Antoon Pelsser

Pricing and Hedging Guaranteed Annuity Options via Static Option Replication

In this paper we derive a market value for with-profits Guaranteed Annuity Options using martingale modelling techniques. Furthermore, we show how to construct a static replicating portfolio of vanilla interest rate swaptions that replicates the with-profits Guaranteed Annuity Option. Finally, we illustrate with historical UK interest rate data from the period 1980 until 2000 that the static replicating portfolio would have been extremely effective as a hedge against the interest rate risk involved in the GAO, that the static replicating portfolio would have been considerably cheaper than up-front reserving and also that the replicating portfolio would have provided a much better level of protection than an up-front reserve.

Pricing Swaptions and Coupon Bond Options in Affine Term Structure Models

We propose an approach to find an approximate price of a swaption in affine term structure models. Our approach is based on the derivation of approximate swap rate dynamics in which the volatility of the forward swap rate is itself an affine function of the factors. Hence, we remain in the affine framework and well-known results on transforms and transform inversion can be used to obtain swaption prices in similar fashion to zero bond options (i.e., caplets). The method can easily be generalized to price options on coupon bonds. Computational times compare favorably with other approximation methods. Numerical results on the quality of the approximation are excellent. Our results show that in affine models, analogously to the LIBOR market model, LIBOR and swap rates are driven by approximately the same type of (in this case affine) dynamics.

Efficient, Almost Exact Simulation of the Heston Stochastic Volatility Model

We deal with discretization schemes for the simulation of the Heston stochastic volatility model. These simulation methods yield a popular and flexible pricing alternative for pricing and managing a book of exotic derivatives which cannot be valued using closed-form expressions. For the Heston dynamics an exact simulation method was developed by Broadie and Kaya (2006), however we argue why its practical use is limited. Instead we focus on efficient approximations of the exact scheme, aimed to resolve the disadvantages of this method; one of the main bottlenecks in the exact scheme is the simulation of the Non-central Chi-squared distributed variance process, for which we suggest an efficient caching technique. At first sight the creation of a cache containing the inverses of this distribution might seem straightforward, however as the parameter space of the inverse Non-central Chi-squared distribution is three-dimensional, the design of such a direct cache is rather complicated, as pointed out by Broadie and Andersen. Nonetheless, for the case of the Heston model we are able to tackle this dimensionality problem and show that the three-dimensional inverse of the non-central chi-squared distribution can effectively be reduced to a one dimensional cache. The performed analysis hence leads to the development of three new efficient simulation methods (the NCI, NCI-QE and BK-DI scheme). Finally, we conclude with a comprehensive numerical study of these new schemes and the exact scheme of Broadie and Kaya, the almost exact scheme of Smith, the Kahl-Jackel scheme, the FT scheme of Lord et al. and the QE-M scheme of Andersen. From these results, we find that the QE-M scheme is the most efficient, followed closely by the NCI-M, NCI-QE-M and BK-DI-M schemes, whilst we observe that all other considered schemes perform a factor 6 to 70 times less efficient than the latter four methods.

Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility

We consider the pricing of long-dated insurance contracts under stochastic interest rates and stochastic volatility. In particular, we focus on the valuation of insurance options with long-term equity or foreign exchange exposures. Our modeling framework extends the stochastic volatility model of Schobel and Zhu (1999) by including stochastic interest rates. Moreover, we allow all driving model factors to be instantaneously correlated with each other, i.e. we allow for a general correlation structure between the instantaneous interest rates, the volatilities and the underlying stock returns. As insurance products often incorporate long-term exposures, they are typically more sensitive to changes in the interest rates, volatility and currencies. Therefore, having the flexibility to correlate the underlying asset price with both the stochastic volatility and the stochastic interest rates, yields a realistic model which is of practical importance for the pricing and hedging of such long-term contracts. We show that European options, typically used for the calibration of the model to market prices, and forward starting options can be priced efficiently and in closed-form by means of Fourier inversion techniques. We extensively discuss the numerical implementation of these pricing formulas, allowing for a fast and accurate valuation of European and forward starting options. The model will be especially useful for the pricing and risk management of insurance contracts and other exotic derivatives involving long-term maturities.

Forward versus Spot Interest-Rate Models of the Term Structure: An Empirical Comparison

Valuation theory for derivatives based on interest rates and bond prices continues to be in flux. A wide variety of models have been introduced over the years, and many are still actively used. Some are based on modeling the behavior of spot interest rates, with enough structure that the model is constrained to be consistent with the current market term structure. Others begin with the observed term structure and model behavior of the forward rates embedded in it. Moraleda and Pelsser conduct a comparison of five of the most common interest rate models, including some of each type. The data used include U.S. spot interest rates for maturities out to ten years and dollar cap and floor prices. The authors find that the models of the spot rate appear to outperform those based on forward rates, with the Black-Karasinski model doing the best overall.

Fast drift-approximated pricing in the BGM model

This paper shows that the forward rates process discretized by a single time step together with a separability assumption on the volatility function allows for representation by a low-dimensional Markov process. This in turn leads to e±cient pricing by for example finite differences. We then develop a discretization based on the Brownian bridge especially designed to have high accuracy for single time stepping. The scheme is proven to converge weakly with order 1. We compare the single time step method for pricing on a grid with multi step Monte Carlo simulation for a Bermudan swaption, reporting a computational speed increase of a factor 10, yet pricing sufficiently accurate.

Mathematical Foundation of Convexity Correction

A broad class of exotic interest rate derivatives can be valued simply by adjusting the forward interest rate. This adjustment is known in the market as convexity correction. Various ad hoc rules are used to calculate the convexity correction for different products, many of them mutually inconsistent. In this research paper we put convexity correction on a firm mathematical basis by showing that it can be interpreted as the side-effect of a change of probability measure. This provides us with a theoretically consistent framework to calculate convexity corrections. Using this framework we review various expressions for LIBOR in arrears and diff swaps that have been derived in the literature. Furthermore, we propose a simple method to calculate analytical approximations for general instances of convexity correction.

Level–Slope–Curvature – Fact or Artefact?

The first three factors resulting from a principal components analysis of term structure data are, in the literature, typically interpreted as driving the level, slope and curvature of the term structure. Using slight generalizations of theorems from total positivity, we present sufficient conditions under which level, slope and curvature are present. These conditions have the nice interpretation of restricting the level, slope and curvature of the correlation surface. It is proven that the Schoenmakers–Coffey correlation matrix also brings along such factors. Finally, we formulate and corroborate a conjecture that the order present in correlation matrices cause slope.

On the applicability of the wang transform for pricing financial risks

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Modeling Non-Monotone Risk Aversion Using SAHARA Utility Functions

We develop a new class of utility functions, SAHARA utility, with the distinguishing feature that it allows absolute risk aversion to be non-monotone and implements the assumption that agents may become less risk averse for very low values of wealth. The class contains the well-known exponential and power utility functions as limiting cases. We investigate the optimal investment problem under SAHARA utility and derive the optimal strategies in an explicit form using dual optimization methods. We also show how SAHARA utility functions extend the class of contingent claims that can be valued using indifference pricing in incomplete markets.