Model and with Applications to Convertible Bonds

Volume 2009, Article ID 945923,17pages doi:10.1155/2009/945923

Research Article

Valuation of Game Options in Jump-Diffusion

Model and with Applications to Convertible Bonds

Lei Wang and Zhiming Jin

College of Science, National University of Defense Technology, ChangSha 410073, China

Correspondence should be addressed to Lei Wang,[email protected] Received 17 November 2008; Accepted 6 March 2009

Recommended by Lean Yu

Game option is an American-type option with added feature that the writer can exercise the option at any time before maturity. In this paper, we consider some type of game options and obtain explicit expressions through solving Stefanfree boundaryproblems under condition that the stock price is driven by some jump-diffusion process. Finally, we give a simple application about convertible bonds.

Copyrightq2009 L. Wang and Z. Jin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetΩ,F,Pbe a probability space hosting a Brownian motion W {Wt : t ≥ 0}and an independent Poisson processN {Nt:t≥0}with the constant arrival rateλ, both adapted to some filtrationF{Ft}_t≥0satisfying usual conditions. Consider the Black-Scholes market.

That is, there is only one riskless bondBand a risky assetS. They satisfy, respectively, dB_trB_tdt, t≥0,

dStS_t−

μdtσdWt−y0

dNt−λdt 1.1

for some constantsμ ∈ R, r, σ > 0 andy₀ ∈ 0,1. Note that the absolute value of relative jump sizes is equal toy₀, and jumps are downwards. It can be comprehended as a downward tendency of the risky asset price brought by bad news or default and so on. From It ˆo formula we can obtain

S_tS₀exp

μ−1

2σ²λy₀

tσW_t 1−y₀Nt. 1.2

Suppose thatX {Xt : t ≤ T} andY {Yt : t ≤ T}be two continuous stochastic processes defined onΩ,F,F,Psuch that for all 0≤ t ≤ T, X_t ≤ Y_ta.s.. The game option is a contract between a holder and writer at timet 0. It is a general American-type option with the added property that the writer has the right to terminate the contract at any time before expiry timeT. If the holder exercises first, then he/she may obtain the value ofXat the exercise time and if the writer exercise first, then he/she is obliged to pay to the holder the value ofY at the time of exercise. If neither has exercised at timeT andT <∞, then the writer pays the holder the valueXT. If both decide to claim at the same time then the lesser of the two claims is paid. In short, if the holder will exercise with strategyτand the writer with strategyγ, we can conclude that at any moment during the life of the contract, the holder can expect to receiveZτ, γ Xτ1_τ≤γYγ1_γ<τ. For a detailed description and the valuation of game options, we refer the reader to Kifer1 , Kyprianou2 , Ekstr ¨om3 , Baurdoux and Kyprianou4 , K ¨uhn et al.5 , and so on.

It is well known that in the no-arbitrage pricing framework, the value of a contract contingent on the assetSis the maximum of the expectation of the total discounted payoff of the contract under some equivalent martingale measure. Since the market is incomplete, there are more than one equivalent martingale measure. Following Dayanik and Egami6 , let the restriction toFtof every equivalent martingale measure P^α in a large class admit a Radon-Nikodym derivative in the form of

dP^α dP

Ft

ηt, dη_tη_t−

βdW_t α−1

dN_t−λdt

, t≥0, η₀1

1.3

for some constantsβ ∈ R andα > 0. The constantsβandαare known as the market price of the diffusion risk and the market price of the jump risk, respectively, and satisfy the drift condition

μ−rσβ−λy0α−1 0. 1.4

Then the discounted value process{e^−rtS_t : t ≥ 0}is aP^α,F-martingale. By the Girsanov theorem, the process{W_t^αWt−βt:t≥0}is a Brownian motion under the measureP^α, and {Nt:t≥0}is a homogeneous Poisson process with the intensityλ^αλαindependent of the Brownian motionW^αunder the same measure. The infinitesimal generator of the processS under the probability measureP^αis given by

A^αfx

rλ^αy₀ x∂f

∂x1

2σ²x²∂²f

∂x² λ^α f

x

1−y₀

−fx

, 1.5

on the collection of twice-continuously differentiable functionsf·. It is easily checked that A^α−rfx 0 admits two solutionsfx x^k1 andfx x^k2, wherek₁ < 0 < 1 k₂ satisfy

1

2σ²kk−1

rλ^αy₀ k−

rλ^α λ^α

1−y₀k0. 1.6

Suppose thatP^α_xis the equivalent martingale measure forSunder the assumption thatS0 x for a specified market priceα·of the jump risk, and denoteE^α_xto be expectation underP^α_x. The following theorem is the Kifer pricing result.

