# compute sharpe ratio for options?

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?

• There is no difference whatsoever. As long as you realize returns on any asset then you can calculate a risk adjusted measure of return, in your case Sharpe Ratio. – Matthias Wolf Oct 31 '13 at 4:44
• ok i have updated my question with an example, please let me know your thoughts – godzilla Oct 31 '13 at 12:21
• You can calculate the value of your portfolio on a daily (or hourly) basis. These values you can use to obtain returns. – Bob Jansen Oct 31 '13 at 12:59
• Your edit makes your question even more confusing: 100 call options (each contract representing 10 shares) and a strike price of 25 pounds makes that a cost of 25k pounds not 250k. Same with the proceeds you receive upon selling the delivered shares at market price. Also, you should read up on the basics of risk adjusted return measures. You need to generate a string of returns before you can calculate the variation of such returns. So, I still do not understand what you try to achieve here??? – Matthias Wolf Oct 31 '13 at 14:22
• Agree with @MatWolf. I now think this question does not fit in our format and acted accordingly. – Bob Jansen Oct 31 '13 at 16:11

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.

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 $\sigma_p$.

• i have updated my question please take a look and let me know your thoughts – godzilla Oct 31 '13 at 12:21

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 ...

• hello there, i have updated my question with an example, if would be much appreciated if you can verify whether or not my approach is correct! – godzilla Oct 31 '13 at 12:20

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