# Risk neutral modelling of a stock

Suppose a stock $$S$$ follows $$dS(t) = \alpha(t)S(t)dt + \sigma(t)S(t)dW(t),$$ where $$W(t)$$ is a Brownian motion under $$P$$. Also suppose there is a short rate process $$r(t)$$. My question would be is it possible to price a stock using the risk-neutral framework, i.e. can I say $$S(t) = E^{Q}[e^{-\int_t^Tr(s)ds}S(T) \mid \mathcal{F}_t]$$ for some $$T$$? More specifically, say I am currently at time $$t=0$$ and I simulate $$S$$ under $$Q$$ (basically change the drift from $$\alpha(t)$$ to $$r(t)$$) $$N$$ times up to time $$T$$ and I want to compute what would be a price $$S(t)$$ for some integer $$t > 0$$. Can I just average $$S(T)$$ over $$N$$ and discount up to time $$t$$?

If this is a valid approach furthermore assume that I computed $$S(t+1)$$ using the same method and I am about to decide in which security to invest for a $$t+1$$ horizon. Then, the rate of return, $$S(t+1)/S(t)-1,$$ is $$r(t)$$. For any other stock, say $$\tilde S$$ with different drift but the same diffusion, the rate of return under risk neutral measure would be again $$r(t)$$ and obviously this simulation would not give me useful information for my investment decision. I could however model both of them under $$P$$ and then pick the one that has higher expected return. Could you elaborate why risk neutral modelling does not work for portfolio choice problem?

• Consider accepting the answer if your question has been answered. – Sanjay Mar 6 '19 at 19:24

The risk neutral measure is used to price assets (e.g. derivatives) and not to base your investment decisions on.

In the first part of you question your simulation gives you the Risk-Neutral expectation of the stock at time $$T$$. If you want the expectation at time $$t$$, then why don't you just simulate from time 0 up to time $$t$$? (I might have misunderstood the question)

For the second part of your question: Risk-Neutral Measure is constructed such that there is no Arbitrage opportunities in the market

Could you elaborate why risk neutral modelling does not work for portfolio choice problem? That question you have (more or less) already answered yourself. $$Q$$ probabilities are not the "real-world" probabilities and the purpose of $$Q$$ is not to forecast the development of the stock

Furthermore, it is redundant to "price" a stock under $$Q$$ because the fair value of the stock is always given by the market price of the stock.

• thanks for your answer. Of course, I could simulate stock up to time $T$. I can also simulate it up to time $T-1$ and get two risk-neutral, i.e. fair, prices. Then, return between $T-1$ and $T$ would be $r(T-1)$. And obviously it makes sense since I simulated a stock such that its discounted value is a martingale. I have troubles with interpretation. If I say that risk-neutral framework gives me prices that should be prevalent at the market, then it says that the prevalent return should be equal to the risk free rate. But then I would never buy a stock and will long the bond. – tosik Feb 27 '19 at 12:33
• Also, if I model it under $P$ does it mean that I allow for arbitrage? – tosik Feb 27 '19 at 15:03
• If you model under P, create derivatives based on time $T$ expectation of the stock and sell those derivatives then yes. – Sanjay Feb 28 '19 at 15:14