Theorem 1.1. Suppose that for allx >0 E^α_x

sup

0≤t≤Te^−rtYt

<∞ 1.7

and ifT ∞thatP^α_xlim_t↑∞e^−rtY_t0 1. LetSt,T be the class ofF-stopping times valued int, T , andS ≡ S_0,∞, then the price of the game option is given by

Vx inf

γ∈S0,T

sup

τ∈S0,T

E^α_x

e^−rτ∧γZτ,γ

sup

τ∈S0,T

γ∈Sinf0,T

E^α_x

e^−rτ∧γZτ,γ

. 1.8

Further the optimal stopping strategies for the holder and writer, respectively, are τ^∗inf

t≥0 :V St

Xt

∧T, γ^∗inf

t≥0 :V St

Yt

∧T. 1.9

2. A Game Version of the American Put Option (Perpetual Israeli δ-Penalty Put Option)

In this case, continuous stochastic processes are, respectively, given by X_t

K−S_t

, Y_t

K−S_t

δ, 2.1 where K > 0 is the strike-price of the option, δ > 0 is a constant and can be considered as penalty for terminating contract by the writer. For the computation of the following, let us first consider the case of the perpetual American put option with the same parameterK.

From Jin7 we know that the price of the option is V^Ax sup

τ∈SE^α_x e^−rτ

K−S_τ

2.2

with the superscript A representing American. Through martingale method we have the following.

Theorem 2.1. The price of the perpetual American option is given by

V^Ax

⎧⎪

⎨

⎪⎩

K−x x∈

0, x^∗ , K−x^∗x

x^∗ _k1

x∈ x^∗,∞

, 2.3

wherex^∗k1K/k1−1, the optimal stopping strategy is τ_∗inf

t≥0 :St≤x^∗

. 2.4

Proposition 2.2. V^Ax is decreasing and convex on 0,∞, and under equivalent martingale measureP^α_x, one has that{e^−rtV^ASt:t≥0}and{e^{−rt∧τx∗}V^AS_t∧τx∗:t≥0}are supermartingale and martingale, respectively.

Now, let us consider this game option. It is obvious that for the holder, in order to obtain the most profit, he will exercise whenSbecomes as small as possible. Meanwhile, he must not wait too long for this to happen, otherwise he will be punished by the exponential discounting. Then the compromise is to stop whenSis smaller than a given constant. While for the writer, a reasonable strategy is to terminate the contract when the value of the assetS equals toK. Then only the burden of a payment of the formδe^−rτis left. For this case, if the initial value of the risky asset is belowKthen it would seem rational to terminate the contract as soon asShitsK. On the other hand, if the initial value of the risky asset is aboveK, it is not optimal to exercise at once although the burden of the payment at this time is onlyδ. A rational strategy is to wait until the last moment thatSt≥Kin order to prolong the payment.

However, it should be noted that the value of theδmust not be too large, otherwise it will be never optimal for the writer to terminate the contract in advance.

Theorem 2.3. Letδ^∗V^AK K−x^∗K/x^∗k1, one has the following.

1Ifδ≥δ^∗, then the price of this game option is equal to the price of the perpetual American put option, that is, it is not optimal for the writer to terminate the contract in advance.

