^{1}

^{*}

^{2}

^{*}

This paper presents two transform methods for pricing contingent claims namely the fast Fourier transform method and the fast Hilbert transform method. The fast Fourier transform method utilizes the characteristic function of the underlying instrument’s price process. The fast Hilbert transform method is obtained by multiplying a square integrable function
*f* by an indicator function associated with the barrier feature in the real domain. This is also obtained by taking the Hilbert transform in the Fourier domain. We derived closed-form solutions for European call options in a double exponential jump-diffusion model with stochastic volatility. We developed fast and accurate numerical solutions by means of the Fourier transform method. The comparison of the probability densities of the double exponential jump-diffusion model with stochastic volatility, the Black-Scholes model and the double exponential jump-diffusion model without stochastic volatility showed that the double exponential jump-diffusion model with stochastic volatility has better performance than the two other models with respect to pricing long term options. An analysis of the fast Fourier transform method revealed that the volatility of volatility
*σ* and the correlation coefficient
*ρ* have significant impact on option values. It was also observed that these parameters
*σ* and
*ρ* have effect on long-term option, stock returns and they are also negatively correlated with volatility. These negative correlations are important for contingent claims valuation. The fast Fourier transform method is useful for empirical analysis of the underlying asset price. This method can also be used for pricing contingent claims when the characteristic function of the return is known analytically. We considered the performance of the fast Hilbert transform method and Heston model for pricing finite-maturity discrete barrier style options under stochastic volatility and observed that the fast Hilbert transform method gives more accurate results than the Heston model as shown in Table 3.

The Black-Scholes model is the first successful attempt to explain the dynamics of pricing options. But some of its assumptions, like constant volatility or log-normal distribution of underlying price of the asset, do not find justification in the markets. Also strong assumptions in the Black-Scholes model makes it impossible to apply in practice since financial asset returns are not normally distributed. They have fatter tails than the normal distribution proposed and extreme observations are much more frequent in high-frequency financial data. The common big returns that are larger than six-standard deviations should appear less than once in a million years if the Black-Scholes framework were accurate. Squared returns, as a measure of volatility, display positive autocorrelation over several days, which contradict the constant volatility assumption. Therefore stochastic volatility is needed for option pricing [

The Black-Scholes model and its extensions constitute the major developments in modern finance. Much of the recent literature on option valuation has successfully applied Fourier analysis to determine option prices such as [

The Bates [

Zeng et al. [

Zeng and Kwok [

In this paper, we focus on the performance of the two transform methods under consideration for the valuation of contingent claims.

The paper is outlined as follows: Section 2 gives a brief overview of Bates model in the theory of option pricing. Section 3 presents the method of the fast Fourier transform for the valuation of contingent claims. Section 4 presents the fast Hilbert transform method for the valuation of timer options (barrier style options). In Section 5, we present some numerical experiments to illustrate the performance of these transforms. Section 6 concludes the paper.

The geometric Brownian motion (Wiener process) is the building block of modern finance. In the Black-Scholes model, the underlying asset price is assumed to follow the dynamics of the geometric Brownian motion of the form:

where,

S_{t}: the underlying asset price, r: the risk-free interest rate,_{t}: the Brownian motion or Wiener process and t: the maturity time.

The solution to (1) is obtained as follows;

Using the Ito’s lemma

From (1), _{t} is assumed to follow the process in (1) but we are interested in the process followed by logS_{t}. Let

Differentiating u with respect to the underlying price of the asset S_{t} and maturity time t we have

Substituting (3) and (4) into (2) yields

Integrating (5) from 0 to t, we have that

The empirical facts, however, do not confirm the model assumptions. Financial returns in this model exhibit much fatter tails in other models than in the Black-Scholes model.

Bates proposed a model with stochastic volatility and jumps. This model is the combination of the Merton and Heston models.

