Table of Contents
ISRN Applied Mathematics
Volume 2011, Article ID 518172, 7 pages
Research Article

Fourier Transform of Lookback Option Price

Department of Mathematics, China Jiliang University, Hangzhou 310018, China

Received 4 September 2011; Accepted 9 October 2011

Academic Editor: H. Y. Chung

Copyright © 2011 Cheng Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The Fourier transform of the damped price of Lookback option under B-S model is presented. Thus, the Lookback option across a range of strikes can be simultaneously priced via FFT algorithm. FFT algorithm is more efficient than both Monte Carlo simulation method and the integral of the usual pricing formula. In addition, by FFT algorithm, investors can easily capture the sensitivity of option prices when the strike prices vary as to make reasonable investment decisions.

1. Introduction

Recently, the Fourier transform of option prices is of great interest to many researchers. Carr and Madan [1] use damped option price method to get the Fourier integral representation of standard European call and put option value. Motivated by the work of Carr and Madan [1], we use similar methods to give the Fourier transform of the damped price of Lookback option.

Although the Lookback option price has the explicit integral formula, the FFT algorithm is more efficient than the usual integral computations. In Section 1, we first get the characteristic function of the log maximum of stock prices on a time interval by a large amount of calculations, then we use the characteristic function to obtain Fourier transform of the damped price of Lookback option. Although the formulation is somewhat cumbersome, it is composed only of elementary functions, and it is readily applied on desktop computers. In Section 2, FFT algorithm is outlined to calculate the inversion of the Fourier transform obtained in Section 1. Thus, the Lookback option prices for a range of strikes can be obtained by only one FFT computation. In Section 3, we make a simple numerical experiment.

Here, we need the characteristic function of the log maximum of stock prices on a time interval, so we only consider the Black-Scholes model.

As usual, we assume that the stock price under the equivalent martingale measure (EMM) 𝑄 satisfies𝑑𝑆(𝑡)=𝑟𝑆(𝑡)𝑑𝑡+𝜎𝑆(𝑡)𝑑𝑊(𝑡),(1.1) where 𝑟 is the risk-free rate, 𝜎 is the volatility, and 𝑊(𝑡) is a 𝑄-Brownian motion. The payoff 𝑋 at maturity 𝑇 of a Lookback call option with fixed strike 𝐾 is 𝑆𝑋=max𝐾,0,(1.2) where 𝑆=max0𝑡𝑇𝑆(𝑡).

We first give a lemma on Brownian motion which will be used in the next section.

Lemma 1.1 (see [2]). Suppose that {𝑊(𝑡)}𝑡0 is a standard 𝑄-Brownian motion, 𝑎 is a constant, and (𝑎𝑡+𝑊(𝑡))=max0𝑡𝑇(𝑎𝑡+𝑊(𝑡)), then the joint density function of (𝑎𝑡+𝑊(𝑡),(𝑎𝑡+𝑊(𝑡))) is 𝑓(𝑥,𝑦)=𝑒𝑎𝑥(1/2)𝑎2𝑇2𝜋2𝑦𝑥𝑇3/2𝑒(2𝑦𝑥)2/2𝑇,𝑥<𝑦.(1.3)

2. Fourier Transform of Damped Option Value

From (1.1), 𝑆(𝑡)=𝑆(0)𝑒(𝑟(1/2)𝜎2)𝑡+𝜎𝑊(𝑡).(2.1) So, ln𝑆(𝑡)=ln𝑆(0)+(𝑟(1/2)𝜎2)𝑡+𝜎𝑊(𝑡) and ln𝑆=ln𝑆(0)+((𝑟(1/2)𝜎2)𝑡+𝜎𝑊(𝑡)), where ((𝑟(1/2)𝜎2)𝑡+𝜎𝑊(𝑡))=max0𝑡𝑇((𝑟(1/2)𝜎2)𝑡+𝜎𝑊(𝑡)).

To obtain the characteristic function of ln𝑆, we first calculate the density function of ((𝑟(1/2)𝜎2/𝜎)𝑡+𝑊(𝑡)).

For simplicity of writing, we use 𝑎 to denote (𝑟(1/2)𝜎2)/𝜎, Φ, 𝜑, and 𝜙 to denote the distribution function, density function, and characteristic function of standard normal distribution 𝑁(0,1), respectively, in the following.

