# The Kelly Criterion and Option Trading

(This is based on/cut-and-pasted from a paper that I co-wrote
with Reuben Brooks.)

The Kelly criterion can be used to calculate the optimal
size of a trade. Specifically, it gives the size that increases the trader's account at the fastest possible rate. It is possible that a given trader might
not actually want this. She might want some sort of volatility or draw down
constraint as well, but traders should still understand the ideas and
implications of Kelly sizing. And misunderstanding abounds.

Generally, traders use an approximate form of the Kelly fraction
that only takes into account the first two moments of the return distribution.
While in many cases this can be a useful heuristic, when returns are highly
skewed the approximation breaks down. In particular, for trades with identical
expected returns, the presence of skew can drastically impact the relative
sizing of long and short option positions.

Long option positions have unlimited profit potential and limited loss potential, and the opposite is true for short option positions. Equivalently, option returns are highly skewed. And this significantly impacts how big you should be trading them.

The derivation of the continuous form of the Kelly criterion can
be found in many places. The result is that the optimal trading fraction of
your bankroll is

SORT OF...

This is actually the result of using a Taylor expansion of the
distribution of trade results. So it requires the neglected higher order terms
that depend on skewness and kurtosis to be of lesser and lesser magnitude. When
considering options this isn’t true.

The basic idea is to extend the Taylor expansion to higher order
terms. The equation we need to solve is,

is
the third raw moment of the trade result distribution.

The solution to this equation is

( Considering the limit when skew approaches zero tells us that the relevant root is the negative one.)

Some Issues:

The solution to this equation is

( Considering the limit when skew approaches zero tells us that the relevant root is the negative one.)

Some Issues:

- Clearly if the argument of the square root is negative there will be no real solutions.
- A skewness of zero leads to a singularity.

These go away if we include the fourth order terms: the kurtosis
of option returns. The downside is that the equations become fairly awful and
won’t help develop intuition.

Despite these problems, the third order expansion is important in
practical situations. For example, take a trade with a return of 25.3% and a
volatility of 201% (numbers from an actual trade that we won’t go into here).
The simple, mean-variance approximation gives a Kelly ratio of 0.062. But
negative skewness will significantly reduce this and positive skewness will
significantly increase this.

The size of the effect is shown in Figure One.

The size of the effect is shown in Figure One.

Figure One: The dependence of f on skewness.

This analysis applies to all trades. But few trades have more
non-normal return distributions that options.

Reuben did a
lot of clever algebra and calculus to find closed form expressions for skewness
and kurtosis of options and I’d encourage everyone to read our paper for these
details. However, monte-carlo simulations can give us an idea of the effect. In
Figures Two and Three we show the results of simulations that confirm this.

Figure Two:
The distribution of profits for a short one-year, ATM straddle (10,000 simulations
of GBM with a return of 0%, an implied volatility of 50% and a realized
volatility of 50%).

Figure Three:
The distribution of profits for a long one-year, ATM straddle (10,000
simulations of GBM with a return of 0%, an implied volatility of 50% and a realized
volatility of 50%).

We can clearly see the large skewness.

This means that it is completely correct to trade long and short volatility strategies at different sizes, even when the expected value is identical.

This means that it is completely correct to trade long and short volatility strategies at different sizes, even when the expected value is identical.