3.6 Use of the CEV Model
4.1.1 Summary of Alternative Methods
Beckers (1980)
Beckers estimates β values for 47 stocks, using almost five years’ data for each. The SDE (3.1) yields
Var µ dSt St ¯ ¯ ¯ ¯St ¶ =δ2Stβ−2dt.
With daily data, we can set dt= 1, and so rewrite the above relationship in the form: ln µ s µ St+1 St ¯ ¯ ¯ ¯ St ¶¶ = lnδ+ β−2 2 lnSt
where s(.) is the standard deviation operator. This is effectively the regres- sion equation forβused by Beckers. Estimation of the regression parameters will yield estimates for both δ and β. Application of this technique requires some refinement however, as the dependent variable in the regression is not observable. Using the fact that for daily dataµ is very small, giving
E µ St+1 St ¶ =eµ≈0.
Beckers approximates the standard deviation on the LHS by |ln(St+1/St)|.
case, the mean value of X will be proportional to its standard deviation. This result is proved by Beckers. Thus, using the single realisation of X
instead of the standard deviation of interest, Beckers estimates β using the regression ln ¯ ¯ ¯ ¯ lnSt+1 St ¯ ¯ ¯ ¯ =a+blnSt+wt (4.1) whereβ = 2b+ 2.
He reports low R2 and Durbin-Watson statistics which lead him to con-
clude that his regression is incomplete. However, using this technique, Beck- ers finds that 38 of his 47 cases had β estimates significantly less than two, with three stocks yielding β significantly greater than two, leaving only six stocks consistent with GBM. Also, on the basis of his estimates, Beckers finds evidence to reject the hypothesis that the true β values for the stocks are the same, and concludes that in general, different stocks will have different values of β, just as their volatilities and other characteristics differ.
Throughout his analysis Beckers restricts the CEV model to β : 0≤β <
2. He concludes that only the 33 estimates in this range support the use of the CEV model, rather than utilising evidence from all the estimates which were significantly above or below two.
MacBeth and Merville (1980)
MacBeth and Merville consider six stocks with options traded on the Chicago Board of Trade Options Exchange. They use one year’s daily data to estimate
β for each of the stocks. From the SDE (3.1), the Chi-squared random variable
(dSt−µStdt)2
δ2Sβ
tdt
≡ut ∼χ21
can be formed given values of µand β. A sample of daily returns is used to estimate µ, and the constant δ2 embedded in what they describe as appro-
this sample tested for goodness of fit with a χ2
1 random variable, using the
Chi-squared goodness of fit test. Using this method, MacBeth and Merville were unable to find β values that gave consistency between sample values and theχ2
1 distribution, for any of their share series. They attributed this to
non-normality of the dBt estimates. Poor estimation of µmay contribute to the inability to find appropriateβ values however.
In light of this failed attempt, MacBeth and Merville attempt a regression similar to that of Beckers. The regression equation, found by taking the natural logarithm of the equation above, is
ln{(dSt−µStdt)2)} −lndt= 2 lnδ+βlnSt+ lnχ21.
This is used to obtain point estimates of β, under the assumption that the (central) Chi-squared random variables are uncorrelated. It is noted that E(lnχ2
1)6= 0, but this should not affect estimates for β.
A further regression method is used to obtain confidence intervals for β, using the inverse relationship between β and ut, the Chi-squared random variable. The regression methods are said to yield “credible but imprecise” estimates for β. In fact the confidence intervals are large, with only integer values ofβ considered. MacBeth and Merville’s point and interval estimates are reproduced in Table 4.1.1. Examination of this shows that only one confidence intervalexcludesgeometric Brownian motion as a model for share price evolution.
A final integer estimate for β is found by computing implied δ values for each stock. To do this they select four option prices on each stock at random, and for a chosen value of β, use a numerical search routine to find a value for δ. This is repeated for other (integer) values until the implied
δ’s are approximately equal. This method yields ˆβ’s that do lie within the confidence limits in Table 4.1.1, but are very different from the previously obtained point estimates. The estimates are shown in the final column of the table.
Stock Code Confidence Limits Point Estimate Implied Estimate ATT −2, 6 3·84 −2 AVON −8, −2 −3·63 0 ETKD −1, 5 3·04 0 EXXN −1, 5 1·62 0 IBM −8, 2 −4·16 −4 XERX −4, 2 −1·69 1
Table 4.1: MacBeth and Merville’s β estimates for six stocks.
