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 δ², 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 which leads to stochastic volatility model with price dynamics given by

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 and this is an undesirable property for volatility. Volatility exhibits rather a mean reverting behavior. Therefore a model based on an Ornstein-Uhlenbeck-type process:

was suggested by Stein and Stein [31] . This process, however, admits negative values of the variance v_t.

Heston [10] eliminated these deficiencies in the stochastic volatility model by introducing two Brownian motions; in the modified stochastic volatility model in (13) and in the Ornstein-Uhlenbeck-type process (15), and also with the assumption that

where ρ is the correlation coefficient.

Thus (13) reduces to

and (15) reduce to

To obtain the variance process Heston set in (17) to obtain

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, and are correlated with rate ρ. The term in (18) simply ensures positive volatility.

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 is the volatility of volatility.

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, there is an inverse proportion between underlying asset price and volatility, when, the skewness is close to zero and when, this means that as the underlying asset increases, volatility increases. The conditions, and lead to increase in the heaviness of the right tail and squeezes the left tail as shown in Figure 1(a), Figure 1(b) and Figure 1(c).

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 in (6) we obtain the stochastic differential equation

The characteristic function is given by

where

x₀ is the initial value for the log-price process and v₀ is the initial value for the volatility process.

Bates combined the Merton model and Heston model to obtain (Bates model). This model, described in Bates model, adds jumps to the dynamics of the Heston model. For a non-dividend paying stock, the dynamics of the stock price S_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 and a log-normal distribution of jump sizes independent of and. If J denotes the jump size then

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 is a compound Poisson process with normal distribution of jump magnitudes.

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, since they can be split into diffusion part D and jump part J. Therefore, we have respectively for (28) and (29) below

and

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

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

jumps are assumed to be independent, the characteristic function is the product of Heston model and the jump part in (30). Figure 2 shows that adding jumps makes it easier to introduce curvature into the volatility surface, at least for short maturities.

Remark 1:

Assuming that the previous dynamics represent the evolution of the state process under a risk-neu- tral measure, then the pricing equation of a European contingent claim c on S is given by

where

Equation (32) holds for, with the assumption that. The partial differential equation is usually given along with a proper payoff. In the case of vanilla European call option we have that:

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.

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 be defined as then the following fundamental properties hold as follows:

・ 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 N² to that of, a very considerable reduction. Let p and j be written as binary numbers i.e. and with, then (45) becomes

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 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 be. The characteristic function of this density is given by

The price of a European call option with maturity T and exercise price K denoted by is given 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 with, then the last equation in (56) becomes

Finally we have

Setting, then (57) becomes

The corresponding put values can be obtained by defining, , with Fourier transform

where is the Fourier transform of. A sufficient condition for to be square-integrable is given by being finite. This is equivalent to.

Substituting (58) into (56) with, we have

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

For the put formula to be well defined, is suffices to choose an appropriate in a way that.

The European call values are calculated using (59). Carr and Madan [2] established that if the denominator of (59) vanishes when v = 0, inducing a singularity in the integrand. Since the fast Fourier transform evaluates the integrand at v = 0, the use of the factor is required.

Now, we obtain the desired option price in terms of using Fourier inversion of the form:

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

where

We are interested in (at the money call values), the case where the strikes near the underlying spot price of the asset. This type of options is traded most frequently. The fast Fourier transforms method returns N values of p and we then consider a uniform spacing of size for the log-strikes around the log-spot price S₀ 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. Hence the inte-

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

Remark 2:

・ For an accurate integration with larger values of we apply basic Simpson’s rule weightings into (65) with the condition, then the accurate call price which is the exact application of the fast Fourier transform method is obtained as:

where, and is called the Kronecker delta function expressed as:

・ 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 [2] to mitigate this numerical inconvenience. This approach works with time values only, which is quite similar to the previous approach. But in this context the call price is obtained by means of the Fourier transform of a modified time value, where the modification involves a hyperbolic sine function instead of exponential function.

