An examination of the effectiveness of static hedging in the presence of stochastic volatility

• Date: 01 September 2003
• DOI: http://doi.org/10.1002/fut.10089
• Author: Jason Fink
• Published date: 01 September 2003

ANEXAMINATION OF THE

EFFECTIVENESS OF STATIC

HEDGING IN THE PRESENCE

OF STOCHASTIC VOLATILITY

JASON FINK

Toft and Xuan (1998) use simulation evidence to demonstrate that the

static hedging method of Derman et al. (1995) performs inadequately

when volatility is stochastic. Particularly, the greater the “volatility of

volatility,” the poorer the static hedge. This article presents an alternative

static hedging methodology, denoted the generalized static hedge, that

appears to perform more reliably. Specifically, the value, delta, and vega of

the static hedges closely approximate those values of the barrier option

being hedged. Further, simulation evidence indicates that when volatility

of volatility is large, the standard deviation of simulated cash flows from

the generalized static hedge position is less than the standard deviation of

simulated cash flows from previously defined static hedge positions.

© 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:859–890, 2003

The author would like to thank T. Wake Epps, Kristin Fink, Chris Otrok, Larry Kochard, partici-

pants at the University of Virginia finance research seminar, Chris Mitchell, and an anonymous

referee for valuable comments and suggestions. All remaining errors are the responsibility of the

author.

For correspondence, Jason Fink, MSC 0203 Finance and Business Law Program, James Madison

University, Harrisonburg, Virginia 22807; e-mail: finkjd@jmu.edu

Received February 2002; Accepted February 2003

■Jason Fink is an assistant professor of finance in the Department of Finance and

Business Law at James Madison University in Harrisonburg, Virginia.

The Journal of Futures Markets, Vol. 23, No. 9, 859–890 (2003) © 2003 Wiley Periodicals, Inc.

Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fut.10089

860 Fink

1Carr et al. (1998) provide an interesting, alternative static hedging methodology.

INTRODUCTION

Derman et al. (1995) present a method to hedge statically a large class of

barrier options when an infinite number of European options are avail-

able. This method may be modified slightly to provide an approximate

static hedge with a finite number of options.1Toft and Xuan (1998) pres-

ent simulation evidence that this approximate static hedge, although

effective when volatility is constant, is not nearly so reliable when volatil-

ity is stochastic. There appears to be significant evidence that volatility

follows a stochastic process that is distinct from, although correlated

with, the stochastic process driving stock prices. The findings of Dumas

et al. (1998) support this by demonstrating that use of a flexible, deter-

ministic volatility function, which is a function of asset price and time,

does not exhibit better hedging or pricing performance when compared

to a reasonable implied volatility method. Bakshi et al. (1997) further

conclude that “taking stochastic volatility into account is of first order

importance” in devising an improvement over the Black-Scholes (1973)

formula. Consequently, poor performance when volatility is stochastic is a

significant indictment of the reliability of a static hedge. This article pres-

ents a simple generalized methodology that overcomes this obstacle.

Numerous studies show that stochastic volatility models give more

accurate predictions of option prices than models without this feature.

Since the 1987 crash, implied volatilities at low strike/price ratios, corre-

sponding to out-of-the-money puts and in-the-money calls, have been

much higher than those of at-the-money options. This is the familiar

“volatility skew.” Various stochastic volatility models have been proposed

to explain these and similar biases that existed pre-1987 (Bakshi et al.,

1997; Bates, 1996; Heston, 1993; Heston & Nandi, 2000; Hull &

White, 1987; Scott, 1987, 1997; Stein & Stein, 1991).

Heston (1993) obtains a computationally feasible expression for

European option prices in a stochastic volatility model that allows corre-

lation between the innovations to stock price and volatility. Nandi (1998)

demonstrates empirically the importance of this stock price-volatility

correlation when hedging options on the S&P 500, and finds that the

Heston model produces better hedges than Black-Scholes, but only

when correlation is permitted to be nonzero. Bakshi et al. (1997) find

that a model that allows for discontinuities in price paths does even

better for pricing but that continuous path models with stochastic

volatility are better for hedging.

Examination of the Effectiveness of Static Hedging 861

Given these findings, we will present a generalized static hedging

algorithm assuming that true option prices are given by the Heston

(1993) stochastic volatility model. This assumption is employed by Toft

and Xuan (1998), who establish a static hedge in the presence of

stochastic volatility that uses the implied volatility function of Dumas

et al. (1998). Their results indicate that this method tends to yield vegas

of the static hedges that are about twice the size of the barrier option

being hedged. We find that the generalized static hedging methodology

does not result in this kind of overweighting and, consequently, is better

able to match both the comparative statics and value of the barrier

option.

The next section presents the generalized static hedging method-

ology, and compares that construction with static hedge procedures

based on implied volatility functions. Section III describes Monte Carlo

simulations conducted to determine the effectiveness of the generalized

static hedge and compares those with other approaches. Section IV

examines the effectiveness of the generalized static hedge, as well as an

implied-volatility static hedge, in an environment approximating market

conditions. Section V concludes.

GENERALIZED STATIC HEDGING

TECHNIQUE

The Underlying Model

We assume that the stochastic process that drives the price of the under-

lying asset is given by:

(1)

where dis a continuously paid dividend rate, kdetermines the rate of

mean reversion of the variance, uis its long run mean, and sdetermines

the volatility of the variance. sis often referred to as “volatility of

volatility.” W1and W2are Wiener processes with correlation r. Heston

(1993) derives the following expression for the value of a European call

option when the stock price follows this process:

C(S, v, X, t)SedtP1XertP2

dv(t) k[uv(t)] dt s2v(t) dW2(t)

dS(t)

S(t) (md) dt 2v(t) dW1(t)