Michel Vellekoop

A tree-based method to price American options in the Heston model

We develop an algorithm to price American options on assets that follow the stochastic volatility model defined by Heston. We use an approach which is based on a modification of a combined tree for stock prices and volatilities, where the number of nodes grows quadratically in the number of time steps. We show in a number of numerical tests that we get accurate results in a fast manner, and that features which are essential for the practical use of stock option pricing algorithms, such as the incorporation of cash dividends and a term structure of interest rates, can easily be incorporated.

The Impact of Multiple Structural Changes on Mortality Predictions

Most mortality models proposed in recent literature rely on the standard ARIMA framework (in particular: a random walk with drift) to project mortality rates. As a result the projections are highly sensitive to the calibration period. We therefore analyse the impact of allowing for multiple structural changes on a large collection of mortality models. We find that this may lead to more robust projections for the period effect but that there is only a limited effect on the ranking of the models based on backtesting criteria, since there is often not yet sufficient statistical evidence for structural changes. However, there are cases for which we do find improvements in estimates and we therefore conclude that one should not exclude on beforehand that structural changes may have occurred.

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.

On Option Pricing Models in the Presence of Heavy Tails

We propose a modification of the option pricing framework derived by Borland which removes the possibilities for arbitrage within this framework. It turns out that such arbitrage possibilities arise due to an incorrect derivation of the martingale transformation in the non-Gaussian option models which are used in that paper. We show how a similar model can be built for the asset price processes which excludes arbitrage. However, the correction causes the pricing formulas to be less explicit than the ones in the original formulation, since the stock price itself is no longer a Markov process. Practical option pricing algorithms will therefore have to resort to Monte Carlo methods or partial differential equations and we show how these can be implemented. An extra parameter, which needs to be specified before the model can be used, will give market makers some extra freedom when fitting their model to market data.

Regularity of the Exercise Boundary for American Put Options on Assets with Discrete Dividends

We analyze the regularity of the optimal exercise boundary for the American Put option when the underlying asset pays a discrete dividend at a known time

The Early Exercise Premium for the American Put Under Discrete Dividends

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An integral equation for American put options on assets with general dividend processes

during the lifetime of the option. The ex-dividend asset price process is assumed to follow Black-Scholes dynamics, and the dividend amount is a deterministic function of the ex-dividend asset price just before the dividend date. The solution to the associated optimal stopping problem can be characterized in terms of an optimal exercise boundary which, in contrast to the case when there are no dividends, may no longer be monotone. In this paper we prove that when the dividend function is positive and concave, then the boundary is nonincreasing in a left-hand neighborhood of

Pricing American options with the SABR model

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An optimal investment problem with randomly terminating income

and tends to 0 as time tends to

Optimal investment and consumption when allowing terminal debt

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