Question

Problem 1: Properties of Options

The price of a European put that expires in six months and has a strike price of $100 is $3.59. The underlying stock price is $102, and a dividend of $1.50 is expected in four months. The term structure is flat, with all risk-free interest rates being 8% (cont. comp.) - Solutions explained please

1. What is the price of a European call option on the same stock that expires in six months and has a strike price of $100?
2. Explain in detail the arbitrage opportunities if the European call price is $6.1. How much will be the arbitrage profit?
3. Explain in detail the arbitrage opportunities if the European call price is $8.8. How much will be the arbitrage profit?

Answer 1

a. Put – Call parity

c + X * e ^–rT = (S – PV(D) ) + p

Given Information

Stock Price (S) = $102

Exercise Price (X) = $100

Risk free interest rate (r) = 8%

Put price (p) = $3.59

Maturity term (T) = 6 months

Dividend (D) = $1.5 in 4 months (t)

Present Value of Dividend = PV(D) = D * e ^–rt = 1.5 * e ^(-0.08*4/12) = $1.461

So, c = (S – PV(D) ) + p – (X * e ^–rT )

c = (102 – 1.461) + 3.59 – (100 * e ^(-0.08*6/12) ) = $8.05

b. Price of call, c = (S – PV(D) ) + p – (X * e ^–rT )

Now, if the call price is $6.1 which is lower than $8.05 (put-call parity determined price)

Arbitrage opportunity is to –

- Buy call

- Sell Stock, sell put and invest amount = (X * e ^–rT ) = $96.08

The cash flows are as follows –

At T=0	At T=6 months (if S <100)	At T= 6 months (if S >100)
Transaction	Cash Flow	Transaction	Cash Flow	Transaction	Cash Flow
Buy Call	-6.1	Call unexercised	0	Exercise call and buy stock	-100
Lend (buy bond) (X * e –rT ) = $96.08 at risk free rate	-96.08	Proceeds from bond	100	Proceeds from bond	100
Short Sell Stock* @ ( (S-PV(D) =100.54)	+100.54	Settle short by delivering stock obtained from put	0	Settle short by delivering stock obtained from call	0
Sell put	+3.59	Put exercised obligation to buy stock	-100	Put unexercised	0
Net Cash Flow	+1.95	Net Cash Flow	0	Net Cash Flow	0

*Note – Here taking short position in stock means that investor is not entitled to the dividend being paid at end of 4 months. So, the stock price to short sell today is reduced by present value of the dividend received

So, there is an arbitrage profit of $1.95

c. Price of call, c = (S – PV(D) ) + p – (X * e ^–rT )

Now, if the call price is $8.8 which is higher than $8.05 (put-call parity determined price)

Arbitrage opportunity is to –

- Sell call

- Buy Stock, buy put and borrow amount = (X * e ^–rT ) = $96.08

The cash flows are as follows –

At T=0	At T=6 months (if S <100)	At T= 6 months (if S >100)
Transaction	Cash Flow	Transaction	Cash Flow	Transaction	Cash Flow
Sell Call	8.8	Call unexercised	0	Call exercised and stock sold	100
Borrow (X * e –rT ) = $96.08 at risk free rate	96.08	Repay loan	-100	Repay loan	-100
Buy Stock* @ ( (S-PV(D) =100.54)	-100.54	Sell stock through exercising put	0	Sell stock through exercising call	0
Buy put	-3.59	Exercise put and sell the stock	100	Put unexercised	0
Net Cash Flow	+0.75	Net Cash Flow	0	Net Cash Flow	0

*Note - Here taking long position in stock means that investor is entitled to the dividend being paid at end of 4 months. So, the stock price to buy today is reduced by present value of the dividend received

So, there is an arbitrage profit of $0.75