compute sharpe ratio for options?

Question

Calculating sharpe ratio for shares is a straight forward task: (average returns - risk free ) / standard deviation. However i remain baffled as to how to tackle the task for options, can someone please advise regarding this?

consider the following example to obtain returns for American options:

step 1) 
- shares of xyz cost £21 each
- 100 call option contracts (10 shares each) cost £2000
- expiration date 11-11-2013
- strike price £25
- price of shares goes up to £30 and trader decides to execute option


* total cost: (25 * 1000) + 2000 = £27,000
* returns: 30,000 - 27,000 = £3,000

step 2)
the confusion arises when you factor in that there are no previous returns. Conversely with shares such as aapl i can calculate weekly returns and easily calculate the average returns and standard deviation from these, for example if the following were aapl weekly returns: 
week 1 : 500
week 2 : 480
week 3 : 550
week 4 : 600
week 5 : 650

the average returns would be : 556 and standard deviation would be 70.21. How can i do the same with options? Would i need to go through a similar procedure of going back a date and doing step 1 again?

thanks in advance

Answer 1

It seems to me that you want to use the series of option prices to estimate the Sharpe ratio given the option prices in your sample. If so, the idea is to realise that for each option price you have at different times $t_1, t_2, ...$ you could actually close the position and realise the profit or loss. So, basically if you have the option prices you just compute the return as if you would close the position at that time. This way you obtain the evolution of the returns from which you get your average return R. On these returns you can also compute the standard deviation $\sigma_r$ and what's left is to compute the Sharpe ratio.

I guess this is some kind of homework and you are not working with real money because the method above is not that sound. Anyway, I hope this helps.

Answer 2

I guess it makes more general approach and calculate the Sharpe ratio on the portfolio level. Of course, if you want you could take a portfolio with only one option to get your answer. I don't think that really makes sense because of the dependence of the returns of different assets.

In this general approach calculation is easy, for the ex post Sharpe ratio we have: $$\frac{R_p - R_f}{\sigma_p}$$ where the subscript $p$ siginifies portfolio as you wrote yourself. If you want to do it ex ante just put a hat on $R_p$ and $\sigma_p$.

Answer 3

Now, I think in practice most people just take the realized returns and the formula and plug them in and thats that.

A more sophisticated way would be to model the distribution of stock prices and the volatility surface simultaneously and then transform the distribution via the Black Scholes formula (European options) to get the P&L.

Then, divide by the price to get the distribution of returns, calculate mean and standard deviation and plug this into the sharpe ratio formula.

This method is a LOT more complicated so the question is "is it worth it?".

If you already have a risk model that gives you the P&L distribution of the option, you should just use it.

The problem with historical option returns is, that they are not even close to being iid ...

Answer 4

I simply priced the options based on the volatility,strike price, issue date, expiration date (the greeks) using a binomial price engine and then calculated the returns based around this