2Ifδ < δ^∗, then the price of the game option is

Vx

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

K−x x∈ 0, k_∗

, AxBx^k1 x∈

k_∗, K , δ

x K

_k1

x∈K,∞

2.5

with

A δk_∗^k1− K−k_∗

K^k1

Kk^k_∗¹−k_∗K^k1 , B K K−k_∗

−δk_∗

Kk^k_∗¹−k_∗K^k1 , 2.6 and the optimal stopping strategies for the holder and writer, respectively, are

τ^∗inf

t≥0 :S_t≤k_∗

, γ^∗inf

t≥0 :S_tK

, 2.7

wherek_∗is the (unique) solution in0, Kto the equation δK

1−k₁

x^k1K²k₁x^k1−1−K^1k1 0. 2.8

Before the proof, we will first give two propositions.

Proposition 2.4. Equation2.8has and only has one root in0, K.

Remark 2.5. If we denote the root of2.8in0, Kbyk_∗, then fromProposition 2.4we know thatKK−k_∗−δk_∗>0, thusB >0.

Proposition 2.6. Vxdefined by the right-hand sides of 2.5is convex and decreasing on0,∞.

Proof. From the expression ofVxandRemark 2.5we know thatVxis convex on0, K and K,∞. Thus, we only need to prove the convexity of Vx at the point K, that is, VK≥VK−. Through elementary calculations we obtain

VK− 1

Kk^k_∗¹−k_∗K^k1

δk_∗^k1− K−k_∗

K^k1

KK−k_∗

−δk_∗

k₁K^k1−1 ,

VK δk1

K .

2.9

Then if we can prove that

δk^k_∗¹− K−k_∗

K^k1≤0, 2.10

VK≥VK−will hold. From2.8we can easily find that whenδ δ^∗,k_∗x^∗. Further, asδdecreases the solutionk_∗increases. Especially, whenδ 0,k_∗K. So if 0 < δ < δ^∗, we havex^∗< k_∗< K.

Now let us verify the correctness of2.10. If not, that is,δ > K−k_∗K/k_∗^k1, then from2.8we obtain

K^1k1−K²k1k^k_∗¹⁻¹−K 1−k1

k_∗^k1δ 1−k1

k^k_∗¹ >

K−k_∗ 1−k1

K^k1, 2.11

rearranging it we have

k_∗K^k1−Kk^k_∗¹

1−k₁k₁K k_∗

>0. 2.12

Sincek_∗> x^∗, so 1−k₁k₁K/k_∗>0, whereask_∗K^k1−Kk^k_∗¹<0, which contradicts with2.12.

So the hypothesis is not true, that is,2.10holds, which also implies thatA≤ 0. SoVxis decreasing on0,∞.

Proof ofTheorem 2.3. 1Suppose thatδ≥δ^∗. From the expression ofV^Axwe can easily find that

K−x≤V^Ax≤K−xδ. 2.13

By means ofProposition 2.2and the Doob Optional Stopping Theorem, we have V^Ax inf

γ∈SE^α_x

e^{−rτ∗∧γ}V^A Sτ_∗∧γ

≤inf

γ∈SE^α_x e^−rτ∗

K−Sτ_∗

1_τ∗≤γe^−rγ K−Sγ

δ

1_γ<τ∗

≤inf

γ∈Ssup

τ∈SE^α_x e^−rτ

K−Sτ

1_τ≤γe^−rγ K−Sγ

δ

1_γ<τ sup

τ∈S

infγ∈SE^α_x e^−rτ

K−S_τ

1_τ≤γe^−rγ

K−S_γ δ

1_γ<τ

≤sup

τ∈S E^α_x e^−rτ

K−Sτ

V^Ax.

2.14

That is, the price of the game option is equal to the price of the perpetual American put option.

