American put options we introduce some notations. We assume that the stock follows the jump diffusion model
dSt= (µ − λk)Stdt + σStdWt+ (Y − 1)StdN, (2.1)
where µ is the instantaneous return per year, σ is the instantaneous volatility per year, W is a standard Brownian motion with subscript t implying a variance of t and N is a Poisson process with increments
dN = (
1, with probability λdt
0, with probability (1 − λ)dt, (2.2) where λ is the intensity of the Poisson process. We assume that W and N are indepen-dent processes and that Y is uncorrelated with both of these.
The jump size (Y − 1) is a random nonnegative variable of its density function G(Y ). The expected change due to jumps, k, is therefore given by
k = EQ[Y − 1] =
Z ∞
0
(Y − 1)G(Y )dY. (2.3)
Furthermore, we denote the constant risk-free rate by r, the the continuous dividend yield by q, the time to expiry by τ , the American put option value with underlying S and time to maturity τ by P (S, τ ), and the strike price by K.
2.1
The Continuation Region and the Stopping Region
The main obstacle when dealing with American options is to derive when it is optimal to exercise the option and when it is optimal to hold the option. Even though we do not know the exact point in time when it is optimal to exercise an American option when entering the contract, we can derive a so-called early exercise boundary. This early exercise boundary divides the spectrum of possible future stock price into a stopping region, where it is optimal to exercise the option, and a continuation region, where it is optimal to hold the option. To determine the early exercise boundary we have to think about the two components of every option value. Every option value is the sum of its intrinsic value and its time value. On entering an American option contract the time value is always positive since otherwise immediate exercising would result in a true arbitrage opportunity. Furthermore, throughout the life of an American option the time value can never be negative. Given these properties it is only optimal to exercise when
the option value is equivalent to its intrinsic value, that means when its time value is zero. We can conclude
P (S, τ ) = K − S when S ∈ E
P (S, τ ) > K − S when S ∈ C, (2.4)
where E = [0, a(τ )] is the stopping region, C = (a(τ ), ∞) is the continuation region and a(τ ) the early exercise boundary at time to expiration τ . Gukhal (2001) derived that the early exercise boundary is non-increasing with time to expiration for American put options using fundamental properties of such put options.
Once we have derived the early exercise boundary, we can examine exactly whether it is optimal or it is not optimal to exercise the option given the realized stock price at that very moment. When the stock price after passing from the continuation region into the stopping region reverses to the continuation region, the option contract should be reentered. In the case of a pure diffusion process one does not incur any additional costs. This is because a pure diffusion process is a continuous process, implying that the exact value of the early exercise boundary is attained, when the stock price passes through it. In the case of a jump diffusion process the continuity property is clearly not fulfilled, implying that the early exercise boundary is not necessarily attained. If the stock price jumps from the stopping into the continuation region without attaining the early exercise boundary, the option holder has to pay a value exceeding the intrinsic value to reenter the option and therefore incurs additional costs. These costs are called rebalancing costs in the literature. A jump from the continuation into the stopping region does not cause any additional costs since for a stock price lower than the early exercise boundary the option value is equivalent to its intrinsic value.
Figure 2.1: Rebalancing Costs
Rebalancing costs occurring due to a jump from the stopping region in this case from 60 into the stopping region in this case 90. The rebalancing cost is equal to the difference between the
option value at S = 90 and its intrinsic value. The black dotted line represents the division into stopping region (left of the line) and continuation region (right of the line).
Theoretical results on the IPDE
of American Put Options
In this Chapter we closely follow the theoretical derivations and results on American call options by Chiarella and Ziogas (2009) and extend these to the American put option case.
3.1
The Inhomogeneous IPDE of the American Put
Op-tion
As Merton (1976) has shown by using Itˆo’s Lemma for the continuous part and an equivalent Lemma for the discontinuous part, a call option F written on a stock following a jump diffusion process must fulfill
rF = 1 2σ 2S2∂2F ∂S2 + (r − q − λk)S ∂F ∂S − ∂F ∂τ + λE[F (SY, τ ) − F (S, τ )], (3.1) where E is the risk neutral expectation operator over the random variable Y , subject to the boundary conditions
F (0, τ ) = 0 for ∀τ ≥ 0, (3.2)
F (S, 0) = max(S − K, 0) for ∀τ ≥ 0. (3.3)
In more recent literature further investigating the integro-partial-integration-equation(IPDE) the authors were able to derive results for American put options written on stocks fol-lowing a jump diffusion stock price process.Pham(1997) proved that an American put option written on such a stock price process must fulfill the following IPDE
∂P ∂τ = 1 2σ 2S2∂2P ∂S2 + (r − q − λk)S ∂P ∂S − rP + λ Z ∞ 0
[P (SY, τ ) − P (S, τ )]G(Y )dY,
(3.4)
in the domain S ∈ (a(τ ), ∞) subject the boundary conditions
P (S, 0) = max(K − S, 0) for S ≥ 0, (3.5)
lim
S↓a(τ )P (S, τ ) = K − a(τ ) for τ ≥ 0, (3.6)
lim S↓a(τ ) ∂P (S, τ ) ∂S = −1 for τ ≥ 0, (3.7) ( P (S, τ ) > (K − S)+ if S > a(τ ) P (S, τ ) = K − S if S ≤ a(τ ) for τ ≥ 0, (3.8) 4
where a(τ ) is defined as the early exercise boundary at time to expiration τ .
As a first step to solve this IPDE subject to the above given boundary conditions, we derive Jamshidian’s Representation (Jamshidian(1992)). This representation is defined on the unrestricted domain S ∈ [0, ∞). As a result the IPDE is no longer only defined in an unknown region.
Proposition 1. The solution to the homogeneous IPDE of an American put option in the domain a(τ ) ≤ S < ∞ subject to the boundary conditions (3.6-3.8) is equivalent to the solution of the following inhomogeneous IPDE:
∂P ∂τ = 1 2σ 2S2∂2P ∂S2 + (r − q − λk)S ∂P ∂S − rP + λ Z ∞ 0
[P (SY, τ ) − P (S, τ )]G(Y )dY + H(a(τ ) − S)[rK − qS − λ
Z ∞
a(τ )/S
[P (SY, τ ) − P (S, τ )]G(Y )dY ],
(3.9)
on (τ, S) ∈ [0, T ] × [0, ∞) subject to the payoff function for the put option at expiration (3.5). Here H(x) represents the Heaviside function given by
H(x) = 1x≥0.
Proof. We evaluate the IPDE in the domain S ≤ a(τ ) ≤ K, that is we evaluate it in the stopping region. We denote this term by Φ(S, τ ). When the stock price is in the stopping region at time to expiration τ the American put option is worth its intrinsic value, that is P (S, τ ) = K − S. Given this observation the IPDE simplifies in the stopping region tremendously since in this case ∂P (S, τ )/∂S = −1 and ∂2P (S, τ )/∂S2= 0.
Φ(S, τ ) =H(a(τ ) − S)[(r − q − λk)S − rK + rS + λ
Z ∞
0
[P (SY, τ ) − (K − S)]G(Y )dY ] =H(a(τ ) − S)[S(q + λ(k + 1)) − K(r + λ) + λ
Z ∞
0
P (SY, τ )G(Y )dY ] =H(a(τ ) − S)[qS − rK + λ
Z ∞
0
(SY − K)G(Y )dY + λ
Z ∞
0
P (SY, τ )G(Y )dY ] =H(a(τ ) − S)[qS − rK + λ
Z ∞
0
[P (SY, τ ) − (K − SY )]G(Y )dY ] =H(a(τ ) − S)[qS − rK + λ
Z ∞
a(τ )/S
[P (SY, τ ) − (K − SY )]G(Y )dY ], where the last equality holds since the American put option price is equal to its intrinsic value in the stopping region S ∈ [0, a(τ )].
Gukhal (2001) provides an empirical interpretation of the inhomogeneous term in-volved in equation (3.9). The term (rK − qS) is the interest the option holder receives minus the dividend he has to pay while the stock price is in the stopping region. The integral term is the rebalancing cost arising from a jump from the stopping into the continuation region as discussed in Chapter 2.
3.2
The Fourier Transform and the resulting
inhomoge-neous ODE
To be able to use a method of lines approach to solve for the option price we need to apply the Fourier transform to transform the inhomogeneous IPDE (3.9) into an
inhomogeneous ODE. Before doing so, we need to reformulate the inhomogeneous IPDE (3.9) in such a way that it is defined on the whole domain of R. This is necessary since negative stock prices do not exist, implying that the Fourier transform cannot be applied to P (S, τ ) directly. One way to achieve this is by reformulating equation (3.9) into an equation with constant coefficients and a standardized strike of 1. We set S = Kex,
P (S, τ ) = KV (x, τ ) and b(τ ) = a(τ )/K. Given these changes the inhomogeneous IPDE becomes ∂V ∂τ = 1 2σ 2∂2V ∂x2 + (r − q − λk − σ2 2 ) ∂V ∂x − (r + λ)V + λ Z ∞ 0
V (x + ln(Y ), τ )G(Y )dY + H(ln(b(τ )) − x)((r − qex) − λ
Z ∞
b(τ )e−x
[V (x + ln(Y ), τ ) − (1 − Y ex)]G(Y )dY )
(3.10)
since ∂P/∂S = e−x∂V /∂x, ∂2P/∂S2 = 1/Ke−2x(∂2V /∂x2 − ∂V /∂x) and ∂P/∂τ = K∂V /∂x. The solution is found by solving equation (3.10) on the domain (τ, x) ∈ [0, T ] × R subject to the initial and boundary conditions
V (x, 0) = max(1 − ex, 0) for ∀x ≥ R, (3.11) lim x→∞V (x, τ ) = 0 for τ ≥ 0, (3.12) and lim x↓ln(b(τ ))V (x, τ ) = 1 − b(τ ) for τ ≥ 0. (3.13)
The smooth-pasting condition is incorporated in the inhomogeneous term and therefore is not necessary to be required according toChiarella and Ziogas (2009).
