Approximate pricing of American exchange options with jumps

• Published date: 01 June 2022
• Author: Guanghua Lian,Robert J. Elliott,Petko Kalev,Zhaojun Yang
• Date: 01 June 2022
• DOI: http://doi.org/10.1002/fut.22316

Received: 18 September 2021

|

Accepted: 21 January 2022

DOI: 10.1002/fut.22316

RESEARCH ARTICLE

Approximate pricing of American exchange options

with jumps

Guanghua Lian

1,2

|Robert J. Elliott

3,4

|Petko Kalev

5

|Zhaojun Yang

2

1

Devision of Global Market, UBS Group AG, Central, Hong Kong

2

Department of Finance, Southern University of Science and Technology, Shenzhen, China

3

School of Business, University of South Australia, Adelaide, South Australia, Australia

4

Haskayne School of Business, University of Calgary, Calgary, Alberta, Canada

5

Department of Economics & Finance, La Trobe University, Melbourne, Victoria, Australia

Correspondence

Guanghua Lian, Devision of Global

Market, UBS Group AG, UBS AG, 2 IFC,

8 Finance St, Central, Hong Kong,

Hong Kong.

Email: andy.lian@ubs.com and

guanghua_lian@berkeley.edu

Funding information

National Natural Science Foundation of

China, Grant/Award Number: 72031003;

Australian Research Council; Social

Sciences and Humanities Research

Council of Canada; Philosophy and Social

Science Plan Project of Guangdong

Province in China, Grant/Award Number:

GD19CYJ23

Abstract

This paper presents a numerical method to price American exchange options

based on jump‐diffusion processes. We first derive a closed‐form expression

for the value of European exchange options, then decompose the value

function of an American exchange option into a European counterpart, and an

early exercise premium that is derived analytically. The early exercise

boundary for an American exchange option approximately satisfies an alge-

braic equation that can be quickly numerically solved. Consequently, a for-

mula is obtained for efficiently pricing American exchange options. The

numerical results reveal that our pricing formula is robust and accurate.

KEYWORDS

American options, exchange options, jump diffusion, option pricing

JEL CLASSIFICATION

G13

1|INTRODUCTION

An exchange option gives the option holder a right to exchange a given quantity of one asset for a given quantity of

another asset and, among other areas, it is predominantly used in foreign exchange, fixed‐income, and equitymarkets. In

practice, many financial assets and real investment opportunities can be analyzed as American exchange options. This

paper develops a formula to solve the pricing problem of American exchange options, where the dynamics of the two

underlying stocks are modeled by geometric Brownian motions with jump‐diffusions. In the literature, research on

American exchange options which depend on two underlying assets is yet rare, although there exist many papers

discussing American options written on one asset. Readers with interests in American options written on one asset are

referred to Detemple (2006) who provided a comprehensive survey on American options. Early research on exchange

options focused only on European options. Margrabe (1978) was among the first to derive a pricing formula for a

J Futures Markets. 2022;42:983–1001. wileyonlinelibrary.com/journal/fut © 2022 Wiley Periodicals LLC

|

983

Guanghua Lian has declared that the analyses and conclusions expressed in this paper are those of the author and do not necessarily reflect views of

UBS Group AG, or other members of UBS Group AG.

European style exchange option in a Black–Scholes framework where the stock prices are modeled by correlated geo-

metric Brownian motion processes. Margrabe (1978)'s approach was later generalized by Wong (2008)whorelaxedsome

restrictions on the drift terms of geometric Brownian motion processes. Kirk and Aron (1995) and Bjerksund and

Stensland (2014), respectively, developed approximate formulae for spread options under geometric Brownian motion

processes with no jumps. Although Gerber and Shiu (1996) also discussed the pricing of a perpetual American exchange

option under pure‐diffusion dynamics, the finite‐lived American exchange options remained untouched. Under two

geometric Brownian motion processes, Broadie and Detemple (1997) presented integral equations for early exercise

boundary and option prices for finite‐lived American spread and exchange options. M.‐L. Liu and Liu (2009) and Wong

(2008), respectively, proposed a new method for the pricing of perpetual American exchange options, under pure‐

diffusion dynamics, from the perspective of solving a two‐variable free boundary problem. Cheang and Chiarella (2011)

presented a probabilistic representation for the American style exchange optionunder jump‐diffusion dynamics, buttheir

representation is yet to be solved to provide numerical values which would assist trading or investment decisions. Garces

and Cheang (2021) recently presents a numerical method (method of lines) to determine prices of European and

American exchange options when underlying asset prices are modeled with stochastic volatility and jump‐diffusion

dynamics. Thisarticle proposes a new techniqueto derive an efficient, analyticalpricing formula for finite‐lived American

exchange options, when the underlying asset prices follow jump‐diffusion processes.

