Semistatic hedging and pricing American floating strike lookback options

• Author: Jr‐Yan Wang,Pai‐Ta Shih,Yi‐Ta Huang,San‐Lin Chung
• Date: 01 April 2019
• DOI: http://doi.org/10.1002/fut.21986
• Published date: 01 April 2019

Received: 22 December 2017

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Accepted: 10 November 2018

DOI: 10.1002/fut.21986

RESEARCH ARTICLE

Semistatic hedging and pricing American floating strike

lookback options

San‐Lin Chung

1

|

Yi‐Ta Huang

1

|

Pai‐Ta Shih

1

|

Jr‐Yan Wang

2

1

Department of Finance, National Taiwan

University, Taipei, Taiwan

2

Department of International Business,

National Taiwan University, Taipei,

Taiwan

Correspondence

San‐Lin Chung, Department of Finance,

National Taiwan University, No. 1,

Section 4, Roosevelt Road, Taipei 10617,

Taiwan.

Email: chungsl@ntu.edu.tw

Funding information

Ministry of Science and Technology,

Taiwan, Grant/Award Number: 104‐2410‐

H‐002‐028‐MY3

Abstract

We price an American floating strike lookback option under the Black–Scholes

model with a hypothetic static hedging portfolio (HSHP) composed of

nontradable European options. Our approach is more efficient than the tree

methods because recalculating the option prices is much quicker. Applying put–

call duality to an HSHP yields a tradable semistatic hedging portfolio (SSHP).

Numerical results indicate that an SSHP has better hedging performance than a

delta‐hedged portfolio. Finally, we investigate the model risk for SSHP under a

stochastic volatility assumption and find that the model risk is related to the

correlation between asset price and volatility.

KEYWORDS

American floating strike lookback option, dynamic hedging, model risk, put–call duality, semistatic

hedging, stochastic volatility model

JEL CLASSIFICATION

G13

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INTRODUCTION

The lookback option is a useful exotic option in financial markets which can be used for risk management or trading.

For example, with floating strike puts the investor can sell the underlying stock at the maximum price during the life of

the option.

1

Even though many methods have been developed to price European‐style lookback options,

2

it remains

difficult to accurately price various types of lookback options; indeed, this has been the focus of recent literature. For

example, Kim, Park, and Qian (2011) propose a binomial tree method for lookback options under the jump‐diffusion

process, Kimura (2011) deals with valuation and premium decomposition of American fractional lookback options,

Leung (2013) derives an analytical pricing formula for the European floating strike lookback option under Heston's

(1993) stochastic volatility (SV) model, and Chang and Li (2018) evaluate amnesiac lookback options with Monte Carlo

simulation.

Although Lai and Lim (2004) provide an analytical solution of American fixed strike lookback options, their results

cannot be exploited to price American floating strike lookback options because the equivalence property does not hold

for American‐style floating strike and fixed strike lookback options.

3

Moreover, lattice‐based methods such as Babbs

J Futures Markets. 2019;39:418–434.wileyonlinelibrary.com/journal/fut418

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© 2018 Wiley Periodicals, Inc.

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Kimura (2011) indicates that the lookback feature has been incorporated into equity‐indexed annuities for years. Moreover, Chang and Li (2018) mention that lookback options can also be found in

the gold market, and suggest that the lookback feature in extremely volatile markets lends itself to use in cryptocurrency markets.

2

For example, see Goldman, Sosin, and Gatto (1979), Conze and Viswanathan (1991), Heynen and Kat (1995), and Wong and Kwok (2003), among others.

3

Eberlein and Papapantoleon (2005) provide a symmetric relationship between European‐style floating strike and fixed strike lookback options for assets driven by general Lévy processes. However,

the equivalence property does not hold for American‐style lookback options.

(2000) can be applied to price American floating strike lookback options. We fill the gap in the literature by proposing

two static hedging portfolios for the valuation and the hedge of American floating strike lookback options, respectively.

The basic idea underlying the static replication approach of Derman, Ergener, and Kani (1995) is to form a portfolio

of standard European options to match the boundary conditions before maturity and the terminal condition at maturity

of the exotic option that will be hedged. This approach has been extended to many types of options: American options

(Chung & Shih, 2009; Chung, Shih, & Tsai, 2013), European‐style Asian options (Albrecher, Dhaene, Goovaerts, &

Schoutens, 2005), European‐style installment options (Davis, Schachermayer, & Tompkins, 2001), and models other

than the Black–Scholes (BS) model (e.g., see Andersen, Andreasen, & Eliezer, 2002; Fink, 2003; Nalholm & Poulsen,

2006; Takahashi & Yamazaki, 2009).

For our research question, the difficulty arises from the fact that the static hedging portfolio of an American floating

strike lookback option generally depends on the maximum (or minimum) price observed so far. In other words,

whenever a new maximum price is observed, one must liquidate the original static hedging portfolio and then

immediately create a new hedging portfolio, which is not consistent with the meaning of “static.”To solve this problem,

as suggested by Schroder (1999) and Babbs (2000), we employ the underlying asset price as the numeraire and derive a

new partial differential equation (PDE) where the relative price of an American floating strike lookback option with

respect to the underlying asset price must follow.

By observing this new PDE and its boundary conditions, we discover that the pricing problem of the relative price of

an American floating strike lookback put option (AFSLPO) can correspond to that of an American call option in a

hypothetic world where the risk‐free rate equals

q

and the dividend yield of the underlying asset equals

r

. Note that

q

and

r

denote the dividend yield and risk‐free rate, respectively, in the real world. Therefore, we directly follow Chung

and Shih (2009) to form a static hedging portfolio for the corresponding American call option.

It should be emphasized that the component European options utilized in the above static hedging portfolio do not

exist in the real world. Instead, they are European options under the hypothetic world mentioned above. Thus we term

our portfolio a hypothetic static hedging portfolio (HSHP hereafter). Once the HSHP is obtained, the current relative

price of an AFSLPO (with respect to the current asset price) equals the value of HSHP and one can then multiply this

portfolio value by the current asset price to get the cash price of the AFSLPO.

Although our HSHP is effective for pricing, it is not realistic for hedging. Fortunately, one can follow Fajardo and

Mordecki (2006) to apply the put–call duality under the BS model to transform an HSHP into a portfolio composed of

European options existing in the real world.

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However, it should be noted that the portfolio transformed from an HSHP

is not a static portfolio because it must be rebalanced whenever a new maximum stock price is observed. For this

reason, we term this portfolio a semistatic hedging portfolio (SSHP henceforth).

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One special feature of an SSHP is that

the calls and puts of the newly rebalanced portfolio have different strike prices, depending on the new realized

maximum of the asset price, but with the same portfolio weights.

To sum up, we offer contributions for market participants of AFSLPOs in three aspects. First of all, for pricing

AFSLPOs, our HSHP approach is efficient and generates monotonically convergent option values with the increase of

the number of time points with matched boundaries. Moreover, the proposed approach preserves an attractive feature

of the static hedging approach: The recalculation of option prices is very quick with the passage of time and/or with

changes in the asset prices. Our numerical analysis suggests that the recalculation time is generally less than 5% of the

initial computational time due to the fact that there is no need to solve the static hedging portfolio weights again. In

contrast, the recalculation time of most (if not all) existing numerical methods is the same as the initial computational

time. Secondly, for hedging purposes, one can construct an SSHP for an AFSLPO. We also examine the hedging

performance of the SSHP in terms of the distribution of hedging errors. The numerical analyses indicate that the

hedging performance of our SSHP is far less risky than that of a delta‐hedged portfolio (DHP hereafter). Finally, we

investigate the model risk for our SSHP approach under Heston’s (1993) SV model. We find that there is model risk and

that it becomes serious especially when the asset price and its volatility are positively correlated. This is because a

higher correlation leads to higher costs for rebalancing the SSHP and thus further deterioration in the hedging

performance of our SSHP approach developed under the BS model.

4

The put–call duality is a general formula to explain the relationship between calls and puts with different strikes; see Equations 14 and 15 of Fajardo and Mordecki (2006). Under the BS model, the

put–call duality is reduced to a simple symmetric relationship in which the price of a call option with underlying asset price S, strike price

X

, interest rate

r

, and dividend yield

q

is equal to the price of

an otherwise identical put option with asset price

X

, strike price

S

, interest rate

q

, and dividend yield

r

. Refer to Carr, Ellis, and Gupta (1998) and McDonald and Schroder (1998) for the put–call

symmetry of European and American options, respectively.

5

Our numerical experiments suggest that the average number of portfolio rebalances is generally small for typical parameter values. Therefore, our SSHP is still practical for hedging an AFSLPO.

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