Lemma 2.1. The density function of (𝑎𝑡+𝑊(𝑡)) is 𝑔(𝑦)=2𝑒𝑇𝜋2𝑦𝑎𝑒(𝑦+𝑇𝑎)2/2𝑇𝑎2𝜋𝑇Φ𝑦𝑇𝑎𝑇.(2.2)

Proof. From Lemma 1.1, in the instruction, the density function of (𝑎𝑡+𝑊(𝑡)) is 𝑔(𝑦)=2𝜋1𝑇2/3𝑦𝑒𝑎𝑥(1/2)𝑎2𝑇(2𝑦𝑥)𝑒(2𝑦𝑥)2/2𝑇=𝑑𝑥2𝜋1𝑇2/32𝑦𝑒2𝑦𝑎𝑦𝑒[𝑥(2𝑦+𝑇𝑎)]2/2𝑇𝑑𝑥𝑒2𝑦𝑎𝑦𝑥𝑒[𝑥(2𝑦+𝑇𝑎)]2/2𝑇.𝑑𝑥(2.3)
Since 𝑦𝑥𝑒[𝑥(2𝑦+𝑇𝑎)]2/2𝑇𝑑𝑥=𝑦[]𝑒𝑥(2𝑦+𝑇𝑎)[𝑥(2𝑦+𝑇𝑎)]2/2𝑇+𝑑𝑥𝑦(2𝑦+𝑇𝑎)𝑒[𝑥(2𝑦+𝑇𝑎)]2/2𝑇𝑑𝑥,(2.4) so 𝑔(𝑦)=2𝜋1𝑇2/3𝑒2𝑦𝑎𝑦[]𝑒𝑥(2𝑦+𝑇𝑎)[𝑥(2𝑦+𝑇𝑎)]2/2𝑇𝑑𝑥+𝑇𝑎𝑦𝑒[𝑥(2𝑦+𝑇𝑎)]2/2𝑇=𝑑𝑥2𝑒𝑇𝜋2𝑦𝑎𝑒(𝑦+𝑇𝑎)2/2𝑇𝑎2𝜋𝑇Φ𝑦𝑇𝑎𝑇.(2.5) The proof is completed.

Proposition 2.2. The characteristic function of (𝑎𝑡+𝑊(𝑡)) is 2𝑎1𝑒𝑖𝑢+2𝑎𝑖(𝑢2𝑎𝑖)𝑇𝑎𝜙.𝑇(𝑢2𝑎𝑖)(2.6)

Proof. From Lemma 2.1, the characteristic function of (𝑎𝑡+𝑊(𝑡)) is 2𝑇𝜋𝑒𝑖𝑢𝑦𝑒2𝑦𝑎𝑒(𝑦+𝑇𝑎)2/2𝑇𝑎2𝜋𝑇Φ𝑦𝑇𝑎𝑇=𝑑𝑦2𝑇𝜋𝑒𝑖(𝑢2𝑎𝑖)𝑦𝑒(𝑦+𝑇𝑎)2/2𝑇𝑑𝑦𝑎2𝜋𝑇𝑒𝑖(𝑢2𝑎𝑖)𝑦Φ𝑦𝑇𝑎𝑇.𝑑𝑦(2.7)
Since the first term in the brackets is 𝑒𝑖(𝑢2𝑎𝑖)𝑦𝑒(𝑦+𝑇𝑎)2/2𝑇𝑑𝑦=2𝜋𝑇𝑒𝑖(𝑢2𝑎𝑖)𝑇𝑎𝜙,𝑇(𝑢2𝑎𝑖)(2.8) and the second term in the brackets is 𝑎2𝜋𝑇𝑒𝑖(𝑢2𝑎𝑖)𝑦Φ𝑦𝑇𝑎𝑇𝑑𝑦=𝑎12𝜋𝑇𝑒𝑖(𝑢2𝑎𝑖)𝑖(𝑢2𝑎𝑖)𝑦Φ𝑦𝑇𝑎𝑇|||||1𝑒𝑖(𝑢2𝑎𝑖)𝑖(𝑢2𝑎𝑖)𝑦𝜑𝑦𝑇𝑎𝑇1𝑇𝑑𝑦=𝑎12𝜋𝑇𝑖(𝑢2𝑎𝑖)𝑒𝑖(𝑢2𝑎𝑖)𝑦1𝑇𝜑𝑦𝑇𝑎𝑇𝑑𝑦=𝑎12𝜋𝑇𝑖𝑒(𝑢2𝑎𝑖)𝑖(𝑢2𝑎𝑖)𝑇𝑎𝜙,𝑇(𝑢2𝑎𝑖)(2.9) so the characteristic function of (𝑎𝑡+𝑊(𝑡)) is 2𝑇𝜋2𝜋𝑇𝑒𝑖(𝑢2𝑎𝑖)𝑇𝑎𝜙𝑇(𝑢2𝑎𝑖)𝑎12𝜋𝑇𝑖𝑒(𝑢2𝑎𝑖)𝑖(𝑢2𝑎𝑖)𝑇𝑎𝜙𝑎𝑇(𝑢2𝑎𝑖)=21𝑒𝑖𝑢+2𝑎𝑖(𝑢2𝑎𝑖)𝑇𝑎𝜙.𝑇(𝑢2𝑎𝑖)(2.10) The proof is completed.