If an important piece of information about a company becomes public it may cause a sudden change in the company’s stock price. To cope with this observation, Merton proposed a model that allows discontinuous trajectories of the underlying asset prices. The Merton model is one of the modern pricing models. This model extends (1) by adding jumps to the stock price dynamics, to obtain the modified price dynamics as

where Z_{t} is a compound Poisson process with a log-normal distribution of jump sizes. The jumps follow the same Poisson process N_{t} with intensity λ, which is independent of W_{t}. The log-jump sizes Y_{i} are independent, identically distributed random variables with mean μ and variance δ^{2}, which are independent of both N_{t} and W_{t}. The model becomes incomplete which means that there are many possible ways to choose a risk-neutral measure such that the discounted price process is a martingale. Merton proposed to change the drift of the geometric Brownian motion and to leave the other ingredients unchanged. The underlying price of the asset dynamics is obtained as

Setting

The underlying price (8) takes the form

The jump components add mass to the tail of the returns distribution. Increasing δ adds mass to both tails, while a negative or positive μ implies relatively more mass in the left or right tail. Let the logarithm of the underlying price of the asset process be given by

The characteristic function of X_{t} is of the form;

The Heston model is one of the most widely used stochastic volatility models today. Its attractiveness lies in the powerful duality of its tractability and robustness relative to other stochastic volatility models.

Equation (1) can be modified by replacing the parameter σ with a stochastic process

where v_{t} is the variance process. There are many possible ways of choosing v_{t}.

Hull and White proposed the use of the geometric Brownian motion

However, the geometric Brownian motion tends to increase exponentially as

was suggested by Stein and Stein [_{t}.

Heston [

where ρ is the correlation coefficient.

Thus (13) reduces to

and (15) reduce to

To obtain the variance process Heston set

Equation (18) is known as the Heston variance process.

Where S_{t} and v_{t} denote underlying price of the asset and volatility processes respectively,

As the process reaches the zero bound, the stochastic part becomes zero and the non-stochastic part pushes up the process. The parameter k measures the speed of mean reversion or rate of reversion, θ is the long run mean or the average level of volatility and

It is clear that, in the Heston model one can implore more than one distribution by changing the value of ρ. We define ρ as the correlation between returns and volatility, and hence we can deduce that ρ affects the heavy tails of the distribution. When

The risk neutral dynamics is given in a similar way as in the Black-Scholes model. Taking the exponential of both sides of (11) which have the underlying asset price S_{t} as

Using (11) and the fact that

The characteristic function is given by

where

x_{0} is the initial value for the log-price process and v_{0} is the initial value for the volatility process.

Bates combined the Merton model and Heston model to obtain (Bates model_{t} and its variance v_{t} are given by

Equation (23) can also be written for the case of dividend paying stock as:

where q is the dividend yield paid by the underlying asset price S_{t}, Z_{t} is a compound Poisson process with intensity

The parameters χ and δ determine the distribution of the jumps and the Poisson process is assumed to be independent of the Brownian motions. Under the risk neutral probability we obtain the equation for the logarithm of the underlying asset price with non-dividend and dividend yields respectively as:

where

Since jumps are independent of the diffusion part in Equation (23), then the characteristic function for the log-price process in which the underlying asset price pays no dividend is obtained as:

Similarly, for dividend paying stock we have:

Equations (28) and (29) can be written in the form

and

In Equation (30), we observe that the diffusion part is similar to (22) with difference of

tral correction. Also (12) has a similar structure as the jump part in (30), where

jumps are assumed to be independent, the characteristic function is the product of Heston model

Remark 1:

Assuming that the previous dynamics represent the evolution of the state process

where

Equation (32) holds for

where K is the strike price or exercise price. It should be noted that closed form solutions of problem (32) for vanilla-option payoff do exist. Nevertheless, direct numerical integration of (32) is important when dealing with non-trivial payoff functions.

This section presents some fundamental properties of Fourier transform and the fast Fourier transform method for the valuation of European options.

The Fourier transform is a generalization of the complex Fourier series and is given by

where

are square integrable and characteristic functions respectively. In the form (35) we have the forward (−i) Fourier transform. The inverse (+i) Fourier transform is given by

Let the Fourier transform of

・ Scaling Property

・ Shifting/Translation Property

・ Fourier Transform of Derivatives

This process can be iterated for the n^{th} derivative to yield

Thus, a differentiation converts to multiplication in Fourier space.

・ Convolution Property

Similarly,

・ Linear Property

The fast Fourier transform is an efficient algorithm for computing the discrete Fourier transform of the form;

where N is typically a power of two. Equation (45) reduces the number of multiplications in the required N summations from an order of N^{2} to that of

The fast Fourier transform can be described by the following three steps as

The basic idea of the fast Fourier transform is to develop an analytic expression for the Fourier transform of the option price and to get the price by means of Fourier inversion.

The Fast Fourier Transform Method for the Valuation of European Call OptionThe Fast Fourier transform method is a numerical approach for pricing options which utilizes the characteristic function of the underlying instrument’s price process. The Fast Fourier transform method assumes that the characteristic function of the log-price is given analytically.

Consider the valuation of European call option. Let the risk neutral density of

The price of a European call option with maturity T and exercise price K denoted by

where p is the log of the strike price K i.e.

Substituting (50) into (49) yields

From (52), it is clearly seen that European call price given by (51) is not square integrable function. We consider a modified version of (51) given by

Equation (53) is square integrable in p over the entire real line. Using (36) and (37), we have that

Substituting (54a) into (53) and solving further, then we obtain a new call value given by

(56) is a direct Fourier transform and lends itself to an application of fast Fourier transform method. (54) is computed as follows:

Since

Finally we have

Setting

The corresponding put values can be obtained by defining

where

Substituting (58) into (56) with

Similarly, for the price of put option we have that:

For the put formula to be well defined, is suffices to choose an appropriate

The European call values are calculated using (59). Carr and Madan [

Now, we obtain the desired option price in terms of

Using basic trapezoidal rule, (60) can be computed numerically as:

where

We are interested in (at the money call values)_{0} of the form:

Equation (63) gives us log-strike levels ranging from −a to a, where

Substituting (63) and (64) into (61), we have

Now, the fast Fourier transforms method can be applied to x_{i} in (45) provided that

gration (60) is an application of the summation (45).

Remark 2:

・ For an accurate integration with larger values of

where

・ For in-the-money and at-the-money options prices, call values are calculated by an exponential function to obtain square integrable function whose Fourier transform is an analytic function of the characteristic function of the log-price. Unfortunately, for very short maturities, the call value approaches to its non-analytic intrinsic value causing the integrand in the inversion formula of Fourier transforms to vary above and below a mean value and therefore remains tedious to be integrated numerically. We use the alternative approach called the “Time Value Method” proposed by Carr and Madan [

Let

where

The prices of OTM options can be obtained by the inversion formula of the Fourier transform of (69) of the form

By substituting (68) into (69) and writing in terms of characteristic functions then (69) becomes

There are no issues regarding the integral of this function in (71) as

where

Solving (72) further and replace

The use of the fast Fourier transform for calculating out-of-the-money option prices is similar to (66). The only differences are that they replace the multiplication by

with a division by

where

We derive a closed-form solution of a European call option pricing under double exponential jump-diffusion model with stochastic volatility. The corresponding European put option can be obtained easily by means of put- call parity. For this purpose, we need the following results.

Lemma 3.3.1

Suppose that the variance process v_{t} follows a square root process of the form:

and s_{1}, s_{2} are any complex, one has

Remark 3:

This result shows that (76) holds because of the affine structure of the variance process.

Lemma 3.3.2 [

Suppose the underlying price of the asset follows:

and z is any complex, then

where

Lemma 3.3.3 [

Suppose

where

Theorem 3.3.4

Let k denote the log of the strike price K, ^{k}. Assume that, under martingale probability measure P^{*}, the underlying asset price S_{t} with dividend paying stock q and its components are given by:

_{T}, _{T} given by

then the initial call value

where

Proof:

From the risk-neutral theory, we have for the case of dividend yield q, the call price of the form

Introducing a change of martingale probability measure from P^{*} to Q^{*} by a Radon-Nikodym derivative, we get

We can write that

Because of the no-arbitrage condition, we can obtain

From the Fourier transform formula, the probability density for this model is given by

Hence,

Therefore, (88) becomes

From (84), (86), (89) and (96), we can obtain the required Theorem 3.3.4.

