Dynkin's Games and Israeli Options

Abstract

We start by briefly surveying a research on optimal stopping games since their introduction by Dynkin more than 40 years ago. Recent renewed interest to Dynkin’s games is due, in particular, to the study of Israeli (game) options introduced in 2000. We discuss the work on these options and related derivative securities for the last decade. Among various results on game options we consider error estimates for their discrete approximations, swing game options, game options in markets with transaction costs, and other questions.

1. Introduction

Optimal stopping games were introduced in 1969 by Dynkin in [1] as an extension of the optimal stopping problem which has been already actively studied since 1950. Optimal stopping and, in particular, its game version was often discussed on Dynkin’s undergraduate seminar at Moscow State University in the end of 1960 which resulted in papers [2–5].

The original setup of optimal stopping games consisted of a probability space (Ω,ℱ,ℙ), a filtration of 𝜎-algebras {ℱ_𝑡}, ℱ_𝑡⊂ℱ with either 𝑡∈ℕ={0,1,2,…} (discrete time case) or 𝑡∈ℝ₊=[0,∞) (continuous time case), {ℱ_𝑡}-adapted payoff process {𝑋_𝑡}, and a pair of {ℱ_𝑡}-adapted 0-1 valued “permission” processes 𝜑_𝑡^(𝑖),𝑖=1,2, such that the player 𝑖 is allowed to stop the game at time 𝑡 if and only if 𝜑_𝑡^(𝑖)=1. If the game is stopped at time 𝑡 then the first player pays to the second one the sum 𝑋_𝑡. Clearly, if 𝜑¹_𝑡≡1 and 𝜑_𝑡⁽²⁾≡0, we arrive back at the usual optimal stopping problem. Observe that in the one-player optimal stopping problem the goal is maximization of the payoff, and the corresponding supremum always exists (may be infinite), so only optimal or almost optimal stopping times remain to be found while in the game version already existence of the game value is the question which should be resolved first, and only then we can look for optimal (saddle point) or almost optimal stopping times of the players.

Few years later Neveu suggested in [6] a very useful generalization of the above setup which turned out to be more convenient for both further study and applications. Namely, now the “permission” processes were dropped off and the players could stop whenever they want, but instead two payoff adapted processes 𝑋_𝑡≥𝑌_𝑡 were introduced. It was prescribed that if the first player stops the game at time 𝑠 and the second one at time 𝑡, then the former pays to the latter the amount 𝑋_𝑠 or 𝑌_𝑡 if 𝑠≤𝑡 or 𝑠>𝑡, respectively. If desired we can have virtual “permission” processes within this setup not by direct regulations but by “market economy” tools. Namely, in order to accomplish this it suffices to prescribe very high payment 𝑋_𝑡 or a very low (may be negative) payment 𝑌_𝑡 where we “forbid” to stop the game by the first player or by the second one, respectively.

We observe that from a bit different perspective differential games with stopping times were studied in the 1970 in a series of papers (see [7, 8] and references there). Game versions of optimal stopping of a Markov process and of a diffusion were considered in [9, 10], respectively. It seems that the term “Dynkin’s game” appeared first in [11].

Israeli or game options were introduced first in [12] though some special callable derivative security LION was discussed before in [13] in a kind of game framework without any rigorous justification. An option or a contingent claim is a certain contract and an American option enables its buyer (holder) to exercise it at any time up to the maturity. A game option gives additionally the right to the option seller (writer, issuer) to cancel it early paying for this a prescribed penalty. The rationale behind this provision comes from an idea that essentially any contract stipulates conditions for a way out so that the financial market should not be an exception.

The classical approach to pricing of options is based on hedging arguments. Namely, the price is defined as a minimal initial amount of a self-financing portfolio which can provide protection (hedging) against any exercising strategy of the option buyer. So, somehow heuristically, this leads to the infimum over the seller’s strategies and to the supremum over the buyer’s strategies; that is, we arrive at a game-type infsup representation which still should be rigorously justified.

The structure of this paper is as follows. In Section 2 we briefly survey main results concerning Dynkin’s games. In Section 3 we discuss the up-to-date research on game options and related derivative securities. In Sections 4 and 5 we exhibit more special results concerning discrete approximations of game options and game options in markets with transaction costs, respectively.

2. Dynkin’s Games

The general modern setup for a Dynkin’s game consists of a probability space (Ω,ℱ,ℙ), a right continuous filtration of complete 𝜎-algebras {ℱ_𝑡}, and three {ℱ_𝑡}-adapted stochastic processes 𝑋_𝑡,𝑌_𝑡, and 𝑍_𝑡 so that when the first player stops the game at time 𝑠 and the second one stops at time 𝑡, then the former pays to the latter the amount 𝐻(𝑠,𝑡)=𝑋_𝑠𝕀_𝑠<𝑡+𝑌_𝑡𝕀_𝑠>𝑡+𝑍_𝑡𝕀_𝑠=𝑡, (1) where 𝕀_Γ=1 if an event Γ occurs and =0, otherwise. We allow the time 𝑡 to run either along nonnegative integers ℕ or along nonnegative reals ℝ₊ up to some horizon 𝑇≤∞ when the game is stopped and the first player pays to the second one the amount 𝐻(𝑇,𝑇)=𝑋_𝑇=𝑌_𝑇=𝑍_𝑇, (2) where in case 𝑇=∞, we assume that 0=𝑋_∞=lim_𝑡→∞𝑋_𝑡=𝑌_∞=lim_𝑡→∞𝑌_𝑡=𝑍_∞=lim_𝑡→∞𝑍_𝑡. (3) In the continuous time case; that is, when 𝑡 runs over ℝ₊, the processes 𝑋_𝑡, 𝑌_𝑡, and 𝑍_𝑡 are supposed to be right continuous.

Next, assume that for any 𝑡∈[0,𝑇], 𝑌_𝑡≤𝑍_𝑡≤𝑋_𝑡ℙ-almostsurely,𝔼sup_{0≤𝑡≤𝑇}𝑌_𝑡+𝑍_𝑡+𝑋_𝑡<∞. (4)(5) Denote by 𝒯_𝑠,𝑡.𝑠≤𝑡 the collection of all stopping times 𝜏 with values between 𝑠 and 𝑡 (i.e., nonnegative random variables such that {𝜏≤𝑢}∈ℱ_𝑢 for all 𝑢). Introduce the upper and the lower values of the game starting at time 𝑡≤𝑇 by 𝑉_𝑡=𝑒𝑠𝑠inf_{𝜎∈𝒯𝑡𝑇}𝑒𝑠𝑠sup_{𝜏∈𝒯𝑡𝑇}𝔼𝐻(𝜎,𝜏)∣ℱ_𝑡,𝑉_𝑡=𝑒𝑠𝑠sup_{𝜏∈𝒯𝑡𝑇}𝑒𝑠𝑠inf_{𝜎∈𝒯𝑡𝑇}𝔼𝐻(𝜎,𝜏)∣ℱ_𝑡. (6) It turns out that we can choose these processes {𝑉_𝑡} and {𝑉_𝑡} to be right upper semicontinuous which is a sufficient regularity in order to proceed here.

Theorem 1. Under the above conditions 𝑉=_𝜏^def𝑉_𝜏=𝑉_𝜏 almost surely for any stopping time 𝜏∈𝒯_0𝑇 and, in particular, the Dynkin’s game has a value 𝑉=𝑉₀=𝑉₀=𝑉₀. (7) Furthermore, for any 𝜀>0 the stopping times 𝜎_𝜀=inf𝑡≤𝑇∶𝑉_𝑡≥𝑋_𝑡,𝜏−𝜀_𝜀=inf𝑡≤𝑇∶𝑉_𝑡≤𝑌_𝑡+𝜀, (8) are 𝜀-optimal, that is, for any 𝜎,𝜏∈𝒯_0𝑇, 𝔼𝐻𝜎_𝜀,𝜏−𝜀≤𝔼𝐻𝜎_𝜀,𝜏_𝜀≤𝔼𝐻𝜎,𝜏_𝜀+𝜀. (9) Under additional regularity conditions (say, 𝑋_𝑡, 𝑌_𝑡, and 𝑍_𝑡 are continuous stochastic processes), the inequality (9) remains true for 𝜀=0 with some 𝜎₀,𝜏₀; that is, there exists a saddle point for the Dynkin’s game above. In the discrete time case we have also the following backward recursive (dynamical programming) relation: 𝑉_𝑛=min𝑋_𝑛,max𝑌_𝑛,𝔼𝑉_𝑛+1∣ℱ_𝑛. (10)

The theorem above follows from [6, 14–16] in the discrete time case and from [17–19] in the continuous time case. Observe that (6), (7), and (9) imply 𝑉₀−𝔼𝐻𝜎_𝜀,𝜏_𝜀≤𝜀. (11)

If the condition (4) does not hold true, then the above game value may not exist (i.e., 𝑉₀>𝑉₀) if the players are restricted to usual (pure) stopping times, and to have the game value, they should be allowed to use randomized stopping times (see [20–24]). Other results on Dynkin's games leading to randomized stopping times can be found in [25–28].

Remark 2. We observe that randomized stopping times used in the above mentioned papers in order to obtain Dynkin's game value without the condition (4) look somewhat different from randomized stopping times we employ in Section 5 in order to study game options in markets with transaction costs. Namely, the above papers deal with randomized stopping times having (in the discrete time case) the form 𝜆(𝑝)=min{𝑛≥0∶𝐴_𝑛≤𝑝_𝑛}, where 𝑝=(𝑝₀,𝑝₁,…) is an adapted to the filtration {ℱ_𝑛} process with 𝑝_𝑛∈[0,1] for all 𝑛 and 𝐴₀,𝐴₁,𝐴₂,… is a sequence of independent identically uniformly distributed on [0,1] random variables independent of payoff processes. Sometimes, it is assumed additionally (see [21]) that 𝐴_𝑛 is ℱ_𝑛+1-measurable and independent of ℱ_𝑛. If 𝑊=(𝑊₀,𝑊₁,𝑊₂,…) is an adapted stochastic process, then we can write 𝑊_𝜆(𝑝)=^∞_𝑛=0𝜓_𝑛𝑊_𝑛where𝜓_𝑛=𝕀_{{𝐴𝑛≤𝑝𝑛}}^𝑛−1_𝑗=0𝕀_{{𝐴𝑗>𝑝𝑗}}. (12) On the other hand, randomized stopping times employed in Section 5 are determined by an adapted nonnegative sequence 𝜒=(𝜒₀,𝜒₁,𝜒₂,…) such that ∑^∞_𝑗=0𝜒_𝑗=1 and for an adapted stochastic process 𝑊 as above we write 𝑊_𝜒=∑^∞_𝑛=0𝜒_𝑛𝑊_𝑛. Here {𝜒_𝑛} is an adapted sequence but not necessarily indicators of events while the above sequence {𝜓_𝑛} is not adapted (unless the filtration is properly enlarged) and it consists of indicators of events. Still, with respect to the enlarged filtration, 𝜆(𝑝) is a usual (pure) stopping time while randomized stopping times of Section 5 look rather differently. Nevertheless, it turns out that these two approaches to randomized stopping times are essentially equivalent if ∏_𝑛≥0(1−𝑝_𝑛)=0 (see [29] in the discrete time case and the corresponding discussion in [22] for the continuous time case).

Among other works on Dynkin’s games we can mention results concerning nonzero-sum games (see [30–33]), Dynkin’s games with asymmetric information (see [34]), more than 2 person optimal stopping games (see [35–37]), optimal stopping games driven by Markov processes (see [2, 9, 38, 39]), Dynkin’s games via backward stochastic differential equations with reflection (see [40–42]) and via Dirichlet forms (see [43]), and some other results on Dynkin's and similar games (see [44–55]).

