Game options

Finance Stochast. 4, 443–463 (2000) c Springer-Verlag 2000
Game options Yuri Kifer Institute of Mathematics, The Hebrew University, Givat Ram 91904 Jerusalem, Israel (e-mail: [email protected]) Abstract. I introduce and study new derivative securities which I call game options (or Israeli options to put them in line with American, European, Asian, Russian etc. ones). These are contracts which enable both their buyer and seller to stop them at any time and then the buyer can exercise the right to buy (call option) or to sell (put option) a specified security for certain agreed price. If the contract is terminated by the seller he must pay certain penalty to the buyer. A more general case of game contingent claims is considered. The analysis is based on the theory of optimal stopping games (Dynkin’s games). Game options can be sold cheaper than usual American options and their introduction could diversify financial markets. Key words: American option pricing, optimal stopping game JEL Classification: G13, C73 Mathematics Subject Classification (1991): 90A09, 60J40, 90D15 1 Introduction A standard (B , S )-securities market consists of a nonrandom (riskless) component Bt , which is described as a savings account (or price of a bond) at time t with an interest r, and of a random (risky) component St , which can be described as the price of a stock at time t. Both discrete time t ∈ Z+ = {0, 1, 2, . . . } Dedicated to E.B.Dynkin on his 75th birthday Partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). Manuscript received: June 1999; final version received: November 1999 444 Y. Kifer and continuous time t ∈ R+ = {t ≥ 0} models can be considered. A standard American option is a contract which enables its buyer to exercise it, i.e. to sell (put option) or to buy (call option) the stock for a specific price K , at any time t which amounts to the gain (K − St )+ in the put and (St − K )+ in the call option cases. The problem of fair pricing of American options leads to the optimal stopping of certain stochastic processes (see [Ka1,2], [My], [SKKM1,2] and references there). In this paper I introduce game options in which the seller of an option can cancel the contract at any time t. In this case the buyer’s gain is the sum (K − St )+ + δt in the put and (St − K )+ + δt in the call option case where δt ≥ 0 is certain penalty paid by the seller. The pricing of these options leads to a game version of the optimal stopping problem introduced in the discrete time case by Dynkin [Dy] (for a continuous time version see [Ki]) but for financial applications it is more appropriate to employ another more general set up studied in [Ne] and [Oh] in the discrete time case and considered in the continuous time case in [BF1] (Markov case) and in [LM] (general case). The formal set up consists of a probability space (Ω, F , P ) together with a stochastic process St ≥ 0, t ∈ Z+ , or t ∈ R+ describing the price of a unit of stock, of a family of complete σ-algebras Ft ⊂ F such that Ft is generated by all Su , 0 ≤ u ≤ t, and of two right continuous with left limits stochastic payoff processes Xt ≥ Yt ≥ 0 adapted to the filtration {Ft , t ∈ Z+ or t ∈ R+ }. A game contingent claim (GCC) is a contract between a seller A and a buyer B which enable A to cancel (terminate) it and B to exercise it at any time t up to a maturity date (horizon) T when the contract is terminated anyway. If B exercises the contract at time t then he gets from A the payment Yt but if A cancels before B exercises then A should pay to B the sum Xt . If A cancels and B exercises at the same time t then A pays to B the sum Yt . It turns out (see Remarks 2.2 and 3.4) that if, instead, in the latter case A pays to B in the amount Xt all results remain the same provided there is no penalty at maturity date. Assuming that clairvoyance is not possible A and B have to use only stopping times with respect to the filtration {Ft } as their cancellation and exercise times. The difference δt = Xt − Yt ≥ 0 is interpreted as the penalty which A pays to B for the contract cancellation. What is the fair price V ∗ that B should pay to A for such contract? In accordance with the modern ideology of option pricing based on hedging it is natural to require that V ∗ should be the minimal capital which enables A to invest it into a skillfully managed self-financing portfolio which will cover his liability to pay to B up to a cancellation stopping time σ no matter what exercise time B chooses. Namely, for any initial capital Z0 > V ∗ the seller A should be able to choose a stopping time σ and to manage a self-financing portfolio having a wealth Zt at time t and being redistributed in discrete times or continuously between the savings account and the stock shares so that Zt is sufficient for payment to B provided the latter exercises GCC at the time t ∈ [0, σ]. Thus hedging in GCC consists in a choice of both a hedging investment policy and of a cancellation time of the contract. I shall show that this leads to the zero sum optimal stopping game of two players with the payoffs e −rt Xt and e −rt Yt . Game options 445 If A is not allowed to terminate the contract before the maturity time T then we arrive at an American Contingent Claim (ACC). The same can be achieved in the framework of my model if the penalty is chosen large enough, for instance, if Xt = Yt + δ and δ > sup EYτ . More precise conditions which ensure that 0≤τ ≤T GCC becomes ACC can be given, as well. On the other hand, I could modify the above model so that B is not allowed to terminate the contract until the maturity date T in the spirit of European (game) options. This also can be considered in the framework of my model if I take Yt = 0 for t < T and Yt = YT > 0 for t = T . Observe, that if the penalty δ0 is zero then either A or B should terminate the contract at once and the price V ∗ equals Y0 . It follows from Theorems 2.1 and 3.1 that the price V ∗ is a continuous increasing function of penalty which varies, thus, from Y0 to sup0≤τ ≤T E (e −rτ Yτ ). In the next section I consider the discrete time case where the stock evolution is described by the popular binomial CRR-model introduced in [CRR]. In Sect. 3 I deal with the continuous time situation where the stock evolution is described by the geometric Brownian motion. The payoff functions Xt and Yt are supposed to be right continuous and having left limits. In particular, one can take Yt = (K − St )+ or Yt = (St − K )+ and Xt = Yt + δt . These cases are naturally to be denominated put and call game options, respectively, with a penalty process δt , t ≥ 0. Other payoff functions leading to exotic game options can be considered, as well. In Sect. 4 I discuss the case when Yt and Xt have the form β t Y (St ) and β t X (St ), β ≤ 1. In this paper I consider only basic problems concerning extension of the option pricing theory to game options and many problems still remain to deal with. First one can consider a multidimensional case of several stocks which can be treated in the same way. Incomplete markets also can be treated in my framework employing optional decompositions of supermartingales and superhedging similarly to [Kr]. Next, the model may include transaction costs, portfolio constraints and uncertainty (random environments) which are important in real stock exchange trading but, of course, complicate the study. Furthermore, it is important for applications to find convenient formulas and algorithms for computation of prices of game options. For the binomial CRR-model computations are not difficult, especially in the Markov case considered in Sect. 4 using recursive formulas there. On the other hand, the geometric Brownian motion model leads to quite nontrivial free boundary problems and variational inequalities but I show in Sects. 3 and 4 that discrete time approximations of continuous time models yield methods for their computation. Essentially, any contract in modern life presumes explicitly or implicitly a cancellation option by each side which then has to pay some penalty, and so it is natural to introduce a buyback option to contingent claims, as well. Moreover, already existing corporate bonds which are both putable by the buyer and callable by the issuer give an example of GCC but, usually, they are more difficult to evaluate than GCC’s considered in this paper since such bonds include various restrictions on possible exercise times and they may depend on other 446 Y. Kifer value processes of underlying securities. Game options are safer for an investment company which issues them, and so it can sell them cheaper than usual American options. Recent problems of hedge funds in some emerging markets may justify introduction of such derivative securities with a buyback option as this may protect the issuing company from collapsing and, on the other hand, to ensure some compensation for the buyer. Moreover, this provides the whole range of GCC depending on the agreed penalty: from high penalty and high price GCC which, actually, coincide with usual American contingent claims to low penalty and low price GCC which provide only little protection for the buyer. In addition, GCCs contain some elements of games of chance and require more advanced trading techniques which may be attractive for some investors who could exploit their more sophisticated skills. Introduction of GCCs could help also to diversify financial markets. As a market name for such contracts I suggest to call them Israeli contingent claims (Israeli options) to put them in line with American, Asian, European, Russian etc. ones. After this paper was submitted my attention was drawn to [MS] where a particular case of a GCC was discussed on a heuristic level without indicating any connection to Dynkin’s games and with a different motivation of introducing a model which can be considered as a simplified version of a Liquid Yield Option Note (LYON) which was really traded on markets. Karatzas informed me recently that a pre-publication version of [CK] contained a remark indicating on the possibility of continuous time GCCs, suggesting their connection with backward stochastic differential equations with two reflecting barriers, and asserting that their price is given by the value of a Dynkin game. This remark does not appear in the printed version of [CK] and neither version contains any precise definitions and proofs concerning financial applications of results from [CK]. Moreover, they seem to had in mind the dual approach to the price process which should dominate possible payoffs at any time, and so if the price of a GCC is defined in a natural way via hedging, as I do in this paper, then the results of [CK] do not imply directly the price formula obtained here. In addition, the approach of [CK] relying on results about backward stochastic differential equations is more complex than the arguments considered here, it requires also stronger assumptions on payoff processes Xt , Yt and, of course, it does not work in the discrete time case. A preprint version of the present paper has appeared at the end of 1997 though I discussed with several people the idea of game options a couple of years earlier. All rights on commercial use of game (Israeli) contingent claims considered in this paper are reserved with the author. 2 Discrete time Let Ω = {1, −1}N be the space of finite sequences ω = (ω1 , ω2 , . . . , ωN ); ωi = 1 or = −1 with the product probability P = {p, q}N , q = 1 − p so that p(ω) =