Recall that 𝑎=(𝑟(1/2)𝜎2)/𝜎 and ln𝑆=ln𝑆(0)+((𝑟(1/2)𝜎2)𝑡+𝜎𝑊(𝑡)), then from Proposition 2.2, we can easily obtain the characteristic function of ln𝑆(denoted by 𝜙) as stated in the following theorem.

Theorem 2.3. The characteristic function of ln𝑆 is 𝜙(𝑢)=2𝑒𝑖𝑢ln𝑆0𝑎1𝑒𝑖𝜎𝑢+2𝑎𝑖(𝜎𝑢2𝑎𝑖)𝑇𝑎𝜙.𝑇(𝜎𝑢2𝑎𝑖)(2.11)
Let 𝑘 denote the log strike price, that is, 𝑘=ln𝐾, and 𝐶(𝑘) the Lookback call option price at time-0. To obtain a square-integrable function, one uses the damped option price [1]; that is, let 𝑐(𝑘)=𝑒𝛼𝑘𝐶(𝑘)=𝑒𝛼𝑘𝑟𝑇𝐸𝑄𝑆𝑒𝑘+,(2.12) for 𝛼>0. The discussion for the choice of 𝛼 in Carr and Madan [1] is applicable here. In Section 3, one makes a simple numerical experiment for 𝛼.

We write Ψ(𝑢) as the Fourier transform of 𝑐(𝑘), 𝜌(𝑠) as the density function of ln𝑆, and 𝜙(𝑢) as the characteristic function of ln𝑆. Then, we haveΨ(𝑢)=𝑒𝑖𝑢𝑘𝑐(𝑘)𝑑𝑘=𝑒𝑟𝑇𝑒𝑖𝑢𝑘𝑒𝛼𝑘𝑘𝑒𝑠𝑒𝑘𝜌(𝑠)𝑑𝑠𝑑𝑘=𝑒𝑟𝑇𝜌(𝑠)s𝑒𝑠+𝛼𝑘𝑒(1+𝛼)𝑘𝑒𝑖𝑢𝑘=𝑒𝑑𝑘𝑟𝑇𝜙(𝑢(𝛼+1)𝑖)𝛼2+𝛼𝑢2.+𝑖(2𝛼+1)𝑢(2.13) From Theorem 2.3, we haveΨ(𝑢)=2𝑒𝑟𝑇𝑒𝑖(𝑢(𝛼+1)𝑖)ln𝑆0𝐴𝑒𝑖(𝜎(𝑢(𝛼+1)𝑖)2𝑎𝑖)𝑇𝑎𝜙𝑇(𝜎(𝑢(𝛼+1)𝑖)2𝑎𝑖)𝛼2+𝛼𝑢2,+𝑖(2𝛼+1)𝑢(2.14) where 𝐴=((𝑖𝜎(𝑢(𝛼+1)𝑖)+𝑎)/(𝑖𝜎(𝑢(𝛼+1)𝑖)+2𝑎)).

