**Adjusting Full Kelly for Catastrophic Loss, with examples taken from Options Trading**

**Kelly’s Criterion Revisted**

Following Kelly’s Criterion if we would make repeated bets on the same positive expectation gamble, how much of our bankroll should we risk each time?

From before the fraction to bet is:

or as some put it

**Example
**Say the gamble is to flip a fair coin, the payoff odds are 2:1. How much should we bet?

The expected or average payoff per play is

Upon a win, for a $1 bet we get our bet back and $2 besides, so the gain per unit bet is 2; hence the Kelly fraction is

of our bankroll.

**Virtues of Kelly**

*• *our investment is compounded so we get exponential growth (or decay)

*• *our growth of capital is maximized

*• *we can not be wiped out – e.g. for an even money 60/40 investment, *f *= 0*.*2, upon a loss we still have 80% of our bankroll.

**Example**

Starting with $10,000, after 11 losses in a row we would still have $860, or 8.6% of our initial bankroll, the probability of 11 losses in a row is 1/25,000 or 0.00004, (*conversely the probability of 11 consecutive losses not happening is 99.99996%*).

**Kelly applied to Option Trading**

While stock investments are more free-form, the option trading has common grounds with gambling:

- fixed terms
- a definite time horizon
- a payoff settlement at expiration

Hence with the proper statistics, we can use the Kelly criterion to determine optimal investment levels while protecting against a string of reverses.

**Gathering Statistics
**We need to identify the characteristic features of a specific option play we customarily make. Then record the particulars and results of that play over many implementations. Steve Lenz and Steve Papale of OptionVue provided this data.

**OptionVue Credit Spread
**The data we receied was for over 75 trades

- number of trades that gained money was 66, and 9 lost money
- average gain per winning trade: $659.12
- average loss per losing trade: $1,799.06

**I regard the average loss per losing trade as the “bet size”.**

**Credit Spread Risk Fraction
**We calculate the following needed for “edge over odds”

win probability.

*Therefore *gain per unit bet

*
*Hence expectation = (0

*.*366)

*∗ 0.*88

*–*(1)

*∗ 0.*12 = 0

*.*202.

And so the Kelly risk fraction is

**Accounting for Catastrophic Loss**

But we can do more with the data. Note that we have “maximum loss” information. This can be regarded as catastrophic loss and taken into account. Let **p** be the probability of a win, **q** the probability of a loss, and **r** the probability of a catastrophic loss. Let **γ ***(greek letter gamma) *be the gain per unit bet and **λ ***(greek letter lambda) *the size of the catastrophic loss per unit bet.

We now rederive the Kelly fraction.

**Kelly Fraction for Catastrophic Loss**

The expectation is now

**E = γp – q – λr**.

The expected growth rate (from the previous talk) is

**E(g) = p log(1 + γf) + q log(1 – f) + r log(1 – λf)**

And the optimal fraction is the root of the quadratic equation

**0 = E – f (pγ(1 + λ) + q(γ – λ) + rλ(γ – 1)) + γλf ^{2}
**

**Credit Spreads with Catastrophic Loss**

The new parameters are now

- prob. of a win
*p*=*.*88 - prob. of an avg loss q

- prob. of a catastrophic loss
*r*

- gain per unit bet
*γ*

- catastrophic loss per unit bet
*λ*

**OptionVue Credit Spreads with Catastrophic Loss**

Substituting the values and solving the quadratic equation

**E – f (pγ(1 + λ) + q(γ – λ) + rλ(γ – 1)) + γλf ^{2} = 0**

we get

*f *= *.*458; or about 46%.

*Courtesy: Ron Shonkwiler (shonkwiler@math.gatech.edu)*