Question

A share in the firm Tamar is trading in the market at £2.90 today. In six months from now its price will either increase by £1.00 or decrease by £0.75. In the following six months, it will again either increase in value by £1.00 or decrease in value by £0.75. The six-month interest rate on a risk-free asset is 1% and it will remain at that value for the whole year. Tamar will not pay any dividends during this time.
(a) Price a European put option on Tamar’s stock with one year to expiry and a strike price of £3.25.

(b) Price a 1-year American put option on Tamar’s stock with one year to expiry and a strike price of £3.25.

(c) Provide a theoretical explanation of the difference between your results for parts (a) and (b). Would you expect a European call option and an American call option on the same asset to be valued differently? Why/why not?

(d) Explain how time to maturity and volatility of the underlying asset price affect the prices of put options.
(Total 25 marks)

Answer 1

The Stock lattice and Put option (both European and American) value at expiration is as shown below :

		4.9	0
		D
	3.9
	B
2.9		3.15	0.1
A		E
	2.15
	C
		1.4	1.85
		F
t=0	t=6	t=12	Value of put at expiration

a)

For Tree at node B, u=4.9/3.9 =1.25641 , d=3.15/3.9 =0.807692

Risk neutral probability p= (1.01-0.807692)/(1.25641-0.807692) =0.450857

So, value of European Option at Node B = (0.450857*0+0.549143*0.1)/1.01 = 0.054371

For Tree at node C, u=3.15/2.15 =1.4651163 , d=1.4/2.15 =0.6511628

Risk neutral probability p= (1.01-0.6511628)/(1.4651163-0.6511628) =0.4408571

So, value of European Option at Node C = (0.440857*0.1+0.559143*1.85)/1.01 = 1.0678218

For Tree at node A, u=3.9/2.9 =1.344828 , d=2.15/2.9 =0.741379

Risk neutral probability p= (1.01-0.741379)/(1.344828-0.741379) =0.445143

So, value of European Option at Node A= (0.445143*0.054371+0.559143*1.0678218)/1.01 = 0.6106

So, value of European Option is £0.61

b) In case of American option, we have to also calculate the value in case of early exercise at each node and then take the higher value . Calculated value is the same as above if value at nodes ahead do not change

So, at node B, exercise value = max(3.25-3.9,0) = 0

So, american option value = max(exercise value, calculated value) = 0.054371

At node C, exercise value = max(3.25-2.15,0) = 1.1

So, american option value = max(exercise value, calculated value) = 1.1

value of American Option at Node A= max ( (0.445143*0.054371+0.559143*1.1)/1.01 , 3.25-2.9) = 0.6283

So, value of American Option is £0.63

c) The American Put option can be exercised at any point during the maturity and hence has a greater value as it gives the flexibility of early exercise and it is sometimes adantageous to early exercise the option (like at Node C) . Thus its value is more than the European Put option

It is never optimal to early exercise an American Call option on a non-dividend paying (during the maturity) stock , hence its value is expected to be the same as that of the European Call option. This is because if the stock is not to be carried, due to time value, the option has more value than if it is exercised, hence it is better to sell the option rather than exercising and then selling the stock & in case the stock is to be carried forward, paying early does not make sense as you can pay the same amount later.

d) Longer maturity increases the prices of put options . This happens because there is more time for the stock price to move up and down and hence , it is more likely that at some point in time, the stock price will decrease. Higher volatility also increases the prices of Put options as stock prices move up and down more , thus creating more value to option holder. .