Computational Methods in Economics and Finance

HW 10

10.1. A chooser option is an option that give its holder the right to choose

among other options at some future date prior to expiration. The

standard chooser option gives its holder the right to choose either a

put or a call option. Speci¯cally, the holder of a chooser option can

choose at time T1 between a put and a call option, both with strike

price K expiring at time T2 > T1.

A chooser option can be thought of as a derivative asset with a terminal

date of T1 and a payout of max(P(T1; T2);C(T1; T2)), where P(t; T) is

the time t value of a put expiring at time T and C is de¯ned similarly

for a call.

Suppose that the underlying asset is described by

dS = rSdt + ¾SdW

(i.e., geometric Brownian motion). Speci¯cally let r = 0:05, ¾ = 0:5,

K = 1, T1 = 1 and T2 = 1:5 (with now being t = 0). Write a Mat-

lab program that prices the chooser option. Plot the chooser option,

together with the prices of the put and the call options.

Comment on how the chooser option priced here compares with the

values of a put and a call with T2 periods until expiration.

10.2. A coupon bond is a bond that makes payments while the bond is alive

as well as at its terminal date. Continuous coupons are like dividend

°ows. Most coupons, however, are paid discretely. Price a 10 year

bond using the CIR interest rate model that pays 0:05 (5%) times its

face value at the start of every year, starting one year after it is issued

(assume a face value of 1). Use the parameters · = 0:1, ® = 0:05 and

¾ = 0:08.

A callable bond is like an option in that it can be exercised early but

the seller has the right to determine whether it is exercised. Price the

above coupon bond with the extra feature that the seller can call the

bond once a year right after paying the coupon. Essentially this means

that the exercise value is 1:05. If the bond is worth more than this

alive, it is optimal to call the bond.

Write a Matlab program that prices the coupon bond both with and

without the callable feature. Note that finsolve could be used to solve

this problem but it is ine±cient (you should make sure you understand

why). You should, therefore, write your own code to solve this problem,

preferably in the form of a function that can be reused for other bond

pricing problems. Your code should be robust so that features such as

the bond's horizon and how often it is callable can be easily altered.

HW 10

10.1. A chooser option is an option that give its holder the right to choose

among other options at some future date prior to expiration. The

standard chooser option gives its holder the right to choose either a

put or a call option. Speci¯cally, the holder of a chooser option can

choose at time T1 between a put and a call option, both with strike

price K expiring at time T2 > T1.

A chooser option can be thought of as a derivative asset with a terminal

date of T1 and a payout of max(P(T1; T2);C(T1; T2)), where P(t; T) is

the time t value of a put expiring at time T and C is de¯ned similarly

for a call.

Suppose that the underlying asset is described by

dS = rSdt + ¾SdW

(i.e., geometric Brownian motion). Speci¯cally let r = 0:05, ¾ = 0:5,

K = 1, T1 = 1 and T2 = 1:5 (with now being t = 0). Write a Mat-

lab program that prices the chooser option. Plot the chooser option,

together with the prices of the put and the call options.

Comment on how the chooser option priced here compares with the

values of a put and a call with T2 periods until expiration.

10.2. A coupon bond is a bond that makes payments while the bond is alive

as well as at its terminal date. Continuous coupons are like dividend

°ows. Most coupons, however, are paid discretely. Price a 10 year

bond using the CIR interest rate model that pays 0:05 (5%) times its

face value at the start of every year, starting one year after it is issued

(assume a face value of 1). Use the parameters · = 0:1, ® = 0:05 and

¾ = 0:08.

A callable bond is like an option in that it can be exercised early but

the seller has the right to determine whether it is exercised. Price the

above coupon bond with the extra feature that the seller can call the

bond once a year right after paying the coupon. Essentially this means

that the exercise value is 1:05. If the bond is worth more than this

alive, it is optimal to call the bond.

Write a Matlab program that prices the coupon bond both with and

without the callable feature. Note that finsolve could be used to solve

this problem but it is ine±cient (you should make sure you understand

why). You should, therefore, write your own code to solve this problem,

preferably in the form of a function that can be reused for other bond

pricing problems. Your code should be robust so that features such as

the bond's horizon and how often it is callable can be easily altered.

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