We define the Fourier transform as
F {V (x, τ )} = ˆV (η, τ ) = Z ∞
−∞
eiηxV (x, τ )dx (3.14) and the inverse formula
F−1{ ˆV (η, τ )} = 1 2π
Z ∞
−∞
e−iηxV (η, τ )dηˆ (3.15) Applying the Fourier transform to the inhomogeneous IPDE (3.10) defined on the unre-stricted region x ∈ R allows us to transform the inhomogeneous IPDE into an inhomo-geneous ordinary differential equation for ˆV . The solution of this inhomogeneous ODE can be readily found.
Proposition 2. The Fourier transformed version of the IPDE (3.10) satisfies the ODE ˆ FJ(η, τ ) = ∂ ˆV ∂τ + [ σ2η2 2 + φiη + (r + λ) − λA(η)] ˆV , (3.16) where φ = r − q − λk − σ2/2, ˆ FJ(η, τ ) = F {FJ(x, τ )} (3.17) with FJ(x, τ ) = H(ln(b(τ )) − x)[(r − qex) (3.18) − λ Z ∞ b(τ )e−x
and
A(η) = Z ∞
0
e−iη ln(Y )G(Y )dY. (3.19)
The solution to this ODE is given by ˆ
V (η, τ ) = ˆV (η, 0) exp{−ζ(η)τ } + Z τ
0
exp{−ζ(η)(τ − ξ)} ˆFJ(η, ξ)dξ (3.20)
where ˆV (η, 0) = F {V (x, 0)} and ζ(η) = σ2η2/2 + φiη + r + λ − λA(η).
Proof. Before starting the proof, it is worthwhile noticing that V (x, τ ) and ∂V (x, τ )/∂x do not approach zero as x → ∞. This implies that without solving this problem we cannot apply the Fourier transform to V (x, τ ) since integrability is not guaranteed. Fortunately, Carr and Madan (1999) and Lee et al. (2004) solved this problem by in-troducing a damping function of the form e−αx with α > 0 and applying the Fourier transform to the dampened option price U (x, τ ) = e−αxV (x, τ ). Applying the Fourier transform is then allowed since U (x, τ ) and ∂U (x, τ )/∂x now approach 0 when x → ∞ insuring the integrability. From the solution of the dampened option price it is possible to recover V (x, τ ). Since this is only a prior step to our proof and readily proven in the stated papers we for now assume that the Fourier transform can be applied to V (x, τ ) directly.
Using the linearity of the Fourier transform which is implied by the linearity of the Riemann integral we can term by term derive the Fourier transform. The only term we have to evaluate explicitly for the Fourier transform is the inhomogeneous term.
F {λ Z ∞
0
V (x + ln(Y ), τ )G(Y )dY } = λ Z ∞ −∞ eiηx Z ∞ 0
V (x + ln(Y ), τ )G(Y )dY dx = λ Z ∞ −∞ eiη(z−ln(Y )) Z ∞ 0 V (z, τ )G(Y )dY dz = λ Z ∞ 0
e−iη ln(Y )G(Y )dY Z ∞
−∞
eiηzV (z, τ )dz = λA(η) ˆV (x, τ ),
where we performed a change of integration variable z = x + ln(Y ) and used Fubini’s Theorem to change the order of integration.
For all other terms we use the well known property F {∂nV /∂xn} = (−iη)nF {V } to
derive expression (3.20).
The first term on the right hand side of solution (3.20) represents the Fourier trans-form of the IPDE of an European put. Therefore, the second term represents the Fourier transform of the early exercise premium. The option price can be easily recovered by determining the Fourier inverse of equation (3.20). Due to the linearity of the Fourier inverse, which again is implied by the linearity of the Riemann integral, we can decom-pose the American put option price into an early exercise premium and the European put option equivalent.
V (x, τ ) = F−1{ ˆV (η, 0) exp{−ζ(η)τ } + Z τ 0 exp{−ζ(η)(τ − ξ)} × ˆFJ(η, ξ)dξ} := VE(x, τ ) + VP(x, τ ) = 1 K[PE(S, τ ) + E(S, τ )] = 1 KP (S, τ ) (3.21)
Proposition 3. The price of the European put option, PE(S, τ ) is given by PE(S, τ ) = ∞ X n=0 e−λτ(λτ )n n! E (n) Q [PBS(SXne −λkτ , r, τ, σ2)], (3.22) where PBS(S, r, τ, σ2) = Ke−rτN (−d2(S, K, r, q, τ, σ2)) (3.23) − Se−qτN (−d1(S, K, r, q, τ, σ2)), d1(S, β, r, q, τ, σ2) = ln(S/β) + [r − q + (σ2/2)]τ σ√τ , (3.24) and d2(S, β, r, q, τ, σ2) = d1(S, β, r, q, τ, σ2) − σ √ τ . (3.25)
N represents the cumulative standard normal distribution function. We define Xn:= Y1Y2...Yn, X0 := 1 and E(n)Q {f (Xn)} = Z ∞ 0 Z ∞ 0 ... Z ∞ 0
f (Xn)G(Y1)G(Y2)...G(Yn)dY1dY2...dYn (3.26)
= Z ∞
0
f (Xn)G(n)(Xn)dXn. (3.27)
The Y1, Y2, ..., Yn are independent draws from the jump size distribution G(Y).
Proof. The proof of this proposition closely follows the one for the corresponding Eu-ropean call option given inChiarella and Ziogas (2009). Therefore, for the parts of the proof that are identical to the call option case only the main steps are highlighted and the reader is referred to Chiarella and Ziogas (2009) for all details on the proof. We begin the proof by noticing that the convolution theorem for the Fourier transform is an efficient way to find the Fourier inverse of the European part in equation (3.21). The convolution theorem is given by
F { Z ∞
−∞
f (x − u, τ )g(u, τ )du} = ˆF (η, τ ) ˆG(η, τ ) (3.28)
where ˆF and ˆG are Fourier transforms of f (x, τ ) and g(x, τ ) respectively, with respect to x.
By defining ˆF (η, τ ) = exp{−ζ(η)τ } and G(η, τ ) = ˆV (η, 0), deriving f (x, τ ) and g(x, τ ) will allow us to determine VE(x, τ ) since
VE(x, τ ) = F−1{ ˆV (η, 0) exp{−ζ(η)τ }}
= Z ∞
−∞
f (x − u, τ )g(u, τ )du.
Applying the Fourier inverse to ˆF (η, τ ) leads to f (x, τ ) = 1
2π Z ∞
−∞
eλτ A(η)e−[(1/2)σ2η2τ +i(φτ +x)η+(r+λ)τ ]dη.
The Fourier inverse of G(η, τ ) is as Chiarella and Ziogas (2009) point out simply the payoff function which is in the underlying case the one of the put option. Therefore,
g(x, τ ) = g(x, 0) = max(1 − ex, 0).
Chiarella and Ziogas (2009) use a Taylor series expansion for eλτ A(η) and derive that
f (x, τ ) = ∞ X n=0 e−λτ(λτ )n n! × E(n)Q [ e −rτ σ√2πτ exp{− [x + ln(Xn) + φτ ]2 2σ2τ }].
Therefore, using the Convolution Theorem for the Fourier transform leads to
VE(x, τ ) = ∞ X n=0 e−λτ(λτ )n n! × E(n)Q [ e −rτ σ√2πτ Z 0 −∞ (1 − eu) exp{−[x − u + ln(Xn) + φτ ] 2 2σ2τ }du].
Rewriting it in terms of S, that is S = Kex we obtain
PE(S, τ ) = ∞ X n=0 e−λτ(λτ )n n! E (n) Q [I1(S, τ ) − I2(S, τ )] (3.29) with I1(S, τ ) = e−rτ σ√2πτ Z 0 −∞ K exp{−[ln(SXn/K) − u + φτ ] 2 2σ2τ }du, I2(S, τ ) = e−rτ σ√2πτ Z 0 −∞ Keuexp{−[ln(SXn/K) − u + φτ ] 2 2σ2τ }du.
