Now, we consider the issue of the appropriate choice of the coefficient. In order for a modified call value to satisfy square-integrability condition, Carr and Madan (1998) provides a sufficient condition:
Z ∞
5.2 The Fourier Transform Methods of Out-of-the-Money (OTM) Option Pricing
Carr and Madan (1998) point out that the call price approaches its non-analytic intrinsic value causing the integrand in the Fourier inversion to be highly oscillatory, and numerical integration problem becomes slow and difficult. Thus, Carr and Madan (1998) provided an alternative approach which works with only time values.
Let zT (k) be the time value of out-of-the-Money (OTM) options with maturity T at the current time t = 0.
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the put price at time t.Let ζT (ω) be the Fourier Transform of zT (k):
ζT (ω) = Z ∞
−∞
e−iωkzT (k) dk (5.4)
The price of out-of-the-Money options are computed by inverting Fourier Transform of the equation (5.4):
We assume the current S0 = 1 for simplicity which means s0 = 0. Thus, under the risk-neutral measure Q, zT (k) can be computed as:
zT (k) = e−rT Z ∞
−∞
ek− esT 1(sT<k,k<0)+ esT − ek 1(sT>k,k>0) qT (sT|F0) dsT
where k = ln (K), st = ln (St) and qT (st|F0) is the risk-neutral density of the log pricc st conditional on filtration F0.
Thus, the expression of ζT (ω) is:
To facilitate numerical integration, Carr and Madan (1998) considers a Fourier Transform of out-of-the-Money (OTM) option zT (k) modified with dampening function sinh(αk):
ΥT (ω) =
Thus, the time value of out-of-the-Money (OTM) option price zT (k) can be obtained by an inverse Fourier Transform of ΥT (ω):
zT (k) = 1
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5.3 European Option Pricing using the Fast Fourier Transform (FFT)
Carr and Madan (1998) apply the Trapezoid rule for the Fourier integral, the value of C (st, k, r, t, T ) is approximate as:
The Fast Fourier Transform return N values of k and for a regular spacing size of λ where N is a power of 2, i.e., N = 2p, p ∈ N , the value for k is
ku = −b + λ (u − 1) , u = 1, ..., N. (5.7) which corresponds to log strike prices ranging from −b to b and thus
b − (b) = N λ ⇒ b = N λ 2 Substituting (5.7) into (5.6) yields:
C (st, k, r, t, T ) ≈ e−αku
Since the structure of the discrete Fourier Transform (DFT) is:
F (k) =
N
X
j=1
e−i2πN(j−1)(u−1)x (j) , j = 1, ..., N. (5.9)
In order to apply the algorithm of the Fast Fourier Transform (FFT) to evaluate the sum (5.8), we choose
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However, if we have to choose η small in order to obtain a fine grid for integration, then we observe call option prices at strike spacings which are relatively large, with few strikes lying in desired region near stock price. Therefore, we would like to obtain an accurate integration with larger values of η and for this purpose, Carr and Madan (1998) use Simpson’s rule weightings and the restriction (5.10), we may rewrite (5.11) as:
C (st, k, r, t, T )
where δn is the Kronecker delta function that is unity for n = 0 and zero otherwise. For near-maturity OTM options
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Chapter 6
Estimation Method
The maximum likelihood estimation (MLE) is the method for estimating the model pa-rameters. This method has some excellent properties such as unbiased, efficiency, consis-tency, and asymptotics normality. However, if the sample data includes the latent data or unobservable data, we only compute the log-likelihood function from the observable data. Since there are many problems such as the higher cost of computing and many feasibilities for the latent data. In this paper, our models have the unobserved latent vari-ables such as return jump risks, stochastic volatility and volatility jump risks. Thus, we consider the expectation-maximization (EM) algorithm which is a iterative method to fit maximum likelihood function and Bayesian posterior distribution to estimate parameters in our models. On the other hand, the stochastic volatility is unobserved variables which belong to hidden Markov model, the particle filtering algorithm is the common method though sampling the particles to fit this model and the Bayesian posterior distribution, and then we can estimate parameters by EM algorithm.
6.1 Nonlinear Dynamic System
In this section, we introduce the general nonlinear dynamic system with state equation and measure equation which also known as hidden Markov model. We use the Baysian tracking method (Particle filtering) to track the dynamic of the latent variable in this kind of model. First, we define the problem of tracking, consider the state space {xt, t ∈ N } of the target, the state equation is given by
xt= ft(xt−1, vt−1) (6.1)
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where ft: Rn×Rn→ Rnis a nonlinear transition function of the state xt−1, {vt−1, t ∈ N } is an independent and identically distributed process noise sequence. The objective of tracking is to estimate xt from the measure equation, given by
zt = ht(xt, wt) (6.2)
where ht : Rn × Rn → Rn is a nonlinear measurement function, {wt, t ∈ N } is an independent and identically distributed measurement noise sequence. Thus, according to the set of all available measurements z1:t = {zi : i = 1, ..., t} at time t, we can seek filtered estimates of xt. From the Bayesian perspective, the tracking problem is to recursively calculate some degree of belief in the state xk at time t when we give the data z1:t. Therefore, it is required to construct the posterior pdf p (xt|z1:t) with the initial pdf, p (x0|z0) = p (x0). Then, the pdf p (xt|z1:t) may be obtained recursively in two steps:
prediction and update.
In the prediction step from time t − 1 to t, we suppose that the required pdf p (xt−1|z1:t−1) is available at time t. The prediction step involves using the equation (6.1) to obtain prior pdf of the state at time t. Thus, by the Chapman-Kolmogorov equation and the Markov property, we obtain the equation
p (xt−1|z1:t−1) = Z
p (xt|xt−1, z1:t−1) p (xt−1|z1:t−1) dxk−1
= Z
p (xt|xt−1) p (xt−1|z1:t−1) dxk−1
(6.3)
where the probabilistic model of the state evolution, p (xt|xt−1), is the defined by equation (6.1) with noise vt−1.
In the update step at time t, by the equation (6.2) and the state xt, a measurements zk becomes available and this measurements can be used to update the prior by Bayes’ rule:
p (xt|z1:t) = p (zt|xt) p (xt|z1:t−1)
p (zt|z1:t−1) (6.4)
where the normalizing constant p (zt|z1:t) =
Z
p (zt|xt) p (xt|z1:t−1) dxt (6.5) The equation (6.5) depends on the likelihood function p (zt|xt) which is defined by the measurement equation (6.2) with noise wt. In the update step, the measurements zk is used to modify the prior probability density to obtain the required posterior probability
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density of the current state xk. Therefore, the recurrence relations between (6.3) and (6.4) form the basis for the optimal Bayesian solution. However, this recursive progation of the posterior probability is only a conceptual solution, it cannot be determined analytically.
Therefore, we use the particle filters to approximate the optimal Baysian solution.