 N  p k q N −k where k = 12 N + ωi . In this section I consider the CRR-model of i =1 Game options 447 financial market which functions at times n = 0, 1, . . . , N < ∞ and consists of a savings account Bn with an interest rate r, so that Bn = (1 + r)n B0 , B0 > 0, r > 0, (2.1) and of a stock whose price at time n equals Sn = S0 n  (1 + ρk ), S0 > 0, (2.2) k =1 where ρk (ω) = 21 (a + b + ωk (b − a)), ω = (ω1 , ω2 , . . . , ωN ). Thus the “random growth rates” ρk , k = 1, . . . , N form a sequence of independent identically distributed random variables on the probability space (Ω, P ) taking values a and b with probabilities q and p, respectively. As usual, I assume −1 < a < r < b, 0 < p < 1. (2.3) Introduce also the (finite) σ-algebras Fn , n = 0, 1, . . . , N where F0 = {∅, Ω} and Fn , n = 1, 2, . . . , N is generated by the random variables {ρk , k = 1, . . . , n}. Recall, (see, for instance, [SKKM1]) that a portfolio strategy π with an initial capital Z0π = z > 0 and a horizon N is a sequence π = (π1 , . . . , πN ) of pairs πn = (βn , γn ) where βn , γn are Fn−1 -measurable random variables representing the number of units on the savings account and of the stock, respectively, at time n so that the price of the portfolio at time n is given by the formula Znπ = βn Bn + γn Sn . (2.4) A portfolio strategy π is called self-financing if all changes in the portfolio value are due to capital gains or losses but not to withdrawal or infusion of funds. This means that (see [SKKM1]), β1 B0 + γ1 S0 = z and for all n > 1, Bn−1 (βn − βn−1 ) + Sn−1 (γn − γn−1 ) = 0. (2.5) Denote by JnN the finite set of stopping times ξ with respect to the filtration {Fn }0≤n≤N (i.e. {ω : ξ(ω) ≤ k } ∈ Fk , k = n, . . . , N ) with values in {n, n + 1, n + 2, . . . , N }. A Game Contingent Claim (GCC) is a contract between investors A and B consisting of a maturity date N < ∞, of selection of a cancellation time σ ∈ J0N by A, of selection of an exercise time τ ∈ J0N by B and of Fn adapted payoff processes ∞ > Xn ≥ Yn ≥ 0, so that A pledges to pay to B at time σ ∧ τ = min(σ, τ ) the sum def R(σ, τ ) = Xσ Iσ<τ + Yτ Iτ ≤σ (2.6) where IQ = 1 if an event Q occurs and = 0 if not. It turns out (see Remark 2.2 below) that if I replace in (2.6) Iσ<τ by Iσ≤τ and Iτ ≤σ by Iτ <σ then the results below remain the same provided XN = YN . 448 Y. Kifer A hedge against a GCC with a maturity date N is a pair (σ, π) of a stopping π time σ ∈ J0N and a self-financing portfolio strategy π such that Zσ∧n ≥ R(σ, n) for all n = 0, 1, . . . , N . The fair price V ∗ of a GCC is the infimum of V ≥ 0 such that there exists a hedge (σ, π) against this GCC with Z0π = V . Theorem 2.1 Let P ∗ = {p ∗ , 1 − p ∗ }N be the probability on the space Ω with r−a , N < ∞ and E ∗ denotes the corresponding expectation. Then the fair p ∗ = b−a ∗ price V ∗ of the above GCC equals V0N which can be obtained from the recursive −N ∗ relations VNN = (1 + r) YN and for n = 0, 1, . . . , N − 1 ∗ ∗ = min((1 + r)−n Xn , max((1 + r)−n Yn , E ∗ (Vn+1N |Fn ))). VnN (2.7) Moreover, for n = 0, 1, . . . , N , ∗ VnN    = min max E (1 + r) R(σ, τ )Fn σ∈JnN τ ∈JnN  
 ∗ −σ∧τ = max min E (1 + r) R(σ, τ )Fn . ∗
−σ∧τ (2.8) τ ∈JnN σ∈JnN Furthermore, for each n = 0, 1, . . . , N the stopping times ∗ ∗ σnN or k = N } and = min{k ≥ n : (1 + r)−k Xk = VkN (2.9) ∗ ∗ τnN = min{k ≥ n : (1 + r)−k Yk = VkN } ∗ belong to JnN (since VNN = (1 + r)−N YN ) and they satisfy    

  ∗ ∗ ∗ −σnN ∧τ ∗ ∗ ∗ −σ∧τnN ∗   E (1 + r) R(σnN , τ )Fn ≤ VnN ≤ E (1 + r) R(σ, τnN )Fn (2.10) for any σ, τ ∈ Jn,N . Finally, there exists a self-financing portfolio strategy π ∗ ∗ ∗ ∗ such that (σ0N , π ∗ ) is a hedge against this GCC with the initial capital Z0π = V0N ∗ ∗ and such strategy is unique up to the time σ0N ∧ τ0N . Proof. Let π = (π1 , ..., πN ), πn = (βn , γn ) be a self-financing portfolio strategy with Z0π = z > 0 then Mnπ = (1 + r)−n Znπ (see [SKKM1]), Mnπ =z+ n  (1 + r)−k γk Sk −1 (ρk − r), (2.11) k =1 which is a martingale with respect to the filtration {Fn }0≤n≤N and the probability P ∗ . Suppose that (σ, π) is a hedge then by the Optional Sampling Theorem (see π [Ne], Theorem II-2-13) for any τ ∈ J0N I have Z0π = E ∗ ((1 + r)−σ∧τ Zσ∧τ )≥ ∗ −σ∧τ ∗ E ((1+r) R(σ, τ )). Since, by the definition, V is the infimum of such initial capitals Z0π then V ∗ is not less than the right hand side of (2.8). In order to prove the inequality in the other direction, for any σ ∈ J0N set Vnσ = max E ∗ (Uτσ |Fn ) where Ukσ = (1 + r)−σ∧k R(σ, k ), k = 0, 1, . . . , N . τ ∈JnN Game options 449 Observe that Ukσ is Fσ∧k -measurable (and so, Fk −measurable) since both σ ∧ k and R(σ, k ) = Xσ∧k Iσ∧k <k + Yσ∧k Iσ∧k =k are Fσ∧k -measurable. It is easy to check directly and follows from general theorems (see [Ne], Proposition VI-12) that {Vnσ }0≤n≤N is a minimal supermartingale with respect to the filtration {Fn }0≤n≤N such that Vnσ ≥ Unσ , n = 0, 1, . . . , N . Now, proceeding in the standard way via the Doob supermartingale decomposition and the martingale representation (on the simple probability space above) I obtain similarly to Sect. 2 and Sect. 5 in [SKKM1] (see also similar continuous time arguments in the next section) that there exists a self-financing portfolio strategy π σ = (π1σ , ..., πNσ ), πnσ = (βnσ , γnσ ) with the portfolio value process σ Znπ = βnσ Bn + γnσ Sn such that (σ, π) is a hedge. ∗ ∗ by (2.9). Then it is easy to see by the backward in, τnN Next, define σnN duction in n that (2.8) and (2.10) hold true. This follows also considering the Dynkin stopping game with the payoff (1 + r)−σ∧τ R(σ, τ ) and applying results from [Oh] though in the finite horizon case the inductive proof is much simpler. ∗ Now take σ ∗ = σ0N ∈ J0N and construct the corresponding self-financing ∗ σ∗ portfolio strategy π = π , as above, which yields the hedge (σ ∗ , π ∗ ) with the ∗ ∗ ∗ where the last equality initial capital V0σ = max E ∗ ((1 + r)−σ ∧τ R(σ, τ )) = V0N τ ∈J0N ∗ follows from (2.10). This together with the first part of the proof gives V ∗ = V0N . ∗ ∗ ∗ σ∗ It remains to obtain the uniqueness. Set τ = τ0N . Since (σ , π ) is a hedge ∗ ∗ ∗ ∗ σ∗ σ∗ ∗ then M0π = V0σ = E ∗ ((1 + r)−σ ∧τ R(σ ∗ , τ ∗ )) ≤ E ∗ ((1 + r)−σ ∧τ Zσπ∗ ∧τ ∗ ) = ∗ ∗ ∗ σ∗ σ σ σ E ∗ Mσπ∗ ∧τ ∗ = M0π is a martingale. It follows that Zσπ∗ ∧τ ∗ = since Mnπ R(σ ∗ , τ ∗ ). Let now π = (π1 , ..., πN ), πn = (βn , γn ) be another self-financing ∗ portfolio strategy with Z0π = V ∗ = V0σ . Then according to the first part of the proof Mnπ = (1 + r)−n Znπ is a martingale and in the same way as above I again σ∗ σ∗ have Zσπ∗ ∧τ ∗ = R(σ ∗ , τ ∗ ) = Zσπ∗ ∧τ ∗ , and so Mσπ∗ ∧τ ∗ = Mσπ∗ ∧τ ∗ . Since both Mnπ σ∗ σ∗ σ∗ and Mnπ are martingales it follows that Mnπ = Mnπ and Znπ = Znπ for all n ≤ σ ∗ ∧ τ ∗ . Since the representation (2.11) is unique, Sn > 0 and ρn = / r for all ∗ ∗ n then γn = γnσ and βn = βnσ for all n ≤ σ ∗ ∧ τ ∗ completing the proof of Theorem 2.1. ⊓ ⊔ Remark 2.2 Consider a bit more general set up where the payoff function R(σ, τ ) given by (2.6) is replaced by R̂(σ, τ ) = Xσ Iσ<τ + Yτ Iτ <σ + Wσ Iσ=τ where Wn is Fn -measurable, Yn ≤ Wn ≤ Xn , n = 0, 1, . . . , N and WN = YN . It follows from ∗ [Oh] that if V̂nN , n = 0, 1, . . . , N are given by (2.8) with R̂ in place of R then ∗ they will still satisfy the recursive relations (2.7) and since V̂NN = (1 + r)−N YN ∗ ∗ I conclude that V̂nN = VnN for all n = 0, 1, . . . , N . By [Oh] the stopping times ∗ −k ∗ ∗ σ̂nN = min{k ≥ n : (1 + r) Xk = V̂k∗ or k = N } = σnN and τ̂nN = min{k ≥ n : −k ∗ ∗ (1+r) Yk = V̂k } = τnN satisfy (2.10) with R replaced by R̂. Next, I define again π (σ, π) to be a hedge if Zσ∧n ≥ R̂(σ, n). Now the same proof as in Theorem 2.1 ∗ shows that the fair price V̂ ∗ of the GCC with the payoff function R̂ equals V̂0N , ∗ ∗ and so by above, V̂ = V . This is rather interesting since R̂(σ, n) ≥ R(σ, n) and the strict inequality is also possible when σ = n. So, sometimes (take, for instance, N = 1, σ = 0, X0 > Y0 ) a hedging portfolio requires less initial capital 450 Y. Kifer when the payoff function is R than when it is R̂. Still, the fair prices of the corresponding GCC’s are the same in both cases. Remark 2.3 Theorem 2.1 can be extended to the infinite horizon case N = ∞ with the same proof relying on results from [Oh] provided that with P ∗ -probability one lim e −rn Xn = 0. (2.12) n→∞ In this case Ω becomes the space of sequences and all statements above will be true now with probability one with respect to the probability P ∗ = {p ∗ , 1−p ∗ }∞ . Since P ∗ is singular with respect to any other probability {p, 1−p}∞ with p = p ∗ and there is no reason why the stock fluctuations should be connected with this particular probability p ∗ , it seems that this extension of Theorem 2.1 to the case N = ∞ may have financial applications only as an approximation of a very large N case. Still, observe that (2.12) always holds true in the game put option case Yn = (K − Sn )+ provided the penalty δn does not grow too fast so that with probability one lim e −nr δn = 0. Moreover, then (2.12) holds also true for the n→∞ game call option case Yn = (Sn − K )+ since by Jensen’s inequality log(1 + r) > p ∗ log(1 + b) + (1 − p ∗ ) log(1 + a), and so by the law of large numbers with n P ∗ −probability one limn→∞ n −1 log((1 + r)−n Sn ) = limn→∞ n −1 k =1 log(1 + ρk ) − log(1 + r) < 0. Remark 2.4 Similarly to [SKKM1] Theorem 2.1 can be generalized to the case when consumption or infusion of capital is also possible. In this case the price of a portfolio π = (π1 , . . . , πN ), πn = (βn , γn ) after new stock prices at time n were π announced is Znπ = βn Bn + γn Sn but immediately before that Zn−1 = βn Bn−1 + γn Sn−1 + gn (see [SKKM1]) where gn is Fn−1 -measurable. An easy modification of the above proof gives the following formula for the fair price V ∗ of the σ∧τ  corresponding GCC V ∗ = min max E ∗ ((1+r)−σ∧τ R(σ, τ )+ (1+r)−(k −1) gk ) σ∈J0N τ ∈J0N k =1 and this minmax equals maxmin of the same expression. Other, correspondingly modified, assertions of Theorem 2.1 remain true in this case, as well. Remark 2.5 It is easy also to generalize the above set up allowing dependence of r, a and b on time, i.e. assuming that ρk (ω) = 12 (ak + bk + ωk (bk − ak )) and n  Bn = B0 (1 + rk ) where rk , ak , bk ; k = 1, . . . , N are nonrandom sequences k =1 satisfying −1 < ak < rk < bk . Setting pk∗ = rk −ak bk −ak and P ∗ = N  k =1 {pk∗ , 1 − pk∗ }, an easy modification of the proof leads to the corresponding statements of Theorem n  2.1 with (1+r)−n replaced by (1+rk )−1 . In fact, one can consider also random k =1 sequences rk , ak , bk satisfying certain conditions. Game options 451 3 Continuous time I adopt here a popular model of a financial market consisting of a savings account with the time evolution Bt = B0 e rt , r ≥0 B0 > 0, and of a stock whose price St is the geometric Brownian motion
 κ2 St = S0 exp (µ − )t + κWt 2 (3.1) (3.2) where {Wt }t≥0 is the standard one dimensional Wiener process starting at zero and κ > 0, µ are some numbers. In the differential form dBt = rBt dt and dSt = St (µdt + κdWt ), (3.3) where the second equation is the Ito stochastic differential equation. Again, r is interpreted as the interest rate on the savings account and the term µdt + κdWt is responsible for random “risky” fluctuations of the stock price where σ and µ are called volatility and appreciation rate, respectively. Let (Ω, F , P ) be the probability space corresponding to the Wiener process, i.e. Ω is the space of continuous functions ω = (ωt )t≥0 , ω0 = 0, F is the Borel σ-field generated by cylinder sets, and P is the Wiener measure on (Ω, F ). Then Wt (ω) = ωt , t ≥ 0. Denote by Ft W the σ-algebra generated by {Wu , u ≤ t} and by Ft the minimal σ-algebra containing Ft W and P −null subsets of F . Then {Ft }t≥0 is a right-continuous filtration and St = S0 e µt Qt , (3.4) 2 where Qt = e κWt −(κ /2)t , t ≥ 0, is a martingale with respect to the filtration {Ft }t≥0 . Recall, (see, for instance, [SKKM2]) that a self-financing portfolio strategy π with an initial capital Z0π = z > 0 and a horizon T < ∞ is a process π = (πt )0≤t≤T of pairs πt = (βt , γt ) with progressively measurable with respect to the filtration Ft , t ≥ 0 processes βt and γt , t ≥ 0 such that T rt e |βt |dt < ∞ and (γt St )2 dt < ∞ (3.5) 0 0 and the portfolio price T Ztπ at time t ∈ [0, T ] is given by Ztπ =z+ t 0 βu dBu + t γu dSu (3.6) 0 where Bu and Su are the same as in (3.1)–(3.3). I call π {Ft }0≤t≤T -progressively measurable if the processes βt and γt are progressively measurable with respect to this filtration (see [KS], Sect. 1.1). 