Overview
The derivation of the
Black Scholes equation
relied on an argument that the risk of a derivative could be
perfectly hedged. If the price of the derivative is assumed to be a function of the following variables
When the underlying asset follows a standard Ito Process such as
- {% S %} - the stock price
- {% \sigma %} - the stock volatility
- {% \tau %} - time to maturity
- {% r %} - the current level of interest rates
When the underlying asset follows a standard Ito Process such as
{% dS = \mu dt + \sigma dW_t %}
and the price of a call option is assumed to be a function of the given variables
{% Price = C(S,\sigma, \tau, r) %}
The model assumes that the level of interest rates and the stock volatility are constants, then
Ito's lemma
asserts that the derivative prices evolves according to
{% dC = \frac{\partial{C}}{\partial{S}} dS + \frac{\partial{C}}{\partial{t}} dt + \frac{1}{2} \frac{\partial^2{C}}{\partial{S}^2} dt %}
List of Greeks
Greeks are defined to be the various partial derivatives of the derivative price against the variables on which it depends.- Delta ({% \Delta %}) {% \frac{\partial{C}}{\partial{S}} %} - the sensitivity of the price with respect to the underlying price.
- Vega {% \frac{\partial{C}}{\partial{\sigma}} %} - the sensitivity of the option price to the undrlying volatility
- Theta - {% \frac{\partial{C}}{\partial{\tau}} %}
- Rho - {% \frac{\partial{C}}{\partial{r}} %} - the sensitivitiy of the price to the interest rate
Hedging
- Hedging in Theory - the theoretical Black Scholes model shows how the derivative can be hedged in theory, which is essentially accomplished by holding {% \Delta %} amount of the underlying.
- Hedging in Practice - in practice, traders will try to construct a portfolio where some of the other Greeks are set to zero as well. This is because the theory assumes that traders can trade contiuously. In addition, some variables, such as the interest rate, are assumed to be constant.