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Some time ago, I wrote a short unpublished note (mostly for my own benefit) when I was trying to understand the derivation of the Black-Scholes equation in financial mathematics, which computes the price of various options under some assumptions on the underlying financial model.  In order to avoid issues relating to stochastic calculus, Itō’s formula, etc. I only considered a discrete model rather than a continuous one, which makes the mathematics much more elementary.  I was recently asked about this note, and decided that it would be worthwhile to expand it into a blog article here.  The emphasis here will be on the simplest models rather than the most realistic models, in order to emphasise the beautifully simple basic idea behind the derivation of this formula.

The basic type of problem that the Black-Scholes equation solves (in particular models) is the following.  One has an underlying financial instrument S, which represents some asset which can be bought and sold at various times t, with the per-unit price $S_t$ of the instrument varying with t.  (For the mathematical model, it is not relevant what type of asset S actually is, but one could imagine for instance that S is a stock, a commodity, a currency, or a bond.)  Given such an underlying instrument S, one can create options based on S and on some future time $t_1$, which give the buyer and seller of the options certain rights and obligations regarding S at an expiration time $t_1$.  For instance,

1. A call option for S at time $t_1$ and at a strike price P gives the buyer of the option the right (but not the obligation) to buy a unit of S from the seller of the option at price P at time $t_1$ (conversely, the seller of the option has the obligation but not the right to sell a unit of S to the buyer of the option at time $t_1$, if the buyer so requests).
2. A put option for S at time $t_1$ and at a strike price P gives the buyer of the option the right (but not the obligation) to sell a unit of S to the seller of the option at price P at time $t_1$ (and conversely, the seller of the option has the obligation but not the right to buy a unit of S from the buyer of the option at time $t_1$, if the buyer so requests).
3. More complicated options, such as straddles and collars, can be formed by taking linear combinations of call and put options, e.g. simultaneously buying or selling a call and a put option.  One can also consider “American options” which offer rights and obligations for an interval of time, rather than the “European options” described above which only apply at a fixed time $t_1$.  The Black-Scholes formula applies only to European options, though extensions of this theory have been applied to American options.

The problem is this: what is the “correct” price, at time $t_0$, to assign to an European option (such as a put or call option) at a future expiration time $t_1$?  Of course, due to the volatility of the underlying instrument S, the future price $S_{t_1}$ of this instrument is not known at time $t_0$.  Nevertheless – and this is really quite a remarkable fact – it is still possible to compute deterministically, at time $t_0$, the price of an option that depends on that unknown price $S_{t_1}$, under certain assumptions (one of which is that one knows exactly how volatile the underlying instrument is).