452 Y. Kifer Denote by JtT the set of stopping times with respect to the filtration {Fu }0≤u≤T with values in [t, T ]. A game contingent claim (GCC) is a contract between investors A and B consisting of a maturity date T < ∞, of selection of a cancellation time σ ∈ J0T by A, of selection of an exercise time τ ∈ J0T by B and of Ft -adapted right continuous with left limits (RCLL) payoff processes ∞ > Xt ≥ Yt ≥ 0, so that A pledges to pay to B at time σ ∧ τ = min(σ, τ ) the sum R(σ, τ ) = Xσ Iσ<τ + Yτ Iτ ≤σ . (3.7) As in the discrete time case, the exchange between ≤ and < in (3.7) does not influence the final result (see Remark 3.4) provided XT = YT . A hedge against such GCC with a maturity date T is a pair (σ, π) of a stopping time σ ∈ J0T and a {Ft }0≤t≤T -progressively measurable self-financing portfolio π strategy π such that Zσ∧t ≥ R(σ, t) with probability one for each t ∈ [0, T ]. Again ∗ the fair price V of a GCC is the infimum of V ≥ 0 such that there exists a hedge (σ, π) against this GCC with Z0π = V . Set (see Sect. 2 in [SKKM2]), Wtµ−r = Wt + µ−r t κ (3.8) then the process {Wtµ−r }t≥0 is the standard Wiener process with respect to the probability P µ−r whose restrictions Ptµ−r to Ft are equivalent to restrictions Pt of P to Ft and dPtµ−r µ−r 1 Wt (ω) − (ω) = exp − dPt κ 2
µ−r κ 2 t (3.9) (see, for instance, [KS], Sect. 3.5). By (3.3) and (3.8), dSt = St (rdt + κdWtµ−r ) (3.10) which together with (3.6) gives dZtπ = rZtπ dt + κγt St dWtµ−r (3.11) for any self-financing portfolio strategy. It is important to observe that both stochastic differential equations (3.10) and (3.11) do not depend explicitly on µ which is usually not known. Assume that E µ−r sup (e −rt Xt ) < ∞, (3.12) 0≤t≤T where E µ−r is the expectation corresponding to the probability P µ−r . ∗ where {VtT∗ }0≤t≤T Theorem 3.1 The fair price V ∗ of the above GCC equals V0T µ−r is the right continuous process such that with P -probability one, Game options 453 VtT∗ = essinf esssup E σ∈JtT τ ∈JtT µ−r
e −rσ∧τ    R(σ, τ )Ft  
 = esssup essinf E µ−r e −rσ∧τ R(σ, τ )Ft . τ ∈JtT σ∈JtT (3.13) (3.13) Moreover, for each t ∈ [0, T ] and ε > 0 the stopping times ε ∗ σtT = inf{u ≥ t : e −ru Xu ≤ VuT + ε or u = T } and (3.14) τtTε = inf{u ≥ t : e −ru Yu ≥ VuT − ε} ∗ belong to JtT (since VTT = e −rT YT ) and with P µ−r -probability one they satisfy    

  ε ε ε E µ−r e −rσtT ∧τ R(σtT , τ )Ft − ε ≤ VtT∗ ≤ E µ−r e −rσ∧τtT R(σ, τtTε )Ft + ε (3.15) for any σ, τ ∈ Jt,T . Furthermore, for each ε > 0 there exists a self-financing ε portfolio strategy π ε such that (σ0T , π ε ) is a hedge against this GCC with the πε ∗ initial capital Z0 ≤ V0T + ε. Suppose, in addition, that the processes Yt and −Xt are upper semicontinuous from the left, i.e. in our circumstances they may ∗ have only positive jumps at points of discontinuity. Then the stopping times σtT = ε ∗ ε ε ε lim σtT and τtT = lim τtT , which are well defined since σtT and τtT are monotone ε↓0 ε↓0 in ε, with P µ−r -probability one satisfy    

  ∗ ∗ µ−r −rσ∧τtT∗ ∗  µ−r −rσtT ∧τ ∗  (3.16) e R(σ, τtT )Ft e R(σtT , τ )Ft ≤ VtT ≤ E E ∗ 0 0 for any σ, τ ∈ Jt,T . Moreover, σtT ∧ τtT∗ = σtT ∧ τtT0 , where σtT and τtT0 are defined µ−r by (3.14) with ε = 0, and so with P -probability one,  
 0 ∗ µ−r 0 0  −rσtT ∧τtT0 (3.17) e VtT = E R(σtT , τtT )Ft . ∗ Furthermore, there exists a self-financing portfolio strategy π ∗ such that (σ0T , π∗ ) π∗ ∗ µ−r is a hedge against this GCC with the initial capital Z0 = V0T and with P − 0 probability one such strategy is unique in this case up to the time σtT ∧ τtT0 . Proof. Let π be a self-financing portfolio strategy with Z0π = z > 0 then in view of (3.11), Mtπ = e −rt Ztπ satisfies Mtπ = M0π + κ t e −ru γu Su dWuµ−r , t ≤ T < ∞, (3.18) 0 and so {Mtπ }0≤t≤T is a martingale with respect to the filtration {Ft }0≤t≤T . Suppose that (σ, π) is a hedge then by the Optimal Sampling Theorem (see [KS], Sect. 1.3) for any τ ∈ J0T , 454 Y. Kifer π Z0π = E µ−r (e −rσ∧τ Zσ∧τ ) ≥ E µ−r (e −rσ∧τ R(σ, τ )). (3.19) If follows that V ∗ ≥ inf sup E µ−r (e −rσ∧τ R(σ, τ )). σ∈J0T τ ∈J0T (3.20) In order to prove the inequality in the other direction for any σ ∈ J0T set Vtσ = esssup E µ−r (Uτσ |Ft ) (3.21) τ ∈JtT where Utσ = e −rσ∧t R(σ, t) and I observe that Utσ is Fσ∧t −measurable. In the same way as in Lemma from Sect. 6 of [SKKM2] I conclude that {Vtσ }0≤t≤T is a supermartingale. Still, since Utσ is not, in general, right continuous I cannot use this lemma directly to conclude that {Vtσ }0≤t≤T has a RCLL modification. It is known that, in order to establish the latter assertion it suffices to show that the function ϕt = E µ−r Vtσ = sup E µ−r Uτσ , t ∈ [0, T ] τ ∈JtT is right continuous (see [KS], Sect. 1.3). Since Vtσ is a supermartingale it follows that lim ϕs ≤ ϕt . For the opposite inequality, I can still employ the argument s↓t from the lemma cited above relying just on the right lower semicontinuity of Utσ , lim Usσ = e −rσ∧t (Xσ Iσ≤t + Yt It<σ ) ≥ Utσ s↓t which follows since Xu ≥ Yu , u ∈ [0, T ] and both Xu and Yu are right continuous. Thus I can and do assume that {Vtσ }0≤t≤T is a RCLL supermartingale. Moreover, in view of (3.12) the family {Vτσ }τ ∈J0T is uniformly integrable with respect to P µ−r (see Lemma 5.5 in [Ka1]). Hence by the Doob-Meyer decomposition theorem (see [KS], Sect. 1.4) I can write Vtσ = M̃tσ − Aσt , t ∈ [0, T ], Aσ0 = 0 (3.22) where {M̃tσ }0≤t≤T is a RCLL martingale with respect to the filtration {Ft }0≤t≤T and {At }0≤t≤T is a RCLL nondecreasing process such that At is Ft -measurable. In view of the martingale representation theorem (see [KS], Sect. 3.4) there exists a {Ft }0≤t≤T progressively measurable process {γtσ }0≤t≤T such that t e −ru γuσ Su dWuµ−r , (3.23) e −ru γuσ Su dWuµ−r − Aσt . (3.24) M̃tσ = M̃0σ + κ 0 and so for all t ∈ [0, T ], σ Vt = V0σ +κ t 0 Game options 455 Set βtσ = (M̃tσ − e −rt γtσ St )B0−1 , t ∈ [0, T ] and σ Ztπ = e rt M̃tσ = βtσ Bt + γtσ St , (3.25) where π σ = (πtσ )0≤t≤T and πtσ = (βtσ , γtσ ) is the {Ft }0≤t≤T -progressively measurable portfolio strategy with the initial capital Z0π = V0σ , then by (3.10), (3.23), and (3.25), dZtπ = Ztπ dt + κγtσ St dWtµ−r = βtσ dBt + γtσ dSt , (3.26) i.e. (3.6) holds true, and so π σ is self-financing. Choose a sequence of stopping times ηn ↓ σ as n ↑ ∞ taking on only finitely many values then it is easy to check as in Lemma VI-1-5 from [Ne] that (3.21) implies also Vησn ∧t = esssup E µ−r (Uτσ |Fηn ∧t ), τ ∈Jηn ∧tT where, for a stopping time η I denote by JηT the set of stopping times with values between η and T . This together with (3.21) and (3.22) yield that for any t ∈ [0, T ] with P µ−r -probability one, σ Zηπn ∧t = e rηn ∧t M̃ησn ∧t = e rηn ∧t (Vησn ∧t + Aσηn ∧t ) ≥ e rηn ∧t Vησn ∧t =e rηn ∧t esssup E µ−r τ ∈Jηn ∧tT (Uτσ |Fηn ∧t ) ≥e rηn ∧t E µ−r (3.27) (Utσ |Fηn ∧t ) = e rηn ∧t Utσ = e r(ηn ∧t−σ∧t) R(σ, t) ≥ R(σ, t) (3.27) since Utσ is Fσ∧ t −measurable and Fσ∧ t ⊂ Fηn ∧ t . The right continuity of σ Zsπ = e rs M̃sσ in s enables me to pass to the limit in (3.27) as n ↑ ∞ yielding σ π Zσ∧t ≥ R(σ, t), and so (σ, π σ ) is a hedge. Next, extend the payoff processes Xt and Yt beyond T by Xt = XT and Yt = YT for all t > T so that the right continuity, existence of left limits, and upper simicontinuity from the left are preserved. Denote by J0∞ the set of stopping times with values in [0, ∞] with respect to the filtration {Ft }t≥0 which is defined for all t ≥ 0 anyway. Consider a game between two players I and II with the payoff processes e −rt Xt and e −rt Yt so that if I chooses a stopping time σ and II chooses a stopping time τ then I pays to II the sum e −rσ Xσ Iσ<τ + e −rτ Yτ Iτ ≤σ = e −rσ∧τ R(σ, τ ). (3.28) Next, I intend to apply to this game the results from [LM] which were stated there for bounded payoff processes but they remain true for Xt ≥ Yt ≥ 0 satisfying (3.12). It follows from [LM] that  
 def Ṽt∗ = essinf esssup E µ−r e −rσ∧τ R(σ, τ )Ft (3.29) σ∈Jt∞ τ ∈Jt∞ = esssup essinf E τ ∈Jt∞ σ∈Jt∞ µ−r
e −rσ∧τ    R(σ, τ )Ft (3.29) 456 Y. Kifer and for each ε > 0 the stopping times σ̃tε = inf{u ≥ t : e −ru Xu ≤ Vu∗ + ε} and τ̃tε = inf{u ≥ t : e −ru Yu ≥ Ṽu∗ − ε} (3.30) satisfy    

  ε ε E µ−r e −r σ̃t ∧τ R(σ̃tε , τ )Ft − ε ≤ Ṽt∗ ≤ E µ−r e −rσ∧τ̃t R(σ, τ̃tε )Ft + ε (3.31) for any σ, τ ∈ Jt∞ . From the definition of Xt and Yt beyond T it follows easily that Ṽt∗ = VtT∗ for all t ∈ [0, T ] since the player II may only decrease his gain if he stops the game later than T . Then by (3.14) and (3.30), τtTε = τ̃tε ∈ JtT for all t ∈ [0, T ]. ε But then σtT = σ̃tε ∧ T also satisfies (3.31), and so (3.13) and (3.15) follow from (3.29) and (3.31). ε Now take σ ε = σ0T ∈ J0T and construct the corresponding self-financing ε ε portfolio strategy π = π σ , as above, which yields the hedge (σ ε , π ε ) with the initial capital ε V0σ = sup E µ−r (e −rσ τ ∈J0T ε ∧τ ∗ R(σ, τ )) ≤ V0T +ε (3.32) where the last inequality in (3.32) follows from (3.15). Since the fair price V ∗ of the GCC is the infimum of initial capitals for which hedging is possible it ∗ ∗ follows that V ∗ ≤ V0T . + ε. This being true for any positive ε yields V ∗ ≤ V0T ∗ ∗ ∗ On the other hand, by (3.13) and (3.20), V ≥ V0T , i.e. in fact, V = V0T , as required. Next, suppose that −Xt and Yt are left upper semicontinuous. Since σtε and τtε may only grow when ε decreases then def ε ∗ def σtT ∈ JtT and τtT∗ = lim τtTε ∈ JtT . = lim σtT ε↓0 ε↓0 0 ∗ ∧ τtT0 = σtT ∧ τtT∗ follows Letting ε ↓ 0 in (3.15) one arrives at (3.16) and σtT easily too (see Theorem 15 in [LM]). ∗ ∗ Let now σ ∗ = σ0T and π ∗ = π σ so that (σ ∗ , π ∗ ) is the corresponding hedge. ∗ ∗ Then by (3.16) it follows similarly to (3.31) that V0σ ≤ V0T . Since I already ∗ ∗ ∗ ∗ σ ∗ proved that V = V0T and by the definition V0 ≥ V , it follows that V0σ = V ∗ . Finally, I obtain the uniqueness assertion in the same way as in the discrete time case. Namely, I derive using (3.27) that if π = (πt )0≤t≤T , πt = (βt , γt ) is σ∗ ∗ another self-financing portfolio strategy with Z0π = V ∗ = V0σ then Ztπ = Ztπ 0 for all t ≤ σtT ∧ τtT0 . Now the pair βt and γt is uniquely defined by (3.6) and (3.11), completing the proof of Theorem 3.1. ⊓ ⊔ In the general continuous time case direct effective computations of the fair price V ∗ are hardly possible. One possibility is to discretize time and to obtain V ∗ as a limit of values Vj∗ of a sequence of discrete time games for which Vj∗ can be obtained by the backward induction as in (2.7). Namely, let Jk(n) ,T be the Game options 457 set of stopping times from J0T taking on only values jn −1 T for j = k , k + 1, ..., n and denote by Vk∗(n) ,T the value of the corresponding Dynkin game when stopping is allowed only at times jn −1 T , j = k , k + 1, ..., n, i.e. Vk∗(n) ,T = inf sup E µ−r e −rσ∧τ R(σ, τ )|Fkn −1 T (n) σ∈Jk(n) ,T τ ∈Jk ,T (3.33) inf E µ−r e −rσ∧τ R(σ, τ )|Fkn −1 T . sup σ∈Jk(n) ,T τ ∈Jk(n) ,T As in Theorem 2.1 these values satisfy the recursive relations −rkn Vk∗(n) ,T = min e −1 Xkn −1 T , max e −rkn T −1 T Ykn −1 T , E µ−r Vk∗(n) +1,T |Fkn −1 T , (3.34) ∗(n) ∗(n) Vn,T = YT , and so, in principle, one can compute V0,T which is the price of the corresponding GCC when A and B can cancel and exercise only at times jn −1 T , j = 0, 1, ..., n. Proposition 3.2 Assume that with P µ−r −probability one Xt and Yt , t ∈ [0, T ] are continuous processes. Then ∗(n) ∗ . V ∗ = V0T = lim V0,T n→∞ (3.35) (n) Proof. For any τ ∈ J0T denote by dn (τ ) the stopping time from J0,T defined −1 −1 −1 by dn (τ )(ω) = jn T if (j − 1)n T < τ (ω) ≤ jn T . In view of the recursive relations (3.34), as in any discrete bounded time Dynkin’s game, there exist (n) (n) (n) optimal stopping times σ0,T , τ0,T ∈ J0,T such that (n) (n) ∗(n) E µ−r e −rσ0,T ∧dn (τ ) R(σ0,T ∧ dn (τ )) ≤ V0,T ≤ E µ−r e (n) −rdn (σ)∧τ0,T (3.36) (n) R(dn (σ) ∧ τ0,T ) for any σ, τ ∈ J0T . Set ε(n) T (ω) = max(|Xt (ω) − Xs (ω)|, |Yt (ω) − Ys (ω)|). sup s,t∈[0,T ],|t−s|≤n −1 T Then for any σ, τ ∈ J0T , −rdn (σ)∧dn (τ ) e −rσ∧dn (τ ) R(σ ∧ dn (τ )) + ε(n) R(dn (σ) ∧ dn (τ )) T ≥e (3.37) ≥ e −rn −1 T −rdn (σ)∧τ e R(dn (σ) ∧ τ ) − ε(n) T . (n) (n) Now, applying (3.16) with t = 0, σ = σ0,T , τ = τ0,T , applying (3.36) with σ = (n) ∗ ∗ ∗ σ0T , τ = τ0,T , and applying the first inequality in (3.37) with σ = σ0T , τ = τ0T (n) ∗ and the second inequality in (3.37) with σ = σ0,T , τ = τ0T I derive easily that e −rn −1 T ∗(n) ∗ ∗ µ−r (n) V0T − E µ−r ε(n) εT . T ≤ V0,T ≤ V0T + E 458 Y. Kifer µ−r By continuity of Xt and Yt , ε(n) −almost surely as n → ∞ and since T → 0 P µ−r (n) I assume (3.12) it follows that E εT → 0 as n → ∞ yielding (3.35). ⊓ ⊔ Next, I discuss the infinite horizon case. Proposition 3.3 Suppose that the processes Xt ≥ Yt ≥ 0 are defined for all t ∈ [0, ∞) and satify conditions of Theorem 3.1, in particular (3.12), with T = ∞. Assume that P µ−r −almost surely lim e −rt Xt = 0. t→∞ (3.38) Then the conclusion of Theorem 3.1 remains true for T = ∞, i.e. the fair prices ∗ Vt∞ of the GCC with infinite horizon starting at t satisfy (3.13)–(3.17) with T = ∞. ∗ Set V ∗ = V0∞ and let V ∗(n) be the value of the Dynkin game corresponding to the payoff processes e −rt Xt and e −rt Yt but where stopping is allowed only at times jn −1 , j = 0, 1, 2, .... Then V ∗ = lim V ∗(n) . n→∞ (3.39) Proof. The first part of Proposition 3.3 follows in the same way as Theorem 3.1 (cf. [Ka1], Sect. 6) taking into account that (3.38) enables me to apply the results from [LM] again. For the second part, set γN (ω) = supt≥N e −rt Xt (ω). Then, in the same way as in the proof of Proposition 3.2 considered with nN in place of n and with N in place of T I obtain easily that −1 ) ) e −rn V ∗ − E µ−r (ε(nN + γN ) ≤ V ∗(n) ≤ V ∗ + E µ−r (ε(nN + γN ). N N (3.40) Since (3.12) is assumed to be true with T = ∞ then (3.38) implies that E µ−r γN → 0 as N → ∞. Hence, letting in (3.40), first, n → ∞ and then N → ∞ I arrive at (3.39). ⊓ ⊔ Remark 3.4 Suppose that the payoff function R(σ, τ ) given by (3.7) is replaced by R̂(σ, τ ) = Xσ Iσ≤τ + Yτ Iτ <σ . Assume also that XT = YT . Then by Lemma 5 from [LM], (3.13) will remain true when R is replaced by R̂ with the same process VtT∗ . Then (3.15)–(3.17) will hold true, as well, with R̂ in place of R. As in the discrete time case, it follows that the fair price V̂ ∗ of the GCC with the ∗ ∗ payoff function R̂ equals V ∗ = V0T with V0T given by (3.13) if hedging pairs π (σ, π) are required to satisfy Zσ∧t ≥ R̂(σ, t) with P µ−r -probability one for each t ∈ [0, T ]. The proof is the same as in Theorem 3.1 (and even a bit easier since R̂(σ, t) is right continuous). Remark 3.5 Similarly to [SKKM2] and [Ka1] Theorem 3.1 can be generalized to the case when consumption is also possible. If {gt }0≤t≤T is a {Ft }0≤t≤T adapted T consumption process (see [Ka1]) with E µ−r ( |gt |dt) < ∞ then the fair price V ∗ 0 Game options 459 of the GCC in this case will be given by V ∗ = inf sup E µ−r (e −rσ∧τ R(σ, τ )+ σ∈J0T τ ∈J0T σ∧τ e −ru gu du). 0 Remark 3.6 It is not difficult to generalize the set up considering r = rt , µ = µt , and κ = κt in (3.3) being {Ft }0≤t≤T −adapted stochastic processes satisfying certain conditions. In the usual American contingent claim case this was done in [Ka1]. On the other hand, one can deal with rt , µt , and σt being stochastic processes independent of the driving Wiener process Wt , i.e. to consider a (B , S )market in a random dynamical environment. 4 Markov case Taking into account that the stock fluctuation processes {Sn }n≥0 and {St }t≥0 given by (2.2) and (3.2), respectively, are Markov processes one can employ other methods of computations of the fair price V ∗ of a GCC if Xn and Yn or Xt and Yt depend only on Sn or St , correspondingly. I start with the discrete time case. Let Xn = β n X (Sn ) and Yn = β n Y (Sn ), n = 0, 1, 2, . . . , 0 < β ≤ 1 for some Borel functions X and Y on (0, ∞). Particular cases of this situation are Yn = β n (K − Sn )+ and Yn = β n (Sn − K )+ , which are discounted put and call game options, with the penalty process having the form δn = β n δ(Bn ) of just δn = β n δ for some constant δ > 0. In this case it follows from (2.7), (2.8), and the Markov property that there exist Borel ∗ functions vk = vk (x ), k = 0, 1, . . . on (0, ∞) such that VnN = (αβ)−n vN −n (Sn ), −1 where α = (1 + r) , and for n = 0, 1, . . . vn+1 (x ) = U vn (x ), v0 (x ) = Y (x ) (4.1) where U g(x ) = min(X (x ), max(Y (x ), αβEx∗ g(S1 ))) and Ex∗ is the expectation for P ∗ provided S0 = x . This provides recursive formulas for computation of the fair price V ∗ (x ) = vN (x ) of the GCC with the horizon N < ∞ given S0 . By (4.1) the sequence vn , n = 0, 1 . . . is monotone nondecreasing, Y ≤ vn ≤ X and the limit v(x ) = lim vn (x ) = lim U n Y (x ) (4.2) n→∞ n→∞ satisfies the equation U v = v. Moreover, it follows from (2.8) and the equality ∗ , S0 = x that v equals the value of the infinite game between the vN (x ) = V0N players I and II described in Sect. 2 when only finite stopping times are allowed (cf. [El] and [Oh]), and so the fair price V ∗ = V ∗ (x ) of the GCC with the infinite horizon N = ∞ given S0 = x equals v(x ). By (2.9) the corresponding optimal (or rational) stopping times (saddle point) for the GCC with N < ∞ are given by ∗ σnN = min{0 ≤ n ≤ N : X (Sn ) = vN −n (Sn ) or n = N }, ∗ τnN = min{0 ≤ n ≤ N : Y (Sn ) = vN −n (Sn )} (4.3) 460 Y. Kifer and for the N = ∞ case σ ∗ = min{n ≥ 0 : X (Sn ) = v(Sn )}, τ ∗ = min{n ≥ 0 : Y (Sn ) = v(Sn )} provided that with P ∗ -probability one σ ∗ and τ ∗ are finite. ∗ Let Yn = β n Y (Sn ), Xn = β n (Y (Sn ) + δ) and consider the fair price V0N = vN as a function of δ ≥ 0 for each fixed initial stock price S0 = x > 0. It is clear that vN = vN (x , δ) is continuous, nondecreasing, and piecewise linear in δ for every x > 0. Moreover, there exist 0 ≤ δ0 (x ) ≤ δ1 (x ) < ∞ such that vN (x , δ) = Y (x ) for all 0 ≤ δ ≤ δ0 (x ) and when δ ≥ δ1 (x ) then vN (x , δ) equals the fair price of the standard American option (where only B is allowed to exercise) with the horizon N . For δ0 (x ) < δ < δ1 (x ) the graph of vN (x , δ) in δ may have different degrees of complexity depending on parameters of the problem. On Fig. 1 I exhibit the graph of vN (x , δ) in δ for the game put option case where Y (z ) = (10 − z )+ , N = 20, x = 9.65, β = 1, a = −0.1, b = 0.1, r = 0.05. 