Roger Lee

Option Pricing by Transform Methods: Extensions, Unification, and Error Control

We extend and unify Fourier-analytic methods for pricing a wide class of options on any underlying state variable whose characteristic function is known. In this general setting, we bound the numerical pricing error of discretized transform computations, such as DFT/FFT. These bounds enable algorithms to select efficient quadrature parameters and to price with guaranteed numerical accuracy.

Asymptotics of Implied Volatility to Arbitrary Order

In a unified model-free framework that includes long-expiry, short-expiry, extreme-strike, and jointly-varying strike-expiry regimes, we generate implied volatility and implied variance approximations, with rigorous error estimates asymptotically smaller than any given power of L, where L denotes the exogenously given absolute log of an option price that approaches zero. Our results, therefore, sharpen to arbitrarily high order of accuracy (and, moreover, extend to general extreme regimes) the model-free asymptotics of implied volatility. We then apply these general formulas to particular examples: Heston (using a previously known L expansion) and Lévy (using saddlepoint methods to derive L expansions).

The small-time smile and term structure of implied volatility under the Heston model

We characterize the asymptotic smile and term structure of implied volatility in the Heston model at small maturities. Using saddlepoint methods we derive a small-maturity expansion formula for call option prices, which we then transform into a closed-form expansion (including the leading-order and correction terms) for implied volatility. This refined expansion reveals the relationship between the small-expiry smile and all Heston parameters (including the pair in the volatility drift coefficient), sharpening the leading-order result of Forde and Jacquier [Int. J. Theor. Appl. Finance, 12 (2009), pp. 861--876], which found the relationship between the zero-expiry smile and the diffusion coefficients.

Implied Volatility: Statics, Dynamics, and Probabilistic Interpretation

Given the price of a call or put option, the Black-Scholes implied volatility is the unique volatility parameter for which the Black-Scholes formula recovers the option price. This article surveys research activity relating to three theoretical questions: First, does implied volatility admit a probabilistic interpretation? Second, how does implied volatility behave as a function of strike and expiry? Here one seeks to characterize the shapes of the implied volatility skew (or smile) and term structure, which together constitute what can be termed the statics of the implied volatility surface. Third, how does implied volatility evolve as time rolls forward? Here one seeks to characterize the dynamics of implied volatility.

Hedging variance options on continuous semimartingales

We find robust model-free hedges and price bounds for options on the realized variance of [the returns on] an underlying price process. Assuming only that the underlying process is a positive continuous semimartingale, we superreplicate and subreplicate variance options and forward-starting variance options, by dynamically trading the underlying asset and statically holding European options. We thereby derive upper and lower bounds on values of variance options, in terms of Europeans.

Variation and share-weighted variation swaps on time-changed Lévy processes

For a family of functions G, we define the G-variation, which generalizes power variation; G-variation swaps, which pay the G-variation of the returns on an underlying share price F; and share-weightedG-variation swaps, which pay the integral of F with respect to G-variation. For instance, the case G(x)=x2 reduces these notions to, respectively, quadratic variation, variance swaps, and gamma swaps.We prove that a multiple of a log contract prices a G-variation swap, and a multiple of an FlogF contract prices a share-weighted G-variation swap, under arbitrary exponential Lévy dynamics, stochastically time-changed by an arbitrary continuous clock having arbitrary correlation with the Lévy driver, under integrability conditions.We solve for the multipliers, which depend only on the Lévy process, not on the clock. In the case of quadratic G and continuity of the underlying paths, each valuation multiplier is 2, recovering the standard no-jump variance and gamma-swap pricing results. In the presence of jump risk, however, we show that the valuation multiplier differs from 2, in a way that relates (positively or negatively, depending on the specified G) to the Lévy measure’s skewness.In three directions this work extends Carr–Lee–Wu, which priced only variance swaps. First, we generalize from quadratic variation to G-variation; second, we solve for not only unweighted but also share-weighted payoffs; and third, we apply these tools to analyze and minimize the risk in a family of hedging strategies for G-variation.

How Leverage Shifts and Scales a Volatility Skew: Asymptotics for Continuous and Jump Dynamics

To model leveraged investments such as leveraged ETFs, define the beta-leveraged product on a positive semimartingale S to be the stochastic exponential of beta times the stochastic logarithm of S.In various asymptotic regimes, we relate rigorously the implied volatility surfaces of the beta-leveraged product and the underlying S, via explicit shifting/scaling transformations. In particular, a family of regimes with jump risk admit a shift coefficient of -3/2, unlike the previously conjectured 1/2 shift. The 1/2, we prove, holds in a family of continuous (including fBm-driven) stochastic volatility regimes at short expiry and at small volatility-of-volatility. In another regime, which does not admit a simple spatial shifting/scaling rule, we find an expiry scaling together with a spatial transformation.