Question

2. Consider the N-step binomial asset pricing model with 0 < d<1< u (a) Assume N-3. Sİ,-100, r-0.05, u-1.10, and d-0.90. Calculate the price at time (b) If the observed market price of the option in part (a) is $25 give a specific arbitrage trading (c) Suppose you wish to earn a profit of $100,000 from implementing your arbitrage trading zero, VO, of the European call-option with strike price K = 87.00. strategy to take advantage of any potential mis-pricing. strategy from part (b). What are the sizes of your posion in the asset, option, and cash (bank account at tme zero?

Answer 1

a.

Factors of Option Binomial Model –

There are three parameters of Option Binomial Pricing Model

• up factor (u)
• down factor (d)
• probability (P)

up factor and down factor used to calculate rise in price and fall in price of underlying assets in one period. Probability is measure probability of rise in price and (1-P) is probability of price fall.

As per Risk-Neutral Probability

$P = \frac{e^{rt}-d}{u-d}$

where,

r = rate of interest

t= time in each period

d = Price down factor

u = Price up factor

P = Probability of Price going up

(1-P) = probability of Price going down

We have following information -

Step (N) = 3

Spot Price of Stock (S0) = 100

Strike Price (K) = 87

r = 0.05

u = 1.1

d = 0.9

For calculation please refer to below spread sheet.

B27 ▲ A Three Period Binomial Price tree Binomial Parameters Initial Stock Price (S1) Strike Price (K) Maturity One Period(t) risk free interest rate 1.1 0.9 0.756355482 0.243644518 1.051271096 100 Lu 87 3 months 1month (1-P) 5% Stock Price 10 0 Call Payoff at 3 133.10 $ 46.10 121.00 110.00 108.90$ 21.90 100.00 99.00 89.10 $ 16 17 90.00 2.10 81.00 72.90 $ 19 20 Value of Europeon Call 0 23 46.10 25 38.24 31.28 21.90 27 28 29 25.29 16.24 12.04 2.10 1.51

Formula Reference-

Value of European Call (V0) = $ 25.29

b.

European call option under binomial pricing theorem is $ 25.29 and current market price of same option is $ 25. This shows current market price of call option is not in equilibrium. Hence, There is arbitrage opportunity exist. Arbitrage an opportunity where arbitrager earn certain profit without making any investment and taking risk.

Arbitrage Strategy in above case -

In above case, Market Price of Call option is under-priced. Hence, to gain profit arbitrage follow below strategy

Today-

• Sale the stock at Spot price i.e $ 100
• Buy Call option at Market Price i.e $ 25
• Deposit Proceed from sale from stock to Bank at risk free interest rate i.e 0.05 & t=3

On Maturity-

• withdraw money from bank with interest
• Exercise the call option and buy the stock at strike price i.e 87
• Arbitrage Profit = Strike Price - Proceed from bank at maturity

C.

To earn $ 100,000 arbitrage Profit - Position of Assets,call option,cash

It is assumed that call option has size of 1 stock.

Firstly, calculate the arbitrage profit for one call option-

For one call option premium to be paid = $ 25

Sale Proceed from selling one stock at spot price = $ 100

After paying call premium from sale proceed deposit remaining amount to bank = $ 75

Interest factor (0.05,3) = ert = e0.05*3 = 1.1618

Amount received from bank at the end of period = $ 75 * 1.1618 = $ 87.135

Exercise the call option and pay to buy stock i.e strike price = $ 87

Remaining Amount would be Arbitrage Profit = 87.135 - 87 = $ 0.135

Thus, Size of Position to earn $ 100,000 arbitrage - as under

Long Call option = 100,000/0.135 = 740,741 (rounded)

Short Stock = 740,741

Cash in Bank = (740,741*100) - (740,741*25)

= 74,074,100 - 18,518,525

= $ 55,555,575

With above position, Arbitrage profit would be -

= (55,555,575*1.1618) - (740,741*87)

= 64,544,467 - 64,444,467

= $ 100,000