We now evaluate I1 and I2
I1(S, τ ) = e−rτ σ√2πτ Z 0 −∞ K exp{−[ln(SXn/K) − u + φτ ] 2 2σ2τ }du = e −rτ σ√2πτ Z 0 −∞ K exp{−[u − β] 2 2σ2τ }du = Ke−rτ√1 2π Z −β σ√τ −∞ exp{−x 2 2 }dx = Ke−rτN [−d2(SXne−λkτ, K, r, q, τ, σ2)],
where β = ln(SXn/K) + φτ . Recall that we defined φ = r − q − λk − σ2/2. The only
mathematical tool we used in the calculation above apart from reformulating is a change of integration variable of x = (u − β)/(σ√τ ) in the third equality.
Now we evaluate I2: I2(S, τ ) = e−rτ σ√2πτ Z 0 −∞ Keuexp{−[ln(SXn/K) − u + φτ ] 2 2σ2τ }du = Ke−rτeβeσ2τ /2 1 σ√2πτ Z 0 −∞ exp{−[u − (β + σ 2τ )]2 2σ2τ }du = SXne−λkτe−qτ 1 √ 2π Z −β+σ2τ σ√τ −∞ exp{− x 2 2σ2τ}dx = SXne−λkτe−qτN [−d1(SXne−λkτ, K, r, q, τ, σ2)].
In this derivation we completed the square by adding and subtracting exp{β +σ22τ} and then performing a change of integration variable x = (u − (β + σ2τ ))/(σ√τ ).
Substituting I1 and I2 into valuation formula (3.29) yields the statement of the propo-sition PE(S, τ ) = ∞ X n=0 e−λτ (λτ )n n! E (n) Q [PBS(SXne −λkτ, r, τ, σ2)].
We continue by deriving the early exercise premium term of equation (3.21). Proposition 4. The early exercise premium, E(S, τ ), in equation (3.21) is given by
E(S, τ ) = ∞ X n=0 Z τ 0 e−λ(τ −ξ))(λ(τ − ξ))n n! (3.30) × E(n)Q [H(SXne−λk(τ −ξ), a(ξ), r, r, (τ − ξ), σ2; P (., ξ))]dξ, where H(S, a(ξ), R, r, (τ − ξ), σ2; P (., ξ)) (3.31)
= E(D)[S, a(ξ), R, r, (τ − ξ), σ2] − λE(J )[S, a(ξ), r, (τ − ξ), σ2; P (., ξ)],
E(D)[S, a(ξ), R, r, τ, σ2] (3.32)
= RKe−rτN [−d2(S, a(ξ), r, q, τ, σ2)] − qSe−qτN [−d1(S, a(ξ), r, q, τ, σ2)],
E(J )[S, a(ξ), r, τ, σ2; P (., ξ)] (3.33) = e−rτ Z ∞ 1 G(Y ) Z a(ξ) a(ξ)/Y
[P (ωY, ξ) − (K − ωY )]κ(S, ω, r, q, τ, σ2)dωdY,
and κ(S, ω, r, q, τ, σ2) = 1 ωσ√2πτ exp{− 1 2d 2 2(S, ω, r, q, τ, σ2)}. (3.34)
In equation (3.31) and (3.32) a new variable R is introduced. This variable is introduced for later use in Chapter 4.
Proof. To prove Proposition 4 we need to perform similar steps as in Proposition 3. We begin by recalling VP(x, τ ) VP(x, τ ) = F−1{ Z τ 0 exp{−ζ(η)(τ − ξ)} × ˆFJ(η, ξ)dξ} = Z τ 0 F−1{exp{−ζ(η)(τ − ξ)} × ˆFJ(η, ξ)}dξ,
where the second equality holds due to the linearity of the Fourier inverse.
Again we want to use the Convolution Theorem for Fourier transforms (3.28). Therefore we need to derive f (x, τ ) and g(x, τ ), such that
F { Z ∞
−∞
f ((x − u), τ )g(u, τ )du} = ˆF (η, τ ) ˆG(η, τ )
We recall that due to how we defined ˆFJ we obtain
g(x, ξ) = F−1{ ˆFJ(η, ξ)} = FJ(x, ξ).
Furthermore, we can use results from the previous Proposition by substitute τ − ξ for τ in the Taylor series expansion. Thereby, we obtain f (x, ξ)
f (x, ξ) = F−1{exp{−ζ(η)(τ − ξ)}} = ∞ X n=0 e−λ(τ −ξ)(λ(τ − ξ))n n! × E(n)Q [ e −r(τ −ξ) σp2π(τ − ξ)exp{− [x + ln(Xn) + φ(τ − ξ)]2 2σ2(τ − ξ) }].
Having derived f (x, ξ) and g(x, ξ) we can now apply the Convolution Theorem and obtain VP(x, τ ) = Z τ 0 [ ∞ X n=0 e−λ(τ −ξ)(λ(τ − ξ))n n! Z ∞ −∞ H(ln(b(ξ)) − u) × [r − qeu− λ Z ∞ b(ξ)e−u
[V (u + ln(Y ), ξ) − (1 − Y eu)]G(Y )dY ]
× E(n)Q [ e
−r(τ −ξ)
σp2π(τ − ξ)exp{−
[x − u + ln(Xn) + φ(τ − ξ)]2
2σ2(τ − ξ) }]du]dξ.
This can be rewritten in terms of the stock price S = Kex E(S, τ ) = ∞ X n=0 λn n!E (n) Q [ Z τ 0 (τ − ξ)ne−λ(τ −ξ)[I3(S, τ ) − I4(S, τ ) − I5(S, τ )]dξ], (3.35) where I3(S, τ ) = re−r(τ −ξ) σp2π(τ − ξ) Z ln(b(ξ)) −∞ K exp{−[ln(SXn/K) − u + φ(τ − ξ)] 2 2σ2(τ − ξ) }du, I4(S, τ ) = qe−r(τ −ξ) σp2π(τ − ξ) Z ln(b(ξ)) −∞ Keuexp{−[ln(SXn/K) − u + φ(τ − ξ)] 2 2σ2(τ − ξ) }du, and I5(S, τ ) =λ e−r(τ −ξ) σp2π(τ − ξ) Z ln(b(ξ)) −∞ exp{−[ln(SXn/K) − u + φ(τ − ξ)] 2 2σ2(τ − ξ) } × Z ∞ b(ξ)e−u
[P (KY eu, ξ) − (K − KY eu)]G(Y )dY du,
where the integration interval of the inner integral of I5(S, τ ) arises due to the nature of
the rebalancing costs, which only occur when the stock price jumps from the stopping into the continuation region. This happens in the American put option case in the interval Y ∈ [b(ξ)e−u, ∞).
We can further simplify the terms I3(S, τ ) and I4(S, τ ). We ultimately obtain
I3(S, τ ) = rKe−r(τ −ξ)N (−d2(SXne−λk(τ −ξ), a(ξ), r, q, τ − ξ, σ2)
and
We can also further simplify I5(S, τ ). By using Fubini’s Theorem we obtain I5(S, τ ) = λe−r(τ −ξ) Z ∞ 1 G(Y ) Z ln(b(ξ)) ln(b(ξ))−ln(Y ) [P (KeuY, ξ) − (K − KY eu)] × 1 σp2π(τ − ξ)exp{− [ln(SXn/K) − u + φ(τ − ξ)]2 2σ2(τ − ξ) }dudY = λe−r(τ −ξ) Z ∞ 1 G(Y ) Z ln(b(ξ)) ln(b(ξ))−ln(Y ) [P (KeuY, ξ) − (K − KY eu)] × Keuκ(SXne−λk(τ −ξ), Keu, r, q, τ − ξ, σ2)dudY,
where κ(S, ω, r, q, τ, σ2) is defined in the proposition.
Performing a change of integration variable of ω = Keu we obtain
I5(S, τ ) = λe−r(τ −ξ) Z ∞ 1 G(Y ) Z a(ξ) a(ξ)/Y [P (ωY, ξ) − (K − Y ω)] × κ(SXne−λk(τ −ξ), ω, r, q, τ − ξ, σ2)dωdY,
where Keu cancels out due to du = 1/Ke−udω.
Substituting I3(S, τ ), I4(S, τ ) and I5(S, τ ) into equation (3.35) we obtain equation
(3.30).
3.3
The Pricing Formula for the American Put Option
Proposition 5. The American put option price, P (S, τ ), is given by
P (S, τ ) = ∞ X n=0 e−λτ(λτ )n n! E (n) Q [PBS(SXne −λkτ , r, τ, σ2)] + ∞ X n=0 E(n)Q [ Z τ 0 e−λ(τ −ξ)(λ(τ − ξ))n n! × H(SXne−λk(τ −ξ), a(ξ), r, r, (τ − ξ), σ2; P (., ξ))dξ], (3.36)
where PBS is defined in Proposition 3 and H in Proposition 4.
Proof. This is an immediate result when substituting equation (3.22) and equation (3.30) into equation (3.21).
The representation of the American put price (3.36) is the sum of the European put price derived byMerton(1976) and an early exercise premium equivalent to the interest earned minus dividend paid when the stock price is in the stopping region minus the rebalancing cost stemming from upward jumps into the continuation region.
American put =European put
+Present value of interest paid on strike price in the stopping region +Expected dividend paid in stopping region
+Expected rebalancing cost due to upward jumps from the stopping into the continuation region.