2 Ifδ < δ^∗, according to the foregoing discussion andTheorem 1.1, there exists a numberksuch that the continuation region is

C

x:g₁x< Vx< g₂x

{x:k < x <∞, x /K} 2.15

withg₁x K−x, g₂x K−xδ, k∈0, Ka constant to be confirmed, while the stopping area is

DD1∪D2, 2.16

whereD₁ {x :Vx g₁x} {x :x ≤ k}is the stopping area of the holder,D₂ {x : Vx g₂x}{x:xK}is the stopping area of the writer. For search of the optimalk_∗and the value ofVx, we consider the following Stefanfree boundaryproblem with unknown numberkandV Vx:

Vx K−x, x∈0, k , A^α−r

Vx 0, x∈k, K∪K,∞, 2.17

and additional conditions on the boundarykandKare given by

limx↓kVx K−k, lim

x→KVx δ, lim

x↓k

∂Vx

∂x −1, lim

x↑∞Vx 0. 2.18

By computing Stefan problem we can easily obtain the expression ofVx denote it byVxdefined by the right-hand sides of2.5, while from2.18we can obtain2.8.

Proposition 2.4implies that this equation has and only has one root in0, K, denote it byk_∗. Accordingly, we can obtain the expression2.6ofA andBand optimal stopping strategy τ^∗ for the holder. Now we must prove that the solution of the Stefan problem gives, in fact,

the solution to the optimal stopping problem, that is,Vx Vx. For that it is sufficient to prove that

a∀τ ∈ S,E^α_xe^{−rτ∧γ∗}Zτ,γ^∗≤Vx;

b∀γ ∈ S,E^α_xe^{−rτ∗∧γ}Z_τ∗,γ ≥Vx;

cE^α_xe^{−rτ∗∧γ∗}Z_τ^∗_,γ^∗ Vx.

First, fromProposition 2.6we know thatVxis a convex function on0,∞such that K−x ≤Vx≤K−xδ. 2.19

SinceVx∈C¹0, K∩C²0, K\{k_∗}, forx∈0, K, we can apply It ˆo formula to the process {e^{−rt∧γ∗}VS_t∧γ^∗:t≥0}and have

e^{−rt∧γ∗}V S_t∧γ^∗

Vx _t∧γ∗

0

e^−ru

A^α−r V S_u

du _t∧γ∗

0

e^−ruσS_uV S_u

dW_u^α

_t∧γ∗

0

e^−ru V S_u−

1−y₀

−V S_u−

dN_u−λ^αdu .

2.20

Note that in 0, K, A^αVx−rVx ≤ 0, while the last two integrals of 2.20 are local martingales, then by choosing localizing sequence and apply the Fatou lemma, we obtain

E^α_xe^{−rτ∧γ∗}V S_τ∧γ∗

≤Vx, 2.21

whereas

Zτ,γ^∗ K−Sτ

1_τ≤γ^∗ K−Sγ^∗

δ

1γ^∗<τ

K−S_τ

1_τ≤γ^∗δ1_γ^∗_<τ

≤V S_τ∧γ∗

.

2.22

For the inequality we have used2.19, hence from2.21we have

E^α_xe^{−rτ∧γ∗}Zτ,γ^∗≤Vx. 2.23

It is simple for the case thatx∈K,∞and the method is the same as before. Thus, we obtain a.

The proof ofb: apply It ˆo formula to the process{e^{−rτ∗∧t}VSτ^∗∧t: t ≥ 0}and note thatV is only continuous atK, we have

e^{−rτ∗∧t}V Sτ^∗∧t

Vx _τ^∗_∧t

0

e^−ru

A^α−r V Su

du _τ^∗_∧t

0

e^−ruσSuV Su

dW_u^α

_τ^∗_∧t

0

e^−ru V S_u−

1−y₀

−V S_u−

dN_u−λ^αdu e^{−rτ∗∧t} VK−VK−

L^K_τ∗∧t,

2.24

whereL^Kis the local time atKofS. SinceVxis convex on0,∞, henceVK−VK−≥ 0. While ink_∗,∞\ {K},A^α−rVx 0, then using the same method as before we have

E^α_xe^{−rτ∗∧γ}V Sτ^∗∧γ

≥Vx. 2.25

Moreover, since V

Sτ^∗∧γ V

Sτ^∗

1_τ^∗_≤γV Sγ

1_γ<τ^∗

≤ K−Sτ^∗

1_τ^∗_≤γ K−Sγ

δ

1_γ<τ^∗ Zτ^∗,γ,

2.26

we can obtain

E^α_xe^{−rτ∗∧γ}Z_τ^∗_,γ ≥Vx, ∀γ ∈ M. 2.27 The proof ofc: takingτ τ^∗, γ γ^∗, it is sufficient to note that ink_∗, K, we have A^αVx−rVx 0 and

V S_τ∗∧γ^∗

V S_τ∗

1_τ∗≤γ^∗V S_γ∗

1_γ∗<τ^∗

K−Sτ^∗

1_τ^∗_≤γ^∗δ1_γ^∗<τ^∗

Zτ^∗,γ^∗.

