Abstract
This paper applies the fast Fourier transform (FFT) approach, within the Black-Scholes framework, to the valuation of options whose time to maturity can be extended to a future date (extendible options). We determine the valuation of the extendible options as sums of expectations of indicator functions, leading to a semianalytic expression for the value of the options over a range of strikes. Compared to Monte Carlo simulation, numerical examples demonstrate that the FFT is both computationally more efficient and higher in accuracy.
1. Introduction
An extendible option allows its time to expiration to be extended to a future date for a fee with a different strike price at the extended expiration time. Longstaff [1] discusses a number of applications of extendible options, such as extendible bonds and warrants, and applications in real estate, demonstrating that extendible options are important in financial markets. For example, Dias and Rocha [2] price petroleum concessions using a mean-reverting framework including jumps for an extendible option. Extendible options can be classified as a holder- or writer-extendible call option or a holder- or writer-extendible put option. Options that are extended by their writer have no fee and they are more straightforward to price. To the best of our knowledge, Longstaff [1] was the first to study extendible options. Chung and Johnson [3] extend the work of Longstaff [1] to a general case where the holder or the writer can extend the option more than once and provide a general closed-form pricing solution for -time extendible options.
In order to price options, many numerical techniques are available, such as Monte Carlo simulation and the Fourier transform method. The fast Fourier transform (FFT) was first introduced by Cooley and Tukey [4] to obtain the discrete Fourier transform (DFT) and its inverse. Implementing the DFT involves points where, for added efficiency, is a power of and it is essential to have an extensively large data set, which reduces the number of computations from to . Monte Carlo simulation often requires substantially more time to compute option prices, while the FFT, which was first implemented to option pricing by Carr and Madan [5], is often more straightforward and effective. This has been demonstrated by a number of authors, such as Pillay and O’Hara [6] and Ibrahim et al. [7]. Hence, the FFT has gained popularity as an effective option pricing methodology (see [8, 9]), and a significant amount of research has extended this method to price various types of options, for instance Asian options [10], spread options [11], and power options [12]. The FFT technique is widely applicable and has been shown to perform well in terms of its speed and accuracy in some other affine models, such as stochastic volatility, stochastic interest rates, and jumps (see [13–15]).
In this paper, we review the pricing framework for extendible call options under the Black and Scholes [16] model first developed by Longstaff [1]. By following the work of Carr and Madan [5], we apply the FFT method to numerically compute the prices of extendible call options. For an effective implementation of the FFT, we provide an analytical representation of the characteristic functions for the extendible options. The usefulness of the FFT is that this technique gives a range of option prices for a collection of strike prices for a given maturity. In addition, we estimate the price of extendible options via Monte Carlo simulation for comparison.
The paper is organized as follows. Section 2 reviews the closed-form formulas for the holder-extendible call options under the Black-Scholes model as provided in Longstaff [1]. Section 2.1 develops a pricing procedure for holder-extendible call options using the FFT approach where the characteristic functions are given, while Section 2.2 applies the Monte Carlo simulation using an Euler scheme and a Milstein scheme. Section 3 presents the numerical results, and Section 4 concludes the paper.
2. Extendible Option
In this section, we derive a closed-form solution for holder-extendible call options as given by Longstaff [1], which is the focus of this paper. An analogous approach can be used to value extendible put options.
Let be a probability space on which a Brownian motion, , is defined, where , , is the natural filtration generated by the Brownian motion. Suppose that is a risk-neutral measure under which the asset price process , is governed by the following dynamics: Within the Black-Scholes framework, the volatility and the risk-free interest rate are assumed to be constant. Using Itô’s lemma, the process followed by the logarithmic asset price is given by This demonstrates that follows a generalized Wiener process with a constant drift and a constant volatility . Therefore, the change in between time and is normally distributed with a mean of and a variance of or Therefore, has a log-normal distribution. The valuation formula for a vanilla call option with a given underlying asset price , strike price , expiration time , risk-free interest rate , and volatility is given as follows: where
Let be the value of an extendible call option with time to expiration of . At the time to expiration , the holder of the extendible call can(1)let the call expire worthless if , or(2)exercise the call and get if , or(3)make a payment of an additional premium to extend the call to with a new strike of if ,where is the underlying asset price and strike price at time is the strike price at time , and Longstaff [1] refers to and as critical values, where .
If at expiration time the option is worth more than the extendible value with a new strike price of for a fee of for extending the expiration time to , then it is best to exercise; that is, . Otherwise, it is best to extend the expiration time of the option to and exercise when it is worth more than zero; that is, . Moreover, the holder of the option should be impartial between extending and not exercising at value and impartial between exercising and extending at value . Therefore, the critical values and are unique solutions of and . See Longstaff [1] and Gukhal [17] for an analysis of the conditions.
The price of the extendible call option is the discounted expectation of the payoff at time and is given by the following: An example illustration of its payoff is provided in Figure 1.
Formulating the value of a call option, at time with a time to expiration extended to , as the discounted conditional expected payoff, yields
or where is the spot price measure, , and .
Corollary 1. The price of an extendible call option with time to expiration and strike price , whose expiration time may be extended to upon a payment of an additional premium , with a new strike price is specified as where denotes the standard univariate cumulative normal distribution function, denotes the standard bivariate cumulative normal distribution function with a correlation coefficient, , and
Longstaff [1] also discusses several special cases of an extendible option, such that if and , then the extendible option reduces to a compound option on with strike price , and if , the value of the extension privilege approaches zero; hence the value of the extendible options is simply (for more discussion on the special cases of the extendible options, see Longstaff [1]).
2.1. Fast Fourier Transform
The FFT approach evaluates an (output) array of the following form: for any complex (input) array . In addition, to implement the FFT method to option pricing, a closed-form representation of the characteristic function is required. For the case of extendible options, we require univariate and bivariate characteristic functions, which are provided by the following lemmas.
Lemma 2. Under the risk-neutral measure , the univariate characteristic function is given by
Lemma 3. Under the risk-neutral measure , the bivariate characteristic function is given by
The univariate characteristic function is used in Carr and Madan [5] to price a European option using FFT within the Black-Scholes dynamics. By employing the Radon-Nikodym derivatives, the characteristic function under measure and the spot price measure are related as follows: Hence, from (10), we consider the following expectations which can be expressed in integral form as follows: where is the probability density of under the risk-neutral probability . It is clear that integrals (17), (20), and (21) are similar, so are integrals (18) and (19). Hence we first apply the FFT to integral (18). Following Carr and Madan [5], we multiply (18) by dampening factors and to ensure square-integrability as follows: Then, by applying the Fourier transform, we have the following: This can be written in closed-form in terms of the characteristic function as follows: Using the inverse Fourier transform, we can recover integral (18): Then, we numerically compute this by invoking the trapezoid rule (see [11]) which approximates the integral by the following sum: where is the distance between the points of the integration grid, . In order to progress, we define a grid of size by where is the distance between the logarithmic critical prices and the logarithmic strikes, respectively, as follows: Hence, we can now evaluate the following sum: .
By choosing this yields the following values of sum on : We take the (input) array as . Hence, the approximation is at different logarithmic critical prices and strike prices such that In an analogous way, by replacing with , we can approximate integral (19). The two-dimensional FFT can also be reduced to a one-dimensional FFT which involves integrals (17), (20), and (21). Therefore for brevity, we exclude the development for the one-dimensional FFT in this paper.
2.2. Monte Carlo Simulation
In this section, we price the extendible call option using Monte Carlo simulation [18]. Given the pricing function of an extendible call option given in (6), the vanilla call option can be priced at time with strike price and expiration time . Thus we only need to compute the asset price until time . Let be the asset price at time on the th path, and let be the vanilla call price with strike price and expiration time at time for the th path; that is, . Recall the closed-form solution for a vanilla call option, so we have where Therefore, the Monte Carlo estimator for the price of an extendible call option is given by where is the number of simulations. Suppose we have with the following process: We discretize the asset path using the Euler scheme as follows: We also apply the Milstein scheme [19] that is strongly convergent with order one such that We let be a partition of the time interval of equal segments with length , where and , for each . We draw a random sample from a normal distribution with mean 0 and volatility 1 for each to simulate (34) and (35), thus generating a sample path for by simulating for . This step is repeated to generate many paths. We then estimate the price of the extendible call option with Monte Carlo simulation.