The early exercise boundary, a(τ ), at time to expiration τ therefore must satisfy K − a(τ ) = ∞ X n=0 e−λτ(λτ )n n! E (n) Q [PBS(a(τ )Xne −λkτ , r, τ, σ2)] + ∞ X n=0 E(n)Q [ Z τ 0 e−λ(τ −ξ)(λ(τ − ξ))n n! × H(a(τ ), a(ξ), r, r, (τ − ξ), σ2; P (., ξ)))dξ], (3.37)
that is the maximal value for which the option value is equivalent to its intrinsic value. In equation (3.37) we observe the main difficulty that arises when dealing with jump diffusion driven stock price processes. The early exercise boundary depends on the unknown American put price P (S, τ ) through E(J ). It is obvious that this is not the case when dealing with pure diffusion processes as λ = 0 leads to no influence of the option value on the early exercise boundary a(τ ). Therefore, the numerical implementation is far more difficult in the jump diffusion case. The detailed discussion on how to implement these dependent integral equations is given in Section 5.1 for the special case of the Merton jump diffusion model.
For later use we also introduce an alternative representation of the jump term of the early exercise premium E(S, τ ).
Proposition 6. E(J ) in equation (3.32) is equivalent to E(J )[S, a(ξ), r, τ, σ2; P (., ξ)] = e−rτ Z ∞ 1 [P (a(ξ)z, ξ) − (K − a(ξ)z)] Z ∞ z
G(Y )κ(SY /a(ξ), z, r, q, τ, σ2)dY dz. (3.38) Proof. Recall equation (3.32)
E(J )[S, a(ξ), r, τ, σ2; P (., ξ)] = e−rτ Z ∞ 1 G(Y ) Z a(ξ) a(ξ)/Y
[P (ωY, ξ) − (K − ωY )]κ(S, ω, r, q, τ, σ2)dωdY.
By performing a change of integration variable z = ωY /a(ξ) we obtain E(J )[S, a(ξ), r, τ, σ2; P (., ξ)] = e−rτ Z ∞ 1 G(Y ) Z Y 1
[P (za(ξ), ξ) − (K − za(ξ))]κ(SY /a(ξ), z), r, q, τ, σ2)dzdY. Using Fubini’s Theorem to change the order of integration we obtain
E(J )[S, a(ξ), r, τ, σ2; P (., ξ)] = e−rτ Z ∞ 1 Z ∞ z
[P (za(ξ), ξ) − (K − za(ξ))]κ(SY /a(ξ), z, r, q, τ, σ2)dY dz.
Even though this representation is far less intuitive from an economic point of view it will yield great advantage when solving the pricing formula. For instance, when we assume lognormally distributed jump sizes as proposed byMerton(1976) the most inner integral can be readily solved analytically. This simplifies the numerical implementation tremendously.
3.4
The Early Exercise Boundary at Expiration
One of the initial values of the numerical implementation in Section 5.1 is the early exercise boundary at expiration. The early exercise boundary at expiration given the stock follows a jump diffusion process differs from the one under a pure diffusion process and therefore needs to be derived.
Proposition 7. The early exercise boundary at expiration of the American put is given by
a(0+) = K min( r + λ R∞
K/a(0+)G(Y )dY
q + λR∞
K/a(0+)Y G(Y )dY
, 1). (3.39)
Proof. Equation (3.38) can be derived by referring to the inhomogeneous term of the inhomogenous IPDE of the American Put Option which is given by
H(a(τ ) − S){rK − qS − λ Z ∞
a(0+)/S
[P (SY, τ ) − (K − SY )]G(Y )dY }.
We set the terms in brackets to 0 and evaluate it at τ = 0 with S = a(0+). As τ = 0 P (SY, τ ) simplifies to max(K − SY, 0) and we obtain
rK − qa(0+) − λ Z ∞
1
[max(K − a(0+)Y, 0) − (K − a(0+)Y )]G(Y )dY = 0. For Y ≤ K/a(0+) the integral is 0 therefore the boundaries of the integral become
rK − qa(0+) − λ Z ∞
K/a(0+)
[max(K − a(0+)Y, 0) − (K − a(0+)Y )]G(Y )dY = 0. Recall that a(τ ) ≤ K for ∀τ ∈ [0, T ] and therefore K/a(0+) ≥ 1.
Since the intrinsic value of the option is zero on the whole domain of the integration interval the equation simplifies to
rK − qa(0+) − λ Z ∞
K/a(0+)
[−(K − a(0+)Y )]G(Y )dY = 0. Solving for a(0+) leads to
a(0+) = K r + λ R∞
K/a(0+)G(Y )dY
q + λR∞
K/a(0+)Y G(Y )dY
.
Since the option is out of the money if a(0+) > K, a(0+) becomes:
a(0+) = K min( r + λ R∞
K/a(0+)G(Y )dY
q + λR∞
K/a(0+)Y G(Y )dY
, 1)
Unfortunately, a general investigation of the behavior of the early exercise boundary at expiration is impossible due to dependence of the integral boundaries on the early exercise boundary at expiration itself. Nevertheless, we can prove that at least one solution exists if the jump size distribution is assumed to be continuous. For this purpose define
f (b) = −b + K min(1, r + λ R∞
K/bG(Y )dY
First let us focus on the boundary behavior of f (b). This is given by lim b→0f (b) = K min(1, r q) ≥ 0 and lim b→Kf (b) = −K + K min(1, r + λR∞ 1 G(Y )dY q + λR∞ 1 Y G(Y )dY ) ≤ 0.
Given this boundary behavior and the fact that the integral is a continuous operator as well as that the integrand is assumed to be a continuous distribution function, we can conclude that indeed there is at least one root of f (b).
Investigation of Merton’s Jump
Diffusion Model
In this Chapter we investigate Merton’s jump diffusion model (MJD) given the results of the previous chapter. In the MJD the probability density function of the jump size is assumed to be
G(Y ) = 1
δY√2πexp{−
(ln (Y ) − (γ −δ22))2
2δ2 }. (4.1)
Proposition 8. Given this jump size distribution the integral equation of P (S, τ ) (3.35) becomes P (S, τ ) = ∞ X n=0 e−λ0τ(λ0τ )n n! PBS(S, rn(τ ), τ, ν 2 n(τ ))] + ∞ X n=0 [ Z τ 0 e−λ0(τ −ξ)(λ0(τ − ξ))n n! × H(S, a(ξ), r, rn(τ − ξ), (τ − ξ), νn2(τ − ξ); P (., ξ))dξ], (4.2) where λ0 = λ(k + 1), rn(τ ) = r − λk + nγ/τ and νn2(τ ) = σ2+ nδ2/τ .
Proof. We begin the proof by recalling following notation
E(n)Q [f (Xn)] = Z ∞ 0 f (Xn) 1 Xnδ √ 2πn × exp{−1 2( ln (Xn) − n[γ − (δ2/2)] δ√n ) 2}dX n. (4.3)
We split up the proof in investigating the different parts of the American put, that is the European component, the earned interest and paid dividend component and the rebalancing cost component. We start with the European component. Merton (1976) derived that ∞ X n=0 e−λτ(λτ )n n! E (n) Q [PBS(SXne −λkτ , r, τ, σ2)] = ∞ X n=0 e−λ0τ(λ0τ )n n! PBS(S, rn(τ ), τ, ν 2 n(τ ))]
where λ0 = λ(k + 1), rn(τ ) = r − λk + nγ/τ and νn2(τ ) = σ2+ nδ2/τ , which coincides
with our proposed equation (4.2).
