MATH 476/567 ACTUARIAL RISK THEORY FALL 2016 PROFESSOR WANG. Homework 3 Solution

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1 MAH 476/567 ACUARIAL RISK HEORY FALL 2016 PROFESSOR WANG Homework 3 Solution 1. Consider a call option on an a nondividend paying stock. Suppose that for = 0.4 the option is trading for $33 an option. What is the new price of the option if the stock price increases by $2? From the question, we know that C(S) = 33. herefore C(S + 2) = C(S) + 2 = = A certain stock is currently trading for $95 per share. he annual continuously compounded risk-free interest rate is 6%, and the stock pays dividends with an annual continuously compounded yield of 3%. he price volatility relevant for the Black-Scholes formula is 32%. (1) Find the delta of a call option on the stock with strike price of $101 and time to expiration of 3 years. (2) Find the delta of a put option on the stock with strike price of $101 and time to expiration of 3 years. σ = 0.33 N(d 1 ) = c = e δ N(d 1 ) = 0.58 p = e δ N( d 1 ) = A call option on XYZ stock has a delta of 0.45, and a put option on XYZ stock with same strike and date to expiration has a delta of he stock is currently trading for $ he gamma for both the call and put is (1) What is the value of for the call and the put if the price of the stock moves up $1? (2) What is the value of for the call and the put if the price of the stock drops $1? We know that c = e δ N(d 1 ) p = e δ N( d 1 ) herefore c p = e δ = 1 Hence we can get δ = 0. Since c = N(d 1 ) = 0.45, we can get d 1 = For the gamma, Γ c = 1 Sσ ( 1 2π e d2 1 2 ) = 0.07 herefore σ = d 2 = d 1 σ = and hence N(d 2 ) = Since σ =

2 We can get Ke r = herefore C = SN(d 1 ) Ke r N(d 2 ) = P = C S + Ke r = (1) If the price of the stock moves up $1, C(S + 1) = C + c Γ c 1 2 = 2.28 P (S + 1) = P + p Γ p 1 2 = 2.34 (2) If the price of the stock drops $1, C(S 1) = C + c ( 1) Γ c ( 1) 2 = 1.38 P (S + 1) = P + p ( 1) Γ p ( 1) 2 = A stock has a price of $567 and a volatility of A certain put option on the stock has a price of $78 and a vega of Suddenly, volatility increases to Find the new put option price. From the question, we know that P 0 = 78. herefore P (σ = 0.51) = P 0 + V (51 45) = You are considering the purchase of a 3-month 41.5-strike European call option on a nondividendpaying stock. You are given: (1) he Black-Scholes framework holds. (2) he stock is currently selling for 40. (3) he stock s volatility is 30%. (4) he current call option delta is 0.5. Which of these expressions represents the price of this option? Explain your answer. (You will receive 0-credit without details) (A) (B) (C) (D) (E) From the question, we know that c = N(d 1 ) = 0.5 and hence d 1 = 0. Since σ We can get r = and d 2 = d 1 σ = Hence N(d 2 ) = herefore Here C = SN(d 1 ) Ke r N(d 2 ) = (1 N(0.15)) = N(0.15) N(0.15) = π 2 = 0 e x2

3 So We choose (D). C = e x Show that the delta of a K 1 K 2 call bull spread is equal to the K 1 K 2 put bull spread when the underlying stock pays no dividends. Here K 1 < K 2. he K 1 K 2 call bull spread is C(K 1 ) C(K 2 ). he K 1 K 2 put bull spread is P (K 1 ) P (K 2 ). We need to check C1 C2 = P1 P2 Since c = N(d 1 ) and p = N( d 1 ), therefore Hence c 1 = p P1 P2 = ( C1 1) ( C2 1) = C1 C2 7. A certain stock is currently trading for $41 per share with a stock volatility of 0.3. Certain put options on the stock have a delta of and a price of $ Find the elasticity of such a put option. Ω = S P = = For a European put option on a stock within the Black-Scholes framework, you are given: (1) he stock price is $105. (2) he strike price is $100. (3) he put option will expire in one year. (4) he continuously compound risk-free interest rate is 5.5%. (5) σ = 0.50 (6) he dividend yield δ = 5%. Calculate the volatility of this put option. σ = 0.36 herefore d 2 = d 1 σ = 0.14, N(d 1 ) = and N(d 2 ) = Hence Finally P = Ke r N( d 2 ) Se δ N( d 1 ) = p = e δ N( d 1 ) = σ option = σ Ω = σ ps P = For a European put option on a stock within the Black-Scholes framework, you are given: (1) he stock price is $50. (2) he strike price is $55. (3) he put option will expire in one year. (4) he continuously compound risk-free interest rate is 3%. 3

4 (5) σ = 0.35 (6) he stock pays no dividends. Calculate the volatility of this put option. σ = 0.01 herefore d 2 = d 1 σ = 0.36, N( d 1 ) = and N( d 2 ) = Hence Finally P = Ke r N( d 2 ) Se δ N( d 1 ) = p = e δ N( d 1 ) = σ option = σ Ω = σ ps P = Consider a portfolio that consists of buying a call option on a stock and selling a put option. he stock pays continuous dividends at the yield rate of 5%. he options have a strike of $62 and expire in six months. he current stock price is $60 and the continuously compounded risk-free interest rate is 15%. Find the elasticity of this portfolio. From put-call parity herefore C P = Se δ Ke r = Ω port = S( c p ) C P = e δ S C P = Additional Problems for Math 567 Students (max. points = 20) 11. You compute the delta for a bull spread with the following information: (1) he continuously compounded risk-free rate is 5%. (2) he underlying stock pays no dividends. (3) he current stock price is $50 per share. (4) he stock s volatility relevant for the Black-Scholes equation is 20%. (5) he time to expiration is 3 months. How much does the delta change after 1 month, if the stock price does not change? From the question, r = 0.05, δ = 0, S = 50, σ = 0.2, = 0.25 and t = Since Delta is additive, then spread = Besdies c = N(d 1 ) herefore we can get: σ Call C 50, C 60, C 50, t C 60, t d N(d 1 ) herefore the change of is ( ) + ( ) = Hence the delta increases by

5 12. Given the following information: Call Price Elasticity A $ B $ C $ Find the elasticity of the portfolio consisting of buying one call option on stock A, one call option on stock B and selling one call option on stock C. [Hint]: Recall elasticity for portfolio Ω port = S n i=1 n i i n i=1 n ic i = he cost of the portfolio is n i=1 ( ni C i n i=1 n ic i ) S i C i = he elasticity for portfolio is 3 Ω port = w i Ω i = = i=1 5