The Black-Sholes Formula – 诺贝尔经济奖Scholes, Merton与LTCM S

Key Points:

1) From 1950 an unknown PhD Math Thesis by French Dr. Bachelier (a PhD student of the 20CE last Polymath Henri Poincaré, who was not impressed with the ‘gambling’ math, gave only an above-average marks to the thesis) – invention of “Options” Trading to eliminate risk in stock fluctuations.

2) The technique is called “Dynamic Hedging”…

3) …by applying the Japanese mathematician “Ito Math”.

4) Noble Prize Economics awarded to “ Black-Sholes Formula”

5). Sholes & Merton Company “LTCM” – making tons of money until… 1997 Asian Crisis, bailed out by USA government after loss of billions !

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Black-Scholes Financial Crisis

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