0.1
EXOTIC OPTIONS
Exercise 1 1- Prove that the value of a European Caplet Up-and-In is equal to the value of a European Caplet minus the European Caplet Up-and-Out. 2- In the same spirit, prove that the value of a European Floorlet Down-and-In is equal to the value of a European Floorlet minus the European Floorlet Down-and-Out. Solution 2 1- The strike and the barrier of the caplet-Up and-In are denoted E and H respectively, with E < H. For simplicity purposes, we consider the tenor and the nominal amount equal to 1. The payoff of this product depends on the value of the exercise rate X at expiry. It is Max [0, X − E] .1X≥H which can be decomposed into: · if X ≥ H: P ayoff = X − E · if X < H: P ayoff = 0 The payoff of the caplet Up-and-Out is Max [0, X − E] .1X H: P ayoff = 0 The payoff of the floor Down-and-Out is Max [0, E − X] .1X>H which can be decomposed into: · if X ≤ H: P ayoff = 0 · if H < X < E: P ayof f = E − X 1
· if X ≥ E: P ayof f = 0 When we sum the two payoffs we obtain the following results (what we call payoff total): · if X ≤ H: P ayoff T otal = E − X · if H < X < E: P ayof f T otal = E − X · if X ≥ E: P ayof f T otal = 0 The payoff total is in fact the payoff of a standard floor, which concludes the proof. Exercise 3 The idea of this exercise is to show that the payoff Min(X, Y ) where X and Y are two interest rates may be decompose into an equivalent payoff including a spread option. Show that the payoff Min(X, Y ) is equal to X − Max(0; X − Y ) or Y − Max(0; Y − X). Solution 4 We consider two cases: · if X ≤ Y : X − Max(0; X − Y ) = X = Min(X, Y ) · if X > Y : X − Max(0; X − Y ) = X − (X − Y ) = Y = Min(X, Y ) Using the same argument, we show that Min(X, Y ) = Y − Max(0; Y − X) Exercise 5 On 05/13/02 a firm buys a barrier caplet Up-and-In whose features are the following: · notional amount: $ 10,000,000 · starting date: 06/03/02
· reference rate: 3-month Libor · barrier: 6%
· strike rate: 5%
· day-count: Actual/360
1- What is the payoff of this option for the buyer on 09/03/02 ? 2- Draw the P&L of this caplet considering that the premium paid by the buyer is equal to 0.08% of the nominal amount. 3- What is the advantage and the drawback of this option compared to a classical caplet ?
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Solution 6 1- The payoff of this option for the buyer on 09/03/02 is Payoff = $10, 000, 000 ×
· µ ¶ ¸ 1 92 × Max 0; R 06/03/02, − 5% × 1R(06/03/02, 1 )≥6% 4 360 4
¢ ¡ where R 06/03/02, 14 is the 3-month Libor rate observed on 06/03/02, 92 is the number of
days between the 06/03/02 and the 09/03/02, and 1A = 1 if event A occurs and 0 otherwise. 2- The P&L is given by the following formula P &L = Payoff − $10, 000, 000 × 0.08% It appears in the following graph. 80000 70000 60000 50000
P&L in $
40000 30000 20000 10000 0 3,0% -10000
3,5%
4,0%
4,5%
5,0%
5,5%
6,0%
6,5%
7,0%
7,5%
8,0%
-20000 Value of the 3-month Libor on 06/03/02
3- The advantage of this option compared to a classical caplet is that the buyer will pay a lower premium. The drawback is that he will gain only if the reference rate is equal or above the barrier. Exercise 7 On 05/13/02 a firm buys a barrier floorlet Down-and-In whose features are the following: · notional amount: $ 10,000,000 · starting date: 06/03/02
· reference rate: 3-month Libor · barrier: 4% 3
· strike rate: 5%
· day-count: Actual/360
1- What is the payoff of this option for the buyer ? 2- Draw the P&L of this floorlet considering that the premium paid by the buyer is equal to 0.07% of the nominal amount. 3- What is the advantage and the drawback of this option compared to a classical floorlet ? Solution 8 1- The payoff of this option for the buyer on 09/03/02 is Payoff = $10, 000, 000 ×
· µ ¶¸ 1 92 × Max 0; 5% − R 06/03/02, × 1R(06/03/02, 1 )≤4% 4 360 4
¡ ¢ where R 06/03/02, 14 is the 3-month Libor rate observed on 06/03/02, 92 is the number of
days between the 06/03/02 and the 09/03/02, and 1A = 1 if event A occurs and 0 otherwise. 2- The P&L is given by the following formula P &L = Payoff − $10, 000, 000 × 0.07% It appears in the following graph. 80000 70000 60000 50000
P&L in $
40000 30000 20000 10000 0 2,0% -10000
2,5%
3,0%
3,5%
4,0%
4,5%
5,0%
5,5%
6,0%
6,5%
7,0%
-20000 Value of the 3-month Libor on 06/03/02
3- The advantage of this option compared to a classical floorlet is that the buyer will pay a lower premium. The drawback is that he will gain only if the reference rate is equal or below the barrier. 4
Exercise 9 Same questions as in the previous exercise but now the firm buys a barrier floorlet Down-and-Out. Solution 10 1- The payoff of this option for the buyer on 09/03/02 is · µ ¶¸ 92 1 Payoff = $10, 000, 000 × × Max 0; 5% − R 06/03/02, × 1R(06/03/02, 1 )>4% 4 360 4 ¡ ¢ where R 06/03/02, 14 is the 3-month Libor rate observed on 06/03/02, 92 is the number of
days between the 06/03/02 and the 09/03/02, and 1A = 1 if event A occurs and 0 otherwise. 2- The P&L is given by the following formula P &L = Payoff − $10, 000, 000 × 0.07% It appears in the following graph. 20000
