In order to calculate option prices, one has to specify two things, namely the option to be priced and the stock price model under which one wants to price it.
The pricing method presented above applies to European options that do only depend on the spot price of the stock at expiration. Hence they are completely specified by giving the expiration dateT and the payoff function w(ST). For the new pricing method to be applicable, the bilateral Laplace transform of the payoff function has to exist on a non-degenerate interval. In Section 3.4, we have presented a number of standard and exotic options satisfying these conditions. The corresponding bilateral Laplace transforms can be found in Table 3.1.
The second step is the specification of the stock price model. This should be done by giving the (ex- tended) characteristic functionχ(z) :=EQ[exp(izXT)]of the random variableXT = ln(ST/(erTS0)) which we have identified as the log return on the forward contract to buy the stock at timeT. Below, we give examples of stock price models, together with the corresponding extended characteristic function χ(z)and the strip of regularity of this characteristic function.
The algorithm developed above is applicable to any valid combination of option and stock price models.
Here “valid” means that there exists a constantR ∈IRas in Theorem 3.2, such that Rlies in the strip of regularity of the bilateral Laplace transformL[v](z)and thatiRlies in the strip of regularity of the extended characteristic functionχ(z).
The method was successfully tested with the stock price models given in Table 3.2. All of these models are of the exponential Lévy type, that is, they assume that the log return process(Xt)t∈IR+ is a process with stationary and independent increments. Hence the distribution of the process is uniquely character- ized by each of its one-dimensional marginal distributions. We chooset= 1, that is, we characterize the Lévy process by the distribution ofX1. The first column in Table 3.2 gives the type of this distribution, and the second column gives its characteristic function.
• A normally distributed log returnX1corresponds to the famous geometric Brownian motion model introduced by Samuelson (1965).
• Generalized hyperbolic (GH) distributions were introduced by Barndorff-Nielsen (1978). Eberlein and Prause (1998) used these class of distributions to model log returns on stocks. This generalized earlier work by Eberlein and Keller (1995), where hyperbolically distributed log returns were considered.
• The class of normal inverse Gaussian (NIG) distributions was proposed in Barndorff-Nielsen (1995) and Barndorff-Nielsen (1998) as a model for log returns. NIG distributions constitute a subclass of the class of GH distributions. See also Barndorff-Nielsen (1997).
Type extended characteristic functionχ(z) χ(iR)<∞if...
normal exp iàzt−σ2 2 z2t
−∞< R <∞
GH eiàzt (δp
α2−β2)λt Kλ(δp
α2−β2)t ãKλ δp
α2−(β+iz)2t
δp
α2−(β+iz)2λt β−α < R < β+α
NIG exp(iztà+tδp
α2−β2) exp(tδp
α2−(β+iz)2) β−α < R < β+α
VG exp(iztà)
(1−iθνz+ (σ2ν/2)z2)t/ν
R > θ σ2
1−
r
1 +2σ2 νθ2
R < θ
σ2
1 + r
1 +2σ2 νθ2
Table 3.2: Different models for the stock price: Characteristic functions and admissible values forR.
• Variance gamma (VG) distributions were first proposed by Madan and Seneta (1987) for the mod- eling of log returns on stocks. Madan, Carr, and Chang (1998) generalize this approach to non- symmetric VG distributions.
All of the non-normal models cited above have been shown to capture the distribution of observed market price movements significantly better than the classical geometric Brownian motion model. Moreover, Madan, Carr, and Chang (1998) and Eberlein and Prause (1998) observed a substantial reduction of the smile effect in call option pricing with the non-symmetrical VG and the GH model, respectively.
As a benchmark example, we have used the method described above to calculate the prices of European call options with one day to expiration. The log return distributions employed were those displayed in Table 3.2. The parameters shown in Table 3.3 were generated as follows. First, the parameters of the respective distribution were estimated by maximum likelihood from a dataset of log returns on the German stock index DAX, from June 1, 1997 to June 1, 1999. Then, an Esscher transform was performed on each of these distributions so as to make eLt a martingale. For simplicity, we have always assumed that the interest rate vanishes,r = 0. Hence the results should be not viewed as reasonable option prices, but rather as an illustration of the algorithm. Figure 3.1 shows the prices of a European call option, displayed as a function of the strike price at a fixed stock price of S0 = 1. The option prices were calculated by means of the algorithm described above. Note that this algorithm yields the option price only at a discrete set of values ofK. But we have chosen this set so dense that—for the limited resolution of the plot—the discrete point set looks like a solid line.
Because of the efficient calculation of option prices by the FFT method, it becomes easier to study the behavior of the option pricing function. Figure 3.2 shows the difference of call option prices from the standard Black-Scholes model. Here the W-shape that usually appears for these differences is distorted.
strike price K
option price (GH model)
0.96 0.98 1.00 1.02 1.04
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Figure 3.1: Call option prices for the generalized hyperbolic stock price model. Parameters are given in row “GH” of Table 3.3. Fixed current stock priceS0= 1. One day to expiration.
strike price K
option price difference
0.96 0.98 1.00 1.02 1.04
-0.0004-0.00020.00.00020.0004
NIG VG GH
Figure 3.2: Difference of call option prices between alternative stock price models and Black Scholes prices. The alternative models are GH (exponential generalized hyperbolic Lévy motion), NIG (normal inverse Gaussian), and VG (variance gamma) with parameters as given in Table 3.3. Fixed current stock priceS0 = 1. One day to expiration.
Type Parameters
normal σ= 0.01773, à=−1.572ã10−4
GH α= 127.827, β =−31.689, δ= 7.07ã10−31, à= 0.0089, λ= 2.191 NIG α= 85.312, β =−27.566, δ= 0.0234, à= 0.00784
VG σ= 0.0168, ν = 0.4597, θ=−0.00962, à= 0.009461 Table 3.3: Estimated Parameters for stock price models in Table 3.2
log return x
estimated density
-0.05 0.0 0.05
0510152025
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Figure 3.3: Empirical density of daily log returns on the German stock index DAX, 1 June 1997 to 1 June 1999. Gauss kernel estimate by S-Plus functiondensity.
This distortion is a result of the fact that the empirically observed return distribution for the DAX in the interval 1 June 1997 till 1 June 1999 was relatively asymmetric. See Figure 3.3.
Figure 3.4 shows the difference of NIG and Black-Scholes option prices, seen as a function of the strike priceK and the time to expiration. Note how the difference grows as time to maturity increases. Also note that the shape of the difference curve changes as time to expiration increases. For larger τ, it becomes increasingly bell-shaped. These effects are due to the different standard deviations of the NIG and the normal distribution under the martingale measure: The standard deviation of the NIG distribution is0.0180, while that of the normal distribution is0.0177(see Table 3.3.)
10 20
30 40
50
time to expiration [days]
0.9
1
1.1 strike price K -0.0005
0
0.0005 0.001
option price difference
Figure 3.4: Difference between call option prices in the NIG model minus call option prices in the Black-Scholes model. Fixed initial stock priceS0 = 1; time to maturities between1 and 50(trading) days. Strike prices between0.8and1.2.
On the Choice of the Rate of Decay,R
It turns out that the precision of the calculation crucially depends on the choice of the rate of decay, R.
We have found that for usual parameter valuesR=−25works best for call options. This coincides with the findings of Carr and Madan (1999).