Remark 4:

For the of non-dividend yield, see [

The Hilbert transform of integrable function f is well defined by the following Cauchy principal value integral

Let the Fourier transform of f be defined as

Suppose that

Here

Using the translation property of the Fourier transform, it is very easy to obtain the following identities for one dimensional case if

Thus,

Remark 5:

The Hilbert transform on

It follows directly from the definition of the Hilbert transform that the associated operator is linear. Another slightly less obvious property is that Hilbert transform commutes with translations and positive dilations. For example, let

If we now let R be the reflection operator given by

There is no simple formula for the Hilbert transform of product of two functions. However we consider here the special cases of the Hilbert transform of

Comparing (90) and (100) we have that

Dividing both sides of the last equation by x and rearrange we have that

The Hilbert transform is an anti-self adjoint operator relative to the duality pairing between

Suppose the dynamics of the underlying price of the asset is given by the following under a given equivalent martingale measure

where X_{t} is a stochastic process starting at 0. Consider a European put option with exercise price K and maturity T. The payoff function for the European put option is given by

Let the risk neutral expectation of discounted payoff be denoted by P_{E} which is defined as

Substituting (102) and (103) into (104) yields

Equation (105) is called the value of European put option under the risk neutral expectation where r is the risk free interest rate.

The following results enable the derivation of the fast Hilbert transform method for the valuation of European standard put option:

Theorem 4.2.1 [

Let

Remark 6:

This theorem shows that the cumulative distribution function can be computed from the characteristic function through the Hilbert transform.

Theorem 4.2.2 [

Let X be a random variable such that

Remark 7:

This theorem shows the expectation for the Hilbert transforms representation.

From the two theorems above we obtain the Hilbert representation of European standard put option price as

Similarly European call option price with the same exercise price and maturity can be obtained by means of put call parity of the form;

Substituting the last equation in (108) into (109) we have

The results below summarize the derivation of the Hilbert transform method for the valuation of European options.

Lemma 4.2.3

Suppose the underlying price of the asset satisfies the martingale condition of the form

call and put options with non-dividend yield i.e.

Lemma 4.2.4

Suppose the underlying price of the asset satisfies the martingale condition of the form

call and put options with dividend yield q are given by

Remark 8:

・ The Lemma 4.2.3 and Lemma 4.2.4 give the fast Hilbert transform for the valuation of European call and put options with non-dividend and dividend yields respectively.

・ The fast Hilbert transform method can also be used for the valuation of timer option which is similar to its European counterpart, except with uncertain expiration date. This type of option is referred to as a barrier style option in the volatility space.

This section presents some numerical examples and discussion of results:

Example 1

Consider the pricing of a contingent claim using fast Fourier transform method with the following parameters

The exercise or strike prices and maturities are generated in MATLAB codes.

The results obtained are shown in the figure below:

Example 2 [

We consider European options pricing with double jumps and stochastic volatility using the following parameters:

We compare both the short-term and long-term probability densities of the double exponential-jump diffusion model with a stochastic volatility (SVDEJD), double exponential-jump diffusion model without a stochastic volatility (DEJD) in the context Black-Scholes model (BS) with the maturity time T = 3 months and T = 2 years. The results obtained are shown in the

Example 3

We consider the pricing of “in-the-money (ITM) and at-the-money (ATM)” and “out-of-the-money (OTM)” European call option with the following parameters given:

We examine the effects of correlation coefficient, strike price and volatility of volatility; on option values using the fast Fourier transform method. The results generated are shown in the

Example 4 [

Consider the pricing of finite-maturity discrete timer options via Monte Carlo method based on 20 Million simulation runs and 800 time steps per year under Heston model and the fast Hilbert transform approach with the following parameters:

