Call Put
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Stock Price:
This represents the spot price of an underlying asset. In this case, the asset is a stock. Exercise Price :
This represents the fixed price at which the owner of an option can purchase (for a Call Option) or sell (for a Put Option) an asset. Interest Rate:
This is the risk-free rate at which interest is paid by borrowers for the use of money that they borrow from a lender. This rate is assumed to be the same for all borrowers and lenders. For the Black-Scholes Model, it assumed to be continuous. Dividend Rate:
This is the total dividend expected per share of common stock or preferred stock. For the Black-Scholes Model, it assumed to be continuous. Variance:
This is the volatility of returns of the underlying asset. Time to Expiry:
This represents the amount of time it takes for an option contract to end. It is usually expressed as T-t where T is the original length of option contract and t is amount of the time that passed after contract has started. |
Price (Option) (V):
Price is the value of an asset at a given time $T-t$. The Price for a European Call option is $C = S e^{-\delta(T-t)}\Phi(d_{1}) - E e^{-r(T-t)}\Phi(d_{2}) $. The Price for a European Put option is $P = Ee^{-r(T-t)}\Phi(-d_{2}) - S e^{-\delta (T-t)}\Phi(-d_{1}) $. Delta :
Delta, $\Delta$ , measures the rate of change of an option's value with respect to changes in the underlying asset's price, S. Delta is the defined as $\frac{\partial V}{\partial S}$. $\frac{\partial C}{\partial S} = e^{-\delta (T-t)}\Phi(d_{1}) $. $\frac{\partial P}{\partial S} = -e^{-\delta (T-t)}\Phi(-d_{1}) $. Gamma:
Gamma, $\Gamma$ , measures the rate of change of an option's delta with respect to changes in the underlying asset's price, S. Gamma is the defined as $\frac{\partial^2V}{\partial S^2}$. $\frac{\partial^2 C}{\partial S^2} = \frac{\partial^2 P}{\partial S^2} = \frac{e^{-\delta (T-t)}\phi(d_{1})}{S\sigma\sqrt{T-t}}$. Theta:
Theta, $\theta$ , measures the rate of change of an option's value with respect to changes in elapsing time. Theta is the defined as $\frac{\partial V}{\partial t}$. $\frac{\partial C}{\partial t}= - e^{-\delta (T-t)} \frac{S \phi (d_{1}) \sigma}{ 2 \sqrt[]{T-t} } -rE e^{-r(T-t)} \Phi (d_{2}) + \delta S e^{-\delta (T-t)} \Phi (d_{1})$ $\frac{\partial P}{\partial t}= - e^{-\delta (T-t)} \frac{S \phi (d_{1}) \sigma}{ 2 \sqrt[]{T-t} } +rE e^{-r(T-t)} \Phi (- d_{2}) - \delta S e^{-\delta (T-t)} \Phi (- d_{1})$. Note: Some sources may define Theta as $\theta = \frac{\partial V}{\partial t} \frac{1}{365}$. Rho:
Rho, $\rho$ , measures the rate of change of an option's with respect to changes in elapsing time. Theta is the defined as $\frac{\partial V}{\partial r}$. $\frac{\partial C}{\partial r}= E(T-t)e^{-r(T-t)} \Phi(d_2)$. $\frac{\partial P}{\partial r}= -E(T-t)e^{-r(T-t)} \Phi(-d_2)$. Note: Some sources may define Rho as $\rho = \frac{\partial V}{\partial r} \frac{1}{100}$. Vega:
Vega, $\nu$ , measures the rate of change of an option's value with respect to changes in the underlying asset's volatility, $\sigma$. Vega is the defined as $\frac{\partial V}{\partial \sigma}$. $\frac{\partial C}{\partial \sigma} = \frac{\partial P}{\partial \sigma} = S e^{-\delta (T-t)} \phi (d_{1}) \sqrt[]{T-t} $. Psi:
Psi, $\Psi$ , measures the rate of change of an option's value with respect to changes in the underlying asset's divident rate ($\delta$). Delta is the defined as $\frac{\partial V}{\partial \delta}$. $\frac{\partial C}{\partial \delta} = - S (T-t) e^{-\delta(T-t)}\phi(d_{1}) $. $\frac{\partial P}{\partial \delta} = S (T-t) e^{-\delta(T-t)}\phi(-d_{1}) $. Note: Some sources may define Psi as $\Psi = \frac{\partial V}{\partial \delta} \frac{1}{100}$. |