Black-Scholes Model

The Black-Scholes Equation

\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV=0 &s=3


  • S: the price of the stock
  • V=V(S,t): the price of a derivative(e.g., option)
  • K: the strike price of the option
  • r: the annualized risk-free interest rate
  • \sigma: the standard deviation of the stock’s return
  • t: a time in year

Other notations in the context

  • T: the strike date
  • \tau=T-t: the remained time

The Greeks
  • “The Greeks” measure the sensitivity of the value of a derivative or a portfolio to changes in parameter value(s) while holding the other parameters fixed. – Black-Scholes model – Wikipedia
    • The parameters: S, t, r, \sigma
  • Delta: \Delta=\frac{\partial V}{\partial S}
  • Gamma: \Gamma = \frac{\partial^2 V}{\partial S^2}
  • Theta: \Theta = \frac{\partial V}{\partial t}=-\frac{\partial V}{\partial \tau}
  • How to predict the stock price
    • \frac{\partial S}{\partial t} = \frac{\partial S}{\partial V} \frac{\partial V}{\partial t} = \Theta / \Delta

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