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I have a model that predicts the future price of a stock, and would like to incorporate this information to value the option. Lets take an example. AAPL stock is trading at 151.89 US dollars today (Sep 22, 2017). I would like to compute the fair value of the call option with a strike of 149 that expires on Oct 20, 2017. From the market I can get the current mid price of the option and its implied volatility. However, lets say my model predicts that the value of AAPL on Oct 20, 2017 is normally distributed with a mean of 145 dollars and a standard deviation of 1 dollar. How could I use this information to compute the value of the option? For the purposes of this question, we can assume the dividend yield is 0 and we know the risk free rate. If somebody has a way to do this for a European option instead of an American that would probably be close enough for my purposes.

The purpose of computing the value is so I can buy or sell the option where market price is most different from the value computed above thus maximizing risk adjusted return. The assumption, of course is that my model is accurate.

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    $\begingroup$ If you have a predicted distribution of the underlying, then the corresponding option price (undiscounted) according to that prediction is just the integral of the payoff and the distribution. $\endgroup$
    – will
    Sep 25, 2017 at 21:15
  • $\begingroup$ I think that makes a lot of sense. Would you discount the option back using a rate derived from the variance of the distribution? $\endgroup$
    – quantvoltrader
    Sep 25, 2017 at 21:30
  • $\begingroup$ that depends on your model, but given your question I doubt it. $\endgroup$
    – will
    Sep 25, 2017 at 21:32
  • $\begingroup$ How would you discount the undiscounted option value back to today? Intuitively it seems to me that the bigger the variance of the underlying price prediction is, the less valuable the option would be today since the risk would be higher. $\endgroup$
    – quantvoltrader
    Sep 25, 2017 at 21:37
  • $\begingroup$ I think you need to do some reading on the topic. The reality is that options increase in value as you increase the volatility, which makes sense when you think about the integral i mention in my comment. as for discounting it back to today, it is not related to the variance, it is to do with the value of having money now vs. having it in the future. I i can invest £95 now in a product that will pay me £100 in 1 year (i.e. a zero coupon bond), then if the expected value of your option comes out to £20 in one year, then today that is worth £20 * £95/£100 = £19 today. $\endgroup$
    – will
    Sep 25, 2017 at 21:41

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You cannot incorporate your own prediction into the "fair value" of the option. "Fair value" of the option is the initial amount required to set up the replication hedging portfolio, and is shown to be equal to the discounted expectation of the option payoff under the risk neutral measure, not the subjective measure that comes out of your "personal" model. You can however compute the distribution of return of buying or selling an option under your "personal" model and build your investment strategy accordingly.

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  • $\begingroup$ If I'm reading this correctly, it seems I could value the option just using the predicted distribution of the underlying. Lets say that the call option is European. Then on expiry, the distribution of the option price would be the same as the distribution of the underlying - strike, but it would be cut off, i..e probabilities would be 0 when the underlying is less than the strike price. The value of the option is then the mean of a normal distribution which is cut off on the left? Does the variance of the distribution matter or does it not matter because of risk neutral pricing? $\endgroup$
    – quantvoltrader
    Sep 25, 2017 at 19:46
  • $\begingroup$ Of course in computing an expectation which is "cut off to the left" the variance matters: the higher the variance the higher the expectation. $\endgroup$
    – Alex C
    Sep 26, 2017 at 0:19

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