The log-optima! strategy has an important roie as a universal pricing asset, and the pricing formula is remarkably easy to derive. As before, we assume that there are n risky assets with prices each governed by geometric Brownian motion as d Pi
Since E(dzi) = 0 for all /, the covariances a/y are defined by E(dzt dz.j) — Oij dt, There is also a risk-free asset (asset number 0) with rate of return ly. Any set of weights itfo, W],w2,- , w„ with YI'Lq wi — I defines a portfolio in the usual way The value of this portfolio will also be governed by geometric Brownian motion We denote the corresponding covariances of this process with that of asset i by crf port
As a special case we denote the log-optimal portfolio by the subscript opt, This portfolio has variance denoted by a02pl and covariance with asset / denoted by ay opl.
The ¡x of any asset can be recovered from the log-optimal portfolio by evaluating the covariance of the asset with that optimal portfolio This is essentially a pricing formula because it shows the relation between drift and uncertainty, The pricing formula is stated here (in four different forms):
Log-optimal pricing formula (LOPF) For any stock i there holds
Equivalent!)', we have
Pi — >'f = A.opc (Mop! — >'f) VS - ff - A.opt^opt ~ 5°f
Proof: The result follows from the equation for the log-optimal strategy (15 6); namely,
Hence cr;.op, = E(dz; dzopt) = OijWj = /x/ — //, where the last step is (15,9). This gives (15,7«) The version (15,7/.?) follows from i-v = pj - \crf, To obtain the alternative expressions we apply the first pricing formula [equation (15.7«)] to the log-optimal strategy itself, obtaining /xopt —/y = <x~pl, Equation (15..8a) follows immediately. The version (15 Sb) follows directly from the definition of pLopt |
According to these formulas the covariance of an asset with the log-optimal portfolio completely determines the instantaneous expected excess return of that asset. Equations (15 7«) and (15,8«), in terms of p.— rf, are easy to remember because they mimic the CAPM equation These equations express the excess expected instantaneous return as a single covariance or, in the alternate version, as a beta-type formula.
Example 15.9 (Three stocks again) Consider the three stocks of Example 15 8 Let us determine p\ using (15.7a). The covariance of Si with the log-optimal portfolio is found from
E[dzl(iu1 dzi + u>2dz2 + W3dz3)] = [1,05 x 09+ 1.38 x .02+1 78 x .01] dt = 14d/„ Therefore,
If V is the value of the log-optimal portfolio, we have
= if + ,14 = .24 which is correct since it coincides with the p\ originally assumed.
Equations (15lb) and (15.8b), in terms of v — rf, are perhaps the most relevant equations since v is the actual observed growth rate. Consider (15 86), which is v/ — i f = /3;.op,cr~p( — For stocks with low volatility (that is, with of small) the excess
15.7 the log-optimal pricing formula* 437
FiGURE 15.8 Log return versus beta.
growth rate is approximately proportional to ft opl This parallels the CAPM result Greater risk leads to greater growth. However, for large volatility the —\af term comes into play and decreases u.
Note in particular that if security i is uncorrected with the log-optimal portfolio, its growth rate will be less than the risk-free rate This is because its volatility provides opportunity that a risk-free asset does not.
The volatility term implies that the relation between risk and return is quadratic rather than linear as in the CAPM theory. To highlight this quadratic feature, suppose, as may on average be true, that the a of any stock is proportional to its ft; that is, a = yft, where y is a constant. Then we find
A graph of this function is shown in Figure 15.8. Note that this curve has a different shape than the traditional beta diagram of the CAPM It is a parabola having a maximum value at ftQpl = c«.«!y2
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