We continue with the value associated with the received interest and paid dividends 16
when the option is exercised early. For ease of notation we call this value for now E(1). This value is given by (3.32) evaluated over the sum and integral in (3.30):
E(1)(S, τ ) = ∞ X n=0 { Z τ 0 e−λ(τ −ξ)(λ(τ − ξ))n n! × E(n)Q [E(D)[SXne−λk(τ −ξ), a(ξ), r, r, (τ − ξ), σ2]]dξ}
Using the result ofChiarella and Ziogas (2009) that
E(n)Q [XnN [d1(SXne−λk(τ −ξ), a(ξ), r, q, τ − ξ, σ2)]] = enγN [d1(S, a(ξ), rn(τ − ξ), q, (τ − ξ), νn2(τ − ξ))] and E(n)Q [N [d2(SXne−λk(τ −ξ), a(ξ), r, q, τ − ξ, σ2)]] = N [d2(S, a(ξ), rn(τ − ξ), q, (τ − ξ), νn2(τ − ξ))] we derive E(n)Q [XnN [−d1(SXne−λk(τ −ξ), a(ξ), r, q, τ − ξ, σ2)]] = E(n)Q [Xn(1 − N [d1(SXne−λk(τ −ξ), a(ξ), r, q, τ − ξ, σ2)])] = enγ− enγN [d 1(S, a(ξ), rn(τ − ξ), q, (τ − ξ), νn2(τ − ξ))] = enγN [−d1(S, a(ξ), rn(τ − ξ), q, (τ − ξ), νn2(τ − ξ))] and E(n)Q [N [−d2(SXne−λk(τ −ξ), a(ξ), r, q, τ − ξ, σ2)]] = E(n)Q [1 − N [d2(SXne−λk(τ −ξ), a(ξ), r, q, τ − ξ, σ2)]] = N [−d2(S, a(ξ), rn(τ − ξ), q, (τ − ξ), νn2(τ − ξ))]. Since enγ = (k + 1)n we obtain E(1)(S, τ ) = ∞ X n=0 { Z τ 0 e−λ(τ −ξ)(λ(τ − ξ))n n! × [rKe−r(τ −ξ)N [−d2(S, a(ξ), rn(τ − ξ), q, (τ − ξ), νn2(τ − ξ))] − qSe−λk(τ −ξ)enγN [−d1(S, a(ξ), rn(τ − ξ), q, (τ − ξ), νn2(τ − ξ))]]dξ} = ∞ X n=0 { Z τ 0 e−λ(τ −ξ)(λ(τ − ξ))n n! e −λk(τ −ξ)enγ × [rKe−(r−λk+nγ/(τ −ξ))(τ −ξ)N [−d2(S, a(ξ), rn(τ − ξ), q, (τ − ξ), νn2(τ − ξ))] − qSN [−d1(S, a(ξ), rn(τ − ξ), q, (τ − ξ), νn2(τ − ξ))]]dξ} = ∞ X n=0 { Z τ 0 e−λ(τ −ξ)(λ(τ − ξ))n n! e −λk(τ −ξ)(k + 1)n × [rKe−rn(τ −ξ)(τ −ξ)N [−d 2(S, a(ξ), rn(τ − ξ), q, (τ − ξ), νn2(τ − ξ))] − qSN [−d1(S, a(ξ), rn(τ − ξ), q, (τ − ξ), νn2(τ − ξ))]]dξ} = ∞ X n=0 { Z τ 0 e−λ0(τ −ξ)(λ0(τ − ξ))n n! × E(D)[S, a(ξ), r, r n(τ − ξ), (τ − ξ), νn2(τ − ξ)]dξ}
Now we focus on the rebalancing cost due to upward jumps from the stopping into the continuation region. For ease of notation we define this cost as E(2) for now. This cost
is given by E(2)(S, τ ) =λ ∞ X n=0 { Z τ 0 e−λ0(τ −ξ)(λ0(τ − ξ))n n! × E(n)Q [E(J )[S, a(ξ), rn(τ − ξ), (τ − ξ), νn2(τ − ξ); P (., ξ)]]dξ}
To evaluate the E(n)Q operator, we have to consider E(n)Q {κ(SXne−λk(τ −ξ), ω, r, q, (τ − ξ), σ2)} = Z ∞ 0 1 Xnδ √ 2πnexp{− 1 2( ln (Xn) − n[γ − (δ2/2)] δ√n ) 2} 1 ωσp2π(τ − ξ) × exp{−[(r − q − λk − (σ 2/2))(τ − ξ) + ln(SX n/ω)]2 2σ2(τ − ξ) }dXn,
since this is the only term in E(J ) depending on Xn.
As κ(S, ω, r, q, τ, σ2) is equivalent for American put options and American call options we can useChiarella and Ziogas(2009) derivations directly, that is by using the change of variable x = ln(Xn), we obtain E(n)Q {κ(SXne−λk(τ −ξ), ω, r, q, (τ − ξ), σ2)} = 1 ωp2π(τ − ξ)ν2 n(τ − ξ) × exp{−[ln(S/ω) + (rn(τ − ξ) − q − (ν 2 n(τ − ξ)/2))(τ − ξ)]2 2ν2 n(τ − ξ)(τ − ξ) }
Using the definitions of λ0, rn(τ ) and νn(τ ) we can rewrite PP(2) as
E(2)(S, τ ) = ∞ X n=0 [ Z τ 0 e−λ0(τ −ξ)(λ0(τ − ξ))n n! × E(J )[S, a(ξ), r n(τ − ξ), (τ − ξ), νn2(τ − ξ); P (., ξ)]],
Combining all obtained results and using the linearity of the integral we obtain Propo-sition 8.
The E(J )term in the integral equation (4.2) involves solving a double integral. This increases the numerical burden. Fortunately, by using the alternative representation derived in Proposition 6 and using the explicit jump size distribution function (4.1) we can determine E(J )in such a way that only a single integral has to be solved. This yields an advantage for us in the numerical implementation in Chapter5.1.
Proposition 9. The term E(J ) in equation (4.2) can be rewritten as
E(J )[S, a(ξ), rn(τ − ξ), (τ − ξ), νn2(τ − ξ); P (., ξ)] (4.4) = e−(τ −ξ)rn(τ −ξ) Z 1 0 1 x2[P (a(ξ) 1 x, ξ) − (K − a(ξ) 1 x)] × κ(S/a(ξ),1 x, rn+1(τ − ξ), q, (τ − ξ), ν 2 n+1(τ − ξ)) × N (−D(S/a(ξ),1 x, rn(τ − ξ), q, ν 2 n(τ − ξ), νn+12 (τ − ξ), (τ − ξ), γ, δ))dx,
and D(S/a(ξ), z, rn(τ ), q, νn(τ ), νn+1(τ ), τ, γ, δ) (4.5) = δ 2ln(S/(a(ξ)z)) + [ln(z)ν2 n+1(τ ) + δ2[rn(τ ) − q] − γνn2(τ )]τ νn(τ )νn+1(τ )δτ .
Proof. We begin the proof by recalling the jump term we want to evaluate according to equation (3.36) E(J )[SXne−λk(τ −ξ), a(ξ), r, τ, σ2; P (S, ξ)] =e−rτ Z ∞ 1 [P (a(ξ)z, ξ) − (K − a(ξ)z)] × Z ∞ z
G(Y )κ(SXne−λk(τ −ξ)Y /a(ξ), z, r, q, τ, σ2)dY dz,
which has to be evaluated under the E(n)Q operator. This means we have to evaluate the inner integral under the E(n)Q operator. So our task is to simplify
I(S, z, τ, ξ) =E(n)Q [ Z ∞
z
G(Y )κ(SY Xne−λk(τ −ξ)/a(ξ), z, r, q, (τ − ξ), σ2)dY ]
= 1 δ√2π Z ∞ z 1 Y exp{− 1 2[ ln(Y ) − (γ − δ2/2) δ ] 2} × J (S, z, τ, ξ, Y )dY, (4.6) where J (S, z, τ, ξ, Y ) = 1 zσp2π(τ − ξ) 1 δ√2πn Z ∞ 0 1 Xn exp{−1 2[ ln(Xn) − n(γ − δ2/2) δ√n ] 2} × exp{−1 2[ ln(SY Xn/(a(ξ)z)) + [r − q − λk − σ2/2](τ − ξ)] σp(τ − ξ) ] 2}dX n.
We obtained the second equality in equation (4.6) by using the definition of κ(S, ω, r, q, τ, σ2) and of the distribution function G(Y ).
Since κ(S, ω, r, q, τ, σ2) does not differ between American calls and American puts we use the result of Chiarella and Ziogas (2009)
J (S, z, τ, ξ, Y )
= 1
zνn(τ − ξ)p2π(τ − ξ)
× exp{−[ln(SY /(a(ξ)z)) + (rn(τ − ξ) − q − [ν
2
n(τ − ξ)/2])(τ − ξ)]2
2ν2
n(τ − ξ)(τ − ξ)
},
where rn(τ ) and νn(τ ) are defined in Proposition 8. Given this expression for J (S, z, τ, ξ, Y )
we can rewrite I(S, z, τ, ξ) as I(S, z, τ, ξ) = 1 zν2 n(τ − ξ)p2π(τ − ξ) × 1 δ√2π Z ∞ z 1 Y exp{− 1 2[ ln(Y ) − (γ − δ2/2) δ ] 2}
× exp{−[ln(SY /(a(ξ)z)) + (rn(τ − ξ) − q − [ν
2
n(τ − ξ)/2])(τ − ξ)]2
2ν2
n(τ − ξ)(τ − ξ)
For this new representation of I(S, z, τ, ξ) we can perform a similar approach asChiarella and Ziogas (2009) used to simplify the expression. At this point we highlight that the main difference in this proof between the call and put option is the integration interval, which is in the put option case [z, ∞] and in the call option case [0, z].
