In the last 48 hours I've stumbled upon an idea that links Value at Risk methodology with put call parity and implied volatility from the Black Sholes model. The essence of the idea is that we can use the prices in the derivative market to construct the expected return distribution on the asset from now to the options expiry. After talking to a prof, I've realized although new to me, this idea has been thought up and proven nearly a decade before I was born. The upside is that it works, and seems to beat most of the other main models, like the normality assumption or the historical distribution assumption. An excellent professional paper that explains all this can be found here, but for now I will walk you through my thought process of the solution.
Remember the Black Sholes model? Before the binomial pricing model it was the way to price options. One of it's major flaws was that the formula relied on the normal distribution to describe prices.
In the middle of my corporate finance class last night I remembered that you can use the B/S model backwards to find the implied volatility of an asset. If a stock is trading at $12.34, and you have a call with a strike at $16 that costs $0.49 you know the expectations of the market that the stock will get to $16 by the options expiry (August in this case). Using the b/s model backwards you can solve for the implied volatility (which is the same as saying the standard deviation of returns) of the stock (turns out to be 0.35). Then using this information you can say under the normal distribution the market expects the likelihood that the stock will reach the strike price is 27% (by cracking out that Z table, or =normsdist() in excel).
But this isn't the fun part. Knowing the implied volatility you should be able to use put call parity to find the put price for the same strike. When I was trying this out I noticed that I was always overestimating the price of the put for the same strike. Logically this means that the market believes the stock will go down easier than it will go up, the left tail is fatter than the right tail.
The really cool part about this is by taking puts and calls of all different out of the money strikes you can begin to map out the expected distribution by paying close attention to the deviation from put call parity. You need to use out of the money options to avoid the bias the volatility smile will bring into your model. After mapping the distribution ( a step I still need to learn how to do), you have a distributions that reflects the markets expectations for the asset performance (the stock) from now to the expiry date.
Jackwerth and Rubinstein's Findings
Now I just need to figure out how to efficiently code it for excel, and I'll have a much better risk/reward picture for any asset I'm looking at for investment or analysis at work, assuming it has a well traded derivative tree. Stumbling blocks I dealing with now include
- Making sure my current Black Sholes model is efficent enough to avoid estimation error.
- Getting historical option prices and weeding out low volume offers that will have a significant bid ask spread.
- A way to automate the process using VB in excel to construct a model that can be run over several days to measure shifts in distribution expectations.
- Some more education to understand how the implementation of the methods used in the paper provided above could work in excel.
- I can peg a dynamic and empirically accurate probability on a target price I come up with for a stock.
- I can have a better idea of what the expected Value at Risk likelihoods are.
- By observing changes in the distribution overtime I can start thinking of applications of the second and third order "Greeks" in adding forecasting information to my analysis.
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