3.2 The CEV Share Price Solution
3.2.4 Graphical Examination of S T
The theoretical distribution for CEV share prices atT, given an initial start- ing value att, is given by Equation (3.11), with an alternative form in Equa- tion (3.14). Both these (improper) density functions present problems in terms of computation. In the first case, the Bessel functionIν(z) is not inter- nally computed in SPLUS, and in the second, there is an infinite summation3.
An alternative is due to Schroder (1989), who discusses computation of the CEV option price. Whilst not the focus of his paper, Schroder represents the transition survivor function of ST givenStin terms of a non-central Chi- squared random variable. Johnson, Kotz & Balakrishnan (1995) introduce the non-central Chi-squared distribution on page 433 as in the following result.
Result 3.3 (The Non-central χ2 Distribution). IfU
1, U2, . . . , Uν are in-
dependent unit normal random variables, and δ1, δ2, . . . , δν are constants
then:
ν
X
j=1
(Uj +δj)2 (3.21)
has a non-centralχ2 distribution, withνdegrees of freedom, and non-centrality
parameter: λ= ν X j=1 δ2 j.
The density function of such a variable is given as:
p(y;ν, λ) = exp(− 1 2(y+λ)) 212ν ∞ X j=0 (y)12ν+j−1λj Γ(12ν+j)22jj!. (3.22)
3It does appear that the terms in the summation should converge very quickly to zero,
due to the presence of bothn! and Γ(ν+n+ 1) in the denominator. Together these terms are like n!2 and will quickly dominate the term in the numerator, suggesting that the
Given the definition of Johnson et al., it appearsν ∈Z+is required, how-
ever the moment generating function for a non-central Chi-squared variable is defined for all ν >0. This moment generating function:
MX(t) = (1−2t)− 1 2νexp µ λt 1−2t ¶
uniquely defines the non-central Chi-squared variable, and so non-integer degrees of freedom are permitted. This is noted by Johnson et al. Note also that λ must be positive.
Using the series expansion of the modified Bessel function shown in Equa- tion (3.13) the density function p(y;ν, λ) can be manipulated to give:
p(y;ν, λ) = 1 2(y/λ) 1 4(ν−2)exp©−1 2(λ+y) ª I1 2(ν−2)( p λy) (3.23) which is the same as that given by Johnson et al. in their Equation (29.4). Beginning with Equation (3.23), the equivalence of the two forms of the density function is demonstrated below.
p(y;ν, λ) = 12(y/λ)14(ν−2)e− 1 2(λ+y)I1 2(ν−2)( p λy) = 1 2(y/λ) 1 4(ν−2)e−12(λ+y)(1 2 p λy)12(ν−2) ∞ X j=0 (1 4( √ λy)2)j j!Γ(1 2(ν−2) +j+ 1) = (12)12νy 1 2(ν−2)e− 1 2(λ+y) ∞ X j=0 λjyj 22jj!Γ(1 2ν+j)
The last equation is of course the same as Equation (3.22).
Theorem 3.3 (The CEV Share Price Distribution Function). The tran-
sition distribution function for a CEV share price ST given St,
FST|St(s, τ) =P(ST < s|St) is given by the equation:
FST|St(s, τ) = 0 s <0 G(˜x, 1 2−β) s= 0 Q(2˜x;2−2β,2˜y) s >0
where y˜ = ˜ks2−β, k˜ and x˜ are defined in Equation (3.12), and Q(x;ν, λ)
is the survivor function at x for a non-central Chi-squared variable with ν
degrees of freedom and non-centrality parameter λ.
Proof. Note firstly that ST ≥ 0, so that for s = 0, P(ST < s|St) = 0.
Secondly,P(ST = 0|St) is given by Theorem 3.1, and isG(˜x,2−1β) as required. I will defer the proof for the final case, where s >0, until Section 3.5.2, since the proof is made considerably easier by results which follow.
Thanks to Schroder’s result, and the fact that the non-central Chi-squared cumulative probability function is an internal function in SPLUS, we can again compare the theoretical distribution for share price at a future timeT, to an empirical distribution of simulated values. Using the Euler method, I have simulated 5000 share pricesτ = 1 year into the future, withβ =−1, and additional parameters St = $5, µ = 0.1, σ = 0.3 and n = 250 subintervals. Of these simulated series, 38 had final values ST = 0, giving an estimate of
P(ST = 0) of 500038 = 0.0076. This is less than the true probability of 0.00896, corresponding to an expected number of bankruptcies of 44.8.
A histogram of the remaining 4962 values is shown in Figure 3.4 along with the density function for a GBM process with the same parameters (plot- ted as a broken line), and the true density function (plotted as a solid line) conditional onST >0, calculated using an approximation to Equation (3.14):
fST|St(s, τ) = (2−β)˜k 1 2−βe−x˜−z˜ 100 X n=0 ˜ xn+2−β1 z˜n+ 1−β 2−β n!Γ(n+ 1 + 2−1β)
where the upper limit of the summation has been replaced by 100. In fact, remarkable accuracy can be achieved using only the first 11 terms of the summation, with a maximum difference of only 0.00005555 between the two approximations, where both are calculated at 500 values of ST in the range [0,10.4]. The theoretical curve is a true density function, which is formed by scaling the functionfST|St(s, τ) above as seen in Equation (3.17):
fST|St(s, τ|ST >0) =
fST|St(s, τ)
0 2 4 6 8 10 0.0 0.05 0.10 0.15 0.20 0.25 0.30 Share Price Relative Frequency
Figure 3.4: 5000 realisations of ST, a future CEV price with β = −1, with
St= $5, τ = 1 year, µ= 0.1 and σ = 0.3, the (solid) theoretical density for these prices, and the lognormal density function with the same parameters.
