I discovered how to price European options and stumbled over a term and an equation I didn't understood :
If we assume that investors are indifferent to risk and that expected returns on all assets are equal. In the case of investing in stocks, by risk-neutral ptobability, the payoff from holding the stock, taking into account the up and down state possibilities, would be equal to the continuously coumpounded risk-free rate expected in the next time step, as follows :
James Ma Weiming in Mastering Python for Finance, p76
What is risk-neutral probability q?
I'm not sure of what u and d are but I think it is the probability for the stock to go up or down. and I definitly don't know what is.
On Wikipedia I found :
In mathematical finance, a risk-neutral measure, (also called an equilibrium measure, or equivalent martingale measure), is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure.
And on Investopedia :
Risk-neutral probabilities are probabilities of future outcomes adjusted for risk, which are then used to compute expected asset values. The benefit of this risk-neutral pricing approach is that the once the risk-neutral probabilities are calculated, they can be used to price every asset based on its expected payoff. These theoretical risk-neutral probabilities differ from actual real world probabilities; if the latter were used, expected values of each security would need to be adjusted for its individual risk profile.
In fact, maybe I was wrong. Indeed : the next section was about knowing if this formulas was relevent for futures too.
Indeed, according to the author :
Unlike investing in stocks, investors not have then to make an upfront payment to tale an option in a futures contract. In a risk neutral sense, the expected growth rate from holding a futures contract is zero and the payoff can be written as follows :
Yet, with pu=1,2 and pd=0,8,the up and down probability, I sohould have had : q= 0,5.
But Here is what the author got :