Let represent the time value of out-of-the-money (OTM), that is, for we have the call price for and for we have the put option for. Setting for simplicity, is defined by

where is the risk-neutral density of the log-price S and I denotes the indicator function. Let the Fourier transform of be defined by

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 or, the time value at k tends to zero can get rather steep as T tends to zero and this can make the Fourier transform to be wide and oscillatory. By considering the damping function, the time value follows a Fourier inversion:

where

Solving (72) further and replace by, then we have

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 and the function call to be replaced by a function. Hence the formula for out-of-the-money option price is given by

where, and is called the Kronecker delta as given by (67).

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 s₁, s₂ 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 [13]

Suppose the underlying price of the asset follows:

and z is any complex, then

where

Lemma 3.3.3 [35]

Suppose is the characteristic function of, then

where

Theorem 3.3.4

Let k denote the log of the strike price K, and the desired value of a T-maturity call option with strike e^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:

is the characteristic function of x_T, is the probability density of x_T given by

then the initial call value is written as

where represents the real part.

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 [35] .

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 be the translation operator defined by, and let be the dilation operator given by for. By a simple change of variable, we have that

If we now let R be the reflection operator given by, then we have that

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 and as follows;

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 and dual space where p and q are Holder conjugate and. Symbolically, ,.

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 [32]

Let and be the cumulative distribution function and the characteristic function of a continuous distribution. Suppose that. Then

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 [32]

Let X be a random variable such that and be the characteristic function of X such that. Then

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

. Define the following probabilities,. Then the fast Hilbert transform method for the valuation of European

call and put options with non-dividend yield i.e. are given by and respectively.

Lemma 4.2.4

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

. Define the following probabilities,. Then the fast Hilbert transform method for the valuation of European

call and put options with dividend yield q are given by and respectively.

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.

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 [35]

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 Figure 3.

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 Table 1 and Table 2 for “in-the- money (ITM) and at-the-money (ATM)” and “out-of-the-money (OTM)” European call options respectively.

Example 4 [33]

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 Table 3.

From Figure 1(a), Figure 1(b) and Figure 1(c) can see that conditions, and have effect on the heaviness of the left and right tails. When, there is an inverse proportion between underlying asset price and volatility, when, the skewness is close to zero and when, this means that as the underlying asset increases, volatility increases.

Figure 2 shows that adding jumps makes it easier to introduce curvature into the volatility surface, at least for short maturities.

Figure 3 builds the volatility surface based on the parameters of the model and enhances an intuitive understanding of the Heston model. It can be seen from Figure 3 above that there is change in volatility perception. Volatility smile shape has been changed yet volatility for in-the-money and at-the-money options is high and also for out-of-the money options volatility is low. It can be seen from Figure 4 that the long density curves still show significantly different pricing structures between the Black-Scholes model (BS) and its two counterparts (SVDEJD and DEJD). But, more importantly, the densities of the double exponential jump-diffusion model with stochastic volatility and the double exponential jump-diffusion model without stochastic volatility also exhibit different shapes. The SVDEJD density has higher peak and assigns more weight to both the entire lower tail and far upper tail, but less weight to those payoffs than the DEJD [35] .

Table 1 and Table 2 show the effect of correlation coefficient ρ, volatility of volatility and strike price K on ITM, ATM and OTM options values using fast Fourier transform. The price of option and the time value for six-month call options associated with volatility of volatility and are relatively close as we can see from Table 1 and Table 2 for ITM, ATM and OTM respectively. The effect of correlation coefficient depends on the relationship between the current underlying price of the asset and strike price. For a positive correlation coefficient, the price of out-of-the-money call option becomes lower. For negative correlation coefficient, the price of in-the-money and at-the-money call options becomes higher. When the correlation coefficient is zero, the effect of volatility of volatility is negligible.

Table 3 shows the comparative results analysis of the fast Hilbert transform and Heston model. It also shows that the relative percentage errors between the fast Hilbert transform and Heston model results are always less than 0.2%. This reveals high level of accuracy of the fast Hilbert transform algorithm.