Question

The stