Superhedging Strategies Research Articles

Robust discrete-time super-hedging strategies under AIP condition and under price uncertainty

Robust discrete-time super-hedging strategies under AIP condition and under price uncertainty

On Market Completions Approach to Option Pricing

Option pricing is one of the most important problems of contemporary quantitative finance. It can be solved in complete markets with non-arbitrage option price being uniquely determined via averaging with respect to a unique risk-neutral measure. In incomplete markets, an adequate option pricing is achieved by determining an interval of non-arbitrage option prices as a region of negotiation between seller and buyer of the option. End points of this interval characterise the minimum and maximum average of discounted pay-off function over the set of equivalent risk-neutral measures. By estimating these end points, one constructs super hedging strategies providing a risk-management in such contracts. The current paper analyses an interesting approach to this pricing problem, which consists of introducing the necessary amount of auxiliary assets such that the market becomes complete with option price uniquely determined. One can estimate the interval of non-arbitrage prices by taking minimal and maximal price values from various numbers calculated with the help of different completions. It is a dual characterisation of option prices in incomplete markets, and it is described here in detail for the multivariate diffusion market model. Besides that, the paper discusses how this method can be exploited in optimal investment and partial hedging problems.

Robust estimation of superhedging prices

We consider statistical estimation of superhedging prices using historical stock returns in a frictionless market with $d$ traded assets. We introduce a plug-in estimator based on empirical measures and show it is consistent but lacks suitable robustness. To address this, we propose novel estimators which use a larger set of martingale measures defined through a tradeoff between the radius of Wasserstein balls around the empirical measure and the allowed norm of martingale densities. We then extend our study, in part, to estimation of risk measures, to the case of markets with traded options, to a multi-period setting and to settings with model uncertainty. We also study convergence rates of estimators and convergence of super-hedging strategies.

European Options in a Nonlinear Incomplete Market Model with Default

We study the superhedging prices and the associated superhedging strategies for European options in a nonlinear incomplete market model with default. The underlying market model consists of one ris...

Game options with gradual exercise and cancellation under proportional transaction costs

ABSTRACTGame (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 randomized) 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 strategies and optimal mixed stopping times for both exercise and cancellation are developed and illustrated. Probabilistic dual representations for bid and ask prices are also established.

Robust pricing and hedging around the globe

We consider the martingale optimal transport duality for cadlag processes with given initial and terminal laws. Strong duality and existence of dual optimizers (robust semistatic superhedging strategies) are proved for a class of payoffs that includes American, Asian, Bermudan and European options with intermediate maturity. We exhibit an optimal superhedging strategy for which the static part solves an auxiliary problem and the dynamic part is given explicitly in terms of the static part.

Robust Pricing and Hedging around the Globe

We consider the martingale optimal transport duality for cadlag processes with given initial and terminal laws. Strong duality and existence of dual optimizers (robust semi-static superhedging strategies) are proved for a class of payoffs that includes American, Asian, Bermudan, and European options with intermediate maturity. We exhibit an optimal superhedging strategy for which the static part solves an auxiliary problem and the dynamic part is given explicitly in terms of the static part. In the case of finitely supported marginal laws, solving for the static part reduces to a semi-infinite linear program.

On the Performance of the Comonotonicity Approach for Pricing Asian Options in Some Benchmark Models from Equities and Commodities

In this paper, we investigate the applicability of the comonotonicity approach in the context of various benchmark models for equities and commodities. Instead of classical Lévy models as in Albrecher et al. we focus on the Heston stochastic volatility model, the constant elasticity of variance (CEV) model and Schwartz’ 1997 stochastic convenience yield model. We show how the technical difficulties of inverting the distribution function of the sum of the comonotonic random vector can be overcome and that the method delivers rather tight upper bounds for the prices of Asian Options in these models, at least for strikes which are not too large. As a by-product the method delivers super-hedging strategies which can be easily implemented.

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 [Game Options, Finance. Stoch., 4 (2000), pp. 443--463] 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 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 generalized Dynkin game, recently introduced in [R. Dumitrescu, M.-C. Quenez, and A. Sulem, Elect. J. Probab., 21 (2016), 64], expressed with a nonlinear expectation induced by a nonlinear backward SDE with default jump. We, moreover, study the existence of superhedging strategies. We then address the case of ambiguity on the model---for example ambiguity on the default probability---and characterize the robust seller's price of a ga...

ROBUST FUNDAMENTAL THEOREM FOR CONTINUOUS PROCESSES

AbstractWe study a continuous‐time financial market with continuous price processes under model uncertainty, modeled via a family of possible physical measures. A robust notion of no‐arbitrage of the first kind is introduced; it postulates that a nonnegative, nonvanishing claim cannot be superhedged for free by using simple trading strategies. Our first main result is a version of the fundamental theorem of asset pricing: holds if and only if every admits a martingale measure that is equivalent up to a certain lifetime. The second main result provides the existence of optimal superhedging strategies for general contingent claims and a representation of the superhedging price in terms of martingale measures.

Arbitrage and duality in nondominated discrete-time models

We consider a nondominated model of a discrete-time financial market where stocks are traded dynamically, and options are available for static hedging. In a general measure-theoretic setting, we show that absence of arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of martingale measures. In the arbitrage-free case, we show that optimal superhedging strategies exist for general contingent claims, and that the minimal superhedging price is given by the supremum over the martingale measures. Moreover, we obtain a nondominated version of the Optional Decomposition Theorem.

