The Black-Scholes Equation
$latex \frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV=0 &s=3$
- $latex S$: the price of the stock
- $latex V=V(S,t)$: the price of a derivative(e.g., option)
- $latex K$: the strike price of the option
- $latex r$: the annualized risk-free interest rate
- $latex \sigma$: the standard deviation of the stock’s return
- $latex t$: a time in year
Other notations in the context
- $latex T$: the strike?date
- $latex \tau=T-t$: the remained time
Description
- The Black?Scholes equation is a partial differential equation, which describes the price of the option over time. – Black-Scholes model – Wikipedia
- The Black-Scholes model can be used to determine the price of a European call option. -?Black Scholes Model Definition | Investopedia
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: $latex S, t, r, \sigma$
- Delta: $latex \Delta=\frac{\partial V}{\partial S}$
- Gamma:?$latex \Gamma = \frac{\partial^2 V}{\partial S^2}$
- Theta:?$latex \Theta = \frac{\partial V}{\partial t}=-\frac{\partial V}{\partial \tau}$
- How to predict the stock price
- $latex \frac{\partial S}{\partial t} = \frac{\partial S}{\partial V} \frac{\partial V}{\partial t} = \Theta / \Delta$