Question

A stock's current price on the London Stock Exchange is £8. There are two possible prices at the end of the year: £11 or £8. A call option to buy one share at £10 at the end of the year sells at £2. Suppose that you are told that a riskless portfolio can be created by selling 3 call options, buying 2 stocks today and borrowing £10 today.

(a) What is the risk free interest rate?

(b) Is the interest rate realistic in the UK? What values in the introduction of the question could explain such an interest rate?

Answer 1

All the financials below are in  £

Sl. No.	Item in portfolio	Cash flow today	Payoff at the end of year in Up state	Payoff at the end of year in Down state
1.	Sell 3 call options	= 3 x Price of call option = 3 x 2 = 6	= - 3 x max (Su - K, 0) = - 3 x max (11 - 10, 0) = - 3	=- 3 x max (Sd - K, 0) = - 3 x max (8 - 10, 0) = 0
2.	Buy 2 stocks	= - 2 x S0 = - 2 x 8 = - 16	= 2 x Su = 2 x 11 = 22	= 2 x Sd = 2 x 8 = 16
3.	Borrow 10	= 10	= - 10 x (1 + rf)	= - 10 x (1 + rf)
	Total	0	19 - 10 x (1 + rf)	16 - 10 x (1 + rf)

Part (a)

If it's a riskless portfolio, then in either state it's total Payoff should be positive. Hence,

19 - 10 x (1 + r_f) ≥ 0 ; Hence, r_f ≤ 19 / 10 - 1 = 90%

and

16 - 10 x (1 + r_f) ≥ 0; ; Hence, r_f ≤ 16 / 10 - 1 = 60%

Based on the two limits, the solution is: The risk free rate, r_f ≤ 60%

Part (b)

Needless to say, such an interest rate is unrealistic in any economy, leave aside UK.

There are two indicators at the beginning of the question that could explain such a high interest rate scenario:

1. The call option at strike price of 10 is out of money call option. Still it carries a price of 2. Such a price is very high for an out of money call option.

2. The current stock price is 8 and it's expected to remain at 8 or grow to 11. The capital markets is on uptick. There is no downside from here. This is also possible only in an economy with very high risk free rate.