3. Numerical Results
In this section, we present a numerical comparison between the FFT and the Monte Carlo simulation techniques which have been described in previous sections. (The computations were implemented in MATLAB and conducted on an Intel Core 2 Quad processor Q8200 @ 2.33 GHz machine running under Windows 7 Professional with 4 GB RAM).
We price the extendible call option for a range of underlying asset prices, , and , with the inputs in Table 1. We found that the dampening factor differs for a one-dimensional and a two-dimensional FFT. For the one-dimensional FFT, we choose , while for the two-dimensional FFT, . Table 2 provides the CPU time taken by the FFT, Euler scheme, and Milstein scheme to produce the prices for extendible call options. It can be seen that FFT takes less than half a second, on average, approximately seconds, to compute the prices in a single computation, while the exact solution computed each of the option prices instantaneously. Euler takes slightly longer than a minute, on average seconds, whereas Milstein scheme takes the longest time to produce results, seconds on average. In Table 3, we document the prices obtained using Monte Carlo simulations with Euler and Milstein schemes. The exact solution is used to compute the relative errors between these two schemes. The error is given by the following: where is the Black-Scholes pricing solution and is the estimation from the simulation. From the table, it can be seen that the relative error for the Euler scheme is smaller than the Milstein scheme. We also plot the price convergence of the Monte Carlo simulation when the underlying asset in Figure 2. From Tables 2 and 3, it can be seen that the Euler scheme is more computationally efficient than the Milstein scheme. According to Platen and Bruti-Liberati [20], the Euler scheme has fewer terms and the same order of weak convergence; hence, apart from efficiency issue, the Milstein scheme also has numerical stability issue which can be worse than that of the Euler scheme. We then compare the extendible call prices computed by the FFT and the exact solution. Table 4 shows the prices and percentage differences taken relative to the FFT prices.
4. Conclusion
This paper studies the pricing of extendible options under the Black and Scholes [16] model by developing a pricing framework which utilizes the FFT technique. The availability of a closed-form representation of the characteristic function enables us to implement the FFT. The characteristic function for the extendible options involves a univariate and a bivariate characteristic function. Hence, the application of the FFT technique involves a one-dimensional and a two-dimensional FFT algorithm.
Additionally, we computed the prices of the extendible options using Monte Carlo simulation and used the closed-form pricing solution as a benchmark to compare the numerical efficiency of the FFT and the Monte Carlo simulation. The FFT produces a range of option prices for a range of strike prices for a given maturity. Moreover, the numerical examples show that the FFT is more computationally efficient than the Monte Carlo simulation.
Dias and Rocha [2] show that the framework of extendible options can be applied to model oil prices using mean reversion with jumps. Hence, further development such as pricing extendible options with other stochastic features such as stochastic interest rate can be applied. In the future, we plan to use the FFT to price extendible options under the Heston stochastic volatility model [21].
The work presented here can also be extended to a more general case, that is, for -extensions. The FFT has become well established as an option pricing tool since it was first introduced by Carr and Madan [5]. Fang and Oosterlee [22] have introduced a Fourier-Cosine also known as the COS method to price options, and this method has been shown to be faster under certain circumstances. Hence, another possible development of this work is to price extendible options using the COS method and compare the performance of the two Fourier methods in the case for extendible options.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
Siti Nur Iqmal Ibrahim’s research was supported by Universiti Putra Malaysia and the Ministry of Higher Education, Malaysia. The authors would like to thank the editor and the referee for their constructive comments that led to a significant improvement of the paper.