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Yan Dolinsky

Publications and source records attributed to Yan Dolinsky.

49 records · Page 3Linked to original sources
arXiv2013

Martingale Optimal Transport and Robust Hedging in Continuous Time

The duality between the robust (or equivalently, model independent) hedging of path dependent European options and a martingale optimal transport problem is proved. The financial market is modeled through a risky asset whose price is only assumed to be a continuous function of time. The hedging problem is to construct a minimal super-hedging portfolio that consists of dynamically trading the underlying risky asset and a static position of vanilla options which can be exercised at the given, fixed maturity. The dual is a Monge-Kantorovich type martingale transport problem of maximizing the expected value of the option over all martingale measures that has the given marginal at maturity. In addition to duality, a family of simple, piecewise constant super-replication portfolios that asymptotically achieve the minimal super-replication cost is constructed.

math.PR↗
arXiv2013

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.

q-fin.PR↗
arXiv2012

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.

q-fin.PM↗
arXiv2011

Numerical schemes for $G$--Expectations

We consider a discrete time analog of $G$--expectations and we prove that in the case where the time step goes to 0 the corresponding values converge to the original $G$--expectation. Furthermore we provide error estimates for the convergence rate. This paper is continuation of [4]. Our main tool is a strong approximation theorem which we derive for general discrete time martingales.

math.PR↗
arXiv2011

Duality and Convergence for Binomial Markets with Friction

We prove limit theorems for the super-replication cost of European options in a Binomial model with friction. The examples covered are markets with proportional transaction costs and the illiquid markets. The dual representation for the super-replication cost in these models are obtained and used to prove the limit theorems. In particular, the existence of the liquidity premium for the continuous time limit of the model proposed in [6] is proved. Hence, this paper extends the previous convergence result of [13] to the general non-Markovian case. Moreover, the special case of small transaction costs yields, in the continuous limit, the $G$-expectation of Peng as earlier proved by Kusuoka in [14].

q-fin.CP↗
arXiv2011

Weak Approximation of G-Expectations

We introduce a notion of volatility uncertainty in discrete time and define the corresponding analogue of Peng's G-expectation. In the continuous-time limit, the resulting sublinear expectation converges weakly to the G-expectation. This can be seen as a Donsker-type result for the G-Brownian motion.

math.PR↗
arXiv2010

Applications of weak convergence for hedging of game options

In this paper we consider Dynkin's games with payoffs which are functions of an underlying process. Assuming extended weak convergence of underlying processes $\{S^{(n)}\}_{n=0}^{\infty}$ to a limit process $S$ we prove convergence Dynkin's games values corresponding to $\{S^{(n)}\}_{n=0}^{\infty}$ to the Dynkin's game value corresponding to $S$. We use these results to approximate game options prices with path dependent payoffs in continuous time models by a sequence of game options prices in discrete time models which can be calculated by dynamical programming algorithms. In comparison to previous papers we work under more general convergence of underlying processes, as well as weaker conditions on the payoffs.

math.PR↗
arXiv2010

Shortfall Risk Approximations for American Options in the multidimensional Black--Scholes Model

We show that shortfall risks of American options in a sequence of multinomial approximations of the multidimensional Black--Scholes (BS) market converge to the corresponding quantities for similar American options in the multidimensional BS market with path dependent payoffs. In comparison to previous papers we consider the multi assets case for which we use the weak convergence approach.

q-fin.CP↗
arXiv2010

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 options for the Black--Scholes model our estimates are a significant improvement of those which were obtained in [8] (for game options in a more general setup)

q-fin.CP↗
arXiv2010

Limit Theorems for Partial Hedging Under Transaction Costs

We study shortfall risk minimization for American options with path dependent payoffs under proportional transaction costs in the Black--Scholes (BS) model. We show that for this case the shortfall risk is a limit of similar terms in an appropriate sequence of binomial models. We also prove that in the continuous time BS model for a given initial capital there exists a portfolio strategy which minimizes the shortfall risk. In the absence of transactions costs (complete markets) similar limit theorems were obtained in Dolinsky and Kifer (2008, 2010) for game options. In the presence of transaction costs the markets are no longer complete and additional machinery required. Shortfall risk minimization for American options under transaction costs was not studied before.

q-fin.CP↗
arXiv2009

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.

q-fin.PR↗
arXiv2008

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].

math.PR↗