As an intermediate step we show that Z ∞ z 1 ωexp{− [ln(ω) + β1]2 α1 −[ln(ω) + β2] 2 α2 }dω =r α1α2π α1+ α2 exp{−(β1− β2) 2 α1+ α2 }N (−f (z)),
where f (x) =√2[(α1+ α2) ln(x) + α1β2+ α2β1]/pα1α2(α1+ α2). This can be derived
performing a couple of changes of integration variable. The proof is as following Z ∞ z 1 ωexp{− [ln(ω) + β1]2 α1 −[ln(ω) + β2] 2 α2 }dω = Z ∞ ln(z) exp{−[x + β1] 2 α1 −[x + β2] 2 α2 }dx = exp{−(β1− β2) 2 α1+ α2 } Z ∞ ln(z) exp{−[(α1+ α2)x + β1α2+ β2α1] 2 α1α2(α1+ α2) }dx = √ α1α2 √ 2√α1+ α2 exp{−(β1− β2) 2 α1+ α2 } Z ∞ f (z) exp{−y 2 2 }dy =r α1α2π α1+ α2 exp{−(β1− β2) 2 α1+ α2 }N (−f (z)),
where we used the change of integration variable x = ln(ω) in the first equality, the completion of the square by −(β1−β2)2
α1+α2 in the second equality and the change of
integra-tion variable y =√2[(α1+ α2)x + α1β2+ α2β1]/pα1α2(α1+ α2) in the third equality
and in the last equality a multiplication by 1 in the form of√π/√π. Given this derivation we obtain that
I(S, z, τ, ξ) = 1 zνn+12 (τ − ξ)p2π(τ − ξ)N (−f (z)) × exp{−[ln(S/(a(ξ)z)) + (rn+1(τ − ξ) − q − [ν 2 n+1(τ − ξ)/2])(τ − ξ)]2 2νn+12 (τ − ξ)(τ − ξ) }, where f (z) = δ 2ln(S/(a(ξ)z)) + [ln(z)ν2 n+1(τ − ξ) + δ2[rn(τ − ξ) − q] − γνn2(τ − ξ)](τ − ξ) νn(τ − ξ)νn+12 (τ − ξ)δ(τ − ξ) = D(S/a(ξ), z, rn(τ − ξ), q, νn(τ − ξ), νn+12 (τ − ξ), (τ − ξ), γ, δ)
is simply derived by determining the integration variable after the second change of integration variable, where β1 = −(γ − δ2/2), β2 = ln(S/(a(ξ)z)) + (rn(τ − ξ) − q −
[νn2(τ − ξ)/2])(τ − ξ), α1 = 2δ2 and α2 = 2νn2(τ − ξ)(τ − ξ).
By substituting the derived expression for I(S, z, τ, ξ) into E(n)Q [E(J )[S, a(ξ), r, (τ − ξ), σ2; P (S, ξ)]] we obtain E(J )[S, a(ξ), rn(τ ), (τ − ξ), νn2(τ ); P (., ξ)] = e−τ rn(τ −ξ) Z ∞ 1 [P (a(ξ)z, ξ) − (K − a(ξ)z)] × κ(S/a(ξ), z, rn+1(τ − ξ), q, (τ − ξ), νn+12 (τ − ξ)) × N (−D(S/a(ξ), z, rn(τ − ξ), q, νn2(τ − ξ), νn+12 (τ − ξ), (τ − ξ), γ, δ))dz (4.7)
For the numerical implementation in Section 5.1, more explicitly to be able to use the Gaussian integration, we perform an additional change of integration variable of the outer integral. By a change of integration variable x = 1/z equation (4.7) becomes
E(J )[S, a(ξ), rn(τ ), (τ − ξ), νn2(τ ); P (., ξ)] = e−τ rn(τ −ξ) Z 1 0 1 x2[P (a(ξ) 1 x, ξ) − (K − a(ξ) 1 x)] × κ(S/a(ξ),1 x, rn+1(τ − ξ), q, (τ − ξ), ν 2 n+1(τ − ξ)) × N (−D(S/a(ξ),1 x, rn(τ − ξ), q, ν 2 n(τ − ξ), νn+12 (τ − ξ), (τ − ξ), γ, δ))dx.
The early exercise boundary at time to expiration τ given lognormally distributed jump sizes is the solution of the following integral equation according to equation (3.36)
K − a(τ ) = ∞ X n=0 e−λ0τ(λ0τ )n n! PBS(a(τ ), K, K, rn(τ ), q, τ, ν 2 n(τ ))] + ∞ X n=0 [ Z τ 0 e−λ0(τ −ξ)(λ0(τ − ξ))n n! × H(a(τ ), a(ξ), r, rn(τ − ξ), (τ − ξ), νn2(τ − ξ); P (., ξ))dξ]. (4.8)
4.1
The Delta of the American Put Option
For our numerical implementation in Section5.1we also need to determine the delta of the American put option.
Proposition 10. The delta of the American put option is given by
∆P(S, τ ) (4.9) = ∞ X n=0 e−λ0τ(λ0τ )n n! ∆BS[S, rn(τ ), τ, ν 2 n(τ )] + ∞ X n=0 [ Z τ 0 e−λ0(τ −ξ)(λ0(τ − ξ))n n! × ∆H(S, a(ξ), r, rn(τ − ξ), (τ − ξ), νn2(τ − ξ); P (., ξ))dξ], where ∆H(S, a(ξ), r, rn(τ ), τ, νn2(τ ); P (., ξ)) (4.10) = ∆E(D)(S, a(ξ), r, rn(τ ), τ, νn2(τ )) − λ∆E(J )(S, a(ξ), rn(τ ), τ, νn2(τ ); P (., ξ)), ∆E(D)[S, a(ξ), r, rn(τ ), τ, νn2(τ )] (4.11) = e−qτ[ 1 νn(τ ) √ τN 0 (d1(S, a(ξ), rn(τ ), q, τ, νn2(τ )))(q − r K a(ξ)) − qN (−d1(S, a(ξ), rn(τ ), q, τ, νn2(τ )))],
and ∆E(J )(S, a(ξ), rn(τ ), τ, νn2(τ ); P (., ξ)) (4.12) = e−rn(τ )τ Z 1 0 [P (a(ξ)/x, ξ) − (K − a(ξ)/x)] x2Sν n+1(τ ) √ τ × κ(S/a(ξ), 1/x, rn+1(τ ), q, τ, νn+12 (τ )) × [ δ νn(τ ) √ τ(−N 0 [−D(S/a(ξ), 1/x, rn(τ ), q, νn2(τ ), νn+12 (τ ), τ, γ, δ)]) + d2(S/a(ξ), 1/x, rn+1(τ ), q, τ, νn+12 (τ )) × N [−D(S/a(ξ), 1/x, rn(τ ), q, νn2(τ ), νn+12 (τ ), τ, γ, δ)])]dx.
Proof. All equations can be found by differentiating with respect to S. One should be especially careful with ∆ED since E(D) involves two different interest rates, r and
rn(τ ).
Besides being necessary for the Newton-Raphson method for approximating the early exercise boundary in the numerical implementation in Section 5.1, the delta is also of crucial importance for hedging purposes.
4.2
The Early Exercise Boundary at Expiration
As we mentioned in Chapter 3 we need to determine the early exercise boundary at expiration as an initial value for our numerical implementation in Section 5.1.
Given lognormally distributed jump sizes the early exercise boundary at expiration derived in Proposition 7 can now be further evaluated.
Proposition 11. If the jump sizes are lognormally distributed, the early exercise bound-ary at expiration must satisfy the implicit equation
a(0+) = K min(1, r + λN [(− ln(K/a(0
+)) + (γ − δ2/2)/δ]
q + λ exp (γ)N [(− ln(K/a(0+)) + (γ + δ2/2)/δ]). (4.13)
Proof. We prove Proposition 11 by determining the integral terms of equation (3.38) using the given distribution function G(Y ). Therefore, recall Proposition 7
a(0+) = Kmin( r + λ R∞
K/a(0+)G(Y )dY
q + λRK/a(0∞ +)Y G(Y )dY
, 1).
The integral in the numerator can be estimated as λ Z ∞ K/a(0+) G(Y )dY = λ Z ∞ K/a(0+) 1 δY√2πexp (− (ln(Y ) − (γ − δ22)2 2δ2 )dY = λ Z − ln(K/a(0+))+(γ− δ2 2) δ −∞ 1 √ 2πexp (− x2 2 )dx = λN (− ln(K/a(0 +)) + (γ −δ2 2) δ ),
where we used the substitution of integration variable x = ln(Y )−(γ−
δ2 2)
δ and the
sym-metry of the normal distribution in the second step.
change of integration variable x = (ln(Y ) − (γ − δ2/2))/δ and then an additional change of integration variable z = x − δ. We obtain
λ Z ∞ K/a(0+) Y G(Y )dY = λ Z ∞ K/a(0+) 1 δ√2πexp (− (ln(Y ) − (γ − δ22)2 2δ2 )dY = λ exp (γ − δ2/2) Z − ln(K/a(0+))+(γ+ δ22) δ −∞ 1 √ 2πexp (− x2− 2δx 2 )dx = λ exp (γ) Z − ln(K/a(0+))+(γ+ δ2 2) δ −∞ 1 √ 2πexp (− (x − δ)2 2 )dx = λ exp (γ)N (− ln(K/a(0 +)) + (γ +δ2 2) δ ).
Substituting these results in equation (3.38) we obtain equation (4.12). The existence of an early exercise boundary, that is that
f (b) = −b + Kmin(1, r + λN [(− ln(K/b) + (γ − δ
2/2)/δ]
q + λ exp (γ)N [(− ln(K/b) + (γ + δ2/2)/δ]) (4.14)
has a root follows directly from the argumentation after Proposition 7 since the lognor-mally distribution is a continuous distribution.
