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PS5 IR Derivatives 3

This document describes a problem set involving interest rate derivatives. It provides a binomial tree model for the evolution of short-term interest rates over time. It then asks students to: 1) Calibrate the risk-neutral probabilities in the tree using market data on zero-coupon bond prices. 2) Explain whether a callable or non-callable bond would have a higher price and the meaning of negative convexity. 3) Use the calibrated probabilities to calculate the price of a callable coupon-bearing bond and determine when it would first be called. 4) Given additional information, use the price of a European call option on the callable bond to compute the bond's price.

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19 views2 pages

PS5 IR Derivatives 3

This document describes a problem set involving interest rate derivatives. It provides a binomial tree model for the evolution of short-term interest rates over time. It then asks students to: 1) Calibrate the risk-neutral probabilities in the tree using market data on zero-coupon bond prices. 2) Explain whether a callable or non-callable bond would have a higher price and the meaning of negative convexity. 3) Use the calibrated probabilities to calculate the price of a callable coupon-bearing bond and determine when it would first be called. 4) Given additional information, use the price of a European call option on the callable bond to compute the bond's price.

Uploaded by

Oscar Lin
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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FM405E - Fixed Income Securities and Credit Markets 2022/23

The London School of Economics

Problem Set 5
Interest rate derivatives III

Assume that the annualized, semi-annually compounded six-month rate, or the “short term
rate”, evolves over time according to the tree described in the following diagram.

uuu: r = 9.5%

uu: r = 8%

u: r = 7% uud: r = 7%

r = 6% ud: r = 6%

d: r = 4% udd: r = 4%

dd: r = 3%

ddd: r = 2.5%

t=0 t = 0.5 years t = 1 year t = 1.5 years

Next, consider a bond expiring in two years, paying off coupon rates of an annualized 6% of
the principal of $1 every six months, and callable at any time by the issuer, at par value. Let
this bond be labeled “BCX”.
Answer the following question, using numerical accuracy up to five decimals in Part(i), Part(iii)
and Part (iv):

i (10 points) Suppose that the prices of three zero coupon bonds expiring in one year, eighteen
months and two years are, respectively, 0.94632, 0.91876 and 0.89166. Use these market data

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to calibrate the risk-neutral probabilities of upward movements in the short-term rate implied
by the binomial tree in Figure 1. These risk-neutral probabilities depend only on calendar
time t, not on the specific state of nature at time t. What is the price of a conventional (i.e.
non-callable) coupon-bearing two year bond yielding an annualized 6% of the principal every
six months?

ii (10 points) Suppose that available for trading is a non-callable bond maturing in two years
and paying coupons semiannually, at an annualized rate of 6% of the principal of $1. Would
the price of this bond be less or greater than the price of the “BCX” bond? Explain without
performing any calculations. What does the difference between the two prices represent?
What is negative convexity?

iii (25 points) Given the market data and the calibrated risk-neutral probabilities obtained in
Part (i), proceed with the calculation of the price of the callable coupon bearing bond. In
which node of the tree would the bond be called for the first time, if any? Motivate and
comment your answers.

iv (15 points) Finally, ignore the market data in Part (i), and suppose that the risk-neutral
probabilities of upward movements in the short-term rate are: (a) unknown from time zero
to 0.5 years; (b) 70%, from 0.5 to one year; and (c) 60%, from one to 1.5 years. Suppose
that available for trading is a European call option written on the BCX bond. This option,
which quotes for $1.7226 × 10−3 , expires in 1.5 years, is struck at $ 0.99000, and becomes
worthless as soon as the underlying callable bond is called back by the issuer. You are asked
to compute the price of the BCX bond, given the price of the European option. Is it less or
greater than the price calculated in Part (iii)? Interpret the results.

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