Delta hedging in discrete time under stochastic interest rate

We propose a methodology based on the Laplace transform to compute the variance of the hedging error due to time discretization for financial derivatives when the interest rate is stochastic. Our approach can be applied to any affine model for asset prices and to a very general class of hedging strategies, including Delta hedging. We apply it in a two-dimensional market model, obtained by combining the models of Black-Scholes and Vasicek, where we compare a strategy that correctly takes into account the variability of interest rates to one that erroneously assumes that they are deterministic. We show that the differences between the two strategies can be very significant. The factors with stronger influence are the ratio between the standard deviations of the equity and that of the interest rate, and their correlation. The methodology is also applied to study the Delta hedging strategy for an interest rate option in the Cox-Ingersoll and Ross model, measuring the variance of the hedging error as a function of the frequency of the rebalancing dates. We compare the results obtained to those coming from a classical Monte Carlo simulation

Fractional smoothness and applications in finance

This overview article concerns the notion of fractional smoothness of random variables of the form $g(X_T)$, where $X=(X_t)_{t\in [0,T]}$ is a certain diffusion process. We review the connection to the real interpolation theory, give examples and applications of this concept. The applications in stochastic finance mainly concern the analysis of discrete time hedging errors. We close the review by indicating some further developments.Comment: Chapter of AMAMEF book. 20 pages

Can tests based on option hedging errors correctly identify volatility risk premia?

This paper provides an in-depth analysis of the properties of popular tests for the existence and the sign of the market price of volatility risk. These tests are frequently based on the fact that for some option pricing models under continuous hedging the sign of the market price of volatility risk coincides with the sign of the mean hedging error. Empirically, however, these tests suffer from both discretization error and model mis-specification. We show that these two problems may cause the test to be either no longer able to detect additional priced risk factors or to be unable to identify the sign of their market prices of risk correctly. Our analysis is performed for the model of Black and Scholes (1973) (BS) and the stochastic volatility (SV) model of Heston (1993). In the model of BS, the expected hedging error for a discrete hedge is positive, leading to the wrong conclusion that the stock is not the only priced risk factor. In the model of Heston, the expected hedging error for a hedge in discrete time is positive when the true market price of volatility risk is zero, leading to the wrong conclusion that the market price of volatility risk is positive. If we further introduce model mis-specification by using the BS delta in a Heston world we find that the mean hedging error also depends on the slope of the implied volatility curve and on the equity risk premium. Under parameter scenarios which are similar to those reported in many empirical studies the test statistics tend to be biased upwards. The test often does not detect negative volatility risk premia, or it signals a positive risk premium when it is truly zero. The properties of this test furthermore strongly depend on the location of current volatility relative to its long-term mean, and on the degree of moneyness of the option. As a consequence tests reported in the literature may suffer from the problem that in a time-series framework the researcher cannot draw the hedging errors from the same distribution repeatedly. This implies that there is no guarantee that the empirically computed t-statistic has the assumed distribution. JEL: G12, G13 Keywords: Stochastic Volatility, Volatility Risk Premium, Discretization Error, Model Erro

Is jump risk priced? - What we can (and cannot) learn from option hedging errors : [This version: November 26, 2004]

When options are traded, one can use their prices and price changes to draw inference about the set of risk factors and their risk premia. We analyze tests for the existence and the sign of the market prices of jump risk that are based on option hedging errors. We derive a closed-form solution for the option hedging error and its expectation in a stochastic jump model under continuous trading and correct model specification. Jump risk is structurally different from, e.g., stochastic volatility: there is one market price of risk for each jump size (and not just \emph{the} market price of jump risk). Thus, the expected hedging error cannot identify the exact structure of the compensation for jump risk. Furthermore, we derive closed form solutions for the expected option hedging error under discrete trading and model mis-specification. Compared to the ideal case, the sign of the expected hedging error can change, so that empirical tests based on simplifying assumptions about trading frequency and the model may lead to incorrect conclusions

Evaluating Discrete Dynamic Strategies in Affine Models

We consider the problem of measuring the performance of a dynamic strategy, rebalanced at a discrete set of dates, whose objective is that of replicating a claim in an incomplete market driven by a general multi-dimensional affine process. The main purpose of the paper is to propose a method to efficiently compute the expected value and variance of the hedging error of the strategy. Representing the pay-off the claim as an inverse Laplace transform, we are able to get semi-explicit formulas for strategies satisfying a certain property. The result is quite general and can be applied to a very rich class of models and strategies, including Delta hedging. We provide illustrations for the cases of interest rate models and Heston's stochastic volatility model.

Robust hedging of digital double touch barrier options

In this dissertation, we present basic idea and key results for model-free pricing and hedging of digital double barrier options. Besides we extend this model to the market with non-zero interest rate by allowing some model-based trading. Moreover we apply this hedging strategies to Heston stochastic volatility model and compare its performances with that of delta hedging strategies in such setting. Finally we further interpret these numerical results to show the advantages and disadvantages of these two types of hedging strategies

Stochastic relaxational dynamics applied to finance: towards non-equilibrium option pricing theory

Non-equilibrium phenomena occur not only in physical world, but also in finance. In this work, stochastic relaxational dynamics (together with path integrals) is applied to option pricing theory. A recently proposed model (by Ilinski et al.) considers fluctuations around this equilibrium state by introducing a relaxational dynamics with random noise for intermediate deviations called ``virtual'' arbitrage returns. In this work, the model is incorporated within a martingale pricing method for derivatives on securities (e.g. stocks) in incomplete markets using a mapping to option pricing theory with stochastic interest rates. Using a famous result by Merton and with some help from the path integral method, exact pricing formulas for European call and put options under the influence of virtual arbitrage returns (or intermediate deviations from economic equilibrium) are derived where only the final integration over initial arbitrage returns needs to be performed numerically. This result is complemented by a discussion of the hedging strategy associated to a derivative, which replicates the final payoff but turns out to be not self-financing in the real world, but self-financing {\it when summed over the derivative's remaining life time}. Numerical examples are given which underline the fact that an additional positive risk premium (with respect to the Black-Scholes values) is found reflecting extra hedging costs due to intermediate deviations from economic equilibrium.Comment: 21 pages, 4 figures, to appear in EPJ B, major changes (title, abstract, main text