2.28

The same result is true for the case thatx∈K,∞.

3. Game Option with Barrier

Karatzas and Wang8 obtain closed-form expressions for the prices and optimal hedging strategies of American put options in the presence of an up-and-out barrier by reducing this problem to a variational inequality. Now we will consider the game option connected with this barrier option. Following Karatzas and Wang, the holder may exercise to take the claim of this barrier option

Xt K−St

1_t<τh, 0≤t <∞. 3.1

Hereh >0 is the barrier, whereas

τhinf

t≥0 :St> h

3.2

is the time when the option becomes “knocked-out”. The writer is punished by an amountδ for terminating the contract early

Y_t

K−S_t δ

1_t<τh. 3.3

First, let us consider this type of barrier option. The price is given by V^Bx sup

τ∈S Exe^−rτ K−Sτ

1_τ<τh 3.4

with the superscriptBrepresenting barrier. Similarly to Karatzas and Wang we can obtain the following.

Theorem 3.1. The price of American put-option in the presence of an up-and-out barrier is

V^Bx

⎧⎪

⎪⎨

⎪⎪

⎩

K−x x∈ 0, p_∗

, AxBx^k1 x∈

p_∗, h ,

0 x∈h,∞,

3.5

whereA p_∗−Kh^k1/hp^k_∗¹−p_∗h^k1, B K−p_∗h/hp^k_∗¹−p_∗h^k1, and the optimal stopping strategy is

τ_∗inf

t≥0 :S_t≤p_∗

, 3.6

wherep_∗is the (unique) solution in0, Kto the equation h

1−k1

x^k1Kh k1x^k1−1−Kh^k1 0. 3.7

The proof of the theorem mainly depends on the following propositions and the process will be omitted.

Proposition 3.2. The expression ofV^Bxdefined by3.5is convex and decreasing on0,∞, and under risk-neutral measureP^α_x, one has that{e^−rtV^BSt:t≥0}and{e^{−rt∧τ∗}V^BSt∧τ∗:t≥0}are supermartingale and martingale, respectively.

Proposition 3.3. Equation3.7has and only has one root in0, K.

Now let us consider the game option with barrierh. The price is given by Vx sup

τ∈S inf

γ∈SEx

e^−rτ

K−Sτ

1_τ≤γ·1_τ<τhe^−rγ K−Sγ

δ

1_γ<τ·1_γ<τh

. 3.8

For this game option, the logic of its solution is similar to the former, and based on this consideration, we have the following theorem.

Theorem 3.4. Letδ^∗V^BK K−p_∗hK^k1−Kh^k1/hp^k_∗¹−p_∗h^k1, one has the following.

1Ifδ≥δ^∗, then the price of this game option is equal to the price of American put options in the presence of an up-and-out barrier, that is, it is not optimal for the writer to exercise early.

2Ifδ < δ^∗, then the price of the game option is given by

Vx

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

K−x x∈

0, b_∗ , C₁xC₂x^k1 x∈

b_∗, K , D1xD2x^k1 x∈K, h,

0 x∈h,∞,

3.9

where

C1 δb_∗^k1− K−b_∗

K^k1

Kb^k_∗¹−b_∗K^k1 , C2 K K−b_∗

−δb_∗ Kb^k_∗¹−b_∗K^k1 , D₁ −δh^k1

hK^k1−Kh^k1, D₂ δh hK^k1−Kh^k1,

3.10

andb_∗is the (unique) solution in0, Kto the equation δK

1−k₁

x^k1K²k₁x^k1−1−K^1k1 0, 3.11

and the optimal stopping strategies for the holder and writer, respectively, are

τ^∗inf

t≥0 :S_t≤b_∗

, γ^∗inf

t≥0 :S_tK

. 3.12

Proposition 3.5. The functionVxdefined by3.9is convex and decreasing on0,∞.