3. Game Options and Their Shortfall Risk

A game (Israeli) option (or contingent claim) studied in [12] is a contract between a writer and a holder at time 𝑡=0 such that both have the right to exercise at any stopping time before the expiry date 𝑇. If the holder exercises at time 𝑡, he or she receives the amount 𝑌_𝑡≥0 from the writer and if the writer, exercises at time 𝑡 before the holder he must pay to the holder, the amount 𝑋_𝑡≥𝑌_𝑡 so that 𝛿_𝑡=𝑋_𝑡−𝑌_𝑡 is viewed as a penalty imposed on the writer for cancellation of the contract. If both exercise at the same time 𝑡, then the holder may claim 𝑌_𝑡, and if neither exercised until the expiry time 𝑇, then the holder may claim the amount 𝑌_𝑇. In short, if the writer will exercise at a stopping time 𝜎≤𝑇 and the holder at a stopping time 𝜏≤𝑇, then the former pays to the latter the amount 𝐻(𝜎,𝜏) where 𝐻(𝑠,𝑡)=𝑋_𝑠𝕀_𝑠<𝑡+𝑌_𝑡𝕀_𝑡≤𝑠. (13) We consider such game options in a standard securities market consisting of a nonrandom component 𝑏_𝑡 representing the value of a savings account at time 𝑡 with an interest rate 𝑟 and of a random component 𝑆_𝑡 representing the stock price at time 𝑡. As usual, we view 𝑆_𝑡,𝑡>0, as a stochastic process on a complete probability space (Ω,ℱ,𝑃), and we assume that it generates a right continuous filtration {ℱ_𝑡} and that the payoff processes 𝑋_𝑡 and 𝑌_𝑡 are right continuous processes adapted to this filtration and satisfying the integrability conditions (5).

The classical approach suggests that valuation of options should be based on the notions of a self-financing portfolio and on hedging. We start with a portfolio strategy 𝜋={𝜋_𝑡}_{0≤𝑡≤𝑇} which is a collection of pairs 𝜋_𝑡=(𝛽_𝑡,𝛾_𝑡) so that the portfolio value 𝑊_𝑡 at time 𝑡 equals 𝑊^𝜋_𝑡=𝛽_𝑡𝑏_𝑡+𝛾_𝑡𝑆_𝑡, (14) where the process (𝛽_𝑡,𝛾_𝑡),0≤𝑡≤𝑇, is supposed to be predictable in the discrete time case and progressively measurable in the continuous time case. A portfolio strategy 𝜋 is called self-financing if all changes in the portfolio value are due to capital gains or losses but not due to withdrawal or infusion of funds. This can be expressed by the relations (see [56]) 𝑏_𝑡−1𝛽_𝑡−𝛽_𝑡−1+𝑆_𝑡−1𝛾_𝑡−𝛾_𝑡−1=0for𝑡=1,2,…,𝑇 (15) in the discrete time case and 𝑊^𝜋_𝑡=𝑊^𝜋₀+^𝑡₀𝛽_𝑢𝑑𝑏_𝑢+^𝑡₀𝛾_𝑢𝑑𝑆_𝑢 (16) in the continuous time case. We assume also in the continuous time case that with probability one: ^𝑇₀𝑏_𝑡𝛽_𝑡𝑑𝑡<∞,^𝑇₀𝛾_𝑡𝑆_𝑡²𝑑𝑡<∞. (17) A pair (𝜎,𝜋) of a stopping time 𝜎≤𝑇 and a self-financing portfolio strategy 𝜋 is called a hedge (against the game contingent claim) if 𝑊^𝜋_𝜎∧𝑡≥𝐻(𝜎,𝑡) with probability one for any 𝑡∈[0,𝑇]. Now the fair price of the game option is defined as the infimum of capitals 𝑥 for which there exists a hedge (𝜎,𝜋) with 𝑊^𝜋₀=𝑥. In a complete market (i.e., having a unique martingale measure) this is a widely acceptable fair price of the option while in an incomplete market or in a market with transaction costs this definition leads to what is known as superhedging (see [56]).

Two popular models of complete markets were considered in [12] for pricing of game options. First, the discrete time Cox, Rox, Rubinstein (CRR) binomial model (see [57]) was treated there where the stock price 𝑆_𝑘 at time 𝑘 is equal to 𝑆_𝑘=𝑆₀^𝑘_𝑗=11+𝜌_𝑗,𝑆₀>0, (18) where 𝜌_𝑗,𝑗=1,2,… are independent identically distributed (i.i.d.) random variables such that 𝜌_𝑗=𝑏>0 with probability 𝑝>0 and 𝜌_𝑗=𝑎<0,𝑎>−1 with probability 𝑞=1−𝑝>0. Secondly, [12] deals with the continuous time Black-Scholes (BS) market model where the stock price 𝑆_𝑡 at time 𝑡 is given by the geometric Brownian motion: 𝑆_𝑡=𝑆₀𝜅exp𝛼−²2𝑡+𝜅𝐵_𝑡,𝑆₀>0, (19) where {𝐵_𝑡}_𝑡≥0 is the standard one-dimensional continuous-in-time Brownian motion (Wiener process) starting at zero and 𝜅>0, 𝛼∈(−∞,∞) are some parameters. In addition to the stock which is a risky security, the market includes in both cases also a savings account with a deterministic growth given by the formulas 𝑏_𝑛=(1+𝑟)^𝑛𝑏₀,𝑏_𝑡=𝑏₀𝑒^𝑟𝑡,𝑏₀,𝑟>0 (20) in the CRR model (where we assume in addition that 𝑟<𝑏) and in the BS model, respectively.

Recall (see [56]) that a probability measure describing the evolution of a stock price in a stochastic financial market is called martingale (risk-neutral) if the discounted stock prices ((1+𝑟)^−𝑘𝑆_𝑘 in the CRR model and 𝑒^−𝑟𝑡𝑆_𝑡 in the BS model) become martingales. Relying on the above hedging arguments the following result was proved in [12].

Theorem 3. The fair price 𝑉 of the game option is given by the formulas 𝑉=min_{𝜎∈𝒯0𝑇}max_{𝜏∈𝒯0𝑇}𝐸(1+𝑟)^{−𝜎∧𝜏}𝐻(𝜎,𝜏) (21) in the CRR market (with usual notations 𝑎∧𝑏=min(𝑎,𝑏), 𝑎∨𝑏=max(𝑎,𝑏)) and 𝑉=inf_{𝜎∈𝒯0𝑇}sup_{𝜏∈𝒯0𝑇}𝐸𝑒^{−𝑟𝜎∧𝜏}𝐻(𝜎,𝜏) (22) in the BS market, where the expectations are taken with respect to the corresponding martingale probabilities, which are uniquely defined since these markets are known to be complete (see [56]), 𝑇 is the expiry time, and 𝒯_𝑠𝑡 is the space of corresponding stopping times with values between 𝑠 and 𝑡 taking into account that in the CRR model 𝜎 and 𝜏 are allowed to take only integer values.

Observe that the formulas (21) and (22) represent also the values of corresponding Dynkin’s (optimal stopping) games with payoffs (1+𝑟)^{−𝜎∧𝜏}𝐻(𝜎,𝜏) and 𝑒^{−𝑟𝜎∧𝜏}𝐻(𝜎,𝜏), respectively, when the first and the second players stop the game at stopping times 𝜎 and 𝜏, respectively. The continuous time BS model is generally considered as a better description of the evolution of real stocks, in particular, since the CRR model allows only two possible values (1+𝑏)𝑆_𝑘 and (1+𝑎)𝑆_𝑘 for the stock price 𝑆_𝑘+1 at time 𝑘+1 given its price 𝑆_𝑘 at time 𝑘. The main advantage of the CRR model is its simplicity and the possibility of easier computations of the value 𝑉 in (21), in particular, by means of the dynamical programming recursive relations (see [12]), 𝑉=𝑉_0,𝑁,𝑉_𝑁,𝑁=(1+𝑟)^−𝑁𝑌_𝑁,𝑉_𝑘,𝑁=min(1+𝑟)^−𝑘𝑋_𝑘,max(1+𝑟)^−𝑘𝑌_𝑘,𝐸𝑉_𝑘+1,𝑁∣ℱ_𝑘, (23) where a positive integer 𝑁 is an expiry time and {ℱ_𝑘}_𝑘≥0 is the corresponding filtration of 𝜎-algebras. By this reason it makes sense to study approximations of the BS model by CRR models which we describe in the next section.

Though game options do not appear explicitly yet as a trading security in contemporary financial markets, it became popular recently to employ game options as a framework for the study of convertible (callable) bonds (see [58–65]). A holder of such bond either does nothing or decides to convert it into a predetermined number of stocks which can be considered as a cash payment depending on the current stock price, especially, in a market without transaction costs. On the other hand, the firm which issued this callable convertible bond may redeem it any time at a call price or force its conversion into stocks, and so this situation can be treated within the setup of game options.

Several papers deal with computation of the fair price of game options in special situations when the underlying stock price evolves according to a Markov process which usually, as in the BS model, turns out to be the geometric Brownian motion and when the payoffs depend only on the current stock price, usually just for the put and call options’ payoffs arriving at a study of the free boundary problem with buyer's and seller's exercise boundaries (see [66–73]). For other callable derivative securities which were studied within the game options framework and its generalizations, we refer the reader to [74–79].

In real market conditions an investor (seller) may not be willing for various reasons to tie in a hedging portfolio the full initial capital required for a (perfect) hedge. In this case the seller is ready to accept a risk that his portfolio value at an exercise time may be less than his obligation to pay and he will need additional funds to fulfil the contract. Thus a portfolio shortfall comes into the picture and it is important to estimate the corresponding risk. We consider here a certain type of risk called the shortfall risk which was defined for game options in [80] by 𝑅(𝑥)=inf_()𝜋,𝜎𝑅(),()𝜋,𝜎where𝑅𝜋,𝜎=sup_𝜏𝔼𝐻(𝜎,𝜏)−𝑏₀𝑊^𝜋_𝜎∧𝜏⁺𝑏_𝜎∧𝜏, (24) where the infimum is taken over all self-financing portfolio strategies 𝜋 with an initial capital 𝑥, and in both infimum and supremum the stopping times 𝜎 and 𝜏 do not exceed the option expiration date (horizon) 𝑇. It was shown in [80] that in the discrete time case both the shortfall risk and the corresponding minimizing portfolio strategies and stopping times could be obtained by means of a backward induction (dynamical programming) algorithm. In the continuous time case the situation is more complicated. For the shortfall risk in the American options case [81] obtained existence of minimizing strategies relying on some convex analysis arguments which are not available in the game options case, and so existence of minimizing portfolio strategies and stopping times in (24) remains an open question.

The papers [82, 83] deal with the, so-called, swing game options which are, in fact, multiple exercise game options. This question was studied before for American options in [84] but the option price obtained there was not justified by classical hedging arguments. This justification was done in [82, 83] for multiple exercise game options in the discrete and continuous time cases, respectively, which by simplification yields the result for American options as well. This investigation required the study of Dynkin's games with multiple stopping which did not appear in the literature before. Observe that multiple exercise options may appear in their own rights when an investor wants to buy or sell an underlying security in several instalments at times of his choosing and, actually, any usual American or game option can be naturally extended to the multiexercise setup so that they may emerge both in commodities, energy and in different financial markets. Suppose, for instance, that a European car producer (having most expenses in euros or pounds) plans to supply autos to USA during a year in several shipments and buys a multiple-exercise option which guarantees a favorable dollar-euro (or dollar-pound) exchange rate at time of shipments (of his choice). The seller of such option can use currencies as underlying securities for his hedging portfolio. A multiple exercise option could be cheaper than a basket of usual one-exercise options if the former stipulates certain delay time between exercises which is quite natural in the above example. Furthermore, the acting sides above may prefer to deal with game rather than American multiple-exercise options since the former is cheaper for the buyer and safer (because of cancellation clause) for the seller.

Next, we describe more precisely game swing (multiple-exercise) options in the CRR market where the stock price evolves according to (18). We consider a swing option of the game type which has the 𝑖th payoff, 𝑖≥1, having the form 𝐻^(𝑖)(𝑚,𝑛)=𝑋_𝑖(𝑚)𝕀_𝑚<𝑛+𝑌_𝑖(𝑛)𝕀_𝑛≤𝑚,∀𝑚,𝑛, (25) where 𝑋_𝑖(𝑛),𝑌_𝑖(𝑛) are ℱ_𝑛-adapted and 0≤𝑌_𝑖(𝑛)≤𝑋_𝑖(𝑛)<∞. Thus for any 𝑖,𝑛 there exist functions 𝑓_𝑛^(𝑖),𝑔_𝑛^(𝑖)∶{𝑎,𝑏}^𝑛→ℝ₊ such that 𝑌_𝑖(𝑛)=𝑓_𝑛^(𝑖)𝜌₁,…,𝜌_𝑛,𝑋_𝑖(𝑛)=𝑔_𝑛^(𝑖)𝜌₁,…,𝜌_𝑛. (26) For any 1≤𝑖≤𝐿−1 let 𝐶_𝑖 be the set of all pairs ((𝑎₁,…,𝑎_𝑖),(𝑑₁,…,𝑑_𝑖))∈{0,…,𝑁}^𝑖×{0,1}^𝑖 such that 𝑎_𝑗+1≥𝑁∧(𝑎_𝑗+1) for any 𝑗<𝑖. Such sequences represent the history of payoffs up to the 𝑖th one in the following way. If 𝑎_𝑗=𝑘 and 𝑑_𝑗=1 then the seller cancelled the 𝑗th claim at the moment 𝑘 and if 𝑑_𝑗=0 then the buyer exercised the 𝑗th claim at the moment 𝑘 (may be together with the seller). For 𝑛≥1 denote by Γ_𝑛 the set of all stopping times with respect to the filtration {ℱ_𝑛}^𝑁_𝑛=0 with values from 𝑛 to 𝑁 and set Γ=Γ₀.

