Description

Dane Johnson
1. The Black-Scholes price for a European put option is
P(S,t,K,T,r,q,σ) = Ke−r(T−t)Φ(−d−) − Se−q(T−t)Φ(−d+) where

and

Please note that I will use the following equality throughout

(a)

(**) Lemma 3.15 was used here: Ke−r(T−t)φ(d−) = Se−q(T−t)φ(d+) The result matches the given expression for ∆P in the text.
Using Put-Call parity we have

Substituting our previous result leads to

This result agrees with the given value for the delta of a call option in the text.
(b)

(*)Where ∂S∂ [d+] is given by

Using Put-Call parity

The gamma of a European put option is the same as the gamma of the
European call option (for the same underlying asset).
(c)

For this last line we used

(**) Lemma 3.15 was used here: Ke−r(T−t)φ(d−) = Se−q(T−t)φ(d+)
Using Put-Call parity we have

Substituting our previous result leads to

This agrees with the given value for the theta of a call option in the text.

(**) Lemma 3.15 was used here: Ke−r(T−t)φ(d−) = Se−q(T−t)φ(d+)
Using Put-Call parity we have

Substituting the previous result for leads to

This matches the ρ(C) in the text.