Portfolio Utility Function Example
Consider the following utility function which balances returns on capital with risk, M=G−0.5∗λ∗V where G is expected gain in capital, is a risk aversion parameter and V is the variance of G. We seek to maximize M.
For a given option, since we expect to close the position for theoretical value, G=E(theo−p0)∗q=(E(theo)−p0)∗q, where theo is the theoretical value, p0 is our original trade price and q is our position size. Over a short enough time duration where we can ignore the effect of theta and with a delta-hedged position, E(theo)=theo , therefore G=theo-p0*qG=theo−p0∗q
Next, assuming a delta-hedged position and negligible effect of theta,
and , therefore,
. Assuming implied volatility is normally distributed, the variance of capital is then
, where iv is the annualized implied vol of the option and vov is the annualized vol of implied vol of the option.
You have a capital pool of $1,000,000. Consider option BTC-24SEP21-32000-P with mark iv of 109.59, mark theo of 0.15 btc ($5,830) and vega of 73.68. Assume annualized vol of vol is 200. You want to trade this option with the intention of holding this position for 24 hours. How much should you buy if you want less than 15% probability of losing no more than 1% of your capital on this trade? Again, assuming normal distribution, 15% probability means 1sd change in capital. Therefore sd = 10000. Next, . Solving for q, we get that you should buy 13 options. This gives you total vega risk exposure of $955.25 per iv change of 1 vol point. Given one standard deviation downward vol move of 10.46, your total drawdown is $955.25 * 10.46 = $10,000
We are attempting to maximize . Substituting earlier results, . Differentiating, Setting and solving for p0, we get that . Subbing previous values in, we get that . Another way of saying that is . The market width on this option is $3,956.4, for an edge of $1.978.2. To match that edge, we would set . Applying the same risk aversion factor and maximum 1sd drawdown to another option, BTC-24SEP21-80000-C with vega of 62.36 and iv of 108.71 yields desired position of 15.31 and edge of $1671.2
We can extend the above result to multiple instruments by replacing q with a vector of quantities for each instrument q, edge theo-p0 with a vector of edges e, and Var with a covariance matrix , s.t
We then get that .