Definition 4. A stopping strategy is a sequence 𝑠=(𝑠₁,…,𝑠_𝐿) such that 𝑠₁∈Γ is a stopping time and for 𝑖>1, 𝑠_𝑖∶𝐶_𝑖−1→Γ is a map which satisfies 𝑠_𝑖((𝑎₁,…,𝑎_𝑖−1),(𝑑₁,…,𝑑_𝑖−1))∈Γ_{𝑁∧(1+𝑎𝑖−1)}.

In other words for the 𝑖th payoff both the seller and the buyer choose stopping times taking into account the history of payoffs so far. Denote by 𝒮 the set of all stopping strategies and define the map 𝐹∶𝒮×𝒮→Γ^𝐿×Γ^𝐿 by 𝐹(𝑠,𝑏)=((𝜎₁,…,𝜎_𝐿),(𝜏₁,…,𝜏_𝐿)) where 𝜎₁=𝑠₁, 𝜏₁=𝑏₁ and for 𝑖>1, 𝜎_𝑖=𝑠_𝑖𝜎₁∧𝜏₁,…,𝜎_𝑖−1∧𝜏_𝑖−1,𝕀_𝜎1<𝜏1,…,𝕀_{𝜎𝑖−1<𝜏𝑖−1},𝜏_𝑖=𝑏_𝑖𝜎₁∧𝜏₁,…,𝜎_𝑖−1∧𝜏_𝑖−1,𝕀_𝜎1<𝜏1,…,𝕀_{𝜎𝑖−1<𝜏𝑖−1}. (27) Set 𝑐_𝑘(𝑠,𝑏)=^𝐿_𝑖=1𝕀_{𝜎𝑖∧𝜏𝑖≤𝑘} (28) which is a random variable equal to the number of payoffs until the moment 𝑘.

For swing options the notion of a self-financing portfolio involves not only allocation of capital between stocks and the bank account but also payoffs at exercise times. At the time 𝑘 the writer's decision how much money to invest in stocks (while depositing the remaining money into a bank account) depends not only on his present portfolio value but also on the current claim. Denote by Ξ the set of functions on the (finite) probability space Ω.

Definition 5. A portfolio strategy with an initial capital 𝑥>0 is a pair 𝜋=(𝑥,𝛾), where 𝛾∶{0,…,𝑁−1}×{1,…,𝐿}×ℝ→Ξ is a map such that 𝛾(𝑘,𝑖,𝑦) is an ℱ_𝑘-measurable random variable which represents the number of stocks which the seller buys at the moment 𝑘 provided that the current claim has the number 𝑖 and the present portfolio value is 𝑦. At the same time the sum 𝑦−𝛾(𝑘,𝑖,𝑦)𝑆_𝑘 is deposited to the bank account of the portfolio. One calls a portfolio strategy 𝜋=(𝑥,𝛾)admissible if for any 𝑦≥0, −𝑦𝑆_𝑘𝑏𝑦≤𝛾𝑘,𝑖,𝑦≤−𝑆_𝑘𝑎. (29) For any 𝑦≥0 denote 𝐾(𝑦)=[−𝑦/𝑏,−𝑦/𝑎].

Notice that if the portfolio value at the moment 𝑘 is 𝑦≥0 then the portfolio value at the moment 𝑘+1 before the payoffs (if there are any payoffs at this time) is given by 𝑦+𝛾(𝑘,𝑖,𝑦)𝑆_𝑘(𝑆_𝑘+1/𝑆_𝑘−1), where 𝑖 is the number of the next payoff. In view of independency of 𝑆_𝑘+1/𝑆_𝑘−1 and 𝛾(𝑘,𝑖,𝑦)𝑆_𝑘 we conclude that the inequality (29) is equivalent to the inequality 𝑦+𝛾(𝑘,𝑖,𝑦)𝑆_𝑘(𝑆_𝑘+1/𝑆_𝑘−1)≥0, that is, the portfolio value at the moment 𝑘+1 before the payoffs is nonnegative. Denote by 𝒜(𝑥) the set of all admissible portfolio strategies with an initial capital 𝑥>0. Denote 𝒜=⋃_𝑥>0𝒜(𝑥). Let 𝜋=(𝑥,𝛾) be a portfolio strategy and 𝑠,𝑏∈𝒮. Set ((𝜎₁,…,𝜎_𝐿),(𝜏₁,…,𝜏_𝐿))=𝐹(𝑠,𝑏) and 𝑐_𝑘=𝑐_𝑘(𝑠,𝑏). The portfolio value at the moment 𝑘 after the payoffs (if there are any payoffs at this moment) is given by 𝑊₀^{(𝜋,𝑠,𝑏)}=𝑥−𝐻⁽¹⁾𝜎₁,𝜏₁𝕀_{𝜎1∧𝜏1=0}𝑊,for𝑘>0,_{𝑘(𝜋,𝑠,𝑏)}=𝑊^{(𝜋,𝑠,𝑏)}_𝑘−1+𝕀_{𝑐𝑘−1<𝐿}×𝛾𝑘−1,𝑐_𝑘−1+1,𝑊^{(𝜋,𝑠,𝑏)}_𝑘−1𝑆_𝑘−𝑆_𝑘−1−^𝐿_𝑖=1𝐻^(𝑖)𝜎_𝑖,𝜏_𝑖𝕀_{𝜎𝑖∧𝜏𝑖=𝑘}. (30)

Definition 6. A (perfect) hedge is a pair (𝜋,𝑠) which consists of a portfolio strategy and a stopping strategy such that 𝑊_𝑘^{(𝜋,𝑠,𝑏)}≥0 for any 𝑏∈𝒮 and 𝑘≤𝑁.

As usual, the option price 𝑉^∗ is defined as the infimum of 𝑊≥0 such that there exists a hedge with an initial capital 𝑊. The following result from [82] provides a dynamical programming algorithm for computation of both the option price and the corresponding hedge.

Theorem 7. For any 𝑛≤𝑁 set 𝑋_𝑛⁽¹⁾=𝑋_𝐿(𝑛),𝑌_𝑛⁽¹⁾=𝑌_𝐿(𝑛),𝑉_𝑛⁽¹⁾=min_{𝜎∈Γ𝑛}max_{𝜏∈Γ𝑛}𝐸𝐻^(𝐿)(𝜎,𝜏)∣ℱ_𝑛, (31) and for 1<𝑘≤𝐿, 𝑋_𝑛^(𝑘)=𝑋_{𝐿−𝑘+1}(𝑛)+𝐸𝑉₍₎^(𝑘−1)_{𝑛+1∧𝑁}∣ℱ_𝑛,𝑌_𝑛^(𝑘)=𝑌_{𝐿−𝑘+1}(𝑛)+𝐸𝑉₍₎^(𝑘−1)_{𝑛+1∧𝑁}∣ℱ_𝑛,𝑉_𝑛^(𝑘)=min_{𝜎∈Γ𝑛}max_{𝜏∈Γ𝑛}𝐸𝑋_𝜎^(𝑘)𝕀_𝜎<𝜏+𝑌_𝜏^(𝑘)𝕀_𝜎≥𝜏∣ℱ_𝑛, (32) where 𝐸 is the expectation with respect to the unique martingale measure. Then 𝑉^∗=𝑉₀^(𝐿)=min_𝑠∈𝒮max_𝑏∈𝒮𝐺(𝑠,𝑏), (33) where 𝐺(𝑠,𝑏)=𝐸∑^𝐿_𝑖=1𝐻^(𝑖)(𝜎_𝑖,𝜏_𝑖) and ((𝜎₁,…,𝜎_𝐿),(𝜏₁,…,𝜏_𝐿))=𝐹(𝑠,𝑏). Furthermore, the stopping strategies 𝑠^∗=(𝑠^∗₁,…,𝑠^∗_𝐿)∈𝑆 and 𝑏=(𝑏^∗₁,…,𝑏^∗_𝐿) given by 𝑠^∗₁=𝑁∧min𝑘∣𝑋_𝑘^(𝐿)=𝑉_𝑘^(𝐿),𝑏^∗₁=min𝑘∣𝑌_𝑘^(𝐿)=𝑉_𝑘^(𝐿),𝑠^∗_𝑖𝑎₁,…,𝑎_𝑖−1,𝑑₁,…,𝑑_𝑖−1=𝑁∧min𝑘>𝑎_𝑖−1∣𝑋_𝑘^{(𝐿−𝑖+1)}=𝑉_𝑘^{(𝐿−𝑖+1)},𝑏^∗_𝑖𝑎₁,…,𝑎_𝑖−1,𝑑₁,…,𝑑_𝑖−1=𝑁∧min𝑘>𝑎_𝑖−1∣𝑌_𝑘^{(𝐿−𝑖+1)}=𝑉_𝑘^{(𝐿−𝑖+1)},𝑖>1, (34) satisfy the saddle point inequalities 𝐺𝑠^∗,𝑏≤𝐺𝑠^∗,𝑏^∗≤𝐺𝑠,𝑏^∗∀𝑠,𝑏, (35) and there exists a portfolio strategy 𝜋^∗∈𝒜(𝑉₀^(𝐿)) such that (𝜋^∗,𝑠^∗) is a hedge.

4. Approximations of Game Options and of Their Shortfall Risk

Following [85] we will consider here approximations of the BS model by a sequence of CRR models with the interest rates 𝑟=𝑟^(𝑛) from (20) and with random variables 𝜌_𝑘=𝜌_𝑘^(𝑛) from (18) given by 𝑟=𝑟^(𝑛)=exp𝑟𝑇𝑛𝜌−1,_𝑘=𝜌_𝑘^(𝑛)=exp𝑟𝑇𝑛𝑇+𝜅𝑛^1/2𝜉_𝑘−1, (36) where 𝜉_𝑗=𝜉_𝑗^(𝑛),𝑗=1,2,… are i.i.d. random variables taking on the values 1 and −1 with probabilities 𝑝^(𝑛)=(exp(𝜅𝑇/𝑛)+1)⁻¹ and 1−𝑝^(𝑛)=(exp(−𝜅𝑇/𝑛)+1)⁻¹, respectively. This choice of random variables 𝜉_𝑖,𝑖∈ℕ, determines already the probability measures 𝑃^𝜉_𝑛={𝑝^(𝑛),1−𝑝^(𝑛)}^∞ for the above sequence of CRR models and since 𝐸^𝜉_𝑛𝜌_𝑘^(𝑛)=𝑟^(𝑛), where 𝐸^𝜉_𝑛 is the expectation with respect to 𝑃^𝜉_𝑛, we conclude that 𝑃^𝜉_𝑛 is the martingale measure for the corresponding CRR market and the fair price 𝑉=𝑉^(𝑛) of a game option in this market is given by the formula (21) with 𝐸=𝐸^𝜉_𝑛.

Let 𝑉 be the fair price of the game option in the BS market. It turns out that for a certain natural class of payoffs 𝑋_𝑡 and 𝑌_𝑡 which may depend on the whole path (history) of the stock price evolution (as in integral or Russian-type options), the error |𝑉−𝑉^(𝑛)| does not exceed 𝐶𝑛^−1/4(ln𝑛)^3/4 where 𝐶>0 does not depend on 𝑛 and it can be estimated explicitly. Moreover, we will see that the rational exercise times of our CRR binomial approximations yield near rational (𝐶𝑛^−1/4(ln𝑛)^3/4-optimal stopping times for the corresponding Dynkin’s games) exercise times for game options in the BS market. Since the values 𝑉^(𝑛) and the optimal stopping times of the corresponding discrete time Dynkin’s games can be obtained directly via the dynamical programming recursive procedure (23) our results provide a justification of a rather effective method of computation of fair prices and exercise times of game options with path-dependent payoffs. The standard construction of a self-financing hedging portfolio involves usually the Doob-Meyer decomposition of supermartingales which is explicit only in the discrete but not in the continuous time case. We will see how to construct a self-financing portfolio in the BS market with a small average (maximal) shortfall and an initial capital close to the fair price of a game option using hedging self-financing portfolios for the approximating binomial CRR markets. The latter problem does not seem to have been addressed before [85] in the literature on this subject. Having in mind that hedging self-financing portfolio strategies can be computed only approximately, their possible shortfalls come naturally into the picture and they should be taken into account in option pricing even if a perfect hedging exists theoretically. Note that these results require not only an approximation of stock prices and the corresponding payoffs, but also we have to take care of the different nature of stopping times in (21) and (22).