0.42 0.41 0.4 0.39 0.38 0.37 0.36 00.35 0.05 0.1 0.15 0.2 0.25 del 0.3 Fig. 1. Consider, next, the case β = 1 and Y (x ) = (x − K )+ . Since E ∗ (Sn+1 |Fn ) = Sn (1 + p ∗ b + (1 − p ∗ )a) = (1 + r)Sn = α−1 Sn (4.4) then αn Sn is a martingale, and so αn Y (Sn ) = (αn Sn − αn K )+ is a submartingale (with respect to the probability P ∗ ). Thus, in view of the Optional Sampling Theorem, for the game call option case with a horizon N < ∞ the fair price is given by V ∗ = min E ∗ ((ασ (Sσ − K )+ + δσ )Iσ<N + αN (SN − K )+ Iσ=N ). σ∈J0N (4.5) This corresponds to the well known fact that American call options with an expiration date N < ∞ coincide with the corresponding European call options. In the game call option case it follows that the buyer B should exercise as late as possible, i.e. at the expiration date N if N < ∞. On the other hand, the Game options 461 seller A should choose an optimal cancellation stopping time which minimizes in (4.5) and it is easy to give examples when this stopping time is nontrivial (i.e. nonconstant). For each m = 0, 1 . . . , set CmA = {x : vm (x ) < X (x )}, CmB = {x : vm (x ) > Y (x )}, Cm = CmA ∩ CmB and DmA = {x : vm (x ) = X (x )}, DmB = {x : vm (x ) = Y (x )}, Dm = DmA ∩ DmB so that Cm ∪ Dm = R since Y ≤ vm ≤ X . Since the sequence vn is nondecreasing then assuming that X (x ) > Y (x ) for A B all x I have CnA ⊂ Cn−1 ⊂ . . . ⊂ C0A = R, ∅ = C0B ⊂ . . . ⊂ Cn−1 ⊂ CnB , and A A A B B B Dn ⊃ Dn−1 ⊃ . . . ⊃ D0 = ∅, R = D0 ⊃ . . . ⊃ Dn−1 ⊃ Dn , n = 0, 1, . . . . By (4.3), ∗ σnN = min{0 ≤ n ≤ N : Sn ∈ DNA −n or n = N } and ∗ τnN = min{0 ≤ n ≤ N : Sn ∈ (4.6) DNB −n }, so that A or B should stop when the stock price Sn gets to the domain DNA −n or DNB −n , respectively. In the continuous time case the stock price fluctuations St form the Markov diffusion process solving the stochastic differential equation (3.10). Suppose that Xt = e −βt X (St ), Yt = e −βt Y (St ), β > 0 and T = ∞. In particular one can take Yt = e −βt (K − St )+ or Yt = e −βt (St − K )+ which are discounted put or call game options, respectively, with some penalty process δt = e −βt δ(St ) or even δt = e −βt δ for some constant δ > 0. Though (3.10) is a degenerate at zero equation but considering instead the stochastic differential equation for Lt = log St one can deal with nondegenerate diffusions. Then the fair price V ∗ = V ∗ (x ) of the GCC with infinite horizon given S0 = x being the value of the infinite time optimal stopping game with the payoff processes e −(r+β)t X (St ) and e −(r+β)t Y (St ) can be described via certain variational inequalities (see [BF1]). Numerical schemes for solving this type of variational inequalities were studied in [Di]. Some computational algorithms for variational inequalities emerging in American options were justified in [JLL] and it seems quite plausible that they can be extended to the game options case. A free boundary approach to even a more general problem for nonzero sum games can be found in [BF2], and so computational methods for elliptic (or parabolic, if the finite horizon case is considered) free boundary problems can be employed here, as well. Observe that discretizing time one obtains recursive relations of the form (2.7). Then the value of the continuous time game, and so of the corresponding GCC, can be obtained by letting the discretization step to zero. Namely, let Ut g(x ) = min(X (x ), max(Y (x ), e −(r+β)t Exµ−r g(St ))). (4.7) Suppose that limt→∞ e −(r+β)t X (St ) = 0 P µ−r − almost surely, which holds true if, for instance, β > 0 and X (x )(|x | + 1)−1 is bounded (since e −rt St is a positive martingale). Then V ∗ (x ) = lim lim Unk−1 Y (x ). (4.8) n→∞ k →∞ ∗(n) limk →∞ Unk−1 Y −1 Indeed, V = is allowed only at times jn is the value of the Dynkin game when stopping , j = 0, 1, 2, ... (see [El]), and so (4.8) follows from 462 Y. Kifer Proposition 3.3. In the finite horizon case such discrete time approximations can be carried out also when β = 0 since then one relies on Proposition 3.2 in place of Proposition 3.3. Another possibility is to approximate continuous time GCC’s based on the geometric Brownian motion stock evolution by discrete time GCC’s based on the binomial CRR model stock evolution. For the American options case this together with error estimates was studied in [La] and after an appropriate modification the method there seems to be extendable to the game options case. Observe that Y ≤ V ∗ ≤ X and consider 4 domains D A = {x : X (x ) = V ∗ (x )}, B D = {x : Y (x ) = V ∗ (x )}, C A = {x : X (x ) > V ∗ (x )}, and C A = {x : Y (x ) < V ∗ (x )}. On C = C A ∩ C B the function V ∗ (x ) satisfies the equation 1 2 2 d 2 V ∗ (x ) dV ∗ (x ) + rx κ x = (r + β)V ∗ (x ) 2 2 dx dx (4.9) with the free boundary conditions V ∗ |∂D A = X and V ∗ |∂D B = Y . A more specific analysis of this problem for the case Y (x ) = (x − K )+ or Y (x ) = (K − x )+ and X (x ) = Y (x ) + δ can be carried out, in principle, along the lines of Sect. 8 from [SKKM2] though it is more complicated here. Still, when β = 0 one conclusion follows easily in the game call option case. Namely, (3.10) implies that e −rt St is a martingale, and so in the game call option case e −rt Yt = (e −rt St − e −rt K )+ is a submartingale. Thus, by the Optional Sampling Theorem the fair price V ∗ for the game call option with a finite horizon T < ∞ is given by V ∗ = inf E µ−r ((e −rσ (Sσ − K )+ + δσ )Iσ<T + e −rT (ST − K )+ Iσ=T ) σ∈J0T and the buyer should not exercise before the expiration date though the seller has to find an optimal cancelation time. References [BF1] [BF2] [CK] [CRR] [Di] [Dy] [El] [JLL] [Ka1] [Ka2] Bensoussan, A., Friedman, A.: Non-linear variational inequalities and differential games with stopping times. J. Funct. Anal. 16, 305–352 (1974) Bensoussan, A., Friedman, A.: Nonzero-sum stochastic differential games with stopping times and free boundary problems. Trans. Amer. Math. 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