A simple method for extracting the probability of default from American put option prices

• DOI: http://doi.org/10.1002/fut.22146
• Date: 01 October 2020
• Author: Greg Orosi,Bo Young Chang
• Published date: 01 October 2020

J Futures Markets. 2020;40:1535–1547. wileyonlinelibrary.com/journal/fut © 2020 Wiley Periodicals LLC

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1535

Received: 14 June 2019

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Accepted: 20 May 2020

DOI: 10.1002/fut.22146

RESEARCH ARTICLE

A simple method for extracting the probability of default

from American put option prices

Bo Young Chang

1

|Greg Orosi

2

1

Bank of Canada, 234 Wellington Street,

Ottawa, Ontario, Canada

2

Department of Mathematics and

Statistics, American University of Sharjah,

Sharjah, UAE

Correspondence

Bo Young Chang, Bank of Canada, 234

Wellington Street, Ottawa, ON K1A 0G9,

Canada.

Email: bchang@bankofcanada.ca

Abstract

We present a novel method for extracting the risk‐neutral probability of default

(PD) of a firm from American put option prices. Building on the idea of a

default corridor proposed by Carr and Wu, we derive a parsimonious closed‐

form formula for American put option prices from which the PD can be in-

ferred. The method is easy to implement. Our empirical results based on seven

large US firms for the period 2002–2010 show that, in some cases, our option‐

implied PD can provide a more accurate estimate of default probability than

the estimates implied from credit default swaps.

KEYWORDS

American put option, arbitrage, default, lower bounds, probability of default

1|INTRODUCTION

We introduce a simple method for extracting the probability of a firm's default from the prices of American put options

on the firm's stock. Building on the idea of a default corridor proposed by Carr and Wu (2011; hereafter, referred to as

CW) and the option‐pricing method proposed by Orosi (2015), we first derive a new parsimonious closed‐form pricing

formula for American put options, which incorporates the possibility of default. We then calibrate the parameters of the

proposed pricing formula to the observed put option prices to obtain an estimate of the probability of default (PD).

An important insight of CW is that American put option values are not determined by the dynamics of the stock price

within a certain low‐strike region. These authors show that in this low‐strike region, American put option prices are simply

linear in strike and the slope of this linear segment is all we need to know to estimate a firm's default probability. Although

theoretically appealing, CW's proposed implementation method has an important limitation in that the deep‐out‐of‐the‐

money options their method requires are not quoted in most cases. Our proposed method helps overcome this limitation.

Taylor, Tzeng, and Widdicks (2014) point out that there has been limited use of equity options to directly estimate

the PD. Nonetheless, the advantage of extracting the option‐implied PD is supported by the empirical tests of Camara,

Popova, and Simkins (2012). They find that employing European‐style options to estimate PD is superior to Moody's

Kealhofer, McQuown, and Vasicek (KMV) model (Bohn, Arora, & Korablev, 2005) and other popular approaches.

Chang and Orosi (2017) also develop a methodology for extracting the default probability from the prices of low‐strike

in‐the‐money call options. However, the calibration of a call option‐pricing formula to the observed prices poses a

significant challenge because of the lack of available market quotes on these call options.

Alternatively, one can employ putoptions that are significantlymore liquid at low strikes. However, to our knowledge,

in the current literature, there is no straightforward method available for pricing and calibrating American‐style put

options on a defaultable stock, which poses a significant challenge for extracting the PD from these options. By deriving a

parsimonious pricing formula for American put options that incorporates the possibility of default, we provide a simple

method for estimating a firm's PD from put options.