Numerical Implementation and
Results
5.1
Numerical Implementation
In this Section we discuss the numerical implementation of the pricing formulas derived in Chapter4 for lognormally distributed jump sizes step by step. It closely follows the numerical implementation ofChiarella and Ziogas (2009) as the main challenges in the implementation of the put pricing formulas can be solved by the same mathematical tools as the ones in the implementation of the call pricing formulas.
The underlying numerical implementation is based on the quadrature scheme proposed byKallast and Kivinukk(2003). As they implemented the quadrature scheme for a pure diffusion stock price process, we have to make additions regarding the jump-part of the associated early exercise premium. These will be discussed throughout the following description of the numerical implementation at the steps where additional steps have to be made.
We begin by discretizing the time to maturity in equally sized time steps. We denote the total number of time steps by N and the size of each time step by h = T /N . In this notation τ = ih for i = 0, 1, 2, ..., N . Furthermore, we define Pi(S) := P (S, ih) = P (S, τ )
as the put value and ai := a(ih) = a(τ ) as the early exercise boundary at time to
ma-turity τ .
For the Poisson process which represents the number of jumps we only consider proba-bilities higher than 10−5. The number of jumps that needs to be considered depends on the time to maturity, with increasing number of jumps as the time to maturity increases. As input for the numerical implementation, the early exercise boundary at expiration and the value of the put option at expiration are needed. The former can be determined with formula (4.12). The latter is the intrinsic value of the option as the time-value of the option is zero at expiration.
To determine ai and subsequently Pi(S) we have to first approximate Pi(S) due to
the dependence of the early exercise boundary on the option value. As Chiarella and Ziogas (2009) stated, a good approximation is the option value of the previous step, that is Pi−1(S). We evaluate the option price of the previous time step on a grid of
stock prices. We choose a grid of stock prices depending on the strike price since the highest convexity of the option value is found close to the strike price. For stock prices far out-of-the-money and far in-the-money we use a wider step size, for stock prices close to the strike price we use a finer step size. In the underlying implementation we use a grid with step size of K/8 from 0 to 3K/4 as well as from 5K/4 to 2K and a step size of K/24 from 3K/4 + K/24 to 5K/4 − K/24. This ensures a better approximation with fewer grid points than an evenly spaced stock price grid. Having determined the option values on the grid points we fit a cubic spline on the grid points which allows us to determine option values between grid points.
We now can determine the early exercise boundary at all time steps by Newton Raph-son’s method as done by Kallast and Kivinukk(2003) with two adjustments. The first adjustment concerns the inner integral of the jump term (4.4). We use the Gaussian inte-gration scheme with 20 inteinte-gration points to numerically compute the inner integral. The weights and integration points are taken fromAbramowitz and Stegun (1970)(Chapter 25, p.916). The second adjustment concerns the limit of the integrand in equation (4.8) and the limit of its derivative with respect to S when ξ converges from below to τ . Apart from the limits associated with the jump term (4.4), all limits are readily derived by Kallast and Kivinukk (2003). When determining the limits of the jump term and its derivative we have to be especially careful about the variances in the κ-function and D-function as a they depend themselves on the time to expiration. As the limit of the derivative is not easy to determine we use the derivative of the limit as done byChiarella and Ziogas (2009) and Kallast and Kivinukk (2003). A reasoning why this is allowed is given in the proof of Proposition 12. The limit for the jump term involves the put option value. As a consequence the derivative of the limit depends on the delta of the put option. Therefore, we need to fit another cubic spline for the delta of the option. As the delta of the option depends again on the early exercise boundary itself, we use as an approximation for ∆i(S) the delta of the previous step that is ∆i−1(S) to be consistent
with the approximation of Pi(S). The limits are given in Proposition 12.
After having derived the early exercise boundary ai we determine the value of Pi(S) as
well as of ∆i(S) on the stock price grid by using formula (4.2) and (4.9) respectively.
We perform the same steps for all i-values. We ultimately use Simpson’s rule in (4.9) as done by Kallast and Kivinukk (2003) to determine the option price as well as the option’s delta.
Proposition 12. The limit of the jump term (4.4) with respect to ξ → τ evaluated for S = a(τ ) using the proposed Gaussian integration scheme with 20 integration points for approximating the integral is given by
lim ξ→τE (J )[a(τ ), a(ξ), r n(τ − ξ), (τ − ξ), νn2(τ − ξ); P (., ξ)] (5.1) ≈ 20 X i=1 w[i] 1 2x[i] √ 2πδ2[P (a(τ ) 1 x[i], τ ) − (K − a(τ ) 1 x[i])] (5.2) × exp{−(ln(x[i]) + γ − δ 2/2)2 2δ2 },
and the derivative of this limit is given by
lim ξ→τ∂E (J )[a(τ ), a(ξ), r n(τ − ξ), (τ − ξ), νn2(τ − ξ); P (., ξ)]/∂a(τ ) (5.3) ≈ 20 X i=1 w[i] 1 2x[i]√2πδ2[∆(a(τ ) 1 x[i], τ ) 1 x[i]+ 1 x[i])] exp{− (ln(x[i]) + γ − δ2/2)2 2δ2 }
if a(τ ) > 0, where x[i] and w[i] for i = 1, 2, ..., 20 are the integration points and weights used in the Gaussian integration scheme. Having an approximation rather than having equality in (5.1) and (5.2) stems solely from the approximation of the integral, not from an approximation of any function in the integrand.
We begin with deriving the limit of the κ-function: lim ξ→τκ(a(τ )/a(ξ), 1 x, rn+1(τ − ξ), q, (τ − ξ), ν 2 n+1(τ − ξ)) = lim ξ→τ[ x νn+1(τ − ξ)p2π(τ − ξ) × exp{−1 2d 2 2(a(τ )/a(ξ), 1/x, rn+1(τ − ξ), q, τ, νn+12 (τ − ξ))}] =√x 2πδ2 exp{− (ln(x) + γ − δ2/2)2 2δ2 }, since lim ξ→τd2(a(τ )/a(ξ), 1/x, rn+1(τ − ξ), q, τ, ν 2 n+1(τ − ξ)) = lim ξ→τ ln(a(τ )/a(ξ)) + ln(x) + (rn+1(τ − ξ)) − q − νn+1(τ − ξ)2/2)(τ − ξ) νn+1(τ − ξ) √ τ − ξ = ln(x) + γ − δ 2/2) δ .
We continue with deriving the limit of the D-function lim ξ→τD(a(τ )/a(ξ), 1/x, rn(τ − ξ), q, ν 2 n(τ − ξ), νn+12 (τ − ξ), (τ − ξ), γ, δ) = lim ξ→τ[ δ2ln(a(τ )/(a(ξ)1/x)) νn(τ − ξ)νn+1(τ − ξ)δ(τ − ξ) + [ln(1/x)ν 2 n+1(τ − ξ) + δ2[rn(τ − ξ) − q] − γνn2(τ − ξ)](τ − ξ) νn(τ − ξ)νn+1(τ − ξ)δ(τ − ξ) ] =δ2a0(τ ) lim ξ→τ(−δ 2σ4(τ − ξ) + σ2δ2 2pσ4(τ − ξ)2+ σ2δ2(τ − ξ)) −1 =0 (5.4)
if the early exercise boundary is continuously differentiable. In all derivations we use n = 0 when ξ converges to τ , since the whole mass of the cumulative distribution function of the Poisson process lies in n = 0 in this case. The second equality in (5.3) stems from using L’Hospital’s Rule, which can be applied as both numerator and denominator converge to 0. The second equality holds as continuous differentiability insures a finite derivative which implies finiteness of the first term in the previous equation.
The continuous differentiability of the early exercise boundary under jump diffusion processes is proven byBayraktar and Xing(2009) excluding the early exercise boundary at maturity. Fortunately, we do not need to evaluate the limit for τ = 0 as a(0+) is given by equation (4.12). Therefore, the limits are valid for the purpose of the underlying numerical implementation.
To discuss why we are allowed to interchange the limit and differentiation operator, we refer to the limit representation of the differentiation operator. Therefore, we have to discuss why we are allowed to interchange the two involved limits. To do so we recall equation (4.4): E(J )[S, a(ξ), rn(τ − ξ), (τ − ξ), νn2(τ − ξ); P (., ξ)] = e−(τ −ξ)rn(τ −ξ) Z 1 0 1 x2[P (a(ξ) 1 x, ξ) − (K − a(ξ) 1 x)] × κ(S/a(ξ),1 x, rn+1(τ − ξ), q, (τ − ξ), ν 2 n+1(τ − ξ)) × N (−D(S/a(ξ),1 x, rn(τ − ξ), q, ν 2 n(τ − ξ), νn+12 (τ − ξ), (τ − ξ), γ, δ))dx, (5.5)
Bayraktar (2009) proved that the American put option value is uniformly Lipschitz continuous in S ∈ R+ and uniformly semi-Hlder continuous in τ ∈ [0, T ]. The function
space of Lipschitz continuous functions as well as the function space of semi-Hlder con-tinuous functions are sub spaces of the function space of uniformly concon-tinuous functions. Furthermore, the exponential function exp(x) is uniformly continuous for x ∈ (−∞, 0] and the standard normal distribution N (x) is uniformly continuous since all continuous distribution function are uniformly continuous. The natural logarithm ln(x) is uniformly continuous on x ∈ (0, ∞). Combining all these properties, we can conclude that E(J )
is uniformly continuous in S ∈ (0, ∞). We have shown in the first part of this proof that the limit of E(J ) with respect to ξ → τ exists. Furthermore, the real numbers R equipped with the Euclidean metric is a complete metric space. Therefore, all require-ments of theorem 1 of the paper by Kadelburg and Marjanovic(2005) are fulfilled and hence the interchange of the limit and the differentiation operator is justified under the assumption that a(τ ) > 0. We assume that a(τ ) > 0 for ∀τ ∈ [0, T ] in the conditions of the proposition.
