in this tutorial we will describe the methodology to price an option using the risk-neutral valuation approach suppose the current price of a stock of a company is s 0 over some time period t the stock price could go up to SU with the probability Q or it could go down to SD with the probability 1 minus Q let's say there is an option contract on the stock with strike price equal to K and time to maturity equal to T and let's say that the continuously compounded discount rate over this period is our percent please note that the methodology described here applies equally well whether the rates are continuously compounded or discretely compounded given the values of these parameters when the stock price goes up to su let the health of this particular option be equal to P you similarly if the stock price goes down to SP let the payoff of this option be equal to P D so at time T we have two possible state he up state when the stock price goes up to su or the down state when the stock price goes down to SD question is what is the expected value of the stock price at time T which corresponds to maturity of the option so expected value of the terminal stock price s T equals the stock price and up state times the probability that the up state will occur plus the stock price in downstate times the probability that down state will occur note that in real world investors demand risk premia or taking on additional risk however in risk usual world there is no risk premia as investors are indifferent to risk and therefore expected return on any security is just the risk-free rate hence the expected value of the stock price at time t must equal the current stock price compounded at the risk-free rate over time period t solving for Q the risk-neutral probability of stock price going up we get Q equal to the current stock price compounded at the risk-free rate over time period t minus the stock price in down state divided by the stock price in up state minus the stock price in down state now that we have calculated Q the risk-neutral probability of stock price going up the expected future payoff of this option is as follows for the expected option payoff equals the payoff of the option in upstate times the probability that upstate would occur plus the payoff of the option in downstate times the probability that down state would occur since we're pricing this option in risk neutral world the only applicable discount rate is the risk-free rate of return therefore the value of the option is the expected payoff discounted at the risk-free rate for the price of this option equals the expected option health discounted at the risk-free rate over time period t and this equals the path the option in upstate times the probability that upstate would occur plus the path of the option in downstate times the probability that downstate would occur discounted back at the risk-free rate over time period t so we have learned how to use the risk-neutral valuation approach to price an option in discrete-time framework if there are any questions or comments please feel free to post thank you