This paper suggests perfect hedging strategies of contingent claims under stochastic volatility and random jumps of the underlying asset price. This is done by enlarging the market with appropriate swaps whose payoffs depend on higher-order sample moments of the asset price process. Using European options and variance swaps, as well as barrier options written on the S&P 500 index, the paper provides clear cut evidence that hedging strategies employing variance and higher-order moment swaps considerably improves upon the performance of traditional delta hedging strategies. Inclusion of the third-order moment swap improves upon the performance of variance swap based strategies to hedge against random jumps. This result is more profound for short-term OTM put options.
We revisit the problem of optimally hedging a European contingent claim (ECC) using a hedging portfolio consisting of a risky asset that can be traded at pre-specified discrete times. The objective function to be minimized is either the second-moment or the variance of the hedging error calculated in the market probability measure. The main outcome of our work is to show that unique solutions exist in a larger class of admissible strategies under integrability and non-degeneracy conditions on the hedging asset price process that are weaker than previously thought possible. Specifically, we do not require the hedging asset price process to be square-integrable, and do not use the bounded mean-variance trade off assumption. Our criterion for admissible strategies only requires the cumulative trading gain, and not the incremental trading gains, to be square integrable. We derive explicit expressions for the second-moment and the variance of the hedging error to arrive at the respective optimal hedging strategies. We use the expressions mentioned above to also give explicit solutions to two constrained mean-variance frontier problems, namely, minimizing the variance subject to a lower bound on the mean profit, and maximizing the mean profit subject to an upper bound on the variance. Further, we explain the connections between our solution and that of the previous formulations. Finally, we identify the associated variance-optimal martingale measure and provide an expression for the L2-approximation price of the hedged ECC in that measure.
Since around the turn of the millennium there has been a general acceptance that one of the more practical improvements one may make in the light of the shortfalls of the classical Black-Scholes model is to replace the underlying source of randomness, a Brownian motion, by a Lévy process. Working with Lévy processes allows one to capture desirable distributional characteristics in the stock returns. In addition, recent work on Lévy processes has led to the understanding of many probabilistic and analytical properties, which make the processes attractive as mathematical tools. At the same time, exotic derivatives are gaining increasing importance as financial instruments and are traded nowadays in large quantities in OTC markets. The current volume is a compendium of chapters, each of which consists of discursive review and recent research on the topic of exotic option pricing and advanced Lévy markets, written by leading scientists in this field. In recent years, Lévy processes have leapt to the fore as a tractable mechanism for modeling asset returns. Exotic option values are especially sensitive to an accurate portrayal of these dynamics. This comprehensive volume provides a valuable service for financial researchers everywhere by assembling key contributions from the world's leading researchers in the field. Peter Carr, Head of Quantitative Finance, Bloomberg LP. This book provides a front-row seat to the hottest new field in modern finance: options pricing in turbulent markets. The old models have failed, as many a professional investor can sadly attest. So many of the brightest minds in mathematical finance across the globe are now in search of new, more accurate models. Here, in one volume, is a comprehensive selection of this cutting-edge research. Richard L. Hudson, former Managing Editor of The Wall Street Journal Europe, and co-author with Benoit B. Mandelbrot of The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward
We study the influence of taking liquidity costs and market impact into account when hedging a contingent claim, first in the discrete time setting, then in continuous time. In the latter case and in a complete market, we derive a fully non-linear pricing partial differential equation, and characterizes its parabolic nature according to the value of a numerical parameter naturally interpreted as a relaxation coefficient for market impact. We then investigate the more challenging case of stochastic volatility models, and prove the parabolicity of the pricing equation in a particular case.
We solve the pricing and hedging problem for the generic variance swap on a swap rate. The solution is not limited to a specifc swap rate model approximation. In order to address the absence of arbitrage constraints and to preserve the model complexity, we develop an alternative approach to swap rate approximations.
