A Proposal on

Valuation and Hedging of Derivatives in Incomplete Market

by

Table of Contents

Introduction3

Purpose of the Study3

Contribution of the Study4

Aim of the Study5

Structure of the Study6

Expected Findings6

References8

Valuation and Hedging of Derivatives in Incomplete Market

Introduction

A state-of-the-art option pricing model includes one or more stochastic volatility factors, jumps in returns, and, sometimes, jumps in variance. Although the dynamics of stock prices may vary, all these models produce quite similar prices for European plain-vanilla options when they are calibrated to the volatility smile. Consequently, it is often difficult to empirically identify the underlying model when looking at only a cross-section of option prices. This gives rise to model risk, since similar prices for European options do not necessarily imply that the models have similar implications when it comes to, e.g., hedging, pricing exotic options, or portfolio planning. (Schoutens & Tistaer 2004 66-78)

Exact model identification is hindered by noisy prices, e.g., due to bid-ask spreads. As argued by (2009), two models cannot be distinguished from each other if the maximal pricing difference is less than the noise in the data. Using empirically observed prices and bid-ask spreads, these authors demonstrate that this problem may be found even in the early models of (1973) (henceforth B-S) and (1976). This would not be an issue if differences between models were unimportant. However, these differences can be (and usually are) important in pricing nonredundant derivatives like exotic options, in hedging derivative positions, and in portfolio planning.

Purpose of the Study

In this study, we will first analyze the impact of model misspecification on the performance of different models when European options are to be hedged. Second, we will study whether hedging errors provide any useful economic information about model fit, and we analyze the implications of model risk for risk measurement.

Contribution of the Study

Hedging errors can be useful for identifying model misspecification. Furthermore, model risk has severe implications for risk measurement and can lead to a significant misestimation, specifically underestimation, of the risk to which a hedged position is exposed. (Poulson & Ewald 2009 693-704)

The first contribution of our study is an analysis of the distribution of hedging errors when the model is misspecified. We focus on what we believe to be the most realistic case, that involving omitted risk factors, i.e., we assume that the incorrect hedge model used by the investor contains fewer risk factors than the true data-generating process (true model). In our simulation study, the true data-generating process is assumed to be the one suggested by Bakshi et al. (1997), which has stochastic volatility and jumps in returns, but deterministic interest rates. Based on the (misspecified) calibrated hedge models, we then implement a delta hedge, a delta-vega hedge, and a local minimum-variance hedge. The resulting realized hedging errors over the next time interval are obtained via Monte Carlo simulation. In addition, we simulate the hedging errors when the true model is used. The fact that even these ideal hedging errors are nonzero is due to discrete trading and market incompleteness. They thus will be employed as a natural ...