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7.6 European Put Options The put call option parity formula, in conjunction with the Black-Scholes equation, yields the unique no arbitrage cost of a European (K, 1) put option: P(s, t, K,r,a) = C(s, t, K,r,a) + Ke" - s (7.7) Whereas the preceding is useful for computational purposes, to deter- mine monotonicity and convexity properties of P = P(s, t, K,r, a) it is also useful to use that P(s, 1, K, r, o) must equal the expected return from the put under the risk neutral geometric Brownian motion process. Consequently, with Z being a standard normal random variable, P(s, 1, K.r.o)=e" E[(K-ser-+√iz)+] = E[(Ke™" — se=¹+√iz)+] Now, for a fixed value of Z, the function (Ke-"-se-t+√iz)+ is 1. Decreasing and convex in s. (This follows because (a - bs)+ is, for b> 0, decreasing and convex in s.)

7.6 European Put Options The put call option parity formula, in conjunction with the Black-Scholes equation, yields the unique no arbitrage cost of a European (K, 1) put option: P(s, t, K,r,a) = C(s, t, K,r,a) + Ke" - s (7.7) Whereas the preceding is useful for computational purposes, to deter- mine monotonicity and convexity properties of P = P(s, t, K,r, a) it is also useful to use that P(s, 1, K, r, o) must equal the expected return from the put under the risk neutral geometric Brownian motion process. Consequently, with Z being a standard normal random variable, P(s, 1, K.r.o)=e" E[(K-ser-+√iz)+] = E[(Ke™" — se=¹+√iz)+] Now, for a fixed value of Z, the function (Ke-"-se-t+√iz)+ is 1. Decreasing and convex in s. (This follows because (a - bs)+ is, for b> 0, decreasing and convex in s.)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 28EQ
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Consider pricing of options using the Black-Scholes formula. If you keep all parameters fixed and increase the value of K, what will happen to the value of C and P?

7.6 European Put Options The put call option parity formula, in conjunction with the Black-Scholes equation, yields the unique no arbitrage cost of a European (K, 1) put option: (7.7) P(s, 1, K,r, o) = C(s, t, K,r,a) + Kes Whereas the preceding is useful for computational purposes, to deter- mine monotonicity and convexity properties of P = P(s, 1, K,r, o) it is also useful to use that P(s, 1, K, r, o) must equal the expected return from the put under the risk neutral geometric Brownian motion process. Consequently, with Z being a standard normal random variable, P(s, 1, K,r,a) = e¹¹ E[(K-ser-+√z)+1 = E[(Ke¯" — se¬÷¹+√iz)+] Now, for a fixed value of Z, the function (Ke-"-se-¹+√iz)+ is 1. Decreasing and convex in s. (This follows because (a - bs)+ is, for b> 0, decreasing and convex in s.)
Transcribed Image Text:7.6 European Put Options The put call option parity formula, in conjunction with the Black-Scholes equation, yields the unique no arbitrage cost of a European (K, 1) put option: (7.7) P(s, 1, K,r, o) = C(s, t, K,r,a) + Kes Whereas the preceding is useful for computational purposes, to deter- mine monotonicity and convexity properties of P = P(s, 1, K,r, o) it is also useful to use that P(s, 1, K, r, o) must equal the expected return from the put under the risk neutral geometric Brownian motion process. Consequently, with Z being a standard normal random variable, P(s, 1, K,r,a) = e¹¹ E[(K-ser-+√z)+1 = E[(Ke¯" — se¬÷¹+√iz)+] Now, for a fixed value of Z, the function (Ke-"-se-¹+√iz)+ is 1. Decreasing and convex in s. (This follows because (a - bs)+ is, for b> 0, decreasing and convex in s.)
Exercises 127 2. Decreasing and convex in r. (This follows because (ae" - b)+ is, for a > 0, decreasing and convex in r.) 3. Increasing and convex in K. (This follows because (aK -b)+ is, for a > 0, increasing and convex in K.) Because the preceding properties remain true when we take expecta- tions, we see that P(s, 1, K, r, o) is decreasing and convex in s. P(s, t, K, r, a) is decreasing and convex in r. P(s, 1, K, r, o) is increasing and convex in K. Moreover, because C(s, t, K, r, o) is increasing in o, it follows from (7.7) that P(s, t, K, r, o) is increasing in o. Finally, • P (s, t, K, r, o) is not necessarily increasing or decreasing in t. The partial derivatives of P(s, 1, K, r, o) can be obtained by us- ing (7.6) in conjunction with the corresponding partial derivatives of C(s, 1, K,r,a).
Transcribed Image Text:Exercises 127 2. Decreasing and convex in r. (This follows because (ae" - b)+ is, for a > 0, decreasing and convex in r.) 3. Increasing and convex in K. (This follows because (aK -b)+ is, for a > 0, increasing and convex in K.) Because the preceding properties remain true when we take expecta- tions, we see that P(s, 1, K, r, o) is decreasing and convex in s. P(s, t, K, r, a) is decreasing and convex in r. P(s, 1, K, r, o) is increasing and convex in K. Moreover, because C(s, t, K, r, o) is increasing in o, it follows from (7.7) that P(s, t, K, r, o) is increasing in o. Finally, • P (s, t, K, r, o) is not necessarily increasing or decreasing in t. The partial derivatives of P(s, 1, K, r, o) can be obtained by us- ing (7.6) in conjunction with the corresponding partial derivatives of C(s, 1, K,r,a).
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