A stock is currently selling for $40. The stock pays no dividends. Given that the volatility relevant for the Black-Scholes equation is 30% and the continuously compounded risk-free interest rate is 8%. Consider a $40-strike purchased put on the stock with time to expiration in 6 months. What are the delta, gamma, vega, theta, and rho? [answer: = 0.384139, = 0.0450, V = 0.1080, = 0.005, = 0.0898]

where

= V_option /S

= S = ²V_option /S² .

= V/r

Call = (T t)Ke^{r(T t)}N(d2)

= V/t

d₁ =[ln(St/K) + (r + 0.5²)(T t)]/ [(T t)]

d₂= d₁ (T t)

V C(S, t) = Se^{(T t)}N(d₁) Ke^{r(T t)}N(d₂)

V P (S, t) = Ke^{r(T t)}N(d₂) Se^{(T t)}N(d₁)