Option Profit and Loss Attribution and Pricing: A New Framework

• Author: LIUREN WU,PETER CARR
• Date: 01 August 2020
• DOI: http://doi.org/10.1111/jofi.12894
• Published date: 01 August 2020

THE JOURNAL OF FINANCE •VOL. LXXV, NO. 4 •AUGUST 2020

Option Profit and Loss Attribution and Pricing:

A New Framework

PETER CARR and LIUREN WU∗

ABSTRACT

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 in-

vestment risk to variation in the underlying security price and the option’s implied

volatility. Taking risk-neutral expectation and demanding no dynamic arbitrage re-

sult 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.

A foolish consistency is the hobgoblin of little minds.

—Ralph Waldo Emerson

DIFFERENT MODELING FRAMEWORKS SERVE DIFFERENT purposes. A major focus

of the existing option pricing literature is to derive option values that are inter-

nally consistent across all strikes and maturities. The literature specifies the

full dynamics of the underlying security price, including the full dynamics of

its instantaneous variance rate, and performs valuation of all options by taking

risk-neutral expectations of their terminal payoffs. The full dynamics specifi-

cation creates a single reference distribution of the relevant terminal random

variable, which is then used to take the expectation. Under this approach, even

if the assumed dynamics are wrong, the valuations on the option contracts re-

main consistent with one another relative to this erroneous reference.

∗Peter Carr is with New York University. Liuren Wu is in the Department of Economics and

Finance, Baruch College. Wewould like to thank Stefan Nagel (the Editor), the associate editor, and

two anonymous referees. Wewould also like to thank Yun Bai; David Gershon; Kris Jacobs; Danling

Jiang; Aaron Kim; Chun Lin; Jason Roth; Angel Serrat; Stoyan Stoyanov; seminar participants

at Baruch College, Credit Suisse, RMIT, Stony Brook University, City University of New York

Graduate Center, and the 2017 6th IFSID Conference on Derivatives for comments. Liuren Wu

gratefully acknowledges the support of a grant from the City University of New York PSC-CUNY

Research Award Program. We have read The Journal of Finance’s disclosure policy and have no

conflicts of interest to disclose.

Correspondence: Liuren Wu, Department of Economics and Finance, Baruch College, One

Bernard Baruch Way, Box B10-225, New York,NY 10010; email: liuren.wu@baruch.cuny.edu.

DOI: 10.1111/jofi.12894

C2020 the American Finance Association

2271

2272 The Journal of Finance R

It is good to be consistent, but it is not good to be wrong. Unfortunately,

the assumed dynamics of the underlying security price and its instantaneous

volatility often deviate strongly from reality. For example, to price long-dated

options, this approach needs to make projections on the underlying security

price and its instantaneous volatility far into the future. The accuracy of long-

dated projections is understandably low, and seemingly innocuous stationarity

assumptions on the instantaneous volatility dynamics often generate much

lower price variation (Giglio and Kelly (2018)) and much flatter implied volatil-

ity smiles (Carr and Wu (2003)) in long-term contracts than actually observed

in the data.

In practice, as long as one does not hold the contracts to maturity,one does not

necessarily need to make long-run predictions to trade long-dated contracts—

An investor can hold a very long-dated contract for a very short period of

time. In this case, the investor is less concerned about the terminal payoff

than about the factors that drive profit and loss over the short holding period.

In fact, the standard recommended practice of marking financial securities to

market makes it vitally important that investors understand the magnitude

and sources of daily value fluctuations, regardless of their intended holding

period. The process of attributing an investment’s profit and loss (P&L) on

a given date to different risk exposures is commonly referred to as the P&L

attribution process.