The results obtained are shown in

From

K | ||||||
---|---|---|---|---|---|---|

10 | 39.9917 | 40.0061 | 39.9974 | 39.9721 | 39.9574 | 39.9347 |

20 | 30.9783 | 31.0157 | 30.9487 | 30.9585 | 30.9181 | 30.8975 |

30 | 23.2417 | 23.2645 | 23.2079 | 23.1979 | 23.1730 | 23.1271 |

40 | 17.1381 | 17.1100 | 17.1282 | 17.0895 | 17.1183 | 17.0685 |

50 | 12.5756 | 12.4911 | 12.6052 | 12.5492 | 12.6352 | 12.6091 |

60 | 9.2644 | 9.1442 | 9.3294 | 9.2743 | 9.3948 | 9.4051 |

70 | 6.8318 | 6.6941 | 6.9255 | 6.8819 | 7.0188 | 7.0680 |

80 | 5.1232 | 4.9858 | 5.2324 | 5.2043 | 5.3407 | 5.4192 |

90 | 3.8049 | 3.6758 | 3.9242 | 3.9136 | 4.0424 | 4.1468 |

100 | 2.8751 | 2.7605 | 2.9957 | 2.9996 | 3.1152 | 3.2345 |

K | ||||||
---|---|---|---|---|---|---|

10 | 40.4751 | 40.4830 | 40.4754 | 40.4715 | 40.4681 | 40.4686 |

20 | 31.0556 | 31.0412 | 31.0418 | 31.0298 | 31.0034 | 30.9899 |

30 | 23.2531 | 23.2751 | 23.2122 | 23.2210 | 23.1881 | 23.1456 |

40 | 17.1308 | 17.1024 | 17.0839 | 17.1220 | 17.1132 | 17.0653 |

50 | 12.5614 | 12.4767 | 12.5361 | 12.5916 | 12.6223 | 12.5975 |

60 | 9.2479 | 9.1277 | 9.2586 | 9.3134 | 9.3792 | 9.3905 |

70 | 6.8147 | 6.6769 | 6.8652 | 6.9086 | 7.0023 | 7.0521 |

80 | 5.1064 | 4.9690 | 5.1878 | 5.2157 | 5.3243 | 5.4130 |

90 | 3.7884 | 3.6583 | 3.8973 | 3.9079 | 4.0262 | 4.1310 |

100 | 2.8592 | 2.7446 | 2.9839 | 2.9799 | 3.0995 | 3.2191 |

K | ρ | Fast Hilbert transform approach | Heston model | Relative percentage error |
---|---|---|---|---|

90 | −0.5 | 17.6905 | 17.6927 | −1.24E−02 |

90 | 0 | 17.5517 | 17.5551 | −1.94E−02 |

90 | 0.5 | 17.4910 | 17.4882 | 1.60E02 |

100 | −0.5 | 12.3996 | 12.4099 | −7.82E−02 |

100 | 0 | 12.2804 | 12.2909 | −8.54E−02 |

100 | 0.5 | 12.2647 | 12.2692 | −3.67E−02 |

110 | −0.5 | 8.4147 | 8.4313 | −1.97E−01 |

110 | 0 | 8.3503 | 8.3634 | 1.57E01 |

110 | 0.5 | 8.3716 | 8.3774 | −6.92E−02 |

In this work we consider the performance measure of fast Fourier transform for the valuation of contingent claims. The fast Fourier transform method is used because of its advantages when compared to the analytic solution. Using the fast Fourier transform with risk neutral approach provides simplicity in calculations. Heston model is one of the most popular stochastic volatility option pricing models. This model is motivated by the widespread evidence that volatility is stochastic and that the distribution of risky asset return has tails heavier than that of a normal distribution.

The stochastic volatility model incorporates several important features of stock returns. We derive a closed form solution for at-the-money; in-the-money and out-of-the-money European call options using fast Fourier transform method. We also derived a closed form solution for pricing European option under double exponential jump-diffusion with stochastic volatility model. The comparison of the probability densities of the SVDEJD, the Black-Scholes model and the double exponential jump-diffusion model without stochastic volatility shows that SVDEJD model has better performance than the than the two other models on pricing long term options. An analysis of the fast Fourier transform method reveals that the volatility of volatility

Finally, we can say that fast Fourier transform method is a technique that increases the speed of computation. It is considerably faster than most available methods such as Heston and Bates models. The Hilbert transform method is good for pricing finite-maturity discrete timer options