Proof. Similar toProposition 2.6, we only need to prove the convexity ofVxat the pointK, that is,

VK−VK−

D2−C2

k1K^k1−1 D1−C1

≥0. 3.13

Through lengthy calculations we know that it is sufficient to show that

δ

hb^k_∗¹−b_∗h^k1

≤

K−b_∗

hK^k1−Kh^k1

. 3.14

Suppose that3.14does not hold, that is,δ >K−b_∗hK^k1−Kh^k1/hb^k_∗¹−b_∗h^k1, then from 3.11we find that

K^1k1−K²k1x^k1−1−K 1−k1

b^k_∗¹ δ 1−k1

b^k_∗¹

>

1−k₁ b^k_∗¹

K−b_∗

hK^k1−Kh^k1

hb^k_∗¹−b_∗h^k1 , 3.15

rearranging it we have h

1−k₁

b^k_∗¹Kh k₁b_∗^k1−1−Kh^k1 <0. 3.16

From3.11, through complex verification we get that whenδ δ^∗,b_∗ p_∗. Furthermore, as δdecreases the solutionb_∗increases, especially whenδ0,b_∗K. So if 0< δ < δ^∗, we have p_∗< b_∗< K. Thus from the property of3.7we know thath1−k₁b_∗^k1Kh k₁b^k_∗¹⁻¹−Kh^k1>0, which contradicts with3.16. So the hypothesis is not true, that is,3.14holds. It is evident thatVxis decreasing.

Remark 3.6. It is obvious that2.8is the same as 3.11, however, their roots not always be equal to each other. Because of these two cases, the scope ofδis different. Penalty with barrier is usually smaller than the other, that is,V^BK< V^AK.

Proof ofTheorem 3.4. 1Suppose thatδ≥δ^∗. FromProposition 3.2we know that

K−x ≤V^Bx≤K−xδ. 3.17

By means of the Doob optional stopping theorem and3.17, we have V^Bx inf

γ∈SE^α_x

e^{−rτ∗∧γ∧τh}V^B Sτ^∗∧γ∧τh

≤inf

γ∈SE^α_x e^−rτ∗

K−Sτ^∗

1_τ^∗_≤γ1_τ^∗<τhe^−rγ K−Sγ

δ

1_γ<τ^∗1_γ<τh

≤inf

γ∈Ssup

τ∈SE^α_x e^−rτ

K−Sτ

1_τ≤γ1_τ<τhe^−rγ K−Sγ

δ

1_γ<τ1_γ<τh sup

τ∈S inf

γ∈SE^α_x e^−rτ

K−S_τ

1_τ≤γ1_τ<τhe^−rγ

K−S_γ δ

1_γ<τ1_γ<τh

≤sup

τ∈SE^α_x e^−rτ

K−Sτ

1_τ<τh V^Bx.

3.18

That is, the price of the game option is equal to the price of American put-options in the presence of an up-and-out barrier.

2Suppose thatδ < δ^∗. Then we may conclude that the holder should search optimal stopping strategy in the class of the stopping times of the formτ_b inf{t≥0 : S_t ≤b}with

b∈0, Kto be confirmed. While the optimal stopping strategy for the writer isγ^∗ inf{t≥ 0 :S_tK}. Considering the following Stefan problem:

Vx K−x, x∈0, b , 3.19 A −rVx 0, x∈b, K∪K, h, 3.20

Vx 0, x∈h,∞, 3.21

limx↓bVx K−b, lim

x→KVx δ, lim

x↑hVx 0, lim

x↓b

∂Vx

∂x −1. 3.22

Through straightforward calculations we can obtain the expression of Vx denote it by Vxdefined by the right-hand sides of3.9. From condition3.22we can obtain3.11.