AMERICAN OPTIONS WITH GRADUAL EXERCISE UNDER PROPORTIONAL TRANSACTION COSTS

American options in a multi-asset market model with proportional transaction costs are studied in the case when the holder of an option is able to exercise it gradually at a so-called mixed (randomized) stopping time. The introduction of gradual exercise leads to tighter bounds on the option price when compared to the case studied in the existing literature, where the standard assumption is that the option can only be exercised instantly at an ordinary stopping time. Algorithmic constructions for the bid and ask prices and the associated superhedging strategies and optimal mixed stopping times for an American option with gradual exercise are developed and implemented, and dual representations are established.

On the Performance of the Comonotonicity Approach for Pricing Asian Option in Some Benchmark Models from Equities and Commodities

In this paper we investigate the applicability of the Albrecher et. al. (2005) comonotonicity approach in the context of various benchmark models for equities and commodities. Instead of classical Levy models as in Albrecher et. al. we focus on the Heston stochastic volatility model, the constant elasticity of variance (CEV) model and Schwartz' 1997 stochastic convenience yield model. We show that the method delivers rather tight upper bounds for the prices of Asian Options in these models and as a by product delivers super-hedging strategies which can be easily implemented.

ON OPTIMAL SUPER-HEDGING AND SUB-HEDGING STRATEGIES

This paper proposes optimal super-hedging and sub-hedging strategies for a derivative on two underlying assets without any specification of the underlying processes. Moreover, the strategies are free from any model of the dependency between the underlying asset prices. We derive the optimal pricing bounds by finding a joint distribution under which the derivative price is equal to the hedging portfolio's value; the portfolio consists of liquid derivatives on each of the underlying assets. As examples, we obtain new super-hedging and sub-hedging strategies for several exotic options such as quanto options, exchange options, basket options, forward starting options, and knock-out options.

Index Options: A Model-Free Approach

This paper contains an overview and an extension of the theory on comonotonicitybased model-free upper bounds and super-replicating strategies for stock index options, as presented in Hobson et al. (2005) and Chen et al. (2008). Whereas these authors only consider index call options, here a uni…ed approach for call and put options is presented. Considering a uni…ed framework gives rise to an e¢ cient algorithm for calculating upper bounds and for determining the corresponding superhedging strategies for both cases. The uni…ed framework also allows to extend several existing results, in particular on the optimality of the superhedging strategies. Several practical issues concerning the implementation of the results are discussed. In particular, a simpli…ed algorithm is presented for the situation where for some of the constituent stock in the index there are no options available.

OPTIMAL SUPERHEDGING UNDER NON-CONVEX CONSTRAINTS — A BSDE APPROACH

We apply theoretical results by Peng on supersolutions for Backward SDEs (BSDEs) to the problem of finding optimal superhedging strategies in a generalized Black–Scholes market under constraints. Constraints may be imposed simultaneously on wealth process and portfolio. They may be non-convex, time-dependent, and random. The BSDE method turns out to be an extremely useful tool for modeling realistic markets: in this paper, it is shown how more realistic constraints on the portfolio may be formulated via BSDE theory in terms of the amount of money invested, the portfolio proportion, or the number of shares held. Based on recent advances on numerical methods for BSDEs (in particular, the forward scheme by Bender and Denk [1]), a Monte Carlo method for approximating the superhedging price is given, which demonstrates the practical applicability of the BSDE method. Some numerical examples concerning European and American options under non-convex borrowing constraints are presented.

The pricing of leverage products: An empirical investigation of the German market for ‘long’ and ‘short’ stock index certificates

This paper describes the first thorough empirical analysis of the pricing of leverage products in the German retail market. These mainly exchange-traded products with an impressive trading volume are frequently advertised as long and short futures contracts, although they are theoretically equivalent to one-sided barrier options. Issuers’ daily quotes for stock index products are compared to (i) theoretical values derived from the prices of Eurex options and to (ii) boundaries obtained from semi-static superhedging strategies. For the vast majority of products, bid and ask quotes significantly exceed both theoretical values and upper hedging boundaries, thus providing almost risk-free profits for the issuers.

When are Static Superhedging Strategies Optimal?

This paper deals with the superhedging of derivatives on incomplete markets, i.e. with portfolio strategies which generate payoffs at least as high as that of a given contingent claim. The simplest solution to this problem is in many cases a static superhedge, i.e. a buy-and-hold strategy generating an affine-linear payoff. We study whether a superhedge can be achieved with less initial capital if we also allow for dynamic trading strategies. The answer to this question depends on the kind of the non-traded risk factors. Our main findings for a stochastic volatility model with unbounded volatility show that there is always an optimal static superhedge. Additionally, there may be infinitely many optimal dynamic superhedges which require the same initial capital. In a model with stochastic jumps, it is always either a dynamic or a static strategy which is optimal, but never both. In a model with a stochastic short rate the properties of the interest rate process are also relevant. When there are no bounds for the interest rate optimal superhedges (if they exist) are always static, since the strategy will never contain an investment in the money market account. On the other hand, when interest rates are either bounded or non-negative either a static or a dynamic strategy is optimal, depending on the respective contingent claim. Our results have important implications for the design of superhedges as they show under which conditions we can restrict the analysis to static strategies. There is no such thing as the incomplete market when it comes to superhedging. Although in continuous-time models the class of possible trading strategies contains much more elements than just static strategies, there is a number of cases where buy-and-hold is as good as or even superior to dynamic strategies.

Superreplication in stochastic volatility models and optimal stopping

In this paper we discuss the superreplication of derivatives in a stochastic volatility model under the additional assumption that the volatility follows a bounded process. We characterize the value process of our superhedging strategy by an optimal-stopping problem in the context of the Black-Scholes model which is similar to the optimal stopping problem that arises in the pricing of American-type derivatives. Our proof is based on probabilistic arguments. We study the minimality of these superhedging strategies and discuss PDE-characterizations of the value function of our superhedging strategy. We illustrate our approach by examples and simulations.