Applications
Content
Applications
Answers to Sample Questions
1. Assume that the stock price of 100 and the strike price of 100 with time to expiration of one
year. Further assume that the risk free rate is 4% and the stock volatility is 30%. What is the call
option price? (hint: use the Black-Scholes model) Answer: 13.7533.
2. Using the assumption of the above question but the stock volatility has dropped to 20%, what
is the change in the call option price? Show that the result is consistent with using a sensitivity
measure .Answer: 3.8283. vega is 38.13 which is approximately equals 3.8283/0.1
3. An interest rate model is arbitrage-free if the pricing of a zero coupon bond by the backward
substitution approach is the same as the discounted value for any interest rate volatilities and spot
curves. Show that the Ho-Lee model is arbitrage-free. (hint: use the Ho-Lee model to show that
the pricing of a 5 year zero coupon bond using the interest rate model is equal to discounting the
$1 payment at the 5 year rate.)
4. Let us consider a European call option with 5 year time to expiration to buy a 3 year bond at a
strike 75. Assuming a flat yield curve of 6% and term structure of spot volatility of 20%. What is
the option price? (hint: use the bond option model) Answer: 6.4982.
5. Using question 8 assumptions, what is the dynamic replication of the option at time 0? How
should the firm hedge the option position? In what sense is this replication “dynamic”? (hint: use
the dynamic hedging model) Answer: short 0.6860 the 5 year bond and buy 0.9263 the 8 year
bond. Buy 0.6860 the 5 year bond and short 0.9263 the 8 year bond for hedging. This hedge
position keeps changing at each state and time of the world, as the model shows, and hence
“dynamic.”
6. Using the assumptions of the above question, what is the price of the European put option?
Can we apply the put call parity to determine the put option price from the call option price?
(hint: use the bond option model.) Answer: Yes. Using the bond option model, you can verify that
the put option value derived by changing the terminal condition and use the backward substation
is the same as that using the put call parity.
7. Consider a constant maturity swap where the tenor is 5 years and receives the 5 year floating
interest rates. Suppose the spot yield curve is flat at 10% (continuously compounding). Suppose
that there is no interest rate risk, what is the value of the swap? Answer: there is no value because
since the yield curve is flat, the 5 year rate equals the short rate. And since there is no interest rate
risk, there is no exchange of interests in the future either.
8. Using the assumptions of the above question, now suppose that there is a term structure of spot
volatilities of 12%. What is the value of the swap on a notional amount of 100? (hint: use
Constant Maturity Swap model) Answer: 0.2785
9. Consider a call option on the yield spread between a 10 year zero coupon bond and the 2 year
zero coupon bond. The strike is 5 basis points. The yield curve is assumed to be flat and the term
structure of spot volatilities is linearly declining at 0.1% annually, starting 10% the first year.
What is the price of the spread option? (Hint: use the spread option model) Answer: 0.000910.
10. Suppose that the interest rate is 5%. The risk free rate is 4%. The debt ratio is 0.6. Beta of the
stock is 1.4, and the risk premium is 3%. What is the weighted average cost of capital of the firm
given the appropriate tax rate is 30%. (Hint: use excel model 120501) (answer: 5.35%)
11. Using the above assumptions, suppose the firm has an earnings before interest and tax of 2
millions. But the firm requires a capital expenditure of $400,000 annually. There is a depreciation
of $200,000 and there is no change of working capital needed, what is the free cashflow of the
firm. (answer: $1.2 million)
12. Using the assumption of question 1 and 2, suppose further that the firm’s free cashflow grows
at 1% annually forever, what is the value of the firm? (answer: $27.67 million)
13. Assume that the current exchange is 1.20 US dollar to a Euro. US and the Euro 3 year spot
rates are 2.5% and 4% respectively. What is the forward currency rate? (Hint: use interest rate
parity model) Answer: 1.1472
14. Using the assumptions of the above question, what is the price of a European call option on
the exchange rate with strike 1.2, expiring in 1 year, assuming that the exchange rate volatility is
14.1%? (Hint: use Garman-Kohlhagen model) Answer: 0.054.
15. Assume a one factor Ho-Lee model with a flat yield curve of 6% and decline term structure of
spot volatility starting 20% in year one and linearly decline 1% a year. Use one year step size in
the construction of the binomial lattice. There are 6 states of the interest rate level at the end of
the 5th year. What are the values of the Arrow Debreu securities for these states, using the forward
measure approach? (hint: use forward measure model) Answer: 0.0210, 0.1092, 0.2269, 0.2358,
0.1225, 0.0254.
16. We use the Ho-Lee model to value a call option on an 8 year zero coupon bond with
expiration 5 years and a strike price of 90. Assume a flat yield curve of 6% and a flat spot
volatility of 20%. We use an annual step size binomial model. What is the distribution of the
forward measure, if we use a 7 year bond as the numeraire. (Hint: use the change of numeraire
excel model) Answer: 0.0244, 0.1345, 0.2962, 0.3260, 0.1794, 0.0395.
17. Use the assumptions of the above question. We can derive the distribution of the forward
measure based on either the 5 year bond or the 7 year bond numeraire. What are the RadonNikodym derivative that translates the one forward measure to the other? (Hint: use RadonNikodym derivative excel model) Answer: 0.88310, .92652, .97208, 1.01988, 1.07003, 1.12264.