Proposition 2.4implies that the root of this equation is unique in0, K, denote it byb_∗and consequentlyτb_∗byτ^∗. Now we only need to prove thatVx Vx. For that it is sufficient to prove that

a ∀τ ∈ S, E^α_xe^{−rτ∧γ∗}Zτ,γ^∗1_τ∧γ^∗<τh≤Vx; 3.23 b ∀γ∈ S, E^α_xe^{−rτ∗∧γ}Z_τ∗,γ1_τ∗∧γ<τh≥Vx. 3.24 cTaking stopping timeττ^∗, γ γ^∗, we have

E^α_xe^{−rτ∗∧γ∗}Z_τ^∗_,γ^∗1_τ^∗_∧γ^∗_<τhVx. 3.25

First, fromProposition 3.5we know thatVxis convex in0,∞and further

K−x ≤Vx≤K−xδ. 3.26

Applying It ˆo formula to the process{e^{−rt∧γ∗∧τh}VS t∧γ^∗∧τh:t≥0}, we have

e^{−rt∧γ∗∧τh}V

S_t∧γ^∗_∧τh

Vx _t∧γ^∗_∧τh

0

e^−ru

A^α−r V Su

du _t∧γ^∗_∧τh

0

e^−ruσSuV Su

dW_u^α

_t∧γ∗∧τh

0

e^−ru V S_u−

1−y₀

−V S_u−

dN_u−λ^αdu .

3.27

It is obvious that whenx∈0, K∪K, h, we haveA −rVx≤0. Since the second and the third integrals of the right-hand sides of3.27are local martingales, so

E^α_xe^{−rτ∧γ∗∧τh}V

S_τ∧γ^∗_∧τh

≤Vx, 3.28

while

E^α_xe^{−rτ∧γ∗∧τh}V

S_τ∧γ^∗_∧τh

E^α_xe^{−rτ∧γ∗}V S_τ∧γ^∗

1_τ∧γ^∗_<τh≥E^α_xe^{−rτ∧γ∗}Z_τ,γ^∗1_τ∧γ^∗_<τh. 3.29

The inequality is obtained from 3.26, and combining3.28 we obtain3.23, that is, a holds.

Applying It ˆo formula to the process{e^{−rτ∗∧t∧τh}VSτ^∗∧t∧τh:t≥0}, we have

e^{−rτ∗∧t∧τh}V

S_τ^∗_∧t∧τh

Vx _τ∗∧t∧τh

0

e^−ruA −rV S_u

du _τ∗∧t∧τh

0

e^−ruσ S_u V

S_u dW_u^α e^{−rτ∗∧t∧τh} VK−VK−

L^K_τ∗∧t∧τh

_τ^∗_∧t∧τh

0

e^−ru V S_u−

1−y₀

−V S_u−

dN_u−λ^αdu .

3.30

The definition ofL^Kis the same asTheorem 2.3. From the convexity ofVxwe know that VK−VK− ≥ 0. Since when x ∈ b∗, K∪K, h,A −rVx 0, so from above expression we have

E^α_xe^{−rτ∗∧γ∧τh}V S_τ∗∧γ∧τh

≥Vx. 3.31

Similarly we have E^α_xe^{−rτ∗∧γ∧τh}V

Sτ^∗∧γ∧τh

E^α_xe^{−rτ∗∧γ}V Sτ^∗∧γ

1_τ^∗_∧γ<τh≤E^α_xe^{−rτ∗∧γ}Zτ^∗,γ1_τ^∗_∧γ<τh. 3.32

From3.31and3.32we know thatbholds. Combiningaandbwe can easily obtain c.

4. A Simple Example: Application to Convertible Bonds

To raise capital on financial markets, companies may choose among three major asset classes:

equity, bonds, and hybrid instruments, such as convertible bonds. As hybrid instruments, convertible bonds has been investigated rather extensively during the recent years. It entitles its owner to receive coupons plus the return of the principle at maturity. However, the holder can convert it into a preset number of shares of stock prior to maturity. Then the price of the bond is dependent on the price of the firm stock. Finally, prior to maturity, the firm may call the bond, forcing the bondholder to either surrender it to the firm for a previously agreed price or else convert it for stock as above. Therefore, the pricing problem has also a game- theoretic aspect. For more detailed information and research about convertible bonds, one is referred to Gapeev and K ¨uhn9 , Sˆırbu et al.10,11 , and so on.

