• Home
  • Textbooks
  • Options, Futures, and Other Derivatives
  • Convexity, timing, and quanto adjustments

Options, Futures, and Other Derivatives

John C. Hull

Chapter 30

Convexity, timing, and quanto adjustments - all with Video Answers

Educators

Chapter Questions

04:37

Problem 1

Explain how you would value a derivative that pays off $100 R$ in 5 years, where $R$ is the 1-year interest rate (annually compounded) observed in 4 years. What difference would it make if the payoff were in (a) 4 years and (b) 6 years?

Umar Sohail Qureshi
Umar Sohail Qureshi
Numerade Educator
03:26

Problem 2

Explain whether any convexity or timing adjustments are necessary when:
(a) We wish to value a spread option that pays off every quarter the excess (if any) of the 5-year swap rate over the 3-month LIBOR rate applied to a principal of $$\$ 100$$. The payoff occurs 90 days after the rates are observed.
(b) We wish to value a derivative that pays off every quarter the 3-month LIBOR rate minus the 3-month Treasury bill rate. The payoff occurs 90 days after the rates are observed.

James Kiss
James Kiss
Numerade Educator
01:34

Problem 3

Suppose that in Example 29.3 of Section 29.2 the payoff occurs after 1 year (i.e., when the interest rate is observed) rather than in 15 months. What difference does this make to the inputs to Black's model?

Ryan Pollard
Ryan Pollard
Numerade Educator

Problem 4

The OIS zero curve is flat at $10 \%$ per annum with annual compounding. Calculate the value of an instrument where, in 5 years' time, the 2 -year swap rate (with annual compounding) is received and a fixed rate of $10 \%$ is paid. Both are applied to a notional principal of $$\$ 100$$. Assume that the volatility of the forward swap rate is $20 \%$ per annum and that the 12-month LIBOR-OIS spread is zero. Explain why the value of the instrument is different from zero.

Check back soon!

Problem 5

What difference does it make in Problem 30.4 if the swap rate is observed in 5 years, but the exchange of payments takes place in (a) 6 years, and (b) 7 years? Assume that the volatilities of all forward rates are $20 \%$. Assume also that the forward swap rate for the period between years 5 and 7 has a correlation of 0.8 with the forward interest rate between years 5 and 6 and a correlation of 0.95 with the forward interest rate between years 5 and 7 .

Check back soon!

Problem 6

The price of a bond at time $T$, measured in terms of its yield, is $G\left(y_T\right)$. Assume geometric Brownian motion for the forward bond yield $y$ in a world that is defined by a numeraire equal to a bond maturing at time $T$. Suppose that the growth rate of the forward bond yield is $\alpha$ and its volatility $\sigma_y$.
(a) Use Itô's lemma to calculate the process for the forward bond price in terms of $\alpha$, $\sigma_y, y$, and $G(y)$.
(b) The forward bond price should follow a martingale in the world considered. Use this fact to calculate an expression for $\alpha$.
(c) Show that the expression for $\alpha$ is, to a first approximation, consistent with equation (30.1).

Check back soon!

Problem 7

The variable $S$ is an investment asset providing income at rate $q$ measured in currency A. It follows the process
$$
d S=\mu_S S d t+\sigma_S S d z
$$
in the real world. Defining new variables as necessary, give the process followed by $S$, and the corresponding market price of risk, in:
(a) A world that is the traditional risk-neutral world for currency A
(b) A world that is the traditional risk-neutral world for currency B
(c) A world that is defined by a numeraire equal to a zero-coupon currency A bond maturing at time $T$
(d) A world that is defined by a numeraire equal to a zero-coupon currency B bond maturing at time $T$.

Check back soon!

Problem 8

A call option provides a payoff at time $T$ of $\max \left(S_T-K, 0\right)$ yen, where $S_T$ is the dollar price of gold at time $T$ and $K$ is the strike price. Assuming that the storage costs of gold are zero and defining other variables as necessary, calculate the value of the contract.

Check back soon!

Problem 9

A Canadian equity index is 400 . The Canadian dollar is currently worth 0.70 U.S. dollars. The risk-free interest rates in Canada and the U.S. are constant at $6 \%$ and $4 \%$, respectively. The dividend yield on the index is $3 \%$. Define $Q$ as the number of Canadian dollars per U.S. dollar and $S$ as the value of the index. The volatility of $S$ is $20 \%$, the volatility of $Q$ is $6 \%$, and the correlation between $S$ and $Q$ is 0.4 . Use DerivaGem to determine the value of a 2 -year American-style call option on the index if:
(a) It pays off in Canadian dollars the amount by which the index exceeds 400 .
(b) It pays off in U.S. dollars the amount by which the index exceeds 400 .

Check back soon!

Problem 10

Consider an instrument that will pay off $S$ dollars in 2 years, where $S$ is the value of the Nikkei index. The index is currently 20,000. The yen/dollar exchange rate is 100 (yen per dollar). The correlation between the exchange rate and the index is 0.3 and the dividend yield on the index is $1 \%$ per annum. The volatility of the Nikkei index is $20 \%$ and the volatility of the yen/dollar exchange rate is $12 \%$. The interest rates (assumed constant) in the U.S. and Japan are $4 \%$ and $2 \%$, respectively.
(a) What is the value of the instrument?
(b) Suppose that the exchange rate at some point during the life of the instrument is $Q$ and the level of the index is $S$. Show that a U.S. investor can create a portfolio that changes in value by approximately $\Delta S$ dollar when the index changes in value by $\Delta S$ yen by investing $S$ dollars in the Nikkei and shorting $S Q$ yen.
(c) Confirm that this is correct by supposing that the index changes from 20,000 to 20,050 and the exchange rate changes from 100 to 99.7 .
(d) How would you delta hedge the instrument under consideration?

Check back soon!

Problem 11

Suppose that the risk-free yield curve is flat at $8 \%$ (with continuous compounding). The payoff from a derivative occurs in 4 years. It is equal to the 5 -year rate minus the 2 -year rate at this time, applied to a principal of $$\$ 100$$ with both rates being continuously compounded. (The payoff can be positive or negative.) Calculate the value of the derivative. Assume that the volatility for all rates is $25 \%$. What difference does it make if the payoff occurs in 5 years instead of 4 years? Assume all rates are perfectly correlated.

Check back soon!

Problem 12

Suppose that the payoff from a derivative will occur in 10 years and will equal the 3 -year U.S. dollar swap rate for a semiannual-pay swap observed at that time applied to a certain principal. Assume that the swap yield curve is flat at $8 \%$ (semiannually compounded) per annum in dollars and $3 \%$ (semiannually compounded) in yen. The forward swap rate volatility is $18 \%$, the volatility of the 10 -year "yen per dollar" forward exchange rate is $12 \%$, and the correlation between this exchange rate and U.S. dollar interest rates is 0.25 . What is the value of the derivative if the swap rate is applied to a principal of (a) $$\$ 100$$ million with a dollar payoff and (b) 100 million yen with a yen payoff? Assume that risk-free rates are $2 \%$ in yen and $6 \%$ in dollars (both semiannually compounded).

Check back soon!

Problem 13

The payoff from a derivative will occur in 8 years. It will equal the average of the 1 -year risk-free interest rates observed at times $5,6,7$, and 8 years applied to a principal of $$\$ 1,000$$. The risk-free yield curve is flat at $6 \%$ with annual compounding and the volatilities of all rates are $16 \%$. Assume perfect correlation between all rates. What is the value of the derivative?

Check back soon!