Deriving Black–Scholes Pricing from a Risk-Neutral Binomial Limit
Summary
The document discusses how to express the price of a European call as a discounted expectation under a risk-neutral measure, starting from a binomial stock-price model and considering its continuous-time limit. The key pricing correction is to discount the terminal payoff by the risk-free rate before taking its expectation. Under the stated Black–Scholes assumptions, the terminal stock price has a lognormal distribution under the chosen risk-neutral measure.
It suggests using properties of that lognormal distribution and the law of the unconscious statistician to evaluate the expected payoff, rather than first deriving the distribution of the discounted payoff itself. A second response sketches the binomial-to-normal limit for log prices. The exchange does not carry out the integral or derive the closed-form call formula, so it offers a route rather than a complete derivation. The argument also assumes a deterministic interest rate and the usual Black–Scholes setup; it does not address more general market models.
Key ideas
- Risk-neutral pricing expresses a European call value as the discounted expected terminal payoff.
- The payoff expectation should be discounted using the risk-free rate over the option’s life.
- Under Black–Scholes assumptions, the risk-neutral terminal stock price is lognormally distributed.
- The lognormal distribution can be used to evaluate the payoff expectation directly.
- The document outlines the derivation but does not show the integral leading to the closed-form formula.
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# Obtain B-S-M from a binomial tree as n goes to infinty using Lebesgue integral
# Obtain B-S-M from a binomial tree as n goes to infinty using Lebesgue integral
My question is simple, consider a European call with payoff max(S_T-K, 0), Let's suppose that the underlying stock follows a binomial tree with up and down factors I know as we take n goes to infinity that the stock is log-normally distributed at time t=T (I know how to derive it). The idea is to derive the B-S-M pricing formula as the expected value of the present value of max(S_t-K, 0) using the Lebesgue integral I write this like: `$\int_{\Omega} max(S_T-K,0) dP$` where P if I am not mistaken is the risk-neutral measure (since usually for a binomial tree we use the risk-neutral probabilities). How can I set P to compute this? Once I have a measure P finding the distribution of max(S_t-K, 0) and then compute its density is right way to continue?. Finally, doing this should I arrive B-S-M pricing formula for a call?
## Answer by minginator (score 0)
https://quant.stackexchange.com/a/78874
If $Q$ is the risk-neutral measure, then take care that you need to discount your payoff appropriately, i.e. if $T$ is the time to expiry, and current time $t = 0$, then you want the integral $$ \int_{\Omega} e^{-rT}\max(S_T-K,0) dQ, $$ or $$ E^{Q}[e^{-rT}\max(S_T-K,0)]. $$ Assuming that you do not have any other assets under consideration, and a non-stochastic interest rate $r$, then you can just take $Q$ to be any probability measure that makes it so that $S_{T}$ has a lognormal distribution with the usual parameters. Rather than working with the distribution of $e^{-rT}\max(S_T-K,0)$ under $Q$, it is probably easier (but still tedious) to apply the law of the unconscious statistician together with some properties of $S_T$ that follows from the assumptions underlying the BSM model. See the first answer to a related question here.
## Answer by Arshdeep (score 0)
https://quant.stackexchange.com/a/79244
By CLT, binomial distribution becomes normal as you increase steps $n$.
$log(S(T))=log(S0)+klog(u)+(n-k)log(d)$ is thus a shifted normal distribution, and so $S(T)$ is lognormal.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.