Correction. Error estimates for binomial approximations of game options

Dynkin Games and Israeli Options

We start briefly surveying research on optimal stopping games since their introduction by E.B.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.

On strong binomial approximation for stochastic processes and applications for financial modelling

This paper considers binomial approximation of continuous time stochastic processes. It is shown that, under some mild integrability conditions, a process can be approximated in mean square sense and in other strong metrics by binomial processes, i.e., by processes with fixed size binary increments at sampling points. Moreover, this approximation can be causal, i.e., at every time it requires only past historical values of the underlying process. In addition, possibility of approximation of solutions of stochastic differential equations by solutions of ordinary equations with binary noise is e…

Hedging of Game Options under Model Uncertainty in Discrete Time

We introduce a setup of model uncertainty in discrete time. In this setup we derive dual expressions for the super--replication prices of game options with upper semicontinuous payoffs. We show that the super--replication price is equal to the supremum over a special (non dominated) set of martingale measures, of the corresponding Dynkin games values. This type of results is also new for American options.

The Continuous Time Nonzero-sum Dynkin Game Problem and Application in Game Options

In this paper we study the nonzero-sum Dynkin game in continuous time which is a two player non-cooperative game on stopping times. We show that it has a Nash equilibrium point for general stochastic processes. As an application, we consider the problem of pricing American game contingent claims by the utility maximization approach.

Rate of convergence for discrete approximation of option prices

In this article, we study the rate of convergence of prices when a model is approximated by some simplified model. We also provide a method how explicit error formula for more general options can be obtained if such formula is available for digital option prices. We illustrate our results by considering convergence of binomial prices to Black-Scholes prices. We also consider smooth convergence in which the approximation does not oscillate for general class of payoff functions.

Recombining tree approximations for Game Options in Local Volatility models

In this paper we introduce a numerical method for optimal stopping in the framework of one dimensional diffusion. We use the Skorokhod embedding in order to construct recombining tree approximations for diffusions with general coefficients. This technique allows us to determine convergence rates and construct nearly optimal stopping times which are optimal at the same rate. Finally, we demonstrate the efficiency of our scheme with several examples of game options.

Game options with gradual exercise and cancellation under proportional transaction costs

Game (Israeli) options in a multi-asset market model with proportional transaction costs are studied in the case when the buyer is allowed to exercise the option and the seller has the right to cancel the option gradually at a mixed (or randomised) stopping time, rather than instantly at an ordinary stopping time. Allowing gradual exercise and cancellation leads to increased flexibility in hedging, and hence tighter bounds on the option price as compared to the case of instantaneous exercise and cancellation. Algorithmic constructions for the bid and ask prices, and the associated superhedging…

Game options in an imperfect market with default

We study pricing and superhedging strategies for game options in an imperfect market with default. We extend the results obtained by Kifer in \cite{Kifer} in the case of a perfect market model to the case of an imperfect market with default, when the imperfections are taken into account via the nonlinearity of the wealth dynamics. We introduce the {\em seller's price} of the game option as the infimum of the initial wealths which allow the seller to be superhedged. We {prove} that this price coincides with the value function of an associated {\em generalized} Dynkin game, recently introduced i…

Error estimates for discrete approximations of game options with multivariate diffusion asset prices

We obtain error estimates for strong approximations of a diffusion with a diffusion matrix $σ$ and a drift b by the discrete time process defined recursively X_N((n+1)/N) = X_N(n/N)+N^{1/2}σ(X_N(n/N))ξ(n+1)+N^{-1}b(XN(n/N)); where ξ(n); n\geq 1 are i.i.d. random vectors, and apply this in order to approximate the fair price of a game option with a diffusion asset price evolution by values of Dynkin's games with payoffs based on the above discrete time processes. This provides an effective tool for computations of fair prices of game options with path dependent payoffs in a multi asset market w…

