or $27. The risk-free interest rate is 10% per annum with continuous compounding. Suppose ST is the stock price at the end of two months. What is the value of a derivative that pays off

2 at this time? ST

At the end of two months the value of the derivative will be either 529 (if the stock price is 23) or 729 (if the stock price is 27). Consider a portfolio consisting of:

???shares

?1?derivativeThe value of the portfolio is either 27??729 or 23??529 in two months. If

27??729?23??529

i.e.,

??50

the value of the portfolio is certain to be 621. For this value of ? the portfolio is therefore riskless. The current value of the portfolio is:

50?25?f

where f is the value of the derivative. Since the portfolio must earn the risk-free rate of interest

(50?25?f)e0?10?2?12?621

i.e.,

f?639?3

The value of the option is therefore $639.3.

This can also be calculated directly from equations (12.2) and (12.3). u?1?08, d?0?92 so that

e0?10?2?12?0?92p??0?6050

1?08?0?92and

f?e?0?10?2?12(0?6050?729?0?3950?529)?639?3

Problem 12.15.

Calculate u, d, and p when a binomial tree is constructed to value an option on a

foreign currency. The tree step size is one month, the domestic interest rate is 5% per annum, the foreign interest rate is 8% per annum, and the volatility is 12% per annum.

In this case a?e(0?05?0?08)?1?12?0?9975

u?e0?121?12?1?0352

d?1?u?0?9660

p?0?9975?0?9660?0?4553

1?0352?0?9660

Problem 12.16.

The volatility of a non-dividend-paying stock whose price is $78, is 30%. The risk-free

rate is 3% per annum (continuously compounded) for all maturities. Calculate values for u, d, and p when a two-month time step is used. What is the value of a four-month European call option with a strike price of $80 given by a two-step binomial tree. Suppose a trader sells 1,000 options (10 contracts). What position in the stock is necessary to hedge the trader’s position at the time of the trade?

u?e0.30?0.1667?1.1303d?1/u?0.8847

e0.30?2/12?0.8847p??0.48981.1303?0.8847The tree is given in Figure S12.3. The value of the option is $4.67. The initial delta is 9.58/(88.16 – 69.01) which is almost exactly 0.5 so that 500 shares should be purchased. 99.6519.6588.169.5878.004.6769.010.0061.050.00 78.000.00 Figure S12.3: Tree for Problem 12.16

Problem 12.17.

A stock index is currently 1,500. Its volatility is 18%. The risk-free rate is 4% per annum (continuously compounded) for all maturities and the dividend yield on the index is 2.5%. Calculate values for u, d, and p when a six-month time step is used. What is the value a 12-month American put option with a strike price of 1,480 given by a two-step binomial tree.

u?e0.18?0.5?1.1357d?1/u?0.8805

e(0.04?0.025)?0.5?0.8805p??0.49771.1357?0.8805The tree is shown in Figure S12.4. The option is exercised at the lower node at the six-month point. It is worth 78.41.

1934.840.001703.600.001500.0078.411320.73159.271162.89317.11 1500.000.00 Figure S12.4: Tree for Problem 12.17

Problem 12.18

The futures price of a commodity is $90. Use a three-step tree to value (a) a nine-month American call option with strike price $93 and (b) a nine-month American put option with strike price $93. The volatility is 28% and the risk-free rate (all maturities) is 3% with continuous compounding.

u?e0.28?0.25?1.1503d?1/u?0.8694 1?0.8694u??0.46511.1503?0.8694The tree for valuing the call is in Figure S12.5a and that for valuing the put is in Figure S12.5b. The values are 7.94 and 10.88, respectively. 136.9843.98119.0826.08103.5214.6290.007.9478.242.2468.020.0059.130.00 90.004.8678.240.00103.5210.5290.0010.8878.2416.8868.0224.98103.524.1690.007.8478.2414.76119.080.00103.520.00136.980.00

Figure S12.5a: Call

59.1333.8 7 Figure S12.5b: Put

Further Questions

Problem 12.19

The current price of a non-dividend-paying biotech stock is $140 with a volatility of 25%. The risk-free rate is 4%. For a three-month time step:

(a) What is the percentage up movement? (b) What is the percentage down movement?

(c) What is the probability of an up movement in a risk-neutral world? (d) What is the probability of a down movement in a risk-neutral world?

Use a two-step tree to value a six-month European call option and a six-month European put option. In both cases the strike price is $150.

(a) u?e0.25?0.25= 1.1331. The percentage up movement is 13.31% (b) d = 1/u = 0.8825. The percentage down movement is 11.75% (c) The probability of an up movement is (e0.04?0.25)?.8825)/(1.1331?.8825)?0.5089 (d) The probability of a down movement is0.4911.

The tree for valuing the call is in Figure S12.6a and that for valuing the put is in Figure S12.6b. The values are 7.56 and 14.58, respectively. 179.7629.76158.6415.00140.007.56123.550.00109.030.00 179.760.00158.644.86140.0014.58123.5524.96109.0340.97 140.0010.00140.000.00

Figure S12.6a: Call

Figure S12.6b: Put

Problem 12.20

In Problem 12.19, suppose that a trader sells 10,000 European call options. How many shares of the stock are needed to hedge the position for the first and second three-month period? For the second time period, consider both the case where the stock price moves up during the first period and the case where it moves down during the first period.

The delta for the first period is 15/(158.64 – 123.55) = 0.4273. The trader should take a long position in 4,273 shares. If there is an up movement the delta for the second period is 29.76/(179.76 – 140) = 0.7485. The trader should increase the holding to 7,485 shares. If there is a down movement the trader should decrease the holding to zero.