# Abstracts

**Risk sensitive utility indifference pricing of perpetual American options under fixed transaction cost**

*Pavel Gapeev (LSE, UK)*

Thursday June 5, 11:30-12:00 | session 7.5 | Options, Futures | room G

The problems of risk sensitive portfolio optimisation under transaction costs have taken a considerable attention in the recent literature on mathematical finance. We study the associated problems of risk sensitive utility indifference pricing for perpetual American options with fixed transaction costs in the classical model of financial market with two tradable assets. Assume that the investors trading in the market must pay transaction costs equal to a fixed fraction of the entire portfolio wealth each time they trade. The objective is to maximise the asymptotic (risk null and risk adjusted) exponential growth rates based on the expected logarithmic or power utility of the difference between the terminal portfolio wealth and a certain amount of the option payoffs. It is shown that the optimal trading policy keeps the number of shares held in the assets unchanged between the transactions. In order to determine the optimal trading times and sizes, we reduce the initial problems to the appropriate (discounted) time-inhomogeneous optimal stopping problems for a one-dimensional diffusion process representing the fraction of the portfolio wealth held by the investor in the risky asset. The optimal trading and exercise times are proved to be the first times at which the risky fraction process exits certain regions restricted by two time-dependent boundaries. Then, certain amounts of assets should be bought or sold or the options should be exercised whenever the risky fraction process hits either the lower or the upper time-dependent curve. The latter are characterised as unique solutions of the associated parabolic-type free-boundary problems for the value functions satisfying the smooth-fit conditions at the curved boundaries. The optimal asymptotic growth rates and trading sizes are specified as parameters maximising the value functions of the resulting optimal stopping problems. We illustrate these results on the examples of the perpetual American call and put as well as the asset-or-nothing options, for which we obtain the utility indifference prices as well as the optimal trading and exercise boundaries in a closed form. We also formulate the same problem under both fixed and proportional transaction costs and solve the associated optimal impulse control problem for the case of the asset-or-nothing perpetual American option.

**Dividend Derivatives**

*Radu Tunaru (University of Kent, UK)*

Thursday June 5, 12:00-12:30 | session 7.5 | Options, Futures | room G

The literature on pricing dividend derivatives is sparse. Dividend derivatives constitute a recent market that has generated increasingly great interest on European derivatives markets. There are futures with a December calendar maturity roll up to ten years and also European call and put options with a wide range of strike prices, for the same maturities. The underlying is the Dow Jones Euro STOXX 50 dividend index and we used dividend derivatives data between December 2008 and February 2012. From the equity derivatives pricing literature it seems conclusive that dividends are stochastic in nature. Hence, it is important to find models that can be easily implemented but that also preserve the stochastic character of dividends.

The first model proposed for pricing dividend derivatives is a jump-diffusion model with beta distributed jump sizes, proposed for the equity dividend index. The jumps are only downwards and the dividend payments are determined also by the evolution of the equity index itself. A Monte Carlo approach is described for pricing vanilla dividend derivatives. It was illustrated that this model can fit the smile of the European call and put dividend index options.

The main result of the paper is related to the stochastic logistic diffusion model, that would be useful to calculate analytically the conditional moments of the cum-dividend underlying variable. The two models developed here for pricing dividend derivative are very different, the first one modeling the dividend payment series while the latter follows the cum-dividend series. Both models rely on the Monte Carlo approach for implementation but there are immediate advantages in doing so since other path dependent derivatives would be priced directly based on the same set of simulations. Both models are specified under the physical measure. The parameters are estimated from the historical time-series.

The term structure of market price of risk is used to fix the martingale pricing measure and it is determined from the dividend futures curves. Then, under this measure, the call and put options are priced consistently for each maturity and across the strike range, fitting the smile (smirk) really well.

**Risk premiums in a multi-factor jump-diffusion model for the joint dynamics of equity options and their underlying**

*Robert Huitema (University of Zurich, Switzerland)*

Thursday June 5, 12:30-13:00 | session 7.5 | Options, Futures | room G

This paper proposes a new approach to measure premiums for volatility and jump risks in option markets. These risks are captured by a multi-factor jump-diffusion model for the joint evolution of the underlying and the implied volatility surface. This market-based approach enables us to carefully test and select the most relevant risk factors in option markets. We extend the approach of Schonbucher (1998) to processes that include jumps and derive a condition that ensures absence of dynamic arbitrage. As this condition is derived under the physical measure, it incorporates a premium for each risk factor in the model. We then interpret the no-arbitrage condition as a noisy measurement of these risk premiums and other latent variables such as the volatility and jump-intensity of the underlying. This allows us to dynamically calibrate these variables to data from several markets using Bayesian filtering methods. The results shed new light on how option risk premiums vary over time and across markets. As our approach provides an accurate and arbitrage-free description of option price dynamics it can also be used for risk management of portfolios of options and for testing dynamic option strategies.