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3.5 Computing the Option Price

3.5.3 CEV Option Prices

In the same way that a call price series can be computed for geometric Brow- nian motion using the Black-Scholes formula, the call price series that cor- responds to a simulated CEV process can be calculated using Schroder’s formula. A difference is that SPLUS cannot evaluate Schroder’s formula when τ is close to zero, because both the quantile at which the non-central Chi-squared survivor function must be evaluated, and the non-centrality pa- rameter become infinite. It appears then that both theQterms will be zero, yielding an option price of 0, not the same as the required (ST−K)+, however a limiting argument will undoubtedly yield the correct result. Computation of Schroder’s formula becomes increasingly slow as the time to maturity nears zero (as would computation of the infinite sums, sincen2must be increasingly

large to compensate for growth in k).

Figure 3.7 displays a realisation of {St}, a CEV process with β = −1,

St = $5, τ = 1, µ = 0.1, σ = 0.3, and 250 subintervals. In addition, series corresponding to {Ct+Ke−rτ} and {Ke−rτ} are shown. The difference be- tween the two latter curves is of course the option price series {Ct}. The graph exhibits the same general qualities shown in the corresponding graphs for the GBM realisations seen in Figures 2.4 and 2.5. A high degree of cor- relation is seen between the series {St} and {Ct}, where every movement in the share price is mirrored by a movement in the option price. In the particular realisation in Figure 3.7, the option matures in-the-money, and so, near exercise when the discounting effect on the exercise price is mini- mal, the option price plus (discounted) exercise price converges to the share

Time to Maturity - Years Price 4.0 4.5 5.0 5.5 6.0 1.0 0.8 0.6 0.4 0.2 0.0 Share price

Option price + pv(exercise price) pv(exercise price)

Figure 3.7: A realisation of a CEV share price with initial value St = $5, and parameters β =1, τ = 1, µ= 0.1, σ = 0.3, and 250 subintervals; also the CEV option prices for this share series added to the present value of the exercise price, withK = $5, andr = 0.06 and time to maturity indicated on the horizontal scale; finally the present value of the exercise price itself.

price, or equivalently, the option price converges to the share price less the (discounted) exercise price.

Both MacBeth & Merville (1980) and Beckers (1980) have computed and analysed CEV prices. In particular they compare option prices for various CEV classes with Black-Scholes option prices. MacBeth and Merville choose CEV processes with β = 0, -2, and -4, aligning the processes to GBM using the relationship

δ =σS1−

β

2

t

whereσ is equivalent to the Black-Scholes volatility parameter, as described earlier.

I have chosen to reproduce part of MacBeth and Merville’s Table 1, using SPLUS and Schroder’s pricing formula seen in Equation (3.44). These figures

are shown in Table 3.1 and correspond to those of MacBeth and Merville except in two cases, shown with an asterisk. These values are given as 0.89 and 0.94 by MacBeth and Merville, instead of 0.88 and 0.95 respectively. The correspondence between the two sets of results indicates that Schroder’s formula is indeed reproducing Ct, correctly and without the need to make decisions regarding the upper and lower limits of summation in Cox’s formula. The fact that two values differ in the second decimal place, indicates that at least twice, the values ofn1andn2 chosen by MacBeth and Merville have not

ensured convergence of the option price seen in Equation (3.36). In pricing a large bundle of options, such differences could amount to significant pricing errors. σ= 0.2 σ = 0.4 τ K β= 0 β =2 β = 0 β =2 40 10·20 10·20 10·28 10·34 30 365 50 1·27 1·27 2·41 2·41 60 0·00 0·00 0·11 0·06 40 10·62 10·64 11·30 11·58 90 365 50 2·36 2·36 4·31 4·33 60 0·06 0·03 ∗0.95 0·73 40 11·32 11·42 12·81 13·35 180 365 50 3·55 3·56 6·28 6·34 60 0·39 0·28 2·35 1·97 40 12·05 12·21 14·15 14·89 270 365 50 4·55 4·57 7·85 7·97 60 ∗0.88 0·69 3·65 3·18

Table 3.1: A section of MacBeth and Merville’s Table 1, of CEV option prices, calculated for the parameters shown, with additional parameters: St = $50 and r= 0.06.

