Question

Suppose you are given the following data:

2-month option on XYZ stock:

Underlying S = 48.1

Strike X = 50

Put price = $2.2

1. What should be the price of call to prevent arbitrage if 2-month interest rate is 6% p.a.?
2. If the actual call price was $1.3 how would you implement an arbitrage opportunity?
3. Compute your payoff at maturity.

Answer 1

a)

As per Put Call Parity, the prices of options with same strike price & expiry date are as follows:

Price of Call + PV of Exercise Price = Spot Price (Current Stock Price) + Price of Put

Interest Rate is assumed as continuous compounding

C + [50*(e^-0.06*2/12)] = 48.1 + 2.2

C + [50*(e^-0.01)] = 50.3

C + [50*0.99(from table)] = 50.3

Therefore, Price of Call = C = 50.3 – 49.5 = $0.8

b)

Actual Price of Call > Theoretical Price. Therefore, Call is Overvalued.

Arbitrage Strategy:

As Call is Overvalued, Sell Call Option, Borrow, Buy Stocks Now and Buy Put Option

c)

Steps for Arbitrage:

Now,

(1) Sell(Write) Call Option at $1.3.

(2) Borrow (Cost of Put + Current Spot Price - Inflow from Call) = (2.2+48.1-1.3) = $49 for 2 months @6%

(3) Buy shares @ $48.1.

(4) Buy Put Option at $2.2.

Balance = 1.3+49-48.1-2.2 = 0

After 2 months,

Case 1: If Stock Price is less than $50, then Exercise Put and Lapse Call. Stock will be sold for $50 under Put contracts.

Case 2: If Stock Price is greater than $50, then Exercise Call and Lapse Put. Stock will be sold for $50 under Call contracts.

Case 3: If Stock Price is equal to $50, then Both Lapse. Stock will be sold for $50 in Market

Therefore, In any Case, we will be able to sell the stock for $50

(5) Sell share @ $50.

(6) Repay the loan along with interest i.e. 49*e^0.01 = 49*1.0101(from table) = 49.4949

Balance = Arbitrage Gain = 50 – 49.4949 = $0.5051