15000
P&L in $
10000
5000
0 3,0%
3,5%
4,0%
4,5%
5,0%
5,5%
6,0%
6,5%
7,0%
7,5%
8,0%
-5000
-10000 Value of the 3-month Libor on 06/03/02
3- The advantage of this option compared to a classical floorlet is that the buyer will pay a lower premium. The drawback is that he will gain only if the reference rate is above the barrier and below the strike rate. Exercise 11 A firm has a floating rate debt of $100,000,000 indexed on the 6-month Libor with 5
a 4-year maturity. The treasurer of this firm anticipates that rates will increase in the future. He buys a 4-year N-cap with a 5% strike, a 6% barrier and a second cap with a 6.5% strike. We assume that the nominal amount and the tenor are respectively equal to Eur 100,000,000 and 6 months. What is the payoff of each N-caplet ? Draw it on a graph. Solution 12 The payoff for each N-caplet is as follows ¤ £ 100, 000, 000 × (L − 5%)+ · 1L≤6% + (L − 6.5%)+ · 1L>6% We draw below the graph of this payoff 2500000
2000000
Pay-Off
1500000
1000000
500000
0 3.50%
4.00%
4.50%
5.00%
5.50%
6.00%
6.50%
7.00%
7.50%
8.00%
8.50%
Libor Level
Exercise 13 A firm holds a bond portfolio indexed on the 6-month Libor with a 10-year maturity. The portfolio manager of this firm anticipates that rates will decrease in the future. He buys a 10-year N-floor with a 7% strike, a 6% barrier and a second floor with a 5% strike. We assume that the nominal amount and the tenor are respectively equal to $ 100,000,000 and 6 months. What is the payoff of each N-floorlet ? Draw the graph of this payoff.
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Solution 14 The payoff of each N-floorlet is given by the following formula ¤ 100, 000, 000 £ × (7% − L1 )+ · 1L1 >6% + (5% − L1 )+ · 1L1 ≤6% 2 We draw below the graph of this payoff 1600000 1400000 1200000
Pay-Off
1000000 800000 600000 400000 200000 0 3.50%
4.00%
4.50%
5.00%
5.50%
6.00%
6.50%
7.00%
7.50%
8.00%
8.50%
Reference Rate Level
Exercise 15 We consider a 6-month Libor incremental fixed swap with $100,000,000 nominal amount where the fixed portion is determined as follows Libor Level
Fixed Portion
7% < Libor
100%
6% < Libor ≤ 7%
80%
5% < Libor ≤ 6%
60%
4% < Libor ≤ 5%
40%
Libor ≤ 4%
0%
The fixed leg is paid annually as the floating leg is received semi-annually. The incremental fixed swap rate is equal to 7.2% as the plain-vanilla swap rate is 6.8%. 1- What is the value of the fixed leg if the 6-month Libor is equal to 6.5% ?
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2- Calculate the financing cost for a firm with a 6-month Libor debt in three different situations: when it does nothing, when it contracts a standard swap and when it contracts an incremental fixed swap. 3- We suppose that the firm contracts an incremental fixed swap. What is its financing cost when the 6-month Libor is respectively 8%, 6.8%, 5.7%, 4.8% and 3.5% ? Solution 16 1- If the Libor is equal to 6.5%, the fixed leg is then equal to fixed leg = $100, 000, 000 × [(80% × 7.2%) + (20% × 6.5%] = $7, 060, 000 2- The results are summarized in the following table Libor Level
Floating Rate Debt
Swapped Floating Rate Debt
Incremental Fixed Swap
7% < Libor
Libor
6.8%
7.2%
6% < Libor ≤ 7%
Libor
6.8%
5.76% + 0.2 × Libor
5% < Libor ≤ 6%
Libor
6.8%
4.32% + 0.4 × Libor
4% < Libor ≤ 5%
Libor
6.8%
2.88% + 0.6 × Libor
Libor ≤ 4%
Libor
6.8%
Libor
3- The results are summarized in the following table Libor Level
Financing Cost
8%
7.2%
6.8%
7.12%
5.7%
6.6%
4.8%
5.76%
3.5%
3.5%
Exercise 17 Consider a firm with a 5-year debt of $100,000,000 nominal amount indexed on the 6-month Libor. The treasurer of this firm wants to lock in the floating rate of its debt. He contracts a subsidized swap: · he enters a standard 6-month Libor swap with a $100,000,000 nominal amount, where he 8
pays the 6.5% fixed rate and receives the 3-month Libor · and sells a 5-year cap with a $100,000,000 nominal amount, the 6-month Libor as reference rate, and 8% as strike. The cap premium, which is paid semi-annually, is equal to 0.5% of the nominal amount. Calculate the financing cost for the firm in the three different situations, when the treasurer contracts a 6-month Libor debt , when he swaps its debt, and when he contracts a subsidized swap. Solution 18 Recall that a subsidized swap is a combination of a plain vanilla swap where the firm pays the fixed rate with the sale of a cap. The results depend on the value of the floating rate on each reset date, superior or inferior to 8%. We summarize them in the following table: Floating-Rate Debt
Swapped Floating-Rate Debt
Subsidized Swap
Libor ≤ 8%
Libor
6.5%
6%
Libor > 8%
Libor
6.5%
Libor -2%
When the 6-month Libor stays below 8%, the financing cost of the subsidized swap is equal to 6.5% − 0.5% = 6%. When the 6-month Libor goes above 8%, the firm has to pay (prorated to the period) Libor-8% to the cap buyer. Its financing cost is then equal to 6% + Libor − 8% = Libor − 2%, which is still below than Libor. Exercise 19 1- We consider the following payoff Payoff = Max(0; X − K) × 1H1 ≤X
