Trying to build option models with price dynamics that better match empirical price behavior, we run into the problem that only a few returns processes, like the standard Black-Scholes lognormal diffusion, lead to closed form solutions for the transition densities. Generally these must be approximated numerically, using one of a variety of approaches. The Euler approximation is probably the most common technique of discretizing the process, but others are also in use, including the Milshtein scheme. Monte Carlo simulation is another approach. Simulation methods may make use of (pseudo-) random numbers, or deterministic quasi-random sequences that may be more efficient. Alternatives include the binomial model, Hermite expansions, and more. In this article, Jensen and Poulsen examine the comparative performance of a large number of approximation techniques in terms of accuracy and execution time. The winner in their tests, by a surprisingly large margin, is the Hermite expansion approach.

In this paper the performance of locally risk-minimizing delta hedge strategies for European options in stochastic volatility models is studied from an experimental as well as from an empirical perspective. These hedge strategies are derived for a large class of diffusion-type stochastic volatility models, and they are as easy to implement as usual delta hedges. Our simulation results on model risk show that these risk-minimizing hedges are robust with respect to uncertainty and misconceptions about the underlying data generating process. The empirical study, which includes the US sub-prime crisis period, documents that in equity markets risk-minimizing delta hedges consistently outperform usual delta hedges by approximately halving the standard deviation of the profit-and-loss ratio.

We use a reflection result to give simple proofs of (well-known) valuation formulas and static hedge portfolio constructions for zero-rebate single-barrier options in the Black–Scholes model. We then illustrate how to extend the ideas to other model types giving (at least) easy-to-program numerical methods and other option types such as options with rebates, and double-barrier and lookback options.

The Black-Scholes model and most other modern derivatives pricing models are based on the idea of replicating an option's payoff by dynamically trading between the underlying asset and riskless borrowing or lending. But dynamic hedging has always worked a lot better in theory than in practice, where transactions costs, stochastic volatility, non-diffusion price jumps, and other deviations from the model's assumptions have to be dealt with. Barrier options in particular present problems because dynamic hedging becomes very difficult when the underlying asset price is near the barrier. An alternative approach that has been explored to some degree in the literature is static hedging, in which the payoff on an option is replicated by a static portfolio of other options. Several different ways to do this have been proposed, though they may entail unrealistic requirements, such as the existence of plain vanilla options with a continuum of strike prices. In this article, Nalholm and Poulsen present an implementa...