Remark 1. In the final step of our numerical implementation we derive the option value through applying Simpson’s rule. This means that we have to determine the limit of E(D) and E(J ) with respect to ξ → τ , but this time for S rather than a(τ ). The limit for E(D)as argued byKallast and Kivinukk (2003) has only to be evaluated if S > a(τ ), in which case the limit is zero. This is because when S ≤ a(τ ) it is optimal to exercise the put option immediately when entering the put option. This implies that the option value is equivalent to its intrinsic value and therefore there is no need to determine the option value through equation (4.2) as the value is readily known.
It follows that we also only have to evaluate the limit of E(J ) for S > a(τ ) as the previous argument stays valid. When S > a(τ ) the limit of the D-function becomes infinity implying a limit of E(J ) of zero.
5.2
Numerical Results
To discuss our results on American put options we consider the same parameter sets as
Chiarella and Ziogas(2009) chose, to insure comparability of the behavior of the early exercise boundaries.
The underlying set of parameters is as following. We consider jump size expectations to be positive, zero or negative, that is we consider γ > 0, γ = 0 or γ < 0. Specifically we investigate γ = ln(1.04), γ = 0 and γ = ln(0.95). Furthermore, we consider an interest rate exceeding the dividend yield as well as vice versa. In particular, we consider r = 0.05 and q = 0.03, and r = 0.03 and q = 0.05. We assume the global variance to be s2 = 0.0593, that is the variance stemming from the diffusion part combined with the variance stemming from the jump part. Therefore, σ2 has to be evaluated for each γ as it is only a fraction of the global variance and dependent on the jump expectation and variance. Apart from these parameters all other parameter are assumed to be equal in all sets to ensure comparability. We choose the strike to be K = 100, the time to maturity to be 0.5 years, the variance of the jump sizes to be δ2 = 0.04 and the jump intensity to be λ = 1. The stock price S is chosen to be 100 when comparing the early exercise boundaries and to be a set of stock prices with values from 50 to 200 when comparing price differences.
Table 5.1: Variance of pure and jump diffusion processes assuming s2 = 0.0593 and a jump size variance of δ2 = 0.04 in case of jump diffusion processes. The variance is derived through using the following formula σ2 = s2−λ[e2γ+δ2
−2eγ+1]. The parameters
are obtained fromChiarella and Ziogas (2009).
σ2 λ γ
0.0593 0
0.0200 1 ln(0.95)
0.0185 1 ln(1)
0.0136 1 ln(1.04)
The validation of the following results is given as I was able to reproduce figure 6 to 9 of
Chiarella and Ziogas(2009) which have been backtested by the Crank-Nicolson scheme solutions with 10000 time steps and 5000 space steps for the true solution. These figures can be found in Appendix 6 in figure (6.2) to (6.5). I also evaluated the call option values and deltas for the stock prices in table (2) and (3) ofChiarella and Ziogas(2009) respectively when using 100 time steps. These values can be found along with the ones of the paper in table (6.1) and (6.2) in the Appendix6. From the option value differences between the schemes we can conclude that 100 time steps lead to a close enough result to understand the impact on the early exercise boundary and the option value when jumps occur. At this point we again want to remind the reader of the similarity of the American call option and the American put option pricing formulas. Furthermore, the results for the American put option has been backtested for the parameter set given in d’Halluin et al. (2005) who also assumes a jump diffusion process with lognormally distributed jump sizes. The results for this parameter set can be found in table (6.3) in the Appendix 6. In the Appendix one can also find figure (6.1), the reproduction of the early exercise boundaries for the pure diffusion case of at-the-money American put options given in Kallast and Kivinukk (2003). Furthermore, I performed a Monte Carlo simulation for all the different parameter sets and checked that exercising in 100% of the cases when the stock is below the early exercise boundary of the American put option is a better strategy than only exercising in 99% of the cases. I simulated 1000000 stock price paths within each Monte Carlo simulation. To illustrate this experiment, the outcome for one parameter set is given in figure (5.1) and table (5.2).
Table 5.2: Monte Carlo Simulation of stock price paths with 1000000 runs with 50 sub time steps for the parameter set r = 0.03, q = 0.05, λ = 1, γ = ln(1), δ2 = 0.04, σ2 = 0.0185, K = 100, and S = 100. The effectiveness stands for the probability of exercising when the stock price is below the early exercise boundary.
Ef f ectiveness V alue
99% 5.737141
100% 5.740301
Now that we have discussed the validation for the derived formulas we proceed by investigating the early exercise boundaries given different parameter sets. Similar to the call option case we observe a large impact of jumps on the early exercise boundary. This shows that early exercising is highly dependent on the stock price process assumption.
Figure 5.1: Sample Stock Price Paths
Sample stock price paths assuming 50 sub time steps,r = 0.03, q = 0.05, λ = 1, γ = ln(1), δ2= 0.04, σ2= 0.0185, K = 100 and S = 100. The red line represents the early exercise
boundary given these parameters.
Figure 5.2: Early Exercise Boundary when r < q
The graph represents the early exercise boundaries of the jump diffusion model with different expected jump sizes as well as of the pure diffusion model of an at-the-money American put
option when r < q. 50 intermediate time steps are used to derive the underlying graphs.
The first observation we can make from figure (5.2) and (5.3) is the difference between the early exercise boundaries of different jump size expectations. In both cases, that is when r < q and r > q, we observe that a smaller jump size expectation leads to a higher early exercise boundary throughout the whole life of the option. This behavior can be easily explained by recalling when rebalancing costs occur. Rebalancing costs occur when the stock price jumps from the stopping region into the continuation re-gion. This jump is associated with a positive jump in the American put case. Therefore, when the jump size expectation is positive the option holder requires an even lower stock price to exercise early than when the jump size expectation is zero or negative since the likelihood of incurring rebalancing costs after having exercised the option is higher.
To understand the reason between different early exercise boundaries of pure and jump diffusion models we have to investigate the two cases, that is r > q and r < q indi-vidually. We begin by discussing the case r > q that is the case of figure (5.3). Close
Figure 5.3: Early Exercise Boundary when r > q
The graph represents the early exercise boundaries of the jump diffusion model with different expected jump sizes as well as of the pure diffusion model of an at-the-money American put
option when r > q. 50 intermediate time steps are used to derive the underlying graphs.
to expiration the early exercise boundary of the pure diffusion model is close to the strike price. As r > q early exercise in the pure diffusion model means earning more interest than having to pay in dividends for all stock prices S below the strike price K. As one approaches time to expiration the stock price at expiration becomes more certain and therefore one should exercise close to expiration if the intrinsic value of the option is positive as otherwise possible interest earnings are missed out. In the case of jump diffusion stock price processes, the presence of jumps removes the certainty of the stock price when approaching expiration. Therefore, one requires a lower stock price for exercising earlier, to insure against potential losses due to an upward jump above the strike price. As discussed before, this results in a lower early exercise boundary when the expected jump size is higher.
As time to maturity increases, we observe that the early exercise boundary of the pure diffusion model decreases fastest. As time to maturity increases, stock price jumps be-come more likely, but they are also more likely to be reversed by either a jump in the opposite direction or by the diffusion process. As a result the negative effect of jumps on the early exercise boundary at expiration decreases with increasing time to maturity implying a flatter slope than in the pure diffusion case.
In the case when r < q, that is the case of figure (5.2), early exercise at expiration in the pure diffusion case is only optimal if the stock price is far lower than the strike since early exercise implies paying a higher dividend yield than the received interest rate. The early exercise boundary at expiration is only slightly negatively affected by jumps. This implies that the early exercise boundary in the pure diffusion case is so low that the stock price is unlikely to actually end up in the stopping region suggesting that the potential rebalancing costs are also relatively small. As time to maturity increases, jumps become more likely and therefore affect the early exercise boundary more. Again we observe the discussed behavior among the different jump size expectations.
We proceed now by discussing American put option price differences between jump dif-fusion processes and pure difdif-fusion processes. Figure (5.4) and (5.5) represent again all the different 6 parameter sets. In Figure (5.4) we investigate the sets in which r < q and in figure (5.5) the ones in which r > q. We determine price differences for a set of stock prices with the lowest being Sl= 50 and the highest being Sh = 200 as below and
above these, extremely small price differences are observed. A positive (negative) price difference is to be interpreted as the jump diffusion model price exceeding (lying below) the pure diffusion price.
