Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2015 / 3 / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii

i

Solution

Let

a = σ τ

, so

d_{+} = d_{-} + a

. Squaring gives

d_{+}^{2} - d_{-}^{2} = 2 a d_{-} + a^{2} = 2 lo g (S / K) + 2 r τ .

Exponentiation proves

S e^{- d_{+}^{2} /2} = K e^{- r τ} e^{- d_{-}^{2} /2} .

At fixed

S, t, T

, only

lo g (S / K)

depends on

K

, so the partial derivatives are

\frac{\partial d _{+}}{\partial K} = \frac{\partial d _{-}}{\partial K} = - \frac{1}{K σ τ} .

ii

Solution

Differentiate the Black-Scholes formula for the European put option with respect to its strike. The chain rule gives

\frac{\partial P}{\partial K} = e^{- r τ} Φ (- d_{-}) - K e^{- r τ} ϕ (d_{-}) \frac{\partial d _{-}}{\partial K} + Sϕ (d_{+}) \frac{\partial d _{+}}{\partial K} .

The final two terms cancel by both identities in part i. Hence the put-price strike derivative is

\frac{\partial P}{\partial K} = e^{- r τ} N (- d_{-}) .

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