The main tool here is the Skorokhod-type embedding (see [86]) of sums of i.i.d. random variables into a Brownian motion (with a constant drift, in this case). This tool was already employed for similar purposes in [87, 88]. The first paper treats an optimal stopping problem which can be applied to an American style option with a payoff function depending only on the current stock price and, more importantly, this function must be bounded and have two bounded derivatives which excludes usual put and call options cases. The second paper deals only with European options and, again, only payoffs (though with some discontinuities) determined by the current stock price are allowed. We observe that the Skorokhod embedding does not provide optimal error estimates in strong approximation theorems and it would be interesting to understand whether other approaches such as the quantile method (see [89–91]) and Stein's method (see [92]) can be employed for approximation of optimal stopping game values with better estimates of errors. Skorokhod embedding does not work also in the multidimensional situation, and for this case another method from [93] was employed in [94] where, actually, more general and not only binomial approximations were considered. More general approximation results for game options were obtained in [95] where only continuity of payoffs was assumed, but as a result no error estimates could be obtained there.

For each 𝑡>0 denote by 𝑀[0,𝑡] the space of Borel measurable functions on [0,𝑡] with the uniform metric 𝑑_0𝑡(𝜐,𝜐)=sup_{0≤𝑠≤𝑡}|𝜐_𝑠−𝜐_𝑠|. For each 𝑡>0, let 𝐹_𝑡 and Δ_𝑡 be nonnegative functions on 𝑀[0,𝑡] such that for some constant 𝐿≥1 and for any 𝑡≥𝑠≥0 and 𝜐,𝜐∈𝑀[0,𝑡], 𝐹_𝑠(𝜐)−𝐹_𝑠(𝜐)+Δ_𝑠(𝜐)−Δ_𝑠(𝜐)≤𝐿(𝑠+1)𝑑_0𝑠(𝜐),𝐹𝜐,_𝑡(𝜐)−𝐹_𝑠(𝜐)+Δ_𝑡(𝜐)−Δ_𝑠(𝜐||)≤𝐿𝑡−𝑠1+sup_{[]𝑢∈0,𝑡}𝜐_𝑢+sup_{[]𝑢∈𝑠,𝑡}𝜐_𝑢−𝜐_𝑠. (37)(38) By (37), 𝐹₀(𝜐)=𝐹₀(𝜐₀) and Δ₀(𝜐)=Δ₀(𝜐₀) are functions of 𝜐₀ only. By (38), 𝐹_𝑡(𝜐)+Δ_𝑡(𝜐)≤𝐹₀𝜐₀+Δ₀𝜐₀+𝐿(𝑡+2)1+sup_{0≤𝑠≤𝑡}𝜐_𝑠. (39)

Next, we consider the BS market on a complete probability space together with its martingale measure 𝑃^𝐵 which exists and is unique as a corollary of the Girsanov theorem (see [56]). Let 𝐵_𝑡,𝑡≥0 be the standard one-dimensional continuous in time Brownian motion with respect to the martingale measure 𝑃^𝐵. Set 𝐵^∗_𝑡𝜅=−2𝑡+𝐵_𝑡,𝑡≥0. (40) Then the stock price 𝑆^𝐵_𝑡(𝑧) at time 𝑡 in the BS market can be written in the form 𝑆^𝐵_𝑡(𝑧)=𝑧exp𝑟𝑡+𝜅𝐵^∗_𝑡,𝑆^𝐵₀(𝑧)=𝑧>0, (41) where 𝑟>0 is the interest rate and 𝜅>0 is the, so called, volatility. We will consider game options in the BS market with payoff processes in the form 𝑌_𝑡=𝐹_𝑡𝑆^𝐵(𝑧),𝑋_𝑡=𝐺_𝑡𝑆^𝐵(𝑧),𝑡∈[0,𝑇],𝑇>0, (42) where 𝐺_𝑡=𝐹_𝑡+Δ_𝑡, 𝐹, Δ satisfies (37) and (38), 𝑆^𝐵(𝑧)=𝑆^𝐵(𝑧,𝜔)∈𝑀[0,𝑇] is a random function taking the value 𝑆^𝐵_𝑡(𝑧)=𝑆^𝐵_𝑡(𝑧,𝜔) at 𝑡∈[0,𝑇], and in the notations 𝐹_𝑡(𝑆^𝐵(𝑧))𝐺_𝑡(𝑆^𝐵(𝑧)) for 𝑡<𝑇 we take the restriction of 𝑆^𝐵(𝑧) to the interval [0,𝑡]. The fair price 𝑉=𝑉(𝑧) of this option with an initial value 𝑧>0 of the stock is given by (22).

Next, we consider a sequence of CRR markets on a complete probability space such that for each 𝑛=1,2,… the stock prices 𝑆_𝑡^(𝑛)(𝑧) at time 𝑡 are given by the formula 𝑆_𝑡^(𝑛)(𝑧)=𝑧exp^{[𝑛𝑡/𝑇]}_𝑘=1𝑟𝑇𝑛𝑇+𝜅𝑛^1/2𝜉_𝑘𝑇,𝑡≥𝑛,𝑆_𝑡^(𝑛)(𝑧)=𝑆₀^(𝑛)(𝑧)𝑇=𝑧>0,𝑡∈0,𝑛, (43) where, recall, 𝜉₁,𝜉₂,… are i.i.d. random variables taking the values 1 and −1 with probabilities 𝑝^(𝑛)=(exp(𝜅𝑇/𝑛)+1)⁻¹ and 1−𝑝^(𝑛)=(exp(−𝜅𝑇/𝑛)+1)⁻¹, respectively. Namely, we consider CRR markets where stock prices 𝑆_𝑚=𝑆^(𝑛)_𝑚/𝑛(𝑧),𝑚=0,1,2,… satisfy (18) with 𝜌_𝑘=𝜌^𝑛_𝑘 given by (36), and, in addition, in place of the interest rate 𝑟 in the first formula in (20) we take the sequence of interest rates 𝑟_𝑛=exp(𝑟𝑇/𝑛)−1, where 𝑟 is the interest rate of the BS market appearing in the second formula of (20) and in (22). We consider 𝑆^(𝑛)(𝑧)=𝑆^(𝑛)(𝑧,𝜔) as a random function on [0,𝑇], so that 𝑆^(𝑛)(𝑧,𝜔)∈𝑀[0,𝑇] takes the value 𝑆_𝑡^(𝑛)(𝑧)=𝑆_𝑡^(𝑛)(𝑧,𝜔) at 𝑡∈[0,𝑇]. For 𝑘=0,1,2,…,𝑛 put 𝑌_𝑘=𝑌_𝑘^(𝑛)(𝑧)=𝐹_{𝑘𝑇/𝑛}𝑆^(𝑛)(𝑧),𝑋_𝑘=𝑋_𝑘^(𝑛)(𝑧)=𝐺_{𝑘𝑇/𝑛}𝑆^(𝑛)(𝑧). (44) Then for each 𝑛 the fair price 𝑉=𝑉^(𝑛)(𝑧) of the game option in the corresponding CRR market with an initial value 𝑧>0 of the stock is given by (21).

Set 𝐻^𝐵_𝑧(𝑠,𝑡)=𝐹_𝑡𝑆^𝐵(𝑧)𝕀_𝑠≥𝑡+𝐺_𝑠𝑆^𝐵(𝑧)𝕀_𝑠<𝑡,𝑄^𝐵_𝑧(𝑠,𝑡)=𝑒^{−𝑟𝑠∧𝑡}𝑅^𝐵_𝑧(),𝐻𝑠,𝑡_𝑧(𝑛)(𝑠,𝑡)=𝐹_𝑡𝑆^(𝑛)(𝑧)𝕀_𝑠≥𝑡+𝐺_𝑠𝑆^(𝑛)(𝑧)𝕀_𝑠<𝑡,𝑄_𝑧^(𝑛)(𝑠,𝑡)=𝑒^{−𝑟𝑠∧𝑡}𝑅_𝑧^(𝑛)(𝑠,𝑡). (45)(46) Denote by 𝒯^𝐵_0𝑇 and 𝒯^𝜉_0𝑛 the sets of stopping times with respect to the Brownian filtration ℱ^𝐵_𝑡,𝑡≥0, with values in [0,𝑇] and with respect to the filtration ℱ^𝜉_𝑘=𝜎{𝜉₁,…,𝜉_𝑘} with values in {0,1,…,𝑛}. Set 𝑉(𝑧)=inf_{𝜎∈𝒯𝐵0𝑇}sup_{𝜏∈𝒯𝐵0𝑇}𝐸^𝐵𝑄^𝐵_𝑧(),𝑉𝜎,𝜏_(𝑛)(𝑧)=min_{𝜁∈𝒯𝜉0𝑛}max_{𝜂∈𝒯𝜉0𝑛}𝐸^𝜉_𝑛𝑄_𝑧^(𝑛)𝜁𝑇𝑛,𝜂𝑇𝑛, (47)(48) where 𝐸^𝐵 and 𝐸^𝜉_𝑛 are the expectations with respect to the probability measures 𝑃^𝐵 and 𝑃^𝜉_𝑛, respectively, and we observe that 𝒯^𝜉_0𝑛 is a finite set so that we can use min and max in (48).

Recall, that we choose 𝑃^𝐵 to be the martingale measure for the BS market and observe that 𝑃^𝜉_𝑛 is the martingale measure for the corresponding CRR market since a direct computation shows that 𝐸^𝜉_𝑛𝜌_𝑘^(𝑛)=𝑟_𝑛. Thus, (47) and (48) give fair prices of the game options in the corresponding markets. We note also that all our formulas involving the expectations 𝐸^𝐵, in particular, (47) giving the fair price 𝑉 of a game option, do not depend on a particular choice of a continuous-in-time version of the Brownian motion since all of them induce the same probability measure on the space of continuous sample paths which already determines all expressions with the expectations 𝐸^𝐵 appearing in this paper.

The following result from [96] provides an estimate for the error term in approximation of the fair price of a game option in the BS market by fair prices of the sequence of game options and prices of Dynkin's games defined above.

Theorem 8. Suppose that 𝑉(𝑧) and 𝑉^(𝑛)(𝑧) are defined by (46)–(48) with functions 𝐹 and 𝐺=𝐹+Δ satisfying (37) and (38). Then there exists a constant 𝐶>0 (which can be explicitly estimated) such that 𝑉(𝑧)−𝑉^(𝑛)(𝑧)≤𝐶𝐹₀(𝑧)+Δ₀(𝑧)+𝑧+1𝑛^−1/4(ln𝑛)^3/4 (49) for all 𝑧,𝑛>0.

We can choose more general i.i.d. random variables 𝜉₁,𝜉₂,… appearing in the definition of 𝑉^(𝑛) as well, but these generalizations do not seem to have a financial mathematics motivation since we want to approximate game options in the BS market by the simplest possible models which are, of course, game options in the CRR market.

