Sep

5

Measuring the risk of a portfolio which includes stocks and options can seem to be an insurmountable problem. Options move with respect to their underlying stock as a function of the option delta, and at the money option with a delta of .50 will move 50% as much as the underlying stock on a point basis. The CAPM theory says that stocks will move with respect to the market as a function of their beta. Empirical evidence says that the alphas are non-persistent, so only the beta need be considered. Thus a stock with a beta of 1.50 will move 150% as much as the underlying market index moves.

So if we wish to construct a portfolio scale metric which will measure the combined market response of a portfolio of stocks and long and short options we would simply multiply the delta and beta by the quantities suitably adjusted.

Suppose we had an IWM Russell 2K ETF call option with a delta of .50 and IWM is at $72 per share. We wish to convert the position to a common metric of an equivalent dollar amount of SP index.

Convert the option to a dollar equivalent IWM by taking 50% of $72 times 100 shares. This gives us $3600 dollars. Let's say the beta of IWM with respect to SP is 1.50. Multiplying $3600 by 150% gives us $5400 worth of pseudo SP index.

For a stock take the stock value of $7200 times the beta of 150% to get a dollar equivalent of $10,800.

Remember that a short position puts a minus sign in front of the above numbers. Also note that a put has a minus sign built in as well. For a short put it is -1 * -1 = +1. After the equivalent risk of each position is calculated then add them up with their signs to find the total of the portfolio.

A final note of caution - this type of analysis is an excellent way to measure linear risk for small movements in the underlying index. However for non-linear assets, such as options one may wish to consider other measures which include gamma and even the third and higher calculus derivatives of the option model.


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