Volatility Trading (Wiley Trading) by Euan Sinclair

Author:Euan Sinclair [Sinclair, Euan]
Language: eng
Format: azw3
Publisher: Wiley
Published: 2013-04-15T16:00:00+00:00

Further, we can calculate the variance of the distribution.

8.25

In this example, we get a standard deviation of 0.137.

What is the Kelly ratio in these circumstances, where clearly we have the extra “risk” of parameter uncertainty? There is a significant chance that we may even be playing in a losing game.

The Kelly ratio is the amount, f, that maximizes the logarithmic gain. We also know, from the work of Chapman (among others), that this corresponds to using the average of the win probability. But we can illustrate this with a simple example.

Consider the simplest case. The winning probability can take one of two values: p1 with probability F1 or p2 with probability F2. Now we can write the growth rate as

8.26

Differentiating this with respect to f and setting equal to zero gives the Kelly ratio as

8.27

This is clearly just

8.28

This doesn't seem particularly interesting, but when we combine this with Equation 8.24 we see that

8.29

If we compare this with the naive value where we estimate p to be w/N we see that

8.30

which implies that the naive estimate will always bias our bets too high.

This effect isn't particularly large and in the limit of large N it disappears completely. In fact after 100 trials we will be off by only 2 percent. Of more importance is the variance of f. How many trials do we need to be reasonably sure that our measured value is close to being correct?

The delta method tells us that the variance of f is given by

8.31

Using Equations 8.25 and 8.31 we get

8.32

So looking back at the case in the figure where w = 6, N = 10: We would estimate that the standard deviation of f would be approximately 0.274. This again diminishes as N increases. If we had observed 60 wins in 100 trials our standard deviation would be only 0.097.

The actual dependence of the standard deviation of the Kelly ratio with sample size conforms to our expectations. More data implies less deviation. For a theoretical win probability of 0.6, the deviation as a function of sample size is shown in Figure 8.10.

Figure 8.10 The Dependence of the Standard Deviation of f as a Function of Sample Size

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.