Rolf Poulsen

Transition Densities of Diffusion Processes: Numerical Comparison of Approximation Techniques

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.

A simple regime switching term structure model

Abstract. We extend the short rate model of Vasicek (1977) to include jumps in the local mean. Conditions ensuring existence of a unique equivalent martingale measure are given, implying that the model is arbitrage-free and complete. We develop efficient numerical methods for computation of zero coupon bond prices, illustrate how the model is easily calibrated to market data and show how other interest rate derivatives can be priced.

Risk Minimization in Stochastic Volatility Models: Model Risk and Empirical Performance

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.

Barrier options and their static hedges: simple derivations and extensions

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.

Monte Carlo Improvement of Estimates of the Mean-Reverting Constant Elasticity of Variance Interest Rate Diffusion

We use Simulation based methods to construct improved estimation procedures for discretely observed diffusions. The benchmark model used for Illustration and comparison is the CKLS model for the short-term interest rate dr = κ(θ — r)dt -f arifW, where W is a Wiener process. Here, κ is the rate of mean-reversion, θ is the long term interest rate level, σ is a scaling parameter, and 7 is the constant elasticity of variance parameter. The likelihood function is unknown, and Standard methods of moments lead to biased and inconsistent estimates of the unknown parameters. Consistent estimates are obtained through a Simulation based correction of the estimating equations (Indirect Inference). The estimation methods are compared in a Monte Carlo study and in applications to different sets of U.S. data. We find that the Simulation based correction reduces the bias in parameter estimates, increases efficiency, and reduces bias in estimated Standard errors.

A two-factor, stochastic programming model of Danish mortgage-backed securities

Abstract Danish mortgage loans have several features that make them interesting: Short-term revolving adjustable-rate mortgages are available, as well as fixed-rate, 10-, 20- or 30-year annuities that contain embedded options (call and delivery options). The decisions faced by a mortgagor are therefore non-trivial, both in terms of deciding on an initial mortgage, and in terms of managing (rebalancing) it optimally. We propose a two-factor, arbitrage-free interest-rate model, calibrated to observable security prices, and implement on top of it a multi-stage, stochastic optimization program with the purpose of optimally composing and managing a typical mortgage loan. We model accurately both fixed and proportional transaction costs as well as tax effects. Risk attitudes are addressed through utility functions and through worst-case (min–max) optimization. The model is solved in up to 9 stages, having 19,683 scenarios. Numerical results, which were obtained using standard soft- and hardware, indicate that the primary determinant in choosing between adjustable-rate and fixed-rate loans is the short–long interest rate differential (i.e., term structure steepness), but volatility also matters. Refinancing activity is influenced by volatility and, of course, transaction costs.

Empirical performance of models for barrier option valuation

In this paper the empirical performance of ve dierent models for barrier option valuation is investigated: the Black-Scholes model, the constant elasticity of variance model, the Heston stochastic volatility model, the Merton jump-diusion model, and the innite activity Variance Gamma model. We use time-series data from the USD/EUR exchange rate market: standard put and call (plain vanilla) option prices and a unique set of observed market values of barrier options. The models are calibrated to plain vanilla option prices, and prediction errors at different horizons for plain vanilla and barrier option values are investigated. For plain vanilla options, the Heston and Merton models have similar and superior performance for prediction horizons up to one week. For barrier options, the continuous-path models (Black-Scholes, constant elasticity of variance, and Heston) do almost equally well, while both models with jumps (Merton and Variance Gamma) perform markedly worse.

Dynamic portfolio optimization with transaction costs and state-dependent drift

The problem of dynamic portfolio choice with transaction costs is often addressed by constructing a Markov Chain approximation of the continuous time price processes. Using this approximation, we present an efficient numerical method to determine optimal portfolio strategies under time- and state-dependent drift and proportional transaction costs. This scenario arises when investors have behavioral biases or the actual drift is unknown and needs to be estimated. Our numerical method solves dynamic optimal portfolio problems with an exponential utility function for time-horizons of up to 40 years. It is applied to measure the value of information and the loss from transaction costs using the indifference principle.

Option Pricing with Excel

We use spreadsheets to illustrate the concepts and techniques of arbitrage-free option pricing. We show how to implement both discrete (binomial) models and continuous (Black-Scholes) models, discuss similarities and differences in the required computational methods, and investigate issues of a practical nature, such as parameter estimation/uncertainty and effects of less-than-perfect hedging.

Financial planning for young households

We analyze the financial planning problems of young households whose main decisions are how to finance the purchase of a house (liabilities) and how to allocate investments in pension savings schemes (assets). The problems are solved using a multi-stage stochastic programming model where the uncertainty is described by a scenario tree generated from a vector auto-regressive process for equity returns and interest rate evolution. We find strong evidence of the importance of taking into account the multi-stage nature of the problem, as well as the need to consider the asset and liability sides jointly.