The Journal of Finance

Option Profit and Loss Attribution and Pricing: A New Framework

Published: 02/19/2020   |   DOI: 10.1111/jofi.12894

PETER CARR, LIUREN WU

This paper develops a new top‐down valuation framework that links the pricing of an option investment to its daily profit and loss attribution. The framework uses the Black‐Merton‐Scholes option pricing formula to attribute the short‐term option investment risk to variation in the underlying security price and the option's implied volatility. Taking risk‐neutral expectation and demanding no dynamic arbitrage result in a pricing relation that links an option's fair implied volatility level to the underlying volatility level with corrections for the implied volatility's own expected direction of movement, its variance, and its covariance with the underlying security return.

The Finite Moment Log Stable Process and Option Pricing

Published: 03/21/2003   |   DOI: 10.1111/1540-6261.00544

Peter Carr, Liuren Wu

We document a surprising pattern in S&P 500 option prices. When implied volatilities are graphed against a standard measure of moneyness, the implied volatility smirk does not flatten out as maturity increases up to the observable horizon of two years. This behavior contrasts sharply with the implications of many pricing models and with the asymptotic behavior implied by the central limit theorem (CLT). We develop a parsimonious model which deliberately violates the CLT assumptions and thus captures the observed behavior of the volatility smirk over the maturity horizon. Calibration exercises demonstrate its superior performance against several widely used alternatives.

Specification Analysis of Option Pricing Models Based on Time‐Changed Lévy Processes

Published: 11/27/2005   |   DOI: 10.1111/j.1540-6261.2004.00667.x

Jing‐zhi Huang, Liuren Wu

We analyze the specifications of option pricing models based on time‐changed Lévy processes. We classify option pricing models based on the structure of the jump component in the underlying return process, the source of stochastic volatility, and the specification of the volatility process itself. Our estimation of a variety of model specifications indicates that to better capture the behavior of the S&P 500 index options, we need to incorporate a high frequency jump component in the return process and generate stochastic volatilities from two different sources, the jump component and the diffusion component.

What Type of Process Underlies Options? A Simple Robust Test

Published: 11/07/2003   |   DOI: 10.1046/j.1540-6261.2003.00616.x

Peter Carr, Liuren Wu

We develop a simple robust method to distinguish the presence of continuous and discontinuous components in the price of an asset underlying options. Our method examines the prices of at‐the‐money and out‐of‐the‐money options as the option's time‐to‐maturity approaches zero. We show that these prices converge to zero at speeds that depend upon whether the underlying asset price process is purely continuous, purely discontinuous, or a combination of both. We apply the method to S&P 500 index options and find the existence of both a continuous component and a jump component in the index.