Beckers compares Square Root and Absolute CEV prices with Black- Scholes prices. In addition to using Equation (3.36) withn1 = 0 andn2 = 995

to obtain the CEV prices, Beckers uses an approximation (attributed to John Cox, but not referenced) for the Square Root CEV case, and the exact formula for the Absolute CEV case, given in Equation (3.39).

Earlier, I described the alignment process used by Beckers, to ensure his CEV prices were comparable to Black-Scholes prices. This involved solving the equation

Var(ST|St) =St2e2µτ(eσ

2

−1)

where the random variableST follows a CEV process withβ <2, forδ[which features in Var(ST|St)], so that all three share price processes considered have the same variance atT. In contrast, MacBeth and Merville’s alignment procedure ensured all processes have the same variance over the interval [t, t+dt].

Using the form for Var(ST|St, β= 1) given in Theorem 3.2, Beckers shows that for a Square Root CEV process,δ is given by the equation:

δ2 = µSte µτ+σ2τ

(eσ2

−1)

eµτ 1 (3.45)

whereσ is the Black-Scholes volatility parameter7. I have used this relation-

ship to give the values seen in Table 3.2.

Months to maturity

σ 1 4 7

0·2 1·2694 1·2828 1·2964 0·3 1·9100 1·9485 1·9877 0·4 2·5579 2·6439 2·7331

Table 3.2: Beckers’ δ values (rounded to 4 d.p.) for the Square Root CEV process, found using Equation (3.45), with additional parameters St = $40, and µ= log(1.05).

Table 3.3 reproduces a section of Beckers’ Table II, for the Square Root CEV case with St = $40 and r = log(1.05). In all but one case, the values given by Schroder’s formula correspond exactly to those given by Beckers, with the single exception when (σ, K, τ) = (0.2,45,124), which reads 0.478 in Beckers’ table.

Months to Maturity σ K 1 4 7 30 10·122 10·495 10·908 35 5·150 5·798 6·478 0.2 40 1·006 2·193 3·063 45 0·019 ∗0.486 1·093 50 0·000 0·059 0·287 30 10·123 10·639 11·307 35 5·235 6·363 7·393 0.3 40 1·471 3·147 4·357 45 0·149 1·248 2·298 50 0·004 0·393 1·082 30 10·136 11·010 12·067 35 5·427 7·121 8·557 0.4 40 1·941 4·145 5·755 45 0·397 2·156 3·670 50 0·043 1·001 2·221

Table 3.3: A section of Beckers’ Table II, showing Square Root CEV prices, with additional parametersSt = $40, and r= log(1.05).

I have been unable to confirm the Absolute CEV prices given by Beckers, and have not attempted to confirm the values of δ he gives in his Appendix B. These values are obtained by solving the equation:

Var(ST|St, β= 0) =St2e2µτ(eσ

2

−1)

for δ, and are reproduced in Table 3.4. These values are of quite a different scale toδvalues given by MacBeth and Merville’s alignment procedure, which are 8, 12, and 16 corresponding to σ values 0.2, 0.3, and 0.4 respectively, indicating the values used by Beckers may have been reported incorrectly.

Note that prices in both Tables 3.1 and 3.3 exhibit the following proper- ties:

7MacBeth and Merville’s method yields δvalues of 1.2649, 1.8974, and 2.5398, corre-

sponding toσ= 0.2, 0.3, and 0.4 respectively. These values are similar to those shown in Table 3.2, however use of these values, instead of Beckers’, results in option prices up to 15% less, with the difference increasing with time to maturity andσ.

Months to maturity

σ 1 4 7

0·2 0·1769 0·1818 0·1868 0·3 0·4006 0·4194 0·4394 0·4 0·7185 0·7727 0·8504

Table 3.4: Beckers’ δ values (rounded to 4 d.p.) for the Absolute CEV process, with additional parametersSt = $40, andµ= log(1.05).

• As K increases, Ct decreases;

• As τ increases, Ct increases;

• As σ increases, Ct increases.

These general trends, observed for the particular values given above, corre- spond to the expected behaviour of call option prices as discussed in Chapter 1.

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