P-measure vs Q-measure

[Joe Wong]

Hi all,

Imagine a scenario in which Company XYZ has promised the following awards of restricted stocks at the end of 2 years based on the ranking of 30-day average closing stock prices among 30 companies in similar industry:

1st - 5th 1000 shares
6th - 10th 500 shares
11th - 20th 100 shares
21st - 30th 0 share

I am trying to estimate the expected number of shares that will be awarded using Monte-Carlo simulation that generates the stock prices of these 30 companies incorporating correlation via Cholesky decomposition. My question is whether I should perform this analysis in risk neutral world (i.e., the Q-measure) and assume all stock prices to grow at the risk-free rate or if I should perform this analysis in the real world (i.e., the P-measure) and assume each stock price to grow at different rates or if the result would be the same. If I perform the analysis in risk neutral world, I am not sure if it makes sense to have the results driven mainly by volatility and not expected growth rate. If I perform the analysis in real world, it is more intuitive but estimating the expected growth rate for 30 companies requires so much subjective judgment that the result would not be any useful. Given that I am merely determining the number of shares and not a payout that needs to be discounted back to present, I am not sure if a option pricing framework is even appropriate.

Please share your thoughts on this. Thanks.

 

[NeedOPT]

Your question simply doesn't make any sense. An "expectation" is ONLY defined under some probability measure. If you want the expectation of the number of shares under P then use P. If you want it under Q then use Q. If you want it under some third measure R which you've decided is appropriate then use R. This is not a meaningful discussion unless you explain what purpose you intend to use the expectation for, and even then there might not be a meaningful answer. The measure Q is often used because you can show that the arbitrage-enforced price of certain types of derivatives follow, miraculously, the discounted expected payoff in the risk-neutral world.

 

[Eugene Krel]

. Given that I am merely determining the number of shares and not a payout that needs to be discounted back to present, I am not sure if a option pricing framework is even appropriate.

Your problem is not relevant to risk-neutrality. Seems to me you just need to figure out probabilities of your 30-day avg closing prices falling within some range and these will come from your simulation.

 

[Joe Wong]

Eugene and NeedOPT, I think you are both right that my question is not relevant to risk-neutrality. I guess my question is more about implementation. In my simulation, should I assume all the stock prices to grow at risk free rate or should I use different growth rate for different stocks that are relevant in my ranking?

 

[Roger Trimble]

If you are using this simulation for estimation, I think you should use P-measure. On the other hand, if you were using this simulation to come up with a value for a contract, I think you might then use Q.