| Game Options is a new type of financial derivative products, similar to the American-style options, but it is essentially different from the ordinary American option:Not only does it give the holder the right to executive the contract, but also gives the writer the right to terminate the contract ahead of time, the option with an application is redeemable convert-ible bonds. in this paper, analysis methods and stochastic analysis theory are used to get an explicit pricing formula for options and the best implementation strategy about the put game option in some jump-diffusion model, which we focus on the research in the case of uniform distribution nature of the positive jump and negative jump, the case of exponential distribution of the positive jump,negative jump, as well as the case of double exponential jump-diffusion model, finally, we obtain the results of numerical analysis to show the sensi-tive relationship of the price of game options and the best implementation of the boundary with the model parameters, as well as to compare the price of game option and the best implementation boundary between the cases of jump and without jump.Game options, also known as Israeli options, it stipulates that the contract can be termi-nated by the holder ahead of time, can also be implemented by the writer in advance. In fact, the game options apply the game theory to the option contract, by means of the additional cost ofδto reduce the losses against the holder as a result of the writer to terminate the contract beforehand; at the same time, the option also gives the writer the right to repurchase the option, therefore, the price of the game option relative to the general American option price should be lower.In 2000, Kifer [6] gives the definition of game options, which is defined as:within the option period [0,T], the game options can be implemented at any time, and if the holder at the time t implements the option contract, then he can receive the fees Xt from the writer; if the writer at the time t to terminate the contract, he shall pay the fees Yt to the holder of. If the writer terminates the contract prior to the holder, then Yt= Xt+δ; if the holder implements the contract prior to the writer, then Yt= Xt. where the practical significance ofδis the additional cost due to the writer to terminate the contract, which can be considered as breach of contract writer to pay compensation to the holder, so Xt≤Yt. setτ,ηrespectively to the times that the holder and the writer enforce the contract, then the holder of the revenue function of R(τ,η)=XτI{τ≤η}+YζI{η＜τ}, where I{τ≤η} and I{η＜τ}, respectively to the Indicator function ofτ≤ηandη＜τ. In [6], Kifer gave a general framework for discussion of game options.In 2004, A.E.Kyprianou in [7] gave the pricing estimate of Israeli option in the first and second special cases, that is, obtained the context of pricing formula in the cases of the perpetual American Israeli put option and permanent Russian type Israeli put option at the cost ofδ. in the same year, E.J.Baurdoux and A.E.Kyprianou [8] then gave a third form of the option pricing. In 2005, Luis Gustavo Hernandez Urena [9] studied in the random interest rate market about the Game option pricing. in 2006, E.Ekstrom [10] studied the price of some properties of the game options by use of stochastic analysis method, In the same year,T.R.Bielecki, S.Creepey, M.Jeanblanc and M.Rutkowski [11] focused on the pricing of the default game options available and the convertible bonds. In 2007, Yan Dolinsky, Yuri Kifer [12] studied the effective hedging issue of the game options when the initial capital value of the portfolio is less than the fair value of options. In 2008, T.R.Bielecki, S.Creepey, M.Jeanblanc and M.Rutkowski studied the estimation and hedging problems of the default game options in the case when considering credit risk under Hazard Process [13]. In 2009, Wang Lei and Jin Zhi-ming [14] studied the pricing of the game options in a jump-diffusion model, which took into account the case the number of jump being random, and the magnitude of the jump being a constanty0.The following are the main theorem results of this paper:Theorem 1.1. When the jump obeys the uniform distribution, letδ*= VA(s)= (K-S0)(K/S0)K1,(1) Ifδ＞δ*, then the price of this game option is equal to the price of the perpetual American put option, that is, to terminate the contract is detrimental for the writer. (2) Ifδ＜δ*, then the price of this game option is With and for the holders and the writer, the optimal stopping time strategies, respectively, are Here k, is the unique solution in (0, K) to the equation as follows:Theorem 1.2. When the jump obeys the negative exponential distribution, Letδ*= V(K), which is shown by (3.16).(1)Ifδ≥δ*then the price of this game option is equal to the price of the perpetual American put option, that is, to terminate the contract is detrimental for the writer.(2)Ifδ＜δ*, then the price of this game option is with and for the holders and the writer, the optimal stopping time strategies, respectively, are Here K* is the unique solution in (0, K) to the equation as follows:Theorem 1.3. When the jump obeys the positive exponential distribution, Letδ*= V(K), which is shown by (3.30).(1)Ifδ≥δ*, then the price of this game option is equal to the price of the perpetual American put option, that is, to terminate the contract is detrimental for the writer.(2)Ifδ＜δ*, then the price of this game option is with and for the holders and the writer,the optimal stopping time strategies,respectively,are Here k* is the unique solution in(0,K)to the equation as follows: A/(α-1)(K/K*)+B/(α-k1)(K/k*)k1+C/(α-k2)(K/k*)k2=δ/(α-k1) (1.9)Theorem 1.4. When the jump obeys the double exponential distribution,Letδ*=V(K),(1)Ifδ≥δ*,then the price of this game option is equal to the price of the perpetual American put option,that is,to terminate the contract is detrimental for the writer.(2)Ifδ＜δ*,then the price of this game option is with and for the holders and the writer, the optimal stopping time strategies, respectively, are Here k, is the unique solution in (0, K) to the equation as follows: and k2＜k1＜1＜k3, Satisfying the following equation:... |
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