Among main examples of options with path-dependent payoff we have in mind integral options where 𝐹_𝑡(𝜐)=^𝑡₀𝑓_𝑢𝜐_𝑢𝑑𝑢−𝐿⁺calloptioncase, (50) or 𝐹_𝑡(𝜐)=𝐿−^𝑡₀𝑓_𝑢𝜐_𝑢𝑑𝑢⁺putoptioncase, (51) where, as usual, 𝑎⁺=max(𝑎,0). The penalty functional may also have here the integral form Δ_𝑡(𝜐)=^𝑡₀𝛿_𝑢𝜐_𝑢𝑑𝑢. (52) In order to satisfy the conditions (37) and (38), we can assume that for some 𝐾>0 and all 𝑥,𝑦,𝑢, 𝑓_𝑢(𝑥)−𝑓_𝑢𝑦+𝛿_𝑢(𝑥)−𝛿_𝑢𝑦,𝑓≤𝐾𝑥−𝑦_𝑢(𝑥)+𝛿_𝑢(𝑥)≤𝐾|𝑥|. (53)

Observe also that the Asian-type (averaged integral) payoffs of the form 𝐹_𝑡(𝜐)=1𝑡^𝑡₀𝑓_𝑢𝜐_𝑢𝑑𝑢−𝐿⁺1or=𝐿−𝑡^𝑡₀𝑓_𝑢𝜐_𝑢𝑑𝑢⁺ (54) do not satisfy the condition (38) if arbitrarily small exercise times are allowed though the latter seems to have only some theoretical interest as it hardly happens in reality. Still, also in this case, the binomial approximation errors can be estimated in a similar way considering separately estimates for small stopping times and for stopping times bounded away from zero. Namely, define 𝑉_𝜀(𝑧) and 𝑉_𝜀^(𝑛)(𝑧) for 𝜀≥0 by (47) and (48), where 𝑄_𝑧^(𝐵)(𝜎,𝜏) and 𝑄_𝑧^(𝑛)(𝜁𝑇/𝑛,𝜂𝑇/𝑛) are replaced by 𝑄_𝑧^(𝐵)(𝜎∨𝜀,𝜏∨𝜀) and 𝑄_𝑧^(𝑛)(𝜁𝑇/𝑛∨𝜀,𝜂𝑇/𝑛∨𝜀), respectively. Assuming that 𝑓_𝑢 and 𝛿_𝑢 are Lipschitz continuous also in 𝑢 (at least for 𝑢 close to 0) in the form |𝑓_𝑠(𝑥)−𝑓_𝑢(𝑥)|+|𝛿_𝑠(𝑥)−𝛿_𝑢(𝑥)|≤𝐾(𝑥+1)|𝑠−𝑢| for some 𝐾>0 and all 𝑠,𝑢,𝑥≥0, we obtain that if 𝜐₀=𝑧 and 𝐹₀(𝜐)=(𝑓₀(𝑧)−𝐿)⁺ or =(𝐿−𝑓₀(𝑧))⁺, then 𝐹_𝑠(𝜐)−𝐹₀(𝑧)≤𝐾𝑠1+sup_{0≤𝑢≤𝑠}𝜐_𝑢+𝐾sup_{0≤𝑢≤𝑠}𝜐_𝑢−𝑧. (55) It is not difficult to see from here that |𝑉(𝑧)−𝑉_𝜀(𝑧)| and |𝑉^(𝑛)(𝑧)−𝑉_𝜀^(𝑛)(𝑧)| do not exceed 𝐶(1+𝑧)√𝜀 for all small 𝜀 and some constant 𝐶. On the other hand, similar to Theorem 8, we see that for some constant 𝐶>0 and all 𝑛,𝜀>0, 𝑉_𝜀(𝑧)−𝑉_𝜀^(𝑛)(𝑧)≤𝐶(1+𝑧)𝜀⁻¹𝑛^−1/4(ln𝑛)^3/4. (56) Choosing 𝜀=𝑛^−1/6√ln𝑛, we obtain that under the above conditions in the case of Asian options, |𝑉(𝑧)−𝑉^(𝑛)(𝑧)| can be estimated by 3𝐶(1+𝑧)𝑛^−1/12(ln𝑛)^1/4.

Another important example of path-dependent payoffs are the, so-called, Russian options where, for instance, 𝐹_𝑡(𝜐)=max𝑚,sup_{[]𝑢∈0,𝑡}𝜐_𝑢,Δ_𝑡(𝜐)=𝛿𝜐_𝑡. (57) Such payoffs satisfy the conditions of Theorem 8. Indeed, (37) is clear in this case and (38) follows since for 𝑡≥𝑠, max𝑚,sup_{[]𝑢∈0,𝑡}𝑣_𝑢−max𝑚,sup_{[]𝑢∈0,𝑠}𝑣_𝑢≤sup_{[]𝑢∈0,𝑡}𝑣_𝑢−sup_{[]𝑢∈0,𝑠}𝑣_𝑢≤sup_{[]𝑢∈𝑠,𝑡}𝑣_𝑢−𝑣_𝑠≤sup_{[]𝑢∈𝑠,𝑡}𝑣_𝑢−𝑣_𝑠. (58)

In order to compare 𝑉(𝑧) and 𝑉^(𝑛)(𝑧) in the case of path-dependent payoffs, we have to consider both BS and CRR markets on one probability space in an appropriate way, and the main tool in achieving this goal will be here the Skorokhod-type embedding (see, for instance, [86], Section 37). In fact, for the binomial i.i.d. random variables 𝜉₁,𝜉₂,… appearing in the setup of the CRR market models above, the embedding is explicit and no general theorems are required, but if we want to extend the result for other sequences of i.i.d. random variables, we have to rely upon the general result. Namely, define recursively 𝜃₀^(𝑛)=0,𝜃^(𝑛)_𝑘+1=inf𝑡>𝜃_𝑘^(𝑛)∶𝐵^∗_𝑡−𝐵^∗_{𝜃𝑘(𝑛)}=𝑇𝑛, (59) where, recall, 𝐵^∗_𝑡=−(𝜅/2)𝑡+𝐵_𝑡. The standard strong Markov property-based arguments (cf. [86], Section 37) show that 𝜃_𝑘^(𝑛)−𝜃^(𝑛)_𝑘−1,𝑘=1,2,… are i.i.d. sequences of random variables such that (𝜃^(𝑛)_𝑘+1−𝜃_𝑘^(𝑛),𝐵^∗_{𝜃(𝑛)𝑘+1}−𝐵^∗_{𝜃𝑘(𝑛)}) are independent of ℱ^𝐵_{𝜃𝑘(𝑛)} (where, recall, ℱ^𝐵_𝑡=𝜎{𝐵_𝑠,𝑠≤𝑡}).

It turns out (see [85, 88]) that 𝐵^∗_𝜃1(𝑛) has the same distribution as 𝑇/𝑛𝜉₁. Set Ξ_𝑘^(𝑛)=𝑇𝑛^𝑘1/2_𝑗=1𝜉_𝑗, (60) then Ξ_𝑘^(𝑛) has the same distribution as 𝐵^∗_{𝜃𝑘(𝑛)}.

Theorem 8 provides an approximation of the fair price of game options in the BS market by means of fair prices of game options in the CRR market which becomes especially useful if we can provide also a simple description of rational (or 𝛿-rational) exercise times of these options in the BS market via exercise times of their CRR market approximations which are, by the definition, optimal (or 𝛿-optimal) stopping times for the Dynkin’s game whose price is given by (48). For each 𝑘=1,2,… introduce the finite 𝜎-algebra 𝒢_𝑘^𝐵,𝑛=𝜎{𝐵^∗_𝜃1(𝑛),𝐵^∗_𝜃2(𝑛)−𝐵^∗_𝜃1(𝑛),…,𝐵^∗_{𝜃𝑘(𝑛)}−𝐵^∗_{𝜃(𝑛)𝑘−1}} which is, clearly, isomorphic to ℱ^𝜉_𝑘=𝜎{𝜉_𝑖,𝑖≤𝑘} considered before since each element of 𝒢_𝑘^𝐵,𝑛 and of ℱ^𝜉_𝑘 is an event of the form 𝐴_𝜄^𝐵,𝑛_(𝑘)=𝐵^∗_{𝜃𝑗(𝑛)}−𝐵^∗_{𝜃(𝑛)𝑗−1}=𝜄_𝑗𝑇𝑛,𝐴,𝑗=1,…,𝑘_{𝜉𝜄(𝑘)}=𝜉_𝑗=𝜄_𝑗,𝑗=1,…𝑘, (61) respectively, where 𝜄^(𝑘)=(𝜄₁,…,𝜄_𝑘)∈{−1,1}^𝑘, 𝜃₀^(𝑛)=0, and 𝐵₀=0. Let 𝒮^𝐵,𝑛 be the set of stopping times with respect to the filtration 𝒢_𝑘^𝐵,𝑛,𝑘=0,1,2,…, where 𝒢₀^𝐵,𝑛={∅,Ω_𝐵} is the trivial 𝜎-algebra, and Ω_𝐵 is the sample space of the Brownian motion. The subset of these stopping times with values in {0,1,…,𝑛} will be denoted by 𝒮^𝐵,𝑛_0,𝑛. For each 𝜄^(𝑛)=(𝜄₁,…,𝜄_𝑛)∈{−1,1}^𝑛 and 𝑘<𝑛 we set 𝜄^(𝑘)=(𝜄₁,…,𝜄_𝑘)∈{−1,1}^𝑘. Denote by 𝒥_0,𝑛 the set of functions 𝜈∶{−1,1}^𝑛→{0,1,…,𝑛} such that if 𝜈(𝜄^(𝑛))=𝑘≤𝑛 and 𝜄̃^(𝑘)=𝜄^(𝑘) for some 𝜄̃^(𝑛)∈{−1,1}^𝑛, then 𝜈(𝜄̃^(𝑛))=𝑘, as well. Define the functions 𝜆_𝜉^(𝑛)∶Ω_𝜉→{−1,1}^𝑛 and 𝜆_𝐵^(𝑛)∶Ω_𝐵→{−1,1}^𝑛 by 𝜆_𝜉^(𝑛)(𝜔)=(𝜉₁(𝜔),…,𝜉_𝑛(𝜔)) and 𝜆_𝐵^(𝑛)(𝜔)=𝑛𝑇𝐵^∗_{𝜃1(𝑛)(𝜔)}(𝜔),𝐵^∗_{𝜃2(𝑛)(𝜔)}(𝜔)−𝐵^∗_{𝜃1(𝑛)(𝜔)}(𝜔),…,𝐵^∗_{𝜃𝑛(𝑛)(𝜔)}(𝜔)−𝐵^∗_{𝜃(𝑛)𝑛−1(𝜔)}(𝜔), (62) where Ω_𝜉 and Ω_𝐵 are sample spaces on which the sequence 𝜉₁,𝜉₂,… and the Brownian motion 𝐵_𝑡 are defined, respectively. It is clear that any 𝜁∈𝒯^𝜉_0𝑛 and 𝜂∈𝒮^𝐵,𝑛_0,𝑛 can be represented uniquely in the form 𝜁=𝜇∘𝜆_𝜉^(𝑛) and 𝜂=𝜈∘𝜆_𝐵^(𝑛) for some 𝜇,𝜈∈𝒥_0,𝑛.

Theorem 9. There exists a constant 𝐶>0 (which can be estimated explicitly) such that if   𝜁^∗_𝑛=𝜇^∗_𝑛∘𝜆_𝜉^(𝑛) and 𝜂^∗_𝑛=𝜈^∗_𝑛∘𝜆_𝜉^(𝑛), 𝜇^∗_𝑛,𝜈^∗_𝑛∈𝒥_0𝑛 are rational exercise times for the game option in the CRR market defined by (43); that is, 𝑉^(𝑛)(𝑧)=min_{𝜁∈𝒯𝜉0𝑛}𝐸^𝜉𝑄_𝑧^(𝑛)𝜁𝑇𝑛,𝜂^∗_𝑛𝑇𝑛=max_{𝜂∈𝒯𝜉0𝑛}𝐸^𝜉𝑄_𝑧^(𝑛)𝜁^∗_𝑛𝑇𝑛𝑇,𝜂𝑛, (63) then 𝜑^∗_𝑛=𝜃_𝜇^(𝑛)_{∗𝑛∘𝜆𝐵(𝑛)} and 𝜓^∗_𝑛=𝜃_𝜈^(𝑛)_{∗𝑛∘𝜆𝐵(𝑛)} are 𝛿_𝑛(𝑧)-rational exercise times for the game option in the BS market defined by (39) and (41); that is, sup_{𝜏∈𝒯𝐵0𝑇}𝐸^𝐵𝑄^𝐵_𝑧𝜑^∗_𝑛,𝜏−𝛿_𝑛(𝑧)≤𝑉(𝑧)≤inf_{𝜎∈𝒯𝐵0𝑇}𝐸^𝐵𝑄^𝐵_𝑧𝜎,𝜓^∗_𝑛+𝛿_𝑛(𝑧), (64) where 𝛿_𝑛(𝑧)=𝐶(𝐹₀(𝑧)+Δ₀(𝑧)+𝑧+1)𝑛^−1/4(ln𝑛)^3/4.

