Pricing Forward Skew Dependent Derivatives. Multifactor Versus Single‐Factor Stochastic Volatility Models

• Author: Jacinto Marabel Romo
• Date: 01 February 2014
• Published date: 01 February 2014
• DOI: http://doi.org/10.1002/fut.21611

PRICING FORWARD SKEW DEPENDENT

DERIVATIVES.MULTIFACTOR VERSUS

SINGLE‐FACTOR STOCHASTIC VOLATILITY MODELS

JACINTO MARABEL ROMO*

Empirical evidence shows that, in equity options markets, the slope of the skew is largely

independent of the volatility level. Single‐factor stochastic volatility models are not flexible

enough to account for the stochastic behavior of the skew. On the other hand, multifactor

stochastic volatility models are able to account for the existence of stochastic skew. Thisstudy

studies the effects of introducing stochastic skew in the valuation of forward skew dependent

exotic options. In particular, I consider cliquet, as well as reverse cliquet structures. The study

also derives a semi‐closed‐form solution for the price of forward‐start options under the

multifactor stochastic specification. The empirical results indicate that the consideration of

additional volatility factors in the context of stochastic volatility models allows us to generate

more flexible smile patterns. This additional flexibility has a relevant impact on the valuation of

forward skew dependent derivatives. In this sense, this study shows that similar calibrations

of single factor and multifactor stochastic volatility models to the current market prices of

plain vanilla options can lead to important discrepancies in the pricing of exotic forward

skew dependent derivatives such as regular cliquet structures and reverse cliquet options.

© 2013 Wiley Periodicals, Inc. Jrl Fut Mark 34:124–144, 2014

1. INTRODUCTION

The assumptions of the Black–Scholes (1973) model imply that the implied volatility surface

should be flat and static. But since the stock market crash on October 1987, equity options

markets have been characterized by a persistent negative dependence of implied volatility with

respect to the strike price. This negative dependence is known as the implied volatility skew.

On the other hand, for foreign currencies out‐of‐the‐money options usually exhibit higher

implied volatilities than at‐the‐money options. This effect is known as the volatility smile.

Furthermore, the implied volatility surface displays term structure and varies stochastically

through time generating vega risk.

There are a number of models that have been proposed to deal with these stylized facts.

Local volatility models postulate that the instantaneous volatility (called local volatility) is a

deterministic function of the underlying asset price and time. Within this group we have the

works of Dupire (1994), Derman and Kani (1994), or Rubinstein (1994). Merton (1976),

among others, incorporates the possibility of jumps in the stochastic process for the

underlying asset price. This feature is consistent with the behavior of financial assets and

Jacinto Marabel Romo is an Equity Derivatives Trader, BBVA and a researcher, University Institute for

Economic and Social Analysis, University of Alcalá, Alcalá de Henares, Madrid, Spain. The content of this

paper represents the author’s personal opinion and does not reflect the views of BBVA.

*Correspondence author, Vía de los Poblados s/n, 28033 Madrid, Spain. Tel: þ34‐915379985, Fax: þ34‐

915370913. e‐mail: jacinto.marabel@bbva.com

Received September 2012; Accepted January 2013

The Journal of Futures Markets, Vol. 34, No. 2, 124–144 (2014)

© 2013 Wiley Periodicals, Inc.

Published online 25 February 2013 in Wiley Online Library (wileyonlinelibrary.com).

DOI: 10.1002/fut.21611

generates implied volatility skew. The main problem with these models is that when the jump

size is stochastic and it has a continuous distribution, the model becomes incomplete. Madan

and Milne (1991) and Carr, Geman, Madan, and Yor (2003), among others, consider Lévy

process‐based models. On the other hand, stochastic volatility models leave the constant

instantaneous volatility assumption of the Black–Scholes (1973) model and assume that

volatility follows a stochastic process possibly correlated with the process for the stock price.

Some examples of this approach can be found in Hull and White (1987) and Hagan, Kumar,

Lesniewski, and Woodward (2002), among others. Stochastic volatility models explain the

basic shapes of smile patterns and they also allow for more realistic theories of the term

structure of implied volatility (Lewis, 2000). One of the most celebrated single‐factor

stochastic volatility models is the Heston (1993) model. The main reason is that it allows the

computation of European option prices quite efficiently. One of the parameters of the model is

the correlation between the underlying asset return and its instantaneous variance. A negative

value for this parameter generates implied volatility skew. But the fact that, in single‐factor

stochastic volatility models, this correlation is constant implies that the model is not flexible

enough to account for the stochastic behavior of the skew observed in options markets (see,

e.g., Christoffersen, Heston, & Jacobs, 2009).

A number of models have been proposed to extend the Heston (1993) model in order to

account for the stylized facts that the model is not able to consider. In particular,

Christoffersen et al. (2009) extend the original Heston (1993) framework to generate a two‐

factor stochastic volatility model built upon the square root process. This model accounts for

stochastic correlation between the asset return and its instantaneous variance. Another

extension to the Heston (1993) stochastic volatility model is proposed by da Fonseca,

Grasselli, & Tebaldi (2008). These authors consider a Wishart specification to introduce a

correlation structure between the single asset noise and the volatility factors. This model is

also able to account for stochastic correlation between the asset return and its instantaneous

variance and, hence, is able to generate stochastic skew.

Note that forward skews and volatility of volatility, are outputs in stochastic volatility

models and cannot be adjusted to reproduce market observables. On the other hand, the local

stochastic volatility model combines the local volatility model, to fit the vanilla surface, and the

stochastic volatility model, to model forward skews (see, e.g., Jex, Henderson, & Wang, 1999;

Lipton, 2002; or Ren, Madan, & Qian, 2008). One advantage of the local stochastic volatility

(LSV) model is that it allows for an exact calibration to the market implied volatility surface as

opposite to the stochastic volatility best‐fit calibration. In this sense, since within the LSV

specification the vanilla surface is matched by construction, we have the freedom to specify

the parameters of the stochastic volatility process to match either historical data or market

data for non‐vanilla options. LSV models became popular in foreign exchange markets.

However, in equity markets they are less popular. The reason may be that one disadvantage of

LSV specifications is their tendency to misprice forward volatility products, such as cliquets,

as pointed out by Karasinski and Sepp (2012). In this sense, these authors propose a model

that is able to generate steeper forward skews.

Christoffersen et al. (2009) show that their model provides better empirical fit than the

Heston (1993) model to the market price of European options using multiple cross‐section

data corresponding to the Standard and Poor’s 500 equity index. The main reason is that, in

equity options markets, the slope of the skew is largely independent of the volatility level

(Derman, 1999). A single‐factor stochastic volatility model can generate steep skew or flat

skew at a given volatility level but cannot generate both for a given parameterization. Note

that in a purely cross‐sectional analysis this is not a problem since we can calibrate different

parameters levels for the one‐factor model to account for the time‐varying nature of the

cross‐section. In this sense, it is important to consider how practitioners use the models.

Forward Skew Dependent Derivatives and Stochastic Volatility 125