Angelika Esser

Valuation of Power Options

This chapter deals with the pricing of certain types of exotic options, called “power options” and “powered options”. The special feature of these contracts is that, compared with plain vanilla options, in the first case the stock price in the payoff function is replaced by the stock price raised to some power, and in the latter case the option payoff is raised to some power. These contracts generalize the special case of a piecewise-linear payoff for plain vanilla contracts. Without loss of generality we only deal with calls, since puts can be priced similarly. If the exponent of a power call option is greater (smaller) than one, the payoff and consequently the price of such contracts is greater (smaller) than the corresponding plain vanilla call option.

Modeling Feedback Effects Using Stochastic Liquidity

This chapter deals with the modeling of asset liquidity. One aspect of liquidity includes the price impact involved in acquiring or liquidating a position. Our objective is to study the interaction between the trading strategy of a large investor, the asset price process, and liquidity in one single setup. There is a growing number of theoretical papers investigating the interaction of liquidity and trading strategies of large investors. Part of this literature considers optimal liquidation strategies for large portfolios, for instance Dubil [17], and Almgren and Chriss [1]. Recently, research has focused more and more on the modeling and hedging aspects that are introduced by liquidity and the presence of large traders. Cvitanic and Ma [14], Frey [22], Frey and Patie [23], and Liu and Yong [31] consider liquidity as an exogenously given source of risk. Frey and Stremme [25], Kampovsky and Trautmann [29], Papanicolaou and Sircar [34], and Schonbucher and Wilmott [38] serve as prominent examples taking into account equilibrium setups.

Pricing by Change of Measure and Numeraire

There are several ways to derive the no-arbitrage price of a contingent claim, such as following a replicating portfolio strategy or solving a partial differential equation. Another prominent approach is martingale pricing, which is the method we deal with in this chapter. We briefly review well-known facts on equivalent measures, the Radon-Nikodym derivative, martingale measures, and the change of numeraires following Geman, El Karoui, and Rochet [27]. The only measures we consider within this thesis are the ones equivalent to the physical measure. The ultimate goal in deriving the pricing formula for a claim is to write it in terms of possibly different artificial probabilites. It is known that the choice of different numeraires allows for a convenient computation of the claim’s fair price. This can be seen when looking at the BS formula: The easiest method for the valuation of a standard call is to choose appropriate normalizing assets and corresponding martingale measures. This will be reviewed to motivate the concept of different numeraires.

Motivation and Overview

The entirely theoretical work presented focuses on structural concepts of no-arbitrage pricing of contingent claims and on modeling aspects of asset price dynamics that are affected by stochastic liquidity. In order to clarify the theoretical findings, we present several applications and illustrative examples including well-known scenarios such as the Black-Scholes [7] (henceforth BS) setting.

Comparison of Discrete and Continuous Models

A natural classification of models is to distinguish between discrete and continuous setups. The most prominent representatives are the Cox, Ross, and Rubinstein [13] binomial tree model and the classical BS model, respectively. In this chapter we contrast a discrete against a continuous setting for both complete and incomplete markets. Various examples, including SV models as important representatives of incomplete markets, shed light on the general theory. The discrete models we consider are state- and time-discrete; the continuous models we refer to are diffusion models. The latter are widely studied (see, e.g. Protter [36]), but for certain problems it might be more appropriate to look at a discrete setting, being a better approximation of reality. Furthermore, for complex problems it may be more adequate to start in discrete time, avoiding tedious technical conditions necessary in continuous time. In this context, the question arises whether all properties of the diffusion setup carry over to the discrete scenario. We will see that this is not always the case.

Decentralizing Risk Management in the Case of Quadratic Hedging

This paper deals with the problem of quadratic hedging with limited initial capital. We show (i) that the optimal amount of capital for the quadratic hedge of a portfolio of contingent claims is equal to the sum of optimal investments for the individual hedges of its components and (ii) that the optimal hedging strategies for individual claims add up to the optimal strategy for the total position. These results have the important implication for risk management that in the case of limited capital the quadratic hedge of a contingent claim can be decomposed into two problems: first, the claim is hedged as if the optimal amount of capital was available, and then an additional quadratic hedge is set up for a zero payoff where now the initial capital is given by the (negative) difference between available and optimal capital. Both this additional hedge and the increase in the expected squared hedging error arising from the capital restriction are independent of the original claim to be hedged.