It is well known (see, for instance, [6]) that when payoffs depend only on the current stock price (a Markov case), 𝛿-optimal stopping times of Dynkin’s games can be obtained as first arrival times to domains where the payoff is 𝛿-close to the value of the game (as a function of the initial stock price). For path-dependent payoffs the situation is more complicated, and, in general, in order to construct 𝛿-optimal stopping times, we have to know the stochastic process of values of the games starting at each time 𝑡∈[0,𝑇] conditioned to the information up to 𝑡. It is not clear what kind of approximation of this process can provide some information about 𝛿-rational exercise times, and the convenient alternative method of their construction exhibited in Theorem 9 seems to be important for both the theory and applications. Moreover, this construction is effective and can be employed in practice since 𝜇^∗_𝑛 and 𝜈^∗_𝑛 are functions on sequences of 1’s and −1’s which can be computed (and stored in a computer) using the recursive formulas (23) even before the stock evolution begins. In order to compute 𝜆_𝐵^(𝑛), we have to watch the discounted stock price 𝑆̌^𝐵_𝑡(𝑧)=𝑒^−𝑟𝑡𝑆^𝐵_𝑡(𝑧) evolution of a real stock at moments 𝜃_𝑘^(𝑛) which are obtained recursively by 𝜃₀^(𝑛)=0 and 𝜃^(𝑛)_𝑘+1=inf𝑡>𝜃_𝑘^(𝑛)∶𝑆̌^𝐵_𝑡(𝑧)=𝑒^{±𝜅(𝑇/𝑛)1/2}𝑆̌^𝐵_{𝜃𝑘(𝑛)}(𝑧) (65) and to construct the {1,−1} sequence 𝜆_𝐵^(𝑛)(𝜔) by writing 1 or −1 on 𝑘th place depending on whether 𝑆̌^𝐵_{𝜃𝑘(𝑛)}(𝑧)=𝑒^{𝜅(𝑇/𝑛)1/2}𝑆̌^𝐵_{𝜃(𝑛)𝑘−1}(𝑧) or 𝑆̌^𝐵_{𝜃𝑘(𝑛)}(𝑧)=𝑒^{−𝜅(𝑇/𝑛)1/2}𝑆̌^𝐵_{𝜃(𝑛)𝑘−1}(𝑧), respectively.

Recall (see [56]) that a sequence 𝜋=(𝜋₁,…,𝜋_𝑛) of pairs 𝜋_𝑘=(𝛽_𝑘,𝛾_𝑘) of ℱ^𝜉_𝑘−1-measurable random variables 𝛽_𝑘,𝛾_𝑘,𝑘=1,…,𝑛, is called a self-financing portfolio strategy in the CRR market determined by (18), (20), (36), and (43) if the price of the portfolio at time 𝑘 is given by the formula 𝑊_𝑘^𝜋,𝑛=𝛽_𝑘𝑏_𝑘+𝛾_𝑘𝑆^(𝑛)_{𝑘𝑇/𝑛}(𝑧)=𝛽_𝑘+1𝑏_𝑘+𝛾_𝑘+1𝑆^(𝑛)_{𝑘𝑇/𝑛}(𝑧), (66) and the latter equality means that all changes in the portfolio value are due to capital gains and losses but not due to withdrawal or infusion of funds. A pair (𝜁,𝜋) of a stopping time 𝜁∈𝒯^𝜉_0𝑛 and a self-financing portfolio strategy 𝜋 is called a hedge for (against) the game option with the payoff 𝑅_𝑧^(𝑛) given by (46) if (see [12]) 𝑊^𝜋,𝑛_𝜁∧𝑘≥𝐻_𝑧^(𝑛)𝜁𝑇𝑛,𝑘𝑇𝑛,∀𝑘=0,1,…,𝑛. (67) It follows from [12] that for any 𝜁∈𝒯^𝜉_0𝑛 there exists a self-financing portfolio strategy 𝜋^𝜁 so that (𝜁,𝜋^𝜁) is a hedge. In particular, if we take the rational exercise time 𝜁=𝜁^∗_𝑛 of the writer, then such 𝜋^𝜁 exists with the initial portfolio capital 𝑉^(𝑛)(𝑧). The construction of 𝜋^𝜁 goes directly via the Doob-Meyer decomposition of supermartingales and a martingale representation lemma (see [12, 56]), both being explicit in the CRR market case. In the continuous time BS market we cannot write the corresponding portfolio strategies in an explicit way, and so some approximations are necessary.

Theorem 10. Let 𝜁∈𝒯^𝜉_0𝑛, 𝜋=𝜋^𝜁, and (66) together with (67) hold true with ℱ^𝜉_𝑘-measurable 𝛽_𝑘=𝛽^𝜁_𝑘 and 𝛾_𝑘=𝛾^𝜉_𝑘, so that (𝜁,𝜋^𝜁) is a hedge. Then 𝛽^𝜁_𝑘=𝑓_𝑘∘𝜆_𝜉^(𝑘−1), 𝛾^𝜁_𝑘=𝑔_𝑘∘𝜆_𝜉^(𝑘−1), and 𝜁=𝜇∘𝜆_𝜉^(𝑛) for some uniquely defined functions 𝑓_𝑘, 𝑔_𝑘 on {−1,1}^𝑘−1 and some 𝜇∈𝒥_0𝑛. Let 𝜑=𝜇∘𝜆_𝐵^(𝑛) and set 𝛽^𝜑_𝑡=𝑓_𝑘∘𝜆_𝐵^(𝑘−1) and 𝛾^𝜑_𝑡=𝑔_𝑘∘𝜆_𝐵^(𝑘−1) whenever 𝑡∈(𝜃^(𝑛)_𝑘−1,𝜃_𝑘^(𝑛)]. Then 𝑊^𝐵_𝑡=𝛽^𝜑_𝑡𝑏_𝑡+𝛾^𝜑_𝑡𝑆^𝐵_𝑡(𝑧) (68) is a self-financing portfolio in the BS market and there exists a constant 𝐶>0 such that 𝐸^𝐵sup_{0≤𝑡≤𝑇}𝐻^𝐵_𝑧𝜃_𝜑^(𝑛),𝑡−𝑊^𝐵_{𝜃𝜑(𝑛)∧𝑡}⁺≤𝐶𝐹₀(𝑧)+Δ₀(𝑧)+𝑧+1𝑛^−1/4(ln𝑛)^3/4, (69) where 𝑎⁺=max(𝑎,0). In particular, there exists a self-financing portfolio of this form satisfying (69) with the initial value 𝑉^(𝑛)(𝑧) (which according to (49) is close to the fair price 𝑉(𝑧) of the game option) taking 𝜑^∗=𝜇^∗∘𝜆_𝐵^(𝑛) if 𝜁^∗=𝜇^∗∘𝜆_𝜉^(𝑛) is the rational exercise time and 𝜋=𝜋^𝜁∗ is the corresponding optimal self-financing hedging portfolio strategy for the CRR market.

The inequality (69) estimates the expectation of the maximal shortfall (risk) of certain (nearly hedging) portfolio strategy which can be constructed effectively in applications since the functions 𝑓_𝑙, 𝑔_𝑙, and 𝜇 are determined by a self-financing hedging strategy in the CRR market which can be computed directly and stored in a computer even before the real stock evolution begins or in case of computer memory limitations we can compute these functions each time when needed using corresponding algorithms for the CRR market. The functions 𝜆_𝐵^(𝑛) or, in other words, the sequences from {−1,1}^𝑛 which should be plugged into the functions 𝑓_𝑙, 𝑔_𝑙, and 𝜇 should be obtained in practice by watching the evolution of the discounted stock price 𝑒^−𝑟𝑡𝑆^𝐵_𝑡 at moments 𝜃_𝑘^(𝑛) as described after Theorem 9.

The paper [96] studied approximations of the shortfall risk 𝑅(𝑥) given by (24) for game options in the BS market by the shortfall risks 𝑅_𝑛(𝑥) of game options in the sequence of CRR markets defined above where the initial capital 𝑥 of all portfolios under consideration is kept the same and the payoffs satisfy the same conditions as above. The convergence lim_𝑛→∞𝑅_𝑛(𝑥)=𝑅(𝑥) was proved in [96] but only the one sided error estimate 𝑅(𝑥)≤𝑅_𝑛(𝑥)+𝐶𝑛^−1/4(ln𝑛)^3/4 (70) was obtained there for game options. On the other hand, relying on some convexity arguments, it was possible to obtain for American options two-sided estimates with the same error term.

In [97] similar approximation results as above were extended to barrier game options. Namely, [97] deals with double knock-out barrier option with two constant barriers 𝐿,𝑅 such that 0≤𝐿<𝑆₀<𝑅≤∞ which means that the option becomes worthless to its holder (buyer) at the first time 𝜏_𝐼 the stock price 𝑆_𝑡 exits the open interval 𝐼=(𝐿,𝑅). Thus for 𝑡≥𝜏_(𝐿,𝑅) the payoff is 𝑋_𝑡=𝑌_𝑡=0. For 𝑡<𝜏_(𝐿,𝑅) path-dependent payoffs satisfying (38) and (39) are considered. Such a contract is of potential value to a buyer who believes that the stock price will not exit the interval 𝐼 up to a maturity date and to a seller who does not want to worry about hedging if the stock price will reach one of the barriers 𝐿,𝑅. Such an option is equivalent to the usual game option when the payoffs 𝑋_𝑡 and 𝑌_𝑡 are replaced by 𝑋^𝐼_𝑡=𝑋_𝑡𝕀_{𝑡<𝜏𝐼} and 𝑌^𝐼_𝑡=𝑌_𝑡𝕀_{𝑡<𝜏𝐼}, respectively. Now, these new payoffs lose regularity conditions (38) and (39), but still it turns out that the error estimates in (49) remain true when we approximate the price of the above barrier game options in the BS market by the prices of corresponding barrier game options in the CRR markets as in Theorem 8 above. The results concerning approximation of the shortfall risk turn out to be very similar for barrier game options to the corresponding results for usual game options described above.

When payoffs depend only on the current stock price (and not path-dependent as in (45) and (46)) then in some special cases it is possible to obtain better error estimates for binomial approximations of prices of game options relying on partial differential equations methods in the free boundary problem. In [98] this was done for American put options in the BS market, and in [99] this was extended to game put options with error estimates of order 𝑛^−1/2 in comparison to 𝑛^−1/4(ln𝑛)^3/4 obtained in Theorem 8.

5. Incomplete Markets and Transaction Costs

Both in incomplete markets and in markets with transaction costs there is no one arbitrage free price of each derivative which can be considered as its fair price, and one of approaches in these circumstances is to study superhedging. Game options in incomplete markets were studied in several papers; in particular, in [100] they were studied from the point of view of utility maximization which leads to non-zero-sum Dynkin's games while in [101] they were studied from the point of view of superhedging and arbitrage free prices.

Next, we concentrate in this section on superhedging pricing of game options in discrete markets with transaction costs. The market model here will consist of a finite probability space Ω with the 𝜎-field ℱ=2^Ω of all subspaces of Ω and a probability measure ℙ on ℱ giving a positive weight ℙ(𝜔) to each 𝜔∈Ω. The setup includes also a filtration {∅,Ω}=ℱ₀⊂ℱ₁⊂⋯⊂ℱ_𝑇=ℱ where 𝑇 is a positive integer called the time horizon. It is convenient to denote by Ω_𝑡 the set of atoms in ℱ_𝑡 so that any ℱ_𝑡-measurable random variable (vector) 𝑍 can be identified with a function (vector function) defined on Ω_𝑡 and its value at 𝜇∈Ω_𝑡 will be denoted either by 𝑍(𝜇) or by 𝑍^𝜇.

The market model consists of a risk-free bond and a risky stock. Without loss of generality, we can assume that all prices are discounted so that the bond price equals 1 all the time, and a position in bonds is identified with cash holding. On the other hand, the shares of the stock can be traded which involves proportional transaction costs. This will be represented by bid-ask spreads; that is, shares can be bought at an ask price 𝑆^𝑎_𝑡 or sold at the bid price 𝑆^𝑏_𝑡, where 𝑆^𝑠_𝑡≥𝑆^𝑏_𝑡>0,𝑡=0,1,…,𝑇 are processes adapted to the filtration {ℱ_𝑡}^𝑇_𝑡=𝑜.

