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 !

Time T

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 ‘*academi*c’, assume no financial ‘**Fat Tail’**, but sadly everybody believes in it and uses it blindly, which causes the recent Financial Crisis.

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Reblogged this on Singapore Maths Tuition.

Black-sholes formula used quite common in the corporate world, affecting employees in various ways:

1. When M&A (Merger & Acquisition) happens, say, your company Q being acquired by M company. Your invested Q stock option (ie not yet mature to sell) will be converted to M shares using the Black-Sholes formula.

2. Derivatives Trading: Blindly trading causes a colleague to lose U$30 million in one night; also an oil company CEO put his oil derivatives position exposed to U$ 500 million loss. Both cases due to the Black-Sholes formula, resulting the colleague being fired and the other CEO jailed for 6 years.

…(sorry) your “un-vested” Q stock options …