Advanced Equity Derivatives: Volatility and Correlation (a review)

In Advanced Equity Derivatives: Volatility and Correlation, Sébastien Bossu reviews important concepts and recent developments in option pricing and modeling, including the latest generation of equity derivatives—namely, volatility and correlation derivatives. Readers should have some familiarity with basic equity derivatives pricing and advanced mathematics because this book references the Black–Scholes model and other formulas for exotics, from the most common to cutting edge. This background is also required for readers who want to solve the problems at the end of each chapter.

In 2004, while working as an equity derivatives analyst at J.P. Morgan in London, Bossu found someone selling correlation and buying it back simultaneously, using two different methods. He discovered that with some corrections, this trade led to a pure dynamic arbitrage. One of the two correlation instruments involved in the trade—the correlation swap—was not priced at fair value. In the book, Bossu introduces and assesses his own refined model and the work of others in this field.

Although the European digital and geometric Asian options are priced using the Black–Scholes formula, barrier options, lookback options, forward start options, and multiasset exotics (based on several underlying stocks or indices) are preferably priced using the local volatility model, a stochastic volatility model, or Monte Carlo simulations.

The Black–Scholes model assumes a single constant-volatility parameter to price options. In practice, however, every listed vanilla option has a different implied volatility for each strike and maturity. Developed by Emanuel Derman and Iraj Kani and by Bruno Dupire in the early 1990s, the local volatility model has become the benchmark model to price and hedge a wide range of such equity exotics as digitals, Asians, and barriers. The local volatility model is best visualized on a binomial tree where, instead of using constant volatility at each node to generate the tree of future spot prices, a different volatility parameter is used at each node. The option is then priced using backward induction. Certain payoffs, however, such as forward start options, are better approached using a stochastic volatility model. These analyses require a high-quality, smooth, implied volatility surface as an input, along with the simulation of all intermediate spot prices until maturity, using short time steps.

As implied by its name, a volatility surface is a three-dimensional graph that plots implied volatilities across option strikes and terms to maturity. There are challenges, however, in creating an implied volatility surface. For one thing, graphing implied volatilities against strike prices for a given expiration yields a skewed “smile” instead of the expected flat surface. This anomaly arises because the standard Black–Scholes option-pricing model assumes constant volatility and lognormal distributions of underlying asset returns. Second, implied volatilities derived from listed option prices are available for only a finite number of listed strikes and maturities. Therefore, it is important to be able to interpolate or extrapolate implied volatilities. Linear interpolation has limitations because it produces a cracked smile curve. Alternative techniques, such as cubic splines, are often used to obtain a smooth curve. Extrapolation is also a difficult endeavor. How can one price a five-year option if the longest listed maturity is two years? Fortunately, some researchers have published the models that market makers at a large equity option house use to mark their positions. There are two ways to model the volatility surface:

1. directly, by specifying a functional form, such as a parametric function or an interpolation/extrapolation method, and
2. indirectly, by modeling the behavior of the underlying asset from the geometric Brownian motion posited by the Black–Scholes model.

The most popular parameterization of the smile of fixed maturity is Jim Gatheral’s SVI (stochastic volatility inspired) model. The implied volatility methods include the SABR (stochastic alpha, beta, and rho) model, the Heston model, and the LNV (lognormal variance) model.

The profit or loss on a delta-hedged option position is driven by the spread between two types of volatility: the instant realized volatility of the underlying stock or index and the option-implied volatility. Option traders, who are specialists in volatility, want to trade it directly—hence, the creation of options on volatility itself. Examples include variance swaps and CBOE Volatility Index, or VIX, futures and options. The payoffs on the former are based on realized volatility, whereas those on the latter are based on implied volatility.

The development of multiasset exotic products made it possible (and at times necessary) to trade correlation more or less directly. The first correlation trades were actually dispersion trades in which a long or short position on a multiasset option (such as an index) was offset by a reverse position on single-asset options (individual constituents of the index). The two most popular types of dispersion trades are vanilla dispersions and variance dispersions. In recent years, the concept of local volatility has been extended to multiple assets, leading to local correlation models: LVLC (local volatility with local correlation) and the dynamic local correlation model. LVLC allows pairwise correlation coefficients to depend on time and spot prices. Alex Langnau’s dynamic local correlation model reproduces the index smile very accurately.

Finally, stochastic correlation models provide a more realistic approach to the pricing and hedging of certain types of derivatives, including worst-of options, best-of options, correlation swaps, and correlation options. Various types of stochastic correlation models—from single correlation to tradable average correlation (including the beta-omega, or B-O, model) to correlation matrix, all using some form of Jacobi processes—are discussed in the last chapter.

—C.G.S.