Answers to Sample Questions
1. Assume that the stock price of 100 and the strike price of 100 with time to expiration of one
year. Further assume that the risk free rate is 4% and the stock volatility is 30%. What is the call
option price? (hint: use the Black-Scholes model) Answer: 13.7533.
2. Using the assumption of the above question but the stock volatility has dropped to 20%, what
is the change in the call option price? Show that the result is consistent with using a sensitivity
measure .Answer: 3.8283. vega is 38.13 which is approximately equals 3.8283/0.1
3. An interest rate model is arbitrage-free if the pricing of a zero coupon bond by the backward
substitution approach is the same as the discounted value for any interest rate volatilities and spot
curves. Show that the Ho-Lee model is arbitrage-free. (hint: use the Ho-Lee model to show that
the pricing of a 5 year zero coupon bond using the interest rate model is equal to discounting the
$1 payment at the 5 year rate.)
4. Let us consider a European call option with 5 year time to expiration to buy a 3 year bond at a
strike 75. Assuming a flat yield curve of 6% and term structure of spot volatility of 20%. What is
the option price? (hint: use the bond option model) Answer: 6.4982.
5. Using question 8 assumptions, what is the dynamic replication of the option at time 0? How
should the firm hedge the option position? In what sense is this replication “dynamic”? (hint: use
the dynamic hedging model) Answer: short 0.6860 the 5 year bond and buy 0.9263 the 8 year
bond. Buy 0.6860 the 5 year bond and short 0.9263 the 8 year bond for hedging. This hedge
position keeps changing at each state and time of the world, as the model shows, and hence
“dynamic.”
6. Using the assumptions of the above question, what is the price of the European put option?
Can we apply the put call parity to determine the put option price from the call option price?
(hint: use the bond option model.) Answer: Yes. Using the bond option model, you can verify that
the put option value derived by changing the terminal condition and use the backward substation
is the same as that using the put call parity.
7. Consider a constant maturity swap where the tenor is 5 years and receives the 5 year floating
interest rates. Suppose the spot yield curve is flat at 10% (continuously compounding). Suppose
that there is no interest rate risk, what is the value of the swap? Answer: there is no value because
since the yield curve is flat, the 5 year rate equals the short rate. And since there is no interest rate
risk, there is no exchange of interests in the future either.
8. Using the assumptions of the above question, now suppose that there is a term structure of spot
volatilities of 12%. What is the value of the swap on a notional amount of 100? (hint: use
Constant Maturity Swap model) Answer: 0.2785
9. Consider a call option on the yield spread between a 10 year zero coupon bond and the 2 year
zero coupon bond. The strike is 5 basis points. The yield curve is assumed to be flat and the term
structure of spot volatilities is linearly declining at 0.1% annually, starting 10% the first year.
What is the price of the spread option? (Hint: use the spread option model) Answer: 0.000910.
10. Suppose that the interest rate is 5%. The risk free rate is 4%. The debt ratio is 0.6. Beta of the
stock is 1.4, and the risk premium is 3%. What is the weighted average cost of capital of the firm
given the appropriate tax rate is 30%. (Hint: use excel model 120501) (answer: 5.35%)
11. Using the above assumptions, suppose the firm has an earnings before interest and tax of 2
millions. But the firm requires a capital expenditure of $400,000 annually. There is a depreciation
of $200,000 and there is no change of working capital needed, what is the free cashflow of the
firm. (answer: $1.2 million)
12. Using the assumption of question 1 and 2, suppose further that the firm’s free cashflow grows
at 1% annually forever, what is the value of the firm? (answer: $27.67 million)
13. Assume that the current exchange is 1.20 US dollar to a Euro. US and the Euro 3 year spot
rates are 2.5% and 4% respectively. What is the forward currency rate? (Hint: use interest rate
parity model) Answer: 1.1472
14. Using the assumptions of the above question, what is the price of a European call option on
the exchange rate with strike 1.2, expiring in 1 year, assuming that the exchange rate volatility is
14.1%? (Hint: use Garman-Kohlhagen model) Answer: 0.054.
15. Assume a one factor Ho-Lee model with a flat yield curve of 6% and decline term structure of
spot volatility starting 20% in year one and linearly decline 1% a year. Use one year step size in
the construction of the binomial lattice. There are 6 states of the interest rate level at the end of
the 5th year. What are the values of the Arrow Debreu securities for these states, using the forward
measure approach? (hint: use forward measure model) Answer: 0.0210, 0.1092, 0.2269, 0.2358,
0.1225, 0.0254.
16. We use the Ho-Lee model to value a call option on an 8 year zero coupon bond with
expiration 5 years and a strike price of 90. Assume a flat yield curve of 6% and a flat spot
volatility of 20%. We use an annual step size binomial model. What is the distribution of the
forward measure, if we use a 7 year bond as the numeraire. (Hint: use the change of numeraire
excel model) Answer: 0.0244, 0.1345, 0.2962, 0.3260, 0.1794, 0.0395.
17. Use the assumptions of the above question. We can derive the distribution of the forward
measure based on either the 5 year bond or the 7 year bond numeraire. What are the RadonNikodym derivative that translates the one forward measure to the other? (Hint: use RadonNikodym derivative excel model) Answer: 0.88310, .92652, .97208, 1.01988, 1.07003, 1.12264.
