Question 1 (12 marks)

Part A: (7 marks)

Consider the option on currency HKD against the USD:

• Current spot rate is HKD7.50 for 1 USD

• Risk-free HKD rate of interest is 5% p.a.

• Risk-free USD rate of interest is 2% p.a.

• Volatility (σ) of the currency returns is 20% p.a.

• Maturity of the option is 3 months.

• Strike rate of the option is HKD8.00 for 1 USD

• The currency options are European in nature

Answer the following questions.

(i)       Draw the terminal payoff diagram for the holder of the currency call option on HKD. [1 mark]

(ii)      Draw the terminal payoff diagram for the holder of the currency put option on USD . [1 mark]

(iii)     How  much  does  it  cost to  hold  (i.e.,  buy)  a  call-HKD  option?  Use the  Garman Kohlhagen model.               [2 marks]

(iv)    What is the minimum terminal exchange rate for the holder of the call-HKD option to profit from holding the currency option?                                                       [1 mark]

(v)      How much does it cost to hold (i.e., buy) a put-HKD option? Do not use the Garman Kohlhagen model.                [2 marks]

Question 1

Part B: (5 marks)

Note that Part A and B are not related.

The table below shows the premiums for several currency options (all the options are maturing in 9 months):

The spot currency rate on the day when all the above options are priced is GBP 0.58 for 1 SGD.

i.        Calculate the current premium for a 9-month maturity call option on GBP with strike price of GBP 0.55 for 1 SGD. Show your working.                               [3 marks]

ii.        Suppose that you  hold the  call  option  on  GBP  specified  in  part  (i)  above  until maturity. On maturity, the spot currency rate is GBP 0.52 for 1 SGD. What is your terminal payoff?                                                                                                   [2 marks]

Question 2              (10 marks)

Consider a stock, the current price (S0 ) of which is $30. We model stock-price evolution using a Binomial model. In every three-month period, u = 1.1052 and d = 0.9048. The risk free rate of interest is 5% per annum continuously compounded. The four-step Binomial tree is shown below:

Node Time:

0.0000            0.2500            0.5000            0.7500            1.0000

A European-style exotic derivative has been written on this stock. The derivative has one year to expiry.  Denote  by S1, S2, S3   and S4  the  stock  price  after  three,  six,  nine  and twelve  months respectively. The payoff to the derivative is specified as follows:

(max(S2 , S4 )− min(S1 , S3 ) Payoff =〈   max(S1 , S2 , S3 )− S4

Required:

if S4   30

if S4  < 30

Using a four-step Binomial framework and the risk-neutral approach, calculate the current value of this exotic derivative. Use continuous compounding for all present value calculations. Show all working.

Question 3              (10 marks)

Suppose that the risk-free rate is 12% p.a. compounded continuously. The current price (S0 ) of a stock is $15. We model the evolution of the stock prices using a 3-step Binomial tree approach. In reality, over any four-month period, there is a 60% chance that the stock price will rise by 29.67% and a 40% chance that the stock price will fall by 22.88%. This is illustrated in the figure below:

Node Time:

0.0000            0.3333            0.6667            1.0000

An exotic derivative with one year to maturity is written on this stock. The payoff structure of the derivative is specified as:

(   6 Payoff =〈

St  + 6

if St  > 16

if St  < 16

where St  is the  stock  price  at time  t. The exotic  derivative  is  Bermudan-style which  can  be exercised at t = 4 months and at t = 12 months.

Required:

Use the risk-neutral Binomial approach to calculate the current value of the exotic derivative. Use continuous compounding for all present value calculations. Show all working.

Question 4   (11 marks)

Part A: (5 marks)

Consider the following two strategies:

Strategy A: Long a call option with X1 = $20; write two call options, each with X2 = $30; and long a call option with X3 = $40

Strategy B: Long a put option with X1 = $20; write two put options, each with X2 = $30; and long a put option with X3 = $40

All the options have the same maturity and they are used on the same stock. Further assume that the firm pays no dividend.

Required:

Use ONLY the put-call parity to prove that both strategies A and B have the same cost. You are NOT allowed to use any other methods such as drawing terminal payoff diagram, or setting up terminal payoff table as your proof.

Question 4

Part B: (6 marks)

Note that Part A and B are not related.

Consider an inverse floating rate coupon bond with 1 year remaining to maturity. On maturity, bondholders are expected to receive $100 face value. Coupons are paid quarterly and the current 3-mth LIBOR observed rate is 5.234% p.a. The annual coupon rate is specified as:

Annual coupon rate = 20% p.a. − 3C

where C is the annual 3-mth LIBOR rate. Assume, for simplicity, that the annual 3-mth LIBOR rate will never exceed 6.67% p.a. (so that the annual coupon rate defined above is always a positive number).

The  following  table  shows  the  current  LIBOR  continuously  compounded  rate  with  different maturities:

For example, the 1-mth LIBOR is 5.0% p.a. compounded continuously. You can treat the LIBOR rates  presented  in  table  above  as  the  discount  rates/spot  rates  with  different  maturities.

Required: What is the current price of the inverse floating rate coupon bond? Show all of your workings.

Question 5              (12 marks)

Several months ago, Buddy Inc. issued a unique fixed income security. As of today, the security is maturing in 11 months.

The security pays semi-annual interest, which is equal to X% p.a minus the 6 month LIBOR, where X% is equal to the sum of 6% p.a. and the 3-month LIBOR rate. That is, in every six months, the interest is defined as:

「6%       L3% ]     L6%

|L 2   +    4   」| −    2

where, L6 (quoted on an annual basis, in %) is the 6-month LIBOR and L3 (also quoted on an annual basis, in %) is the 3-month LIBOR. Assume that the 6-month LIBOR will never exceed 6% p.a. At maturity, the company will pay $100 as the face value of the security. Also, assume all other bonds and floating rate notes have a face value of $100, respectively.

Table 1 shows the 3-month and 6-month LIBOR rates observed in the previous months as follows: Table 1

For example, two months ago, the 3-month LIBOR rate was observed at 3.4% p.a.

Table 2 shows the predicted 3-month and 6-month LIBOR rates over the next few months: Table 2

For example, five months from today, the 3-month LIBOR rate is predicted equal to 2.4% p.a.

Table  3  shows  the  current  LIBOR  rates  (assume  continuous  compounding)  with  different maturities over the next 12 months:

Table 3

For example, the current 3-month LIBOR rate is 2.9% p.a. compounded continuously.

Required:

Calculate the current price of the security. Show all working.