Black-Scholes Equation (1997 Nobel Economics)
Use: Pricing Derivatives (Options): calculate the value of an option before it matures.
1/2 (σS)².∂²V/∂S² + rS.∂V/∂S + (∂V/∂T – rV) = 0
Without last 2 terms=> heat equation !
Price S of the commodity
Price V of the derivative
Risk free interest r (govt bond)
Volatility = σ of the stock = standard deviation
Assumptions: (Arbitrage Pricing Theory)
No transaction costs
No limit on short-selling
Possible to borrow/lend at risk-free rate
Market prices behave like Brownian motion: constant in rate of drift and market volatility
Put option: right to sell at a specific time for an agreed price if you wish.
Call option: right to buy at a specific time for an agreed price if you wish.
One Black-Sholes formula each for Put and Call respectively.
Derivative was invented in 1900 by Mr. Bachelier, a French PhD student of Poincaré, the Mathematics of Financial prediction on stock market, based on Brownian Motion Theory of gas from Einstein. Poincaré was not impressed, gave the thesis only mention honorable (credit) , not trés honorable (distinction).
The problem of this Black-Sholes formula is the assumption being too ‘academic’, assume no financial ‘Fat Tail’, but sadly everybody believes in it and uses it blindly, which causes the recent Financial Crisis.