The liquidation value at time 𝑡 of a portfolio (𝛾,𝛿) consisting of an amount 𝛾 of cash (or bond) and 𝛿 shares of the stock equals 𝜃_𝑡𝛾,𝛿=𝛾+𝑆^𝑏_𝑡𝛿⁺−𝑆^𝑎_𝑡𝛿⁻ (71) which in case 𝛿<0 means that a portfolio owner should spend the amount 𝑆^𝑎_𝑡𝛿⁻ in order to close his short position. Observe that fractional numbers of shares are allowed here so that both 𝛾 and 𝛿 in a portfolio (𝛾,𝛿) could be, in principle, any real numbers. By definition, a self-financing portfolio strategy is a predictable process (𝛼_𝑡,𝛽_𝑡) representing positions in cash (or bonds) and stock at time 𝑡,𝑡=0,1,…,𝑇 such that 𝜃_𝑡𝛼_𝑡−𝛼_𝑡+1,𝛽_𝑡−𝛽_𝑡+1≥0∀𝑡=0,1,…,𝑇−1, (72) and the set of all such portfolio strategies will be denoted by Φ.

As before, we consider here a game option which is a contract between its seller and buyer such that both have the right to exercise it at any time up to a maturity date (horizon) 𝑇. In the presence of transaction costs there is a difference whether we stipulate that the option to be settled in cash or both in cash and shares of stock while in the former case an assumption concerning transaction costs in the process of portfolio liquidation should be made. We adopt here the setup where the payments 𝑋_𝑡 and 𝑌_𝑡 are made both in cash and shares of the stock, and transaction costs take place always when a portfolio adjustment occurs. Thus, the payments are, in fact, adapted random 2-vectors 𝑋_𝑡=(𝑋_𝑡⁽¹⁾,𝑋_𝑡⁽²⁾) and 𝑌_𝑡=(𝑌_𝑡⁽¹⁾,𝑌_𝑡⁽²⁾) where the first and the second coordinates represent, respectively, a cash amount to be paid and a number of stock shares to be delivered, and as we allow also fractional numbers of shares, both coordinates can take on any nonnegative real value. The inequality 𝑋_𝑡≥𝑌_𝑡 in the zero transaction costs case is replaced in our present setup by Δ_𝑡=𝜃_𝑡𝑋_𝑡⁽¹⁾−𝑌_𝑡⁽¹⁾,𝑋_𝑡⁽²⁾−𝑌_𝑡⁽²⁾≥0, (73) and Δ_𝑡 is interpreted as a cancellation penalty. We impose also a natural assumption that 𝑋_𝑇⁽¹⁾=𝑌_𝑇⁽¹⁾ and 𝑋_𝑇⁽²⁾=𝑌_𝑇⁽²⁾; that is, on the maturity date there is no penalty. Therefore, if the seller cancels the contract at time 𝑠 while the buyer exercises at time 𝑡, the former delivers to the latter a package of cash and stock shares which can be represented as a 2-vector in the form 𝐻(𝑠,𝑡)=𝐻⁽¹⁾(𝑠,𝑡),𝐻⁽²⁾(𝑠,𝑡)=𝑋_𝑠𝕀_𝑠<𝑡+𝑌_𝑡𝕀_𝑡≤𝑠, (74) where 𝕀_𝐴=1 if an event 𝐴 occurs and 𝕀_𝐴=0 if not. It will be convenient to allow the payment components 𝑋_𝑡⁽¹⁾,𝑋_𝑡⁽²⁾ and 𝑌_𝑡⁽¹⁾,𝑌_𝑡⁽²⁾ to take on any real (and not only nonnegative) values which will enable us to demonstrate complete duality (symmetry) between the seller's and the buyer's positions.

A pair (𝜎,𝜋) of a stopping time 𝜎≤𝑇 and of a self-financing strategy 𝜋=(𝛼_𝑡,𝛽_𝑡)^𝑇_𝑡=0 will be called a superhedging strategy for the seller of the game option with a payoff given by (74) if for all 𝑡≤𝑇, 𝜃_𝜎∧𝑡𝛼_𝜎∧𝑡−𝐻⁽¹⁾(𝜎,𝑡),𝛽_𝜎∧𝑡−𝐻⁽²⁾(𝜎,𝑡)≥0, (75) where, as usual, 𝑐∧𝑑=min(𝑐,𝑑) and 𝑐∨𝑑=max(𝑐,𝑑). The seller's (ask or upper hedging) price 𝑉^𝑎 of a game option is defined as the infimum of initial amounts required to start a superhedging strategy for the seller. Since in order to get 𝛼₀ amount of cash and 𝛽₀ shares of stock at time 0, the seller should spend −𝜃₀−𝛼₀,−𝛽₀=𝛼₀+𝛽⁺₀𝑆^𝑎₀−𝛽⁻₀𝑆^𝑏₀ (76) in cash, we can write 𝑉^𝑎=inf_𝜎,𝜋−𝜃₀−𝛼₀,−𝛽₀∶()𝛼𝜎,𝜋with𝜋=_𝑡,𝛽_𝑡^𝑇_𝑡=0.beingasuperhedgingstrategyfortheseller (77)

On the other hand, the buyer may borrow from a bank an amount 𝜃₀(−𝛼₀,−𝛽₀) to purchase a game option with the payoff (74) and begining with the negative valued portfolio (𝛼₀,𝛽₀) to manage a self-financing strategy 𝜋=(𝛼_𝑡,𝛽_𝑡)^𝑇_𝑡=0 so that for a given stopping time 𝜏≤𝑇 and all 𝑠≤𝑇, 𝜃_𝑠∧𝜏𝛼_𝑠∧𝜏+𝐻⁽¹⁾(𝑠,𝜏),𝛽_𝑠∧𝜏+𝐻⁽²⁾(𝑠,𝜏)≥0. (78) In this case the pair (𝜏,𝜋) will be called a superhedging strategy for the buyer. The buyer’s (bid or lower hedging) price 𝑉^𝑏 of the game option above is defined as the supremum of initial bank loan required to purchase this game option and to manage a superhedging strategy for the buyer. Thus, 𝑉^𝑏=sup_𝜏,𝜋𝜃₀−𝛼₀,−𝛽₀∶()𝛼𝜏,𝜋with𝜋=_𝑡,𝛽_𝑡^𝑇_𝑡=0.beingasuperhedgingstrategyforthebuyer (79) It follows from the representations of Theorem 11 that 𝑉^𝑎≥𝑉^𝑏.

First, we recall the notion of a randomized stopping time (see [102–104] and references there) which is defined as a nonnegative adapted process 𝜒 such that ∑^𝑇_𝑡=0𝜒_𝑡=1. The set of all randomized stopping times will be denoted by 𝒳 while the set of all usual or pure stopping times will be denoted by 𝒯. It will be convenient to identify each pure stopping time 𝜏 with a randomized stopping time 𝜒^𝜏 such that 𝜒^𝜏_𝑡=𝕀_{𝜏=𝑡} for any 𝑡=0,1,…,𝑇, so that we could write 𝒯⊂𝒳. For any adapted process 𝑍 and each randomized stopping time 𝜒 the time-𝜒 value of 𝑍 is defined by (𝑍)_𝜒=𝑍_𝜒=^𝑇_𝑡=0𝜒_𝑡𝑍_𝑡. (80)

Considering a game option with a payoff given by (74) we write also 𝐻𝜒,𝜒=^𝑇_𝑠,𝑡=0𝜒_𝑠𝜒_𝑡𝐻(𝑠,𝑡) (81) which is the seller’s payment to the buyer when the former cancels and the latter exercises at randomized stopping times 𝜒 and 𝜒, respectively. In particular, if 𝜎 and 𝜏 are pure stopping times, then 𝐻𝜒,𝜒^𝜏=^𝑇_𝑠=0𝜒_𝑠𝐻(),𝐻𝜒𝑠,𝜏_𝜎,𝜒=^𝑇_𝑡=0𝜒_𝑡𝐻(𝜎,𝑡). (82) Next, we introduce the notion of an approximate martingale which is defined for any randomized stopping time 𝜒 as a pair (𝑃,𝑆) of a probability measure 𝑃 on Ω and of an adapted process 𝑆 such that for each 𝑡=0,1,…,𝑇, 𝑆^𝑏_𝑡≤𝑆_𝑡≤𝑆^𝑎_𝑡,𝜒^∗_𝑡+1𝑆^𝑏_𝑡≤𝔼_𝑃𝑆^𝜒∗_𝑡+1∣ℱ_𝑡≤𝜒^∗_𝑡+1𝑆^𝑎_𝑡, (83) where 𝔼_𝑃 is the expectation with respect to 𝑃, 𝜒^∗_𝑡=^𝑇_𝑠=𝑡𝜒_𝑠,𝑍^𝜒∗_𝑡=^𝑇_𝑠=𝑡𝜒_𝑠𝑍_𝑠,𝜒^∗_𝑇+1=0,𝑍^𝜒∗_𝑇+1=0. (84) Given a randomized stopping time 𝜒 the space of corresponding approximate martingales (𝑃,𝑆) will be denoted by 𝒫(𝜒) and we denote by 𝒫(𝜒) the subspace of 𝒫(𝜒) consisting of pairs (𝑃,𝑆) with 𝑃 being equivalent to the original (market) probability ℙ.

Next, we introduce some convex analysis notions and notations (see [104, 105] for more details). Denote by Θ the family of functions 𝑓∶ℝ→ℝ∪{−∞} such that either 𝑓≡−∞ or 𝑓 is a (finite) real valued polyhedral (continuous piecewise linear with finite number of segments) function. If 𝑓, 𝑔∈Θ, then, clearly, 𝑓∧𝑔, 𝑓∨𝑔∈Θ. The epigraph of 𝑓∈Θ is defined by epi(𝑓)={(𝑥,𝑦)∈ℝ²∶𝑥≥𝑓(𝑦)}. For any 𝑐≥𝑑 the function ℎ_[𝑑,𝑐](𝑦)=𝑐𝑦⁻−𝑑𝑦⁺, clearly, belongs to Θ. Observe that the self-financing condition (72) can be rewritten in the form 𝛼_𝑡−𝛼_𝑡+1,𝛽_𝑡−𝛽_𝑡+1∈epiℎ_[𝑆^𝑏_𝑡,𝑆^𝑎_𝑡]. (85) For each 𝑓∈Θ and 𝑐≥𝑑 there exists a unique function gr_[𝑑,𝑐](𝑓)∈Θ such that epigr_[𝑑,𝑐]𝑓=epiℎ_[𝑑,𝑐]+epi𝑓. (86) For any 𝑦∈ℝ,𝜇∈Ω_𝑡, and 𝑡=0,1,…,𝑇 define 𝑞^𝑎_𝑡(𝑦)=𝑞^𝑎_𝑡(𝜇,𝑦), 𝑞^𝑏_𝑡(𝑦)=𝑞^𝑏_𝑡(𝜇,𝑦), 𝑟^𝑎_𝑡(𝑦)=𝑟^𝑎_𝑡(𝜇,𝑦), 𝑟^𝑏_𝑡(𝑦)=𝑟^𝑏_𝑡(𝜇,𝑦) by 𝑞^𝑎_𝑡𝑦=𝑋_𝑡⁽¹⁾+ℎ_[𝑆^𝑏_𝑡,𝑆^𝑎_𝑡]𝑦−𝑋_𝑡⁽²⁾,𝑟^𝑎_𝑡𝑦=𝑌_𝑡⁽¹⁾+ℎ_[𝑆^𝑏_𝑡,𝑆^𝑎_𝑡]𝑦−𝑌_𝑡⁽²⁾,𝑞^𝑏_𝑡𝑦=−𝑋_𝑡⁽¹⁾+ℎ_[𝑆^𝑏_𝑡,𝑆^𝑎_𝑡]𝑦+𝑋_𝑡⁽²⁾,𝑟^𝑏_𝑡𝑦=−𝑌_𝑡⁽¹⁾+ℎ_[𝑆^𝑏_𝑡,𝑆^𝑎_𝑡]𝑦+𝑌_𝑡⁽²⁾ (87) with ℎ_[𝑑,𝑐] the same as in (85) and (86). Observe that if 𝑐≥𝑑≥0, then either ℎ_[𝑑,𝑐]≡0 or ℎ_[𝑑,𝑐] is a monotone decreasing function, and so 𝑞^𝑎_𝑡≥𝑟^𝑎_𝑡,𝑞^𝑏_𝑡≤𝑟^𝑏_𝑡. (88) Introduce also 𝐺^𝑎_𝑠,𝑡𝑦=𝐻⁽¹⁾(𝑠,𝑡)+ℎ_[𝑆^𝑏_{𝑠∧𝑡,𝑆}^𝑎_𝑠∧𝑡]𝑦−𝐻⁽²⁾(𝑠,𝑡)=𝑞^𝑎_𝑠𝑦𝕀_𝑠<𝑡+𝑟^𝑎_𝑡𝑦𝕀_𝑡≤𝑠,𝐺^𝑏_𝑠,𝑡𝑦=−𝐻⁽¹⁾(𝑠,𝑡)+ℎ_[𝑆^𝑏_{𝑠∧𝑡,𝑆}^𝑎_𝑠∧𝑡]𝑦+𝐻⁽²⁾(𝑠,𝑡)=𝑞^𝑏_𝑠𝑦𝕀_𝑠<𝑡+𝑟^𝑏_𝑡𝑦𝕀_𝑡≤𝑠. (89) Clearly, the superhedging conditions (75) of the seller and (78) of the buyer are equivalent to 𝛼_𝜎∧𝑡,𝛽_𝜎∧𝑡∈epi𝐺^𝑎_𝜎,𝑡𝛼∀𝑡=0,1,…,𝑇,_𝑠∧𝜏,𝛽_𝑠∧𝜏∈epi𝐺^𝑏_𝑠,𝜏∀𝑠=0,1,…,𝑇, (90) respectively. Observe also that 𝑞^𝑎_𝑡(0)=−𝑞^𝑏_𝑡(0)=𝜃_𝑡𝑋_𝑡⁽¹⁾,𝑋_𝑡⁽²⁾,𝑟^𝑎_𝑡(0)=−𝑟^𝑏_𝑡(0)=𝜃_𝑡𝑌_𝑡⁽¹⁾,𝑌_𝑡⁽²⁾. (91) We recall that 𝑋_𝑇⁽¹⁾=𝑌_𝑇⁽¹⁾ and 𝑋_𝑇⁽²⁾=𝑌_𝑇⁽²⁾, and so 𝑞^𝑎_𝑇=𝑟^𝑎_𝑇 and 𝑞^𝑏_𝑇=𝑟^𝑏_𝑇. In [106] the following results were obtained.