In this paper, we develop a new valuation framework that links the pricing

of a security at a given point in time to its P&L attribution without directly

referring to the terminal payoffs of the investment. The P&L attribution re-

quires the specification of a risk structure for computing the investment’s risk

exposures and magnitudes. We take an investment in a European option as an

example and perform the P&L attribution on the option contract via the explicit

Black-Merton-Scholes (BMS) option pricing formula. Black and Scholes (1973)

and Merton (1973) derive their option pricing formula by assuming constant-

volatility geometric Brownian motion dynamics for the underlying security

price. Their assumption does not match reality as return volatilities tend to

vary strongly over time. Nevertheless, their pricing equation has been widely

used by practitioners as a simple and intuitive representation of the option

value in terms of its major risk sources, that is, variations in the underlying

security price and its return volatility.In addition to the security price and con-

tract terms (such as strike and maturity), the pricing equation takes a volatility

input that can be used to match the observed option price. The volatility input

that matches the observed option price is commonly referred to as the BMS

option implied volatility.

A Taylor series expansion of the BMS option pricing formula attributes the

option investment P&L to partial derivatives in time, the underlying security

price, and the option’s implied volatility. When the underlying security price

and the option’s implied volatility move continuously over time, expanding to

the first order in time and second order in price and volatility is sufficient

to bring the residual error to an order lower than the length of the short

investment horizon.

Option Profit and Loss Attribution and Pricing 2273

Taking the risk-neutral expectation over the P&L attribution via the BMS

pricing formula and demanding no dynamic arbitrage result in a simple pricing

equation that relates the option’s implied volatility level to the underlying

instantaneous volatility as well as corrections due to the implied volatility’s

expected direction, variance, and covariance with the security return.

In contrast to the traditional option pricing approach, which links the values

of all option contracts to a single reference dynamics specification, our new

approach links the current fair value of one option contract’s implied volatility

to current conditional moments of log changes in the security price and the

given contract’s implied volatility. This subtle—but vital—shift in perspective

is due to the use of the option’s own implied volatility rather than the underlying

security’s instantaneous variance rate as a state variable. This new perspective

allows one to completely localize the valuation of an option contract’s implied

volatility level at a given point in time to the moment conditions on the variation

of this particular implied volatility at that point in time.

For comparison, one can regard the traditional option pricing framework as

a bottom-up valuation approach, with the key focus being specifying the ap-

propriate basis that centralizes the valuation of all contracts. The key strength

of this centralized bottom-up approach is that cross-sectional consistency is

maintained through the use of one set of reference dynamics. By contrast, our

new framework is a decentralized top-down approach. By pricing an option con-

tract based only on its own risk-neutral moment conditions, the theory does not

impose cross-sectional consistency across different option contracts. The new

approach is decidedly local in terms of the investment horizon it considers, and

very much decentralized in terms of the contract it values.

Under this new approach, to compare the valuation of two or more distinct

option contracts, one must first compare the risk exposures and magnitudes

of these contracts. One can impose common factor structures on the moment

conditions of different contracts to link the valuations of these contracts. Never-

theless, their commonality,or lack thereof, is left as an empirical determination

or theoretical construction, but not a no-arbitrage condition. Therefore, while

the traditional bottom-up approach is about the search for the omnipotent ba-

sis reference, our new top-down approach is more about accurate forecasts of

the moment conditions for the particular contract. The two approaches do not

directly compete, but rather complement each other. Dynamics specifications

from the traditional approach can provide insights for formulating hypothe-

ses on moment condition estimation, while empirically identified co-movement

patterns in the moment conditions can provide guidance for the specification

of reference dynamics.

To illustrate, we explore the cross-sectional pricing implications of the new

framework under various commonality assumptions. First, we define the at-

the-money option at a given maturity as the particular option whose log strike-

forward ratio is equal to half of the option’s total implied variance, which is the

BMS risk-neutral mean of the log security return to maturity.The fair valuation

of this at-the-money option does not depend on the variance and covariance of

the implied volatility change, but depends only on its risk-neutral drift and the

instantaneous variance level. By imposing a common risk-neutral drift on two