The conclusions drawn from these results are that different stocks in gen- eral have differentβvalues, and thatβis in general less than two. In addition, MacBeth and Merville add that there was evidence in their sample that β
changes over time, and that it can be greater than two. The apparent sup- port for the CEV model provides an explanation for “why practitioners who use the Black-Scholes model to value call options must constantly adjust the variance rate they input to the model.”2
Comments by Manaster (1980) on the analysis just described appear at the end of the paper by MacBeth & Merville. These comments focus mainly on the estimation method ofβ andδ. Manaster suggests that ifβ andδwere estimated jointly, results “even more favourable to the [CEV] model” could be achieved. He objects to the daily re-estimation of ˆδ and the variance of returns, since the CEV model has a changing variance built in and re- estimation of parameters prevents this aspect of the model from being tested. Emanuel and MacBeth (1982)
This paper examines the data used by MacBeth & Merville (1980) in addition to a further year’s data for the same six stocks. Emanuel and MacBeth employ a method that they themselves admit is not optimal. They use the relationship s µ dSt St ¯ ¯ ¯ ¯ St ¶ =δS β−2 2 t
to estimate β, where the standard deviation on the left hand side is given by an at-the-money call’s implied volatility, found using the Black-Scholes model. The Black-Scholes implied volatilities are assumed to be reasonable estimates of share price volatility when calculated using observed at-the- money option prices3 even though the Black-Scholes model may not be ap-
propriate in general. In order to find ˆβ, squared differences between CEV option prices and selected market option prices4 are minimised overβ. Using
this estimation technique, four of the six stocks are found to haveβ estimates greater than two, lending support to the alternative model that they derive. Marsh and Rosenfeld (1983)
Marsh and Rosenfeld’s paper does not model share prices but rather interest rates, using the CEV model. In order to obtain estimates of β for their two interest rate series, Marsh and Rosenfeld construct a likelihood function for each series using a modification of the density function given by Cox, seen in Equation (3.11), withτ = 1 day. This likelihood function is then calculated forβ ∈ {0,1,2}for different values ofδ, where the lognormal density function is used when β = 2. The three different likelihood functions corresponding to the three values ofβ, have maxima at a specific value ofδ. These maxima are then compared, and the model that gives the largest of these selected. This in turn specifies the appropriate estimate of β. Using the interest rate data, of the three models considered, the lognormal (β = 2) model gave the highest likelihood for each of the time series.
Tucker, Peterson and Scott (1988)
Tucker et al. use the CEV model to describe yet another underlying asset, foreign currency exchange rates. In order to estimate β and δ for six major currencies, the log-likelihood function derived by Christie (1982) is used and
3These observations are documented in Mayhew (1995) and Beckers (1981).
4Options with less than 90 days to maturity were omitted from Emanuel and MacBeth’s
maximised with respect to β and δ jointly using the Newton-Raphson pro- cedure. This method is discussed in the following sections, as it is similar to that which I have used.
Five years’ data is analysed for the exchange rates (to US dollars) of each of the six currencies: the British pound, Canadian dollar, Deutsche mark, French franc, Japanese yen, and the Swiss franc. The data period is split into five year-long blocks, and β and δ estimated for each of the five sub- periods. They test the null hypothesis H0 : β = 2 against the alternative
HA : β 6= 2 for each of the six currencies over each of the five subintervals, and find significant evidence to reject on 26 of the 30 occasions. In addition to the support provided for the CEV model, and in particular Emanuel and MacBeth’s model, there was evidence to suggest that β was changing over time for the exchange rates.
Melino and Turnbull (1991)
Melino & Turnbull also use the CEV model to describe the evolution of ex- change rates over time. Spot exchange rate data for five currencies were available for a 712 year period. Like Marsh & Rosenfeld, Melino & Turn- bull use an adapted form of the CEV density function for the process to construct a likelihood function. This is evaluated at the restricted range of
β = 0,12, . . . ,2, for a combination of values for the three other unknown parameters (including δ). In addition to this method, they use an approx- imation like that of Tucker et al. and find that “the maximised value of the (quasi) log-likelihood were virtually identical to those obtained from the continuous time model.”5 Once again though, this is only maximised with re-
spect to the other parameters for fixed values ofβ. The case of lognormality was rejected in four out of the five tests.