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Black-Scholes Model

The Black-Scholes Model is a mathematical model used in financial markets to calculate the theoretical price of options, including put and call options. It was developed by economists Fischer Black and Myron Scholes, with notable contributions from Robert Merton. The Black-Scholes formula for a European call option CC (an option that can only be exercised at the end of its life) is given as:

C=S0eโˆ’qTN(d1)โˆ’Keโˆ’rTN(d2)C = S_0 e^{-qT}N(d_1) - K e^{-rT}N(d_2)

whereas for a European put option PP, it is given as:

P=Keโˆ’rTN(โˆ’d2)โˆ’S0eโˆ’qTN(โˆ’d1)P = K e^{-rT}N(-d_2) - S_0 e^{-qT}N(-d_1)

In these equations:

  • S0S_0 is the current price of the underlying asset.
  • KK is the strike price of the option.
  • TT is the time to maturity of the option.
  • rr is the risk-free interest rate.
  • qq is the rate of continuous dividends paid by the underlying asset.
  • N(โ‹…)N(\cdot) is the standard normal cumulative distribution function.
  • d1d_1 and d2d_2 are calculated as follows:
d1=lnโก(S0K)+(rโˆ’q+ฯƒ22)TฯƒTd_1 = \frac{\ln\left(\frac{S_0}{K}\right) + \left(r - q + \frac{\sigma^2}{2}\right)T}{\sigma\sqrt{T}} d2=d1โˆ’ฯƒTd_2 = d_1 - \sigma\sqrt{T}


  • lnโก(โ‹…)\ln(\cdot) is the natural logarithm function.
  • ฯƒ\sigma is the standard deviation of the asset's returns (volatility).

The model assumes several things about the market and the asset, such as:

  • There are no transaction costs or taxes.
  • The risk-free rate and volatility of the underlying are known and constant. (Usually US government bond)
  • The returns on the underlying asset are normally distributed.
  • The markets are efficient.