In the literature, there are many papers discussing pricing derivatives (but not yet exchange options) with jump

diffusions, for example, Feng and Linetsky (2008), Mina et al. (2015), Yang (2018), W. Liu and Zhu (2019) and among

many others. There is also empirical evidence in the literature (see, e.g., Eraker, 2004, Chang et al., 2019), showing there

are jumps in asset prices. In financial markets, stocks or indices sometimes experience large movements of more than

5%–10% on a single day,for example, as seen in the Flash Crash.The Dow Jones Industrial Average (DJIA) plunged about

1000 points (about 9%) within 13 min during 2:32 p.m. and 2:45p.m. on May 6, 2010, and then recovered 600 points in

20 min.

1

Flash crashes have also occurred in the fixed income and forex markets. First, on October 15, 2014, the yield on

the benchmark 10‐year US Treasury bond experienced “a16‐basis‐point drop and then rebound between 9:33am and

9:45am”.

2

Second, on March 18, 2015, the US dollar lost more than 3% of its value against the Euro and other currencies

in just under 4 min. In addition, market‐wide shocks and crashes other than flash crashes are also observed and studied.

For instance, Bongaerts et al. (20 21) investigate intraday market‐wide stock price crashes to 12 international stock

markets. The authors find some strong evidence of spillover effects of jumps in stock prices, market liquidity, and trading

activity across the majority of studied markets which are to a lesser extent due to “liquidity dry‐ups”but most likely are a

response to macroeconomic announcements. Finally, a more recent example of an intraday shock with a permanent price

impact is the US dollar steadily declining against all other major currencies since the start of the Covid‐19 pandemic in

March 2020.

3

These observations suggest a pure diffusion model alone is not capable of describing important jump

characteristics of asset prices, because, if in a continuous‐diffusion model, it should be very low probabilities for these

observed large movements to occur in such a short time. Adding jump diffusion dynamics may provide a better model,

and it is of importance to include jump‐diffusion feature for the pricing of American exchange options.

The contribution of this study is twofold. First, it extends the existing research in the literature on European exchange

options, and perpetual American exchange options to derive analytical approximations for finite‐horizon American

exchange options, under jump‐diffusion processes. Second, our new method improves significantly the computational

efficiency in comparison with the results of Lindset (2007), which is the most relevant work in the literature. Under the

same framework of jump‐diffusion processes, Lindset (2007)extendedGeskeandJohnson(1984)'s approach to value an

American exchange option, combining a European exchange option and a Bermudan exchange option with two exercise

dates by using Richardson extrapolation as in Geske and Johnson (1984). Our method also provides higher numerical

accuracies for American exchange options when time to maturity is large, as discussed in Section 3.

1

The joint report of US Commodity Futures Trading Commission (CFTC) and Securities and Exchange Commission (SEC) describes the event details

of Flash Crash. At around 2:32 p.m. on May 6, 2010, the trader of a large fundamental broker initiated a “selling”algorithm for the June 2010 E‐mini

S&P500 stock index futures contract, targeting aggressively an execution fixed rate of 9% of the total trading volume based on the previous minute

trade data. Kirilenko et al. (2017) study the trading pattern of intraday intermediaries during the event. See https://www.sec.gov/files/marketevents-

report.pdf.

2

See Joint Staff Report: The US Treasury Market on October 15, 2014 (July 13, 2015), prepared by the US Department of the Treasury, Board of

Governors of the Federal Reserve System, Federal Reserve Bank of New York, US Securities and Exchange Commission, and US Commodity Futures

Trading Commission, at https://home.treasury.gov/system/files/276/joint-staff-report-the-us-treasury-market-on-10-15-2014.pdf.

3

Roach (2021) reports in the Bloomberg Opinion, on January 25, 2021 (https://www.bloomberg.com/opinion/articles/2021-01-25/the-dollar-s-crash-

is-only-just-beginning).

984

|

LIAN ET AL.