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 = 33 + 0.4 2 = 33.8 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 ) = 0.6293 c = e δ N(d 1 ) = 0.58 p = e δ N( d 1 ) = 0.34 3. 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 -0.55. he stock is currently trading for $48.00. he gamma for both the call and put is 0.07. (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 = 0.126. For the gamma, Γ c = 1 Sσ ( 1 2π e d2 1 2 ) = 0.07 herefore σ = 0.1178. d 2 = d 1 σ = 0.2438 and hence N(d 2 ) = 0.4037. Since σ = 0.126 1
We can get Ke r = 49.057. herefore C = SN(d 1 ) Ke r N(d 2 ) = 1.7957 P = C S + Ke r = 2.8527 (1) If the price of the stock moves up $1, C(S + 1) = C + c 1 + 1 2 Γ c 1 2 = 2.28 P (S + 1) = P + p 1 + 1 2 Γ p 1 2 = 2.34 (2) If the price of the stock drops $1, C(S 1) = C + c ( 1) + 1 2 Γ c ( 1) 2 = 1.38 P (S + 1) = P + p ( 1) + 1 2 Γ p ( 1) 2 = 3.44 4. A stock has a price of $567 and a volatility of 0.45. A certain put option on the stock has a price of $78 and a vega of 0.23. Suddenly, volatility increases to 0.51. 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) = 79.38 5. 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) 20 20.453 0.15 (B) 20 16.138 0.15 (C) 20 40.453 0.15 (D) 16.138 0.15 (E) 40.453 0.15 20.453 20.453 From the question, we know that c = N(d 1 ) = 0.5 and hence d 1 = 0. Since σ We can get r = 0.102256 and d 2 = d 1 σ = 0.15. Hence N(d 2 ) = 0.4404. herefore Here C = SN(d 1 ) Ke r N(d 2 ) = 20 40.453(1 N(0.15)) = 40.453N(0.15) 20.453 N(0.15) = 1 0.15 2π 2 = 0 e x2
So We choose (D). C = 16.138 0.15 e x2 20.453 6. 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 0.3089 and a price of $2.886. Find the elasticity of such a put option. Ω = S P = 0.3089 41 2.886 = 4.39 8. 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 ) = 0.6406 and N(d 2 ) = 0.4443. Hence Finally P = Ke r N( d 2 ) Se δ N( d 1 ) = 16.6996 p = e δ N( d 1 ) = 0.342 σ option = σ Ω = σ ps P = 1.075 9. 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
(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 ) = 0.5040 and N( d 2 ) = 0.6406. Hence Finally P = Ke r N( d 2 ) Se δ N( d 1 ) = 8.9917 p = e δ N( d 1 ) = 0.5040 σ option = σ Ω = σ ps P = 0.98 10. 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 = 0.9985 Ω port = S( c p ) C P = e δ S C P = 58.61 Additional Problems for Math 567 Students (max. points = 20) 11. You compute the delta for a 50 60 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 = 1 12. Since Delta is additive, then spread = 50 60 Besdies c = N(d 1 ) herefore we can get: σ Call C 50, C 60, C 50, t C 60, t d 1 0.175-1.6482 0.1429-2.090 N(d 1 ) 0.5695 0.0497 0.5568 0.0183 herefore the change of is (0.5695 0.0497) + (0.5568 0.0183) = 0.0187. Hence the delta increases by 0.02. 4
12. Given the following information: Call Price Elasticity A $ 9.986 4.367 B $ 8.553 4.236 C $ 6.307 5.227 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 9.986 + 8.553 6.307 = 12.232 he elasticity for portfolio is 3 Ω port = w i Ω i = 9.986 8.553 6.307 4.367 + 4.236 5.227 = 3.83 12.232 12.232 12.232 i=1 5