Risk-Neutral Pricing of Stocks, Forwards, and Carry Constraints

# Risk neutral measure doubt


# Risk neutral measure doubt












For a derivative in a complete market, we can say that: $h_0 = E(h_t)$ assuming 0 risk free rate. Is the above relation also valid for a stock/ non derivative i.e. $s_0 = E(s_t)$ under the same risk neutral measure?

## Answer by Richi Wa (score 2)

https://quant.stackexchange.com/a/38306

Theory answer: The rational that gives your second equation is the logic of a forward price. What gives me the pay-off $S_T$ at time $T$? It is the forward contract. It's price $F(0,T)$ at time $0$ is $$ F(0,T) = S_0 (1+r*T) $$ or some other compound interest term. Why is it true? Because you could form an arbitrage portfolio. (Short) sell the stock to day, and earn interest until $T$. Then at $T$ you buy it back and give $S_T$ to the long side of the contract. So in theory this is the arbitrage-free price.

In real trading several issues arise: will I be able to buy back the stock that I shortened? What is the interest I can gain and so forth.

The answer is: yes, the martingale holds for the underlying (non-derivative) as well. As usual, in reality things are more complicated but the theoretical price often is a good starting point.

## Answer by David Addison (score 1)

https://quant.stackexchange.com/a/38305

Your proposition for doubting the risk neutral measure is a large part of the reason why, according to Espen Haug and Nassim Taleb, “Option traders use (very) sophisticated heuristics, never the Black–Scholes–Merton formula”.

According to the text:

> Black, Scholes and Merton did not invent any formula, just found an argument to make a well known (and used) formula compatible with the economics establishment, by removing the “risk” parameter through “dynamic hedging”, (2) option traders use (and evidently have used since 1902) sophisticated heuristics and tricks more compatible with the previous versions of the formula of Louis Bachelier and Edward O. Thorp (that allow a broad choice of probability distributions) and removed the risk parameter using put-call parity, (3) option traders did not use the Black–Scholes–Merton formula or similar formulas after 1973 but continued their bottom-up heuristics more robust to the high impact rare event.

## Answer by Charles Fox (score 0)

https://quant.stackexchange.com/a/41907

No, although you mentioned 0 risk free rate, you did not mention dividends or hard to borrow stock loan fees. These are both forms of convenience yield, which factors into forward pricing as well.

In the opposite case, it is difficult/expensive to store electricity. As a result, you cannot easily buy it in the spot market, hold it, and sell a futures contract. This causes the cash and carry relationship to be a weak inequality rather than an equality.