Second-order BSDEs with general reflection and game options under uncertainty

The aim of this paper is twofold. First, we extend the results of [33] concerning the existence and uniqueness of second-order reflected 2BSDEs to the case of two obstacles. Under some regularity assumptions on one of the barriers, similar to the ones in [10], and when the two barriers are completely separated, we provide a complete wellposedness theory for doubly reflected second-order BSDEs. We also show that these objects are related to non-standard optimal stopping games, thus generalizing the connection between DRBSDEs and Dynkin games first proved by Cvitanic and Karatzas [11]. More prec…

熵正则化BSDE方法用于百慕大期权与博弈

An Entropy Regularized BSDE Approach to Bermudan Options and Games

本文通过熵正则化反射倒向随机微分方程研究百慕大期权和博弈期权的最优停止问题，提出基于策略改进的强化学习算法并证明其收敛性，同时给出神经网络数值求解方案。

具有购买信息选项的平均场博弈

Mean field games with option to buy information

研究连续时间有限时域平均场博弈，其中代表性玩家的目标函数依赖于隐藏状态，玩家可选择付费观察该状态，通过连接最优控制与自由停止模型给出解的表征和显式示例。

Corrigendum for "Second-order reflected backward stochastic differential equations" and "Second-order BSDEs with general reflection and game options under uncertainty"

The aim of this short note is to fill in a gap in our earlier paper [16] on 2BSDEs with reflections, and to explain how to correct the subsequent results in the second paper [15]. We also provide more insight on the properties of 2RBSDEs, in the light of the recent contributions [13, 23] in the so--called $G-$framework.

Tug-of-war, market manipulation and option pricing

We develop an option pricing model based on a tug-of-war game. This two-player zero-sum stochastic differential game is formulated in the context of a multi-dimensional financial market. The issuer and the holder try to manipulate asset price processes in order to minimize and maximize the expected discounted reward. We prove that the game has a value and that the value function is the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the non-linear and completely degenerate infinity Laplace operator.

Game-theoretic derivation of upper hedging prices of multivariate contingent claims and submodularity

We investigate upper and lower hedging prices of multivariate contingent claims from the viewpoint of game-theoretic probability and submodularity. By considering a game between "Market" and "Investor" in discrete time, the pricing problem is reduced to a backward induction of an optimization over simplexes. For European options with payoff functions satisfying a combinatorial property called submodularity or supermodularity, this optimization is solved in closed form by using the Lovász extension and the upper and lower hedging prices can be calculated efficiently. This class includes the opt…

Stopper-Controller Games embedded in Single-Player Control Problems

In 2002, Benjamin Jourdain and Claude Martini discovered that for a class of payoff functions, the pricing problem for American options can be reduced to pricing of European options for an appropriately associated payoff, all within a Black-Scholes framework. This discovery has been investigated in great detail by Sören Christensen, Jan Kallsen and Matthias Lenga in a recent work in 2020. In the present work we prove that this phenomenon can be observed in a wider context, and even holds true in a setup of non-linear stochastic processes. We analyse this problem from both probabilistic and ana…

Error Estimates for Multinomial Approximations of American Options in Merton's Model

We derive error estimates for multinomial approximations of American options in a multidimensional jump--diffusion Merton's model. We assume that the payoffs are Markovian and satisfy Lipschitz type conditions. Error estimates for such type of approximations were not obtained before. Our main tool is the strong approximations theorems for i.i.d. random vectors which were obtained [14]. For the multidimensional Black--Scholes model our results can be extended also to a general path dependent payoffs which satisfy Lipschitz type conditions. For the case of multinomial approximations of American …

Nash equilibria for game contingent claims with utility-based hedging

Game contingent claims (GCCs) generalize American contingent claims by allowing the writer to recall the option as long as it is not exercised, at the price of paying some penalty. In incomplete markets, an appealing approach is to analyze GCCs like their European and American counterparts by solving option holder's and writer's optimal investment problems in the underlying securities. By this, partial hedging opportunities are taken into account. We extend results in the literature by solving the stochastic game corresponding to GCCs with both continuous time stopping and trading. Namely, we …