Now, we will give a simple example of pricing convertible bonds, as the application of pricing game options. Consider the stock process which pays dividends at a certain fixed rated∈0, r, that is,

dS_tS_t−

μ−ddtσdW_t−y₀

dN_t−λdt

. 4.1

Then the infinitesimal generator ofSbecomes

A^αfx

r−dλ^αy0

x∂f

∂x1

2σ²x²∂²f

∂x² λ^α f

x 1−y0

−fx

, 4.2

andA^α−rfx 0 admits two solutionsfx x^k1andfx x^k2withk₁ < 0< 1 < k₂ satisfying

1

2σ²kk−1

r−dλ^αy₀ k−

rλ^α λ^α

1−y_0k

0. 4.3

At any time, the bondholder can convert it into a predetermined number η > 0 of stocks, or continue to hold the bond and collecting coupons at the fixed ratec > 0. On the other hand, at any time the firm can call the bond, which requires the bondholder to either immediately surrender it for the fixed conversion priceK > 0 or else immediately convert it as described above. In short, the firm can terminate the contract by paying the amount max{K, ηS}to the holder. Then, if the holder terminates the contract first by converting the bond intoηstocks, he/she can expect todiscountedreceive

L_{t t}

0

c·e^−rudue^−rtηS_t, 4.4

while if the firm terminate the contract first, he/she will pay the holder

U_{t t}

0

c·e^−rudue^−rt

K∨ηS_t

. 4.5

Then, according toTheorem 1.1, the price of the convertible bonds is given by V^CBx inf

γ∈Ssup

τ∈S E^α_x

Lτ1_τ≤γUγ1_γ<τ sup

τ∈S inf

γ∈SE^α_x

Lτ1_τ≤γUγ1_γ<τ

. 4.6

Note that when c ≥ rK, the solution of4.6 is trivial and the firm should call the bond immediately. This implies that the bigger the coupon ratec, the more the payoffof the issuer, then they will choose to terminate the contract immediately. So we will assume thatc < rK in the following.

Now, let us first consider the logic of solving this problem. It is obvious that ηx ≤ V^CBx≤K∨ηxfor allx >0chooseτ 0 andγ 0, resp.. Note whenS_t≥K/η, L_tU_t,

thenV^CBx ηx for allx ≥ K/η. Hence the issuer and the holder should search optimal stopping in the class of stopping times of the form

γ_ainf

t≥0 :S_t≥a

, τ_binf

t≥0 :S_t≥b

, 4.7

respectively, with numbers 0 < a, b ≤ K/η to be determined. Note when the process S fluctuates in the interval0, K/η, it is not optimal to terminate the contract simultaneously by both issuer and holder. For example, if the issuer chooses to terminate the contract at the first time thatSexceeds some pointa∈ 0, K/η, thenηa < K, and the holder will choose the payoffof coupon rather than converting the bond into the stock, which is a contradiction.

Similarly, one can explain another case. Then only the following situation can occur: either a < bK/η,b < aK/η, orbaK/η.

For search of the optimala_∗, b_∗ and the value ofV^CBx, we consider an auxiliary Stefan problem with unknown numbersa, b,andVx

A^α−r

Vx −c, 0< x < a∧b,

ηx < Vx< ηx∨K, 0< x < a∧b 4.8

with continuous fit boundary conditions

Vb− ηb, Vx ηx 4.9

for allx > b, b≤aK/η, and

Va− K, Vx ηx∨K 4.10

for allx > a, a≤bK/η, and smooth fit boundary conditions

Vb− η if b < a K

η, Va− 0 ifa < b K

η. 4.11

By computing the Stefan problem we can obtain that if

K > k₂ k2−1

c

r, 4.12

thenb_∗< a_∗K/η, and the expression ofVxis given by

Vx ηb_∗ k₂

x b_∗

_k2 c

r 4.13