Investment Class Homework

Investment – Problem Set 9

Q1

1. According to the original 5-year swap with XYZ firm, ABC firm need to pay the LIBOR in return for a fixed 8% rate on notional principal of $10 million.

ABC will receive: 8%*10 million = $800,000

Two years later, if XYZ firm defaults and ABC has to enter a new swap with a 6% swap rate. In other words, if XYZ firm defaults, ABC has to turn to swap with equal payments but lower return.

ABC will receive: 6%*10 million = $600,000

ABC lost 800,000-600,000=$200,000 every year.

Firm ABC is definitely harmed by the default.

1. From the third year, firm ABC suffers from a loss of $200,000 every year. Thus at the end of the second year or at the beginning of the third year, when firm XYZ defaults, firm ABC will have a loss:

200,000/1.06+200,000/1.062+200,000/1.063= $534,602.39

1. If firm ABC goes bankrupt and defaults the swap, firm XYZ will benefit. However, even if firm ABC reorganizes, the swap should still be an asset of ABC. Actually, firm ABC will never default because there is an arbitrage opportunity here. As the ABC hold the 5-year swap, it will gain $200,000 with no cost every year if ABC enters another market swap to pay a fixed 6% in return for the LIBOR.

Q2

If risk free rate rf =0, an American put option should not be exercised early.

Let’s look at the put option with an exercise price of K first:

At time 0, we assume the price of the put option is P.

At time T, the payoff of the put option will be max{K-ST,0}.

However, we can use another strategy to replicate the put option: short a stock and lend K.

At time 0, this strategy will cost K-St.

At time T, this strategy will generate payoff of K-ST.

In contrast, max{K-ST,0}≥ K-ST. Thus there must be P≥ K-St, or there will be an arbitrage.

If we sell the option, we will get P, which is not smaller than the payoff of exercise this option early. Therefore, an American put option should not be exercised early.

Q3

The value of the call option is max{ST-K,0} In the two-state stock price model:

S0=100 with no dividends, K=100

uS0=120 ⇒ Cu=120-100=20   π=50%

dS0=120 ⇒ Cd=0   1-π=50%

Hedge ratio: H = (Cu- Cd)/ (uS0- dS0) =0.5

We need to long one stock and short two calls with the cost 100-2C at time 0:

At time T:

Discount the payoff to time 0, we will have:

80/ (1+10%) =100-2C

C=13.64

Q4

The value of the call option is max{ST-K,0} In the two-state stock price model: