Delta Hedging and Greeks in Financial Mathematics, Summaries of Mathematical finance

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Lecture 14

Sergei Fedotov

20912 - Introduction to Financial Mathematics

Lecture 14

(^1) ∆-Hedging

(^2) Greek Letters or Greeks

Delta for European Call Option

Let us show that ∆ = ∂ ∂CS = N (d 1 ).

First, find the derivative ∆ = ∂ ∂CS by using the explicit solution for the European call C (S, t) = SN (d 1 ) − Ee−r^ (T^ −t)N (d 2 ).

Delta for European Call Option

Let us show that ∆ = ∂ ∂CS = N (d 1 ).

First, find the derivative ∆ = ∂ ∂CS by using the explicit solution for the European call C (S, t) = SN (d 1 ) − Ee−r^ (T^ −t)N (d 2 ).

∂C

∂S

= N (d 1 ) + SN′^ (d 1 )

∂d 1 ∂S − Ee−r^ (T^ −t)N′^ (d 2 )

∂d 2 ∂S

Delta-Hedging Example

Find the value of ∆ of a 6-month European call option on a stock with a strike price equal to the current stock price ( t = 0). The interest rate is 6% p.a. The volatility σ = 0.16.

Delta-Hedging Example

Find the value of ∆ of a 6-month European call option on a stock with a strike price equal to the current stock price ( t = 0). The interest rate is 6% p.a. The volatility σ = 0.16.

Solution: we have ∆ = N (d 1 ) , where d 1 = ln(S 0 /E )+(r +σ^2 / (^2) )T σ(T ) 12

Delta for European Put Option

Let us find Delta for European put option by using the put-call parity:

S + P − C = Ee−r^ (T^ −t).

Delta for European Put Option

Let us find Delta for European put option by using the put-call parity:

S + P − C = Ee−r^ (T^ −t).

Let us differentiate it with respect to S

Delta for European Put Option

Let us find Delta for European put option by using the put-call parity:

S + P − C = Ee−r^ (T^ −t).

Let us differentiate it with respect to S

We find 1 + ∂ ∂PS − ∂ ∂CS = 0.

Therefore ∂ ∂PS = ∂ ∂CS − 1 = N (d 1 ) − 1.

Greeks

The option value: V = V (S, t | σ, r , T ).

Greeks represent the sensitivities of options to a change in underlying parameters on which the value of an option is dependent.

Greeks

The option value: V = V (S, t | σ, r , T ).

Greeks represent the sensitivities of options to a change in underlying parameters on which the value of an option is dependent.

• Delta:

∆ = ∂ ∂VS measures the rate of change of option value with respect to changes in the underlying stock price

• Gamma:

Γ = ∂ (^2) V ∂S^2 =^

∂∆ ∂S measures the rate of change in ∆ with respect to changes in the underlying stock price.

Greeks

The option value: V = V (S, t | σ, r , T ).

Greeks represent the sensitivities of options to a change in underlying parameters on which the value of an option is dependent.

• Delta:

∆ = ∂ ∂VS measures the rate of change of option value with respect to changes in the underlying stock price

• Gamma:

Γ = ∂ (^2) V ∂S^2 =^

∂∆ ∂S measures the rate of change in ∆ with respect to changes in the underlying stock price.

Problem Sheet 6: Γ = ∂ (^2) C ∂S^2 =^

N′(d 1 ) Sσ √ T −t ,^ where^ N

′(d 1 ) = √ 1 2 π e

− d

(^21) (^2).

Greeks

• Vega, ∂ ∂σV , measures sensitivity to volatility σ.

One can show that ∂ ∂σC = S

T − tN′(d 1 ).

Greeks

• Vega, ∂ ∂σV , measures sensitivity to volatility σ.

One can show that ∂ ∂σC = S

T − tN′(d 1 ).

• Rho:

ρ = ∂ ∂Vr measures sensitivity to the interest rate r.