Lecture 14. Sergei Fedotov Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) / 7

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1 Lecture 14 Sergei Fedotov Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) / 7

2 Lecture Hedging 2 Greek Letters or Greeks Sergei Fedotov (University of Manchester) / 7

3 Delta for European Call Option Let us show that = C S = N (d 1). Sergei Fedotov (University of Manchester) / 7

4 Delta for European Call Option Let us show that = C S = N (d 1). First, find the derivative = C S by using the explicit solution for the European call C(S,t) = SN (d 1 ) Ee r(t t) N (d 2 ). Sergei Fedotov (University of Manchester) / 7

5 Delta for European Call Option Let us show that = C S = N (d 1). First, find the derivative = C S 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 = Sergei Fedotov (University of Manchester) / 7

6 Delta for European Call Option Let us show that = C S = N (d 1). First, find the derivative = C S 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 = N (d 1 ) + (SN (d 1 ) Ee r(t t) N (d 2 )) d 1 S. We need to prove See Problem Sheet 6. (SN (d 1 ) Ee r(t t) N (d 2 )) = 0. Sergei Fedotov (University of Manchester) / 7

7 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 = Sergei Fedotov (University of Manchester) / 7

8 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 = Solution: we have = N (d 1 ), where d 1 = ln(s 0/E)+(r+σ 2 /2)T σ(t) 1 2 Sergei Fedotov (University of Manchester) / 7

9 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 = Solution: we have = N (d 1 ), where d 1 = ln(s 0/E)+(r+σ 2 /2)T We find σ(t) 1 2 and d 1 = ( (0.16)2 0.5)) = N (0.3217) Sergei Fedotov (University of Manchester) / 7

10 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). Sergei Fedotov (University of Manchester) / 7

11 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 Sergei Fedotov (University of Manchester) / 7

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). Let us differentiate it with respect to S We find 1 + P S C S = 0. Sergei Fedotov (University of Manchester) / 7

13 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 + P S C S = 0. Therefore P S = C S 1 = N (d 1) 1. Sergei Fedotov (University of Manchester) / 7

14 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. Sergei Fedotov (University of Manchester) / 7

15 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: = V S measures the rate of change of option value with respect to changes in the underlying stock price Sergei Fedotov (University of Manchester) / 7

16 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: = V S 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. Sergei Fedotov (University of Manchester) / 7

17 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: = V S 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 2π e d Sergei Fedotov (University of Manchester) / 7

18 Greeks Vega, V σ, measures sensitivity to volatility σ. Sergei Fedotov (University of Manchester) / 7

19 Greeks Vega, V σ, measures sensitivity to volatility σ. One can show that C σ = S T tn (d 1 ). Sergei Fedotov (University of Manchester) / 7

20 Greeks Vega, V σ, measures sensitivity to volatility σ. One can show that C σ = S T tn (d 1 ). Rho: ρ = V r measures sensitivity to the interest rate r. Sergei Fedotov (University of Manchester) / 7

21 Greeks Vega, V σ, measures sensitivity to volatility σ. One can show that C σ = S T tn (d 1 ). Rho: ρ = V r measures sensitivity to the interest rate r. One can show that ρ = C r = E(T t)e r(t t) N(d 2 ). Sergei Fedotov (University of Manchester) / 7

22 Greeks Vega, V σ, measures sensitivity to volatility σ. One can show that C σ = S T tn (d 1 ). Rho: ρ = V r measures sensitivity to the interest rate r. One can show that ρ = C r = E(T t)e r(t t) N(d 2 ). The Greeks are important tools in financial risk management. Each Greek measures the sensitivity of the value of derivatives or a portfolio to a small change in a given underlying parameter. Sergei Fedotov (University of Manchester) / 7

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