Theorem 11. (I)  Price Representations. In the above notations, 𝑉^𝑎=min_𝜎∈𝒯max_𝜒∈𝒳max_{()∈𝑃,𝑆𝒫𝜒}𝔼_𝑃𝐻⁽¹⁾(𝜎,⋅)+𝑆𝐻⁽²⁾(𝜎,⋅)_𝜒=min_𝜎∈𝒯max_𝜒∈𝒳sup_{()𝜒𝑃,𝑆∈𝒫}𝔼_𝑃𝐻⁽¹⁾(𝜎,⋅)+𝑆𝐻⁽²⁾(𝜎,⋅)_𝜒,𝑉^𝑏=max_𝜏∈𝒯min_𝜒∈𝒳min_{()∈𝑃,𝑆𝒫𝜒}𝔼_𝑃𝐻⁽¹⁾(⋅,𝜏)+𝑆𝐻⁽²⁾(⋅,𝜏)_𝜒=max_𝜏∈𝒯min_𝜒∈𝒳inf_{()𝜒𝑃,𝑆∈𝒫}𝔼_𝑃𝐻⁽¹⁾(⋅,𝜏)+𝑆𝐻⁽²⁾(⋅,𝜏)_𝜒, (92)(93) where 𝐻⁽¹⁾(𝜎,⋅), 𝐻⁽²⁾(𝜎,⋅) and 𝐻⁽¹⁾(⋅,𝜏),𝐻⁽²⁾(⋅,𝜏) denote functions on {0,1,…,𝑇} whose values at 𝑡 are obtained by replacing ⋅ by 𝑡.
(II)  Recurrent Price Computations(i) For any 𝑥∈ℝ, 𝜇∈Ω_𝑇 and 𝜎∈𝒯 define        𝑧^𝜇_𝑇(𝑥)=𝑤^𝜇_𝑇(𝑥)=𝑟^𝑎_𝑇𝜇,𝑥. (94) Next, for 𝑡=1,2,…,𝑇 and each 𝜇∈Ω_𝑡−1 define by backward induction 𝐳^𝜇_𝑡−1=max_{𝜈⊂𝜇,𝜈∈Ω𝑡}𝑧^𝜈_𝑡,𝑤^𝜇_𝑡−1=gr_{[𝑆𝑏𝑡−1(𝜇),𝑆𝑎𝑡−1(𝜇)]}𝐳^𝜇_𝑡−1,𝑧^𝜇_𝑡−1(𝑥)=min𝑞^𝑎_𝑡−1𝜇,𝑥,max𝑟^𝑎_𝑡−1𝜇,𝑥,𝑤^𝜇_𝑡−1(𝑥). (95) Then 𝑧₀(0)=𝑉^𝑎.(ii) For any 𝑥∈ℝ, 𝜇∈Ω_𝑇 and 𝜏∈𝒯 define 𝑢^𝜇_𝑇(𝑥)=𝑣^𝜇_𝑇(𝑥)=𝑟^𝑏_𝑇𝜇,𝑥. (96) Next, for 𝑡=1,2,…,𝑇 and each 𝜇∈Ω_𝑡−1 define by the backward induction 𝐮^𝜇_𝑡−1=max_{𝜈⊂𝜇,𝜈∈Ω𝑡}𝑢^𝜈_𝑡,𝑣^𝜇_𝑡−1=gr_{[𝑆𝑏𝑡−1(𝜇),𝑆𝑎𝑡−1(𝜇)]}𝐮^𝜇_𝑡−1,𝑢^𝜇_𝑡−1(𝑥)=min𝑟^𝑏_𝑡−1𝜇,𝑥,max𝑞^𝑏_𝑡−1𝜇,𝑥,𝑣^𝜇_𝑡−1(𝑥). (97) Then 𝑢₀(0)=−𝑉^𝑏.
(III)  Superhedging Strategies(i) Construct by induction a sequence of (pure) stopping times 𝜎_𝑡∈𝒯 and a self-financing strategy (𝛼,𝛽) such that 𝛼_𝑡,𝛽_𝑡∈epi𝑧_𝑡⧵epi𝑞^𝑎_𝑡𝑜𝑛𝑡<𝜎_𝑡 (98) for each 𝑡=0,1,…,𝑇 in the following way. First, take any ℱ₀-measurable portfolio (𝛼₀,𝛽₀)∈epi(𝑧₀) and set 𝜎₀=0𝑖𝑓𝛼₀,𝛽₀∈epi𝑞^𝑎₀,𝑇𝑖𝑓𝛼₀,𝛽₀∉epi𝑞^𝑎₀. (99) Suppose that an ℱ_𝑡-measurable portfolio (𝛼_𝑡,𝛽_𝑡)∈epi(𝑧_𝑡) and a stopping time 𝜎_𝑡∈𝒯 have already been constructed for some 𝑡=0,1,…,𝑇−1 so that (98) holds true. By (86) and (96), 𝛼_𝑡,𝛽_𝑡∈epi𝑤_𝑡=epiℎ_[𝑆^𝑏_𝑡,𝑆^𝑎_𝑡]+epi𝐳_𝑡𝑜𝑛𝑡<𝜎_𝑡 (100) and so there exists an ℱ_𝑡-measurable portfolio (𝛼_𝑡+1,𝛽_𝑡+1) such that 𝛼_𝑡+1,𝛽_𝑡+1∈epi𝐳_𝑡,𝛼_𝑡−𝛼_𝑡+1,𝛽_𝑡−𝛽_𝑡+1∈epiℎ_[𝑆^𝑏_𝑡,𝑆^𝑎_𝑡]𝑜𝑛𝑡<𝜎_𝑡, (101) and (𝛼_𝑡+1,𝛽_𝑡+1)=(𝛼_𝑡,𝛽_𝑡) on {𝑡≥𝜎_𝑡} which provides the self-financing condition (85) both on {𝑡<𝜎_𝑡} and on {𝑡≥𝜎_𝑡}. By (96) it follows also that (𝛼_𝑡+1,𝛽_𝑡+1)∈epi(𝑧_𝑡+1) on {𝑡<𝜎_𝑡}⊃{𝑡+1<𝜎_𝑡+1}. Set 𝜎_𝑡+1=𝜎_𝑡𝑖𝑓𝑡≥𝜎_𝑡,𝑡+1𝑖𝑓𝑡<𝜎_𝑡𝑎𝑛𝑑𝛼_𝑡+1,𝛽_𝑡+1∈epi𝑞^𝑎_𝑡+1,𝑇𝑖𝑓𝑡<𝜎_𝑡𝑎𝑛𝑑𝛼_𝑡+1,𝛽_𝑡+1∉epi𝑞^𝑎_𝑡+1. (102) Finally, set 𝜎=𝜎_𝑇∈𝒯. Then the pair (𝜎,𝜋) with 𝜋=(𝛼,𝛽) constructed by the above algorithm with (𝛼₀,𝛽₀)=(𝑉^𝑎,0) is a superhedging strategy for the seller.(ii) Construct by induction a sequence of (pure) stopping times 𝜏_𝑡∈𝒯 and a self-financing strategy (𝛼,𝛽) such that 𝛼_𝑡,𝛽_𝑡∈epi𝑢_𝑡⧵epi𝑟^𝑏_𝑡𝑜𝑛𝑡<𝜏_𝑡 (103) for each 𝑡=0,1,…,𝑇 in the following way. First, take any ℱ₀-measurable portfolio (𝛼₀,𝛽₀)∈epi(𝑢₀) and set 𝜏₀=0𝑖𝑓𝛼₀,𝛽₀∈epi𝑟^𝑏₀,𝑇𝑖𝑓𝛼₀,𝛽₀∉epi𝑟^𝑏₀. (104) Suppose that an ℱ_𝑡-measurable portfolio (𝛼_𝑡,𝛽_𝑡)∈epi(𝑢_𝑡) and a stopping time 𝜏_𝑡∈𝒯 have already been constructed for some 𝑡=0,1,…,𝑇−1 so that (98) holds true. By (86) and (98), 𝛼_𝑡,𝛽_𝑡∈epi𝑣_𝑡=epiℎ_[𝑆^𝑏_𝑡,𝑆^𝑎_𝑡]+epi𝐮_𝑡𝑜𝑛𝑡<𝜏_𝑡, (105) and so there exists an ℱ_𝑡-measurable portfolio (𝛼_𝑡+1,𝛽_𝑡+1) such that 𝛼_𝑡+1,𝛽_𝑡+1∈epi𝐮_𝑡,𝛼_𝑡−𝛼_𝑡+1,𝛽_𝑡−𝛽_𝑡+1∈epiℎ_[𝑆^𝑏_𝑡,𝑆^𝑎_𝑡]𝑜𝑛𝑡<𝜏_𝑡, (106) and (𝛼_𝑡+1,𝛽_𝑡+1)=(𝛼_𝑡,𝛽_𝑡) on {𝑡≥𝜏_𝑡} which provides the self-financing condition (85) both on {𝑡<𝜏_𝑡} and on {𝑡≥𝜏_𝑡}. By (98) it follows also that (𝛼_𝑡+1,𝛽_𝑡+1)∈epi(𝑢_𝑡+1) on {𝑡<𝜏_𝑡}⊃{𝑡+1<𝜏_𝑡+1}. Set 𝜏_𝑡+1=𝜏_𝑡𝑖𝑓𝑡≥𝜏_𝑡,𝑡+1𝑖𝑓𝑡<𝜏_𝑡𝑎𝑛𝑑𝛼_𝑡+1,𝛽_𝑡+1∈epi𝑟^𝑏_𝑡+1,𝑇𝑖𝑓𝑡<𝜏_𝑡𝑎𝑛𝑑𝛼_𝑡+1,𝛽_𝑡+1∉epi𝑟^𝑏_𝑡+1. (107) Finally, set 𝜏=𝜏_𝑇∈𝒯. Then the pair (𝜏,𝜋) with 𝜋=(𝛼,𝛽) constructed by the above algorithm with (𝛼₀,𝛽₀)=(−𝑉^𝑏,0) is a superhedging strategy for the buyer.

There are by now very few papers on game options with transaction costs. In [107] it is shown that the cheapest superhedging strategy for a game option in a Black-Scholes market with transaction costs is the buy-and-hold portfolio strategy together with a hitting time of a Borel set. The shortfall risk for a game option in a Black-Scholes market with transaction costs is obtained in [108] as a limit of corresponding expressions for a sequence of binomial models in the spirit of Section 4.

Acknowledgment

This work was partially supported by the ISF Grant no. 82/10.