CFRM 405: Mathematical Methods for Quantitative Finance Homework 3

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+) (1)

where

d+ =

log

S

K

+

r − q +

σ

2

2

T − t

σ

√

T − t

and

d− = d+ − σ

√

T − t =

log

S

K

+

r − q −

σ

2

2

T − t

σ

√

T − t

Compute each of

(a) ∆(P) = ∂P

∂S

(b) Γ(P) = ∂

2P

∂S2

(c) θ(P) = ∂P

∂t

(d) ρ(P) = ∂P

∂r

by taking derivatives of (1). Verify that your answers are correct using put-call parity.

You can find expressions for The Greeks on pages 92 and 93 of the Stefanica text. Verify

that your answer matches the expression for the put option and that put-call parity

gives the expression for the call option. And as always, verify your calculations with

Mathemat

Example

Compute the vega of a European put option.

vega(P) = ∂P

∂σ =

∂

∂σ

Ke−r(T −t)Φ(−d−) − Se−q(T −t)Φ(−d+)

= Ke−r(T −t)φ(−d−)

∂

∂σ(−d−) − Se−q(T −t)φ(−d+)

∂

∂σ(−d+)

= Ke−r(T −t)φ(d−)

∂

∂σ(−d−) − Se−q(T −t)φ(d+)

∂

∂σ(−d+)

Lemma 3.15 states that Ke−r(T −t)φ(d−) = Se−q(T −t)φ(d+), thus

vega(P) = ∂P

∂σ = Se−q(T −t)φ(d+)

∂

∂σ(−d−) − Se−q(T −t)φ(d+)

∂

∂σ(−d+)

= Se−q(T −t)φ(d+)

∂

∂σ(−d−) −

∂

∂σ(−d+)

= Se−q(T −t)φ(d+)

∂

∂σ(d+) −

∂

∂σ(d−)

= Se−q(T −t)φ(d+)

∂

∂σ

d+ − d−

But d− = d+ − σ

√

T − t =⇒ d+ − d− = σ

√

T − t, thus

vega(P) = ∂P

∂σ = Se−q(T −t)φ(d+)

∂

∂σ

σ

√

T − t

vega(P) = ∂P

∂σ = Se−q(T −t)φ(d+)

√

T − t

Check the result using put-call parity:

P = C − Se−q(T −t) + Ke−r(T −t)

vega(P) = ∂P

∂σ =

∂

∂σ

C − Se−q(T −t) + Ke−r(T −t)

=

∂C

∂σ

The vega of a European put option is the same as the vega for a European call option.

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