In Figure 3.4, we see that again there is good agreement between the simulated share prices and theory. Unlike the GBM equivalent seen in Figure 2.3, in this case there are two sources of error, due to sampling error and use of the Euler approximation for the solution to the SDE (3.1). The close fit of the density function to the histogram suggests that aggregate errors in the Euler method of simulating share price under the CEV model have been small.
We can also compare the empirical cumulative distribution function to the theoretical one, which, thanks to Schroder, can be computed directly. These functions are shown in Figure 3.5 where the solid line represents the theoretical curve and the broken line the empirical results from the sample shown in Figure 3.4,with St= $5, τ = 1, µ= 0.1,σ = 0.3 and β =−1.
s P(Share Price < s) 0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0
Figure 3.5: The empirical cumulative distribution function of the 5000 re- alisations of ST shown in Figure 3.4, a future CEV price with β = −1, with parameters St = $5, τ = 1, µ = 0.1 and σ = 0.3, and the theoretical distribution function for these prices.
curve, the empirical and theoretical distributions are virtually indistinguish- able.
Estimating the sample mean and standard deviation for all 5000 reali- sations yields estimates ¯ST = 5.5096, and s(ST) = 1.5516, which are both below the theoretical values of 5.5259 and 1.5782, despite the fact that fewer bankruptcies were observed than expected. These differences are not signifi- cant at the 5% level4.
In Theorem 3.2 I derived a form for the variance of a future CEV share price ST, given St and β. The variance is given in Equation (3.18) and is used to calculate the theoretical standard deviation for the sample above.
4A two-sided test ofH
0: µ= 5e0.1 yields a test statistic of -0.7293, and ap-value of
0.4658. Similarly a two-sided test of H0 : σ2 = 1.5782 yields a test statistic of 4831.91
Using SPLUS, it is possible to calculate the standard deviation of ST given
St by restricting the infinite summation to a finite number of terms. This computation fails when β is close to two, since the quantile at which the gamma density function must be evaluated grows very large. Increasing the shape parameter, and the upper limit of the summation does not solve the problem since both Γ(n+ 1 + i
2−β) terms, wherei = 1,2, become too large, even though their ratio is likely to remain moderate in comparison. Figure 3.6 shows the standard deviation function for−2≤β ≤2, and the additional parameters St = $5, τ = 1 year, µ= 0.1 and σ = 0.3. Due to computation problems the standard deviation for 1.5 < β < 2 has been estimated by linear interpolation using the standard deviation when β = 1.5 found using the approximation, and that for β = 2, given by Equation (2.7).
beta
Standard Deviation of Share Price
-2 -1 0 1 2 1.58 1.60 1.62 1.64 1.66 1.68 1.70
Figure 3.6: The standard deviation of a future CEV share price ST, with
St= $5, τ = 1 year, µ= 0.1 and σ = 0.3, for −2≤β ≤2.
The relationship shown in the graph is not monotonic, with the variance decreasing as β decreases from 2 to approximately -0.50, after which the
variance begins to increase.
3.3
Properties of
C
T- the Exercise Payoff
Many results proved for the GBM case apply to the CEV exercise payoff also. In particular the form for the probability density function of CT given
St, derived in Theorem 2.3 holds, except of course the functions FST|St(s, τ)
and fST|St(s, τ) are different. In addition we saw that E(CT|St) is very sim-
ilar to the price of the option at t, but in future value terms, and with µ
present in the solution instead ofr. In the case of CEV exercise payoffs I will restrict myself to noting the form of the cumulative distribution function of the payoff. A graphical representation of both the theoretical and simulated exercise payoffs follows from Figure 3.5 by settingFST|St(s, τ) = 0 for s < K,
and repositioning the curve, moving the origin tos =K. The shape of both curves will be identical for s > K. This result is stated and proved in the following Theorem.
Theorem 3.4 (The Distribution of CT given St). At exercise, the call
option will deliver
CT =
(
0 ST ≤K
ST −K ST > K
(3.24)
the cumulative distribution function of which is, given St:
FCT|St(c, τ) = 0 c <0 Q(2˜x;2−2β,2˜kK2−β) c= 0 Q(2˜x;2−2β,2˜k(K +c)2−β) c >0 (3.25)
where x˜ and ˜k are defined in Equation (3.12).
Proof. ForK >0:
P(CT = 0|St) =P(ST < K|St) =Q(2˜x;2−2β,2˜kK2−β)
from Theorem 3.3. Likewise, forc > 0
P(CT < c|St) = P(ST −K < c|St) =P(ST < K+c|St)
=Q³2˜x;2−2β,2˜k(K+c)2−β´ again using the distribution function of ST given in Theorem 3.3.