Question

2. Arbitrage on the tree A stock that pays no dividends has price today of 100. In one year's time the stock is worth 110 with probability 0.75, and 85 with probability 0.25. The one-year annually compounded interest rate is 5%. a) Calculate the forward price of the stock for a forward contract with maturity one year. (b) Calculate the price of a one-year European put option with strike 100. (c) Suppose you observe that the put option in part (b) has a market p rice of 4. Determine an arbitrage portfolio and calculate how much profit is generated at time T 1 by this portfolio.

Answer 1

(a) In this question no storage cost or dividend income is given, thus, basic forward price formula will be used

Forward Price = S * e^rt

^S = spot price = 100

e = 2.71828

r = rate of interest = 5% or 0.05

t = time = 12 months or 12/12 = 1

Forward Price = 100 * 2.71828^0.05*1 = 100 * 1.05127 = 105.13

Thus, forward price of the stock for the forward contract will be 105.13

(b) The question will be solved using Binomial Option Pricing Model for one period

upper factor (u) = 1 + [((110-100)/100)*0.75] = 1 + 0.075 = 1.075

down factor (d) = 1 - [((100-85)/100)*0.25] = 1 - 0.0375 = 0.9625

  

625 0.09750.78 1+r-d_1+005-0.9 π= 1.075-0.9625 0.1125

1 - π = 1 - 0.78 = 0.22

Value of put option with strike price 100 at expiration using MAX:

(i) P+ = When price is 100 * u or 100 * 1.075 = 107.50

MAX(0,100-107.50) = 0 ; since options price cannot be negative

(ii) P- = When price is 100 * d or 100 * 0.9625 = 96.25

MAX(0,100-96.25) = 3.75

value of put option (P) =
1+Y
=
0.78(0)+0.22)(2.75) 1+0.05
= 0.79

Thus, value of put option with strike price 100 will be 0.79

(c) As per part (b), put option with strike price 100 should have value 0.79 but it is trading at 4. Thus, it is overpriced and one should sell put option at 4 as well as short stock at 100 to complete arbitrage.

The ratio of arbitrage will be (3.75+0)/(107.50-96.25) = 3.75/11.25=0.33333. It means that for selling 1 put option with strike price 100 at 4, investor should short 0.33333 of the stock. This will be the arbitrage portfolio.

At T=1, the profit generated by arbitrage portfolio in both cases will be:

(i) Expiry price = 96.25

[premium gain/loss] + [stock gain/loss]

[(4)-(100-96.25)] + [(100-96.25)(0.33333)] = [4-3.75] + [3.75*0.33333] = 0.25 + 1.25 = 1.50

(ii) Expiry price = 107.50

[premium gain/loss] + [stock gain/loss]

[4] + [(100-107.50)(0.33333)] = [4] + [-7.50*0.33333] = 4 - 2.50 = 1.50