a.
Factors of Option Binomial Model –
There are three parameters of Option Binomial Pricing Model
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
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.
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-
On 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
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