Yan Dolinsky

Robust Hedging with Proportional Transaction Costs

Duality for robust hedging with proportional transaction costs of path dependent European options is obtained in a discrete time financial market with one risky asset. Investor’s portfolio consists of a dynamically traded stock and a static position in vanilla options which can be exercised at maturity. Only stock trading is subject to proportional transaction costs. The main theorem is duality between hedging and a Monge-Kantorovich type optimization problem. In this dual transport problem the optimization is over all the probability measures which satisfy an approximate martingale condition related to consistent price systems in addition to the usual marginal constraints.

Martingale Optimal Transport in the Skorokhod Space

The dual representation of the martingale optimal transport problem in the Skorokhod space of multi dimensional cadlag processes is proved. The dual is a minimization problem with constraints involving stochastic integrals and is similar to the Kantorovich dual of the standard optimal transport problem. The constraints are required to hold for every path in the Skorokhod space. This problem has the financial interpretation as the robust hedging of path dependent European options.

Hedging with risk for game options in discrete time

We study the problems of efficient hedging of game (Israeli) options when the initial capital in the portfolio is less than the fair option price. In this case a perfect hedging is impossible and one can only try to minimise the risk (which can be defined in different ways) of having not enough funds in the portfolio to pay the required amount at the excercise time. We solve the minimization problems and find via dynamical programming appropriate efficient hedging strategies for discrete time game options in multinomial markets. The approach and some of the results are new also for standard American options.

Error estimates for binomial approximations of game options.

This note deals with the substantial inaccuracies in Lemmas 3.4 and 3.5 [more specifically, inequalities (4.48) and (4.53) of their proofs] and in Theorems 2.2 and 2.3 of [1] related to the important point that if a game option is not exercised or canceled before the expiration (horizon) time then the seller pays no penalty to the buyer, which is natural but does not agree well with the direct extension of payoff formulas beyond the horizon. The arguments in [1] do not require any modification if penalties in the corresponding game options are extended by zero beyond the horizon which, in view of the Lipschitz-type condition (2.2) there, would be a somewhat restrictive requirement since it eliminates the case of a constant (nonzero) penalty. Of course, there is no problem with the argument there if we consider just the American options case. We will deal with Theorems 2.2 and 2.3 later on in this note (warning the reader that our first correction of the proof there contains inaccuracies) and we start with showing that the estimate of Theorem 2.1 remains true if in place of Lemmas 3.4–3.6 we employ the following argument which extends the idea of Lemma 3.6 there. In the notations, the value of a game option in the Black–Scholes market is given by

Hedging of game options with the presence of transaction costs

We study the problem of super-replication for game options under proportional transaction costs. We consider a multidimensional continuous time model, in which the discounted stock price process satisfies the conditional full support property. We show that the super-replication price is the cheapest cost of a trivial super-replication strategy. This result is an extension of previous papers (see [3] and [7]) which considered only European options. In these papers the authors showed that with the presence of proportional transaction costs the super--replication price of a European option is given in terms of the concave envelope of the payoff function. In the present work we prove that for game options the super-replication price is given by a game variant analog of the standard concave envelope term. The treatment of game options is more complicated and requires additional tools. We combine the theory of consistent price systems together with the theory of extended weak convergence which was developed in [1]. The second theory is essential in dealing with hedging which involves stopping times, like in the case of game options.

Approximating stochastic volatility by recombinant trees

A general method to construct recombinant tree approximations for stochastic volatility models is developed and applied to the Heston model for stock price dynamics. In this application, the resulting approximation is a four tuple Markov process. The first two components are related to the stock and volatility processes and take values in a two dimensional Binomial tree. The other two components of the Markov process are the increments of random walks with simple values in \{−1, +1\}. The resulting efficient option pricing equations are numerically implemented for general American and European options including the standard put and calls, barrier, lookback and Asian type pay-offs. The weak and extended weak convergences are also proved.

Binomial Approximations for Barrier Options of Israeli Style

We show that prices and shortfall risks of game (Israeli) barrier options in a sequence of binomial approximations of the Black--Scholes (BS) market converge to the corresponding quantities for similar game barrier options in the BS market with path dependent payoffs and the speed of convergence is estimated, as well. The results are new also for usual American style options and they are interesting from the computational point of view, as well, since in binomial markets these quantities can be obtained via dynamical programming algorithms. The paper continues the study of [11]and [7] but requires substantial additional arguments in view of pecularities of barrier options which, in particular, destroy the regularity of payoffs needed in the above papers.

Binomial approximations of shortfall risk for game options

We show that the shortfall risk of binomial approximations of game (Israeli) options converges to the shortfall risk in the corresponding Black--Scholes market considering Lipschitz continuous path-dependent payoffs for both discrete- and continuous-time cases. These results are new also for usual American style options. The paper continues and extends the study of Kifer [Ann. Appl. Probab. 16 (2006) 984--1033] where estimates for binomial approximations of prices of game options were obtained. Our arguments rely, in particular, on strong invariance principle type approximations via the Skorokhod embedding, estimates from Kifer [Ann. Appl. Probab. 16 (2006) 984--1033] and the existence of optimal shortfall hedging in the discrete time established by Dolinsky and Kifer [Stochastics 79 (2007) 169--195].

Perfect and Partial Hedging for Swing Game Options in Discrete Time

The paper introduces and studies hedging for game (Israeli) style extension of swing options considered as multiple exercise derivatives. Assuming that the underlying security can be traded without restrictions, we derive a formula for valuation of multiple exercise options via classical hedging arguments. Introducing the notion of the shortfall risk for such options we study also partial hedging which leads to minimization of this risk.

Applications of weak convergence for hedging of game options.

In this paper we consider Dynkins games with payoffs which are functions of an underlying process. Assuming extended weak convergence of underlying processes