4Pricing Options
4.1 Derivation of Black–Scholes Equation
This section is based on an article (Sewell 2018) published in the Mathematical Association of America's College Mathematics Journal.
Suppose at time 
your friend owns a certain asset (a stock, for example) whose market price is currently 
He hands you a signed piece of paper on which is written, “At time 
I promise to sell this asset to you for price 
if you want to buy it then.” How much is this piece of paper (a European “call” option) worth to you? The Black–Scholes equation (Black and Scholes 1973) is a partial differential equation that attempts to answer this question and is widely used for pricing “options” such as the one your friend offered you. This equation will be derived here from basic principles, in a way which we believe is unique in that it does not require any background in financial mathematics or stochastic calculus.
If you knew that the market price of this asset was going to be 
at time the paper your friend ...
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