# 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?

• What makes you say that $h_0 = E(h_t)$? It depends on the payoff function of the derivative. Your second statement is true though. Feb 17 '18 at 8:03

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.

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.

• The statement in yellow is not correct, is believed only by its 2 authors and substantially undervalues the contribution of Black, Scholes and Merton. Repeating this statement will not make it true, even if it repeated 1000 times. Mar 20 '18 at 16:28
• @AlexC Samuelson and McKean (1965) (dse.unisalento.it/c/document_library/…) were almost there. They just needed to figure out what to do about the drift. Setting drift to the risk free rate allowed Black-Scholes-Merton to express the problem in a form which had a known solution through the heat (diffusion) equation. Mar 21 '18 at 1:49
• I know this. I have read Samuelson and Boness and another one I can't remember who was also did a PhD on options, plus Bachelier had a good idea also and I browsed through his work many years ago. There was a long history before BSM (there always is in science and technology), but the first users on the floor of the CBOE took the BS formula and started using that. And they knew it was the BS formula, nothing else, they had never heard of the others. BS was the 1st formula good enough for the real world. Mar 21 '18 at 3:25

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.