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?

q= \frac{e^{rt}-d}{u-d}

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 e^{rt} 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 :

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  • 1
    Get a book, e.g one by Hull with the title Option Futures and something and read it properly. It will make it much clearer. Reading bits online will only confuse you.
    – DumbCoder
    Commented Jun 20, 2017 at 10:28
  • @DumbCoder I have the hard copy actually Commented Jun 20, 2017 at 13:42
  • There is some bad nomenclature in that document. A probability can't be 1.2, what the 1.2 is in the process is the upward price movement, meaning that the stock can go up 20% (S * 1.2) or down 20% (S * 0.8).
    – D Stanley
    Commented Jun 21, 2017 at 18:29
  • @DStanley understood, that's clearer Commented Jun 22, 2017 at 9:01

1 Answer 1


You have actually asked several questions, so I think what I'll do is give you an intuition about risk-neutral pricing to get you started. Then I think the answer to many of your questions will become clear.

Physical Probability

There is some probability of every event out there actually occurring, including the price of a stock going up. That's what we call the physical probability. It's very intuitive, but not directly useful for finding the price of something because price is not the weighted average of future outcomes. For example, if you have a stock that is highly correlated with the market and has 50% chance of being worth $20 dollars tomorrow and a 50% chance of being worth $10, it's value today is not $15. It will be worth less, because it's a risky stock and must earn a premium.

When you are dealing with physical probabilities, if you want to compute value you have to take the probability-weighted average of all the prices it could have tomorrow and then add in some kind of compensation for risk, which may be hard to compute.

Risk-Neutral Probability

Finance theory has shown that instead of computing values this way, we can embed risk-compensation into our probabilities. That is, we can create a new set up "probabilities" by adjusting the probability of good market outcomes downward and increasing the probability of bad market outcomes. This may sound crazy because these probabilities are no longer physical, but it has the desirable property that we then use this set of probabilities to price of every asset out there: all of them (equity, options, bonds, savings accounts, etc.). We call these adjusted probabilities that risk-neutral probabilities. When I say price I mean that you can multiply every outcome by its risk-neutral probability and discount at the risk-free rate to find its correct price.

To be clear, we have changed the probability of the market going up and down, not our probability of a particular stock moving independent of the market. Because moves that are independent of the market do not affect prices, we don't have to adjust the probabilities of them happening in order to get risk-neutral probabilities.

Anyway, the best way to think of risk-neutral probabilities is as a set of bogus probabilities that consistently give the correct price of every asset in the economy without having to add a risk premium. If we just take the risk-neutral probability-weighted average of all outcomes and discount at the risk-free rate, we get the price. Very handy if you have them.

Risk-Neutral Pricing

We can't get risk-neutral probabilities from research about how likely a stock is to actually go up or down. That would be the physical probability. Instead, we can figure out the risk-neutral probabilities from prices.

If a stock has only two possible prices tomorrow, U and D, and the risk-neutral probability of U is q, then

Price = [ Uq + D(1-q) ] / e^(rt)

The exponential there is just discounting by the risk-free rate. This is the beginning of the equations you have mentioned. The main thing to remember is that q is not the physical probability, it's the risk-neutral one. I can't emphasize that enough. If you have prespecified what U and D can be, then there is only one unknown in that equation: q. That means you can look at the stock price and solve for the risk neutral probability of the stock going up.

The reason this is useful is that you can same risk-neutral probability to price the associated option. In the case of the option you don't know its price today (yet) but you do know how much money it will be worth if the stock moves up or down. Use those values and the risk-neutral probability you computed from the stock to compute the option's price. That's what's going on here.

To remember: the same risk-neutral probability measure prices everything out there. That is, if you choose an asset, multiply each possibly outcome by its risk-neutral probability, and discount at the risk-free rate, you get its price. In general we use prices of things we know to infer things about the risk-neutral probability measure in order to get prices we do not know.

  • Thank you for this insightful answer ! - Yet why does moves that are independent of the market do not affect prices ? - And do we really need bogus probabilities that consistently give the correct price as far as we already have the price as an input ? Commented Jun 29, 2017 at 8:30
  • So is it useful too to get the prices of tomorrow we don't know yet from the price we know today ? It seems to me to be much more the case as far as we usually know what an asset price is on time t but we don't know what it will be at t+1. Commented Jun 29, 2017 at 8:41
  • I'm not sure what you are saying in this last comment. The exercise you are looking at uses today's stock price to estimate the risk-neutral probability of stock prices tomorrow and uses those probabilities to figure out what the current price of the option should be. Seems useful to me.
    – farnsy
    Commented Jun 29, 2017 at 14:38
  • This answer reminds me of CAPM. Let $E(R_s)=R_f+\beta [E(R_m)+R_f]$. As we can see, we need a compensation for the excess risk we bear, which comes only from the correlation with the market (e.g the systematic risk) as idiosyncratic can be diversified. If we remove this risk premium, by changing to $mathbb{Q}$ then we get as expected value the risk free rate.
    – Aleksander
    Commented Jan 10, 2019 at 6:48
  • 1
    @user10507 Your question is ambiguous in the context of your question. My quote refers to the physical probabilities, which is what people normally mean when they say "probability." You would not pay $15 for an asset that has a $15 expected value under the physical (true) probabilities if it is risky. Why would you? The risk-neutral probability puts greater probability weight on the negative outcome, like 40/60 for example, in order to get the correct price (in that case, $14). But remember 40/60 is not the physical probability--the asset has a 50% true probability of being worth $20.
    – farnsy
    Commented Jun 1, 2020 at 0:32

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