3. Using FFT to Price Lookback Option

𝐶(𝑘) thus can be calculated by taking the Fourier inversion transform 𝑒𝐶(𝑘)=𝛼𝑘2𝜋𝑒𝑖𝑢𝑘=𝑒Ψ(𝑢)𝑑𝑢𝛼𝑘𝜋Re0𝑒𝑖𝑢𝑘,Ψ(𝑢)𝑑𝑢(3.1) where Re denotes the real part of a complex number and the second equality is due to that Ψ(𝑢) is odd in its imaginary part and even in its real part.

The above integral can be computed using FFT. A numerical approximation for 𝐶(𝑘) is𝑒𝐶(𝑘)𝛼𝑘𝜋𝑁𝑗=1𝑒𝑖𝑢𝑗𝑘Ψ𝑢𝑗Δ𝑢,(3.2) where 𝑢𝑗=𝑗Δ𝑢,𝑗=1,2,,𝑁. Discussions on the errors in the numerical computing are presented in Lee [3].

The FFT is an efficient algorithm that computes the sum of the following form:𝑦(𝑛)=𝑁𝑗=1𝑒𝑖(2𝜋/𝑁)𝑗𝑛𝑥(𝑗),𝑛=1,,𝑁.(3.3) The interesting values of the strike price 𝐾 is around the forward price, that is, 𝑒𝑟𝑇𝑆0, so we calculate the sum (3.2) for 𝑘=𝑟𝑇+ln𝑆0+𝑚Δ𝑘,𝑚=(𝑁/2)+1,,𝑁/2.

For (3.2) to be transformed to the form of (3.3), we let 𝑛=𝑚+(𝑁/2) (then 𝑛 ranges from 1 to 𝑁), Δ𝑢Δ𝑘=2𝜋/𝑁, and 𝑁 be an integer power of 2.

Then, (3.2) turns to be𝑒𝐶(𝑘)𝛼𝑘𝜋𝑁𝑗=1𝑒𝑖(2𝜋/𝑁)𝑗𝑛𝑒𝑖𝑗Δ𝑢(𝑟𝑇+ln𝑆0)+𝑖𝑗𝜋Ψ𝑢𝑗Δ𝑢,𝑛=1,,𝑁,(3.4) where 𝑒𝑖𝑗Δ𝑢(𝑟𝑇+ln𝑆0)+𝑖𝑗𝜋Ψ(𝑢𝑗)Δ𝑢 is the counterpart of 𝑥(𝑗) in (3.3). So, (3.4) can be calculated by FFT quickly.

4. Simple Numerical Experiment

In our example, 𝑇=1; 𝑟=0.05; sigma = 0.3; 𝑆0=5; 𝑁=10000; delta𝑢=0.01, and we use matlab software as a computing tool.

Motivated by Lord et al. [4], we let 𝛼 ranges from 1 to 3, increasing 0.1 each step. Then, we calculated the option prices 𝐶(𝑘) against strikes 𝐾=𝑒𝑘 around the forward future price 𝑆0𝑒r𝑇 for every 𝛼. Fix 𝑁=214, and we find that for each 𝛼, the curves of option prices against strikes are almost the same curve, as Figure 1 shows. It shows the results are relatively stable for different 𝛼.

Figure 1: Option prices 𝐶(𝑘) against strikes 𝐾.

We take 𝛼=2 and compare the results with Monte Carlo method. See Figure 2, where 𝑁=214 in FFT algorithm and Monte Carlo method do 104 simulations. Although the results are similar, FFT is much more efficient than Monte Carlo simulation.

Figure 2: The results of FFT and Monte Carlo.

5. Conclusion

Although the Lookback option price has the explicit integral formula, the FFT algorithm is more efficient than the usual integral computations. Also, using FFT calculation once, practitioners can directly capture the price sensitivity of an option with varying strike prices. Using the same technique above, we can also obtain the Fourier transform of Lookback call and put options with floating strikes under Black-Scholes model. For asset prices under Lévy processes, Feng and Linetsky [5] take the asset prices on discrete time points to get the approximate price of Lookback options; Kou [6] give a survey of several discretization method to price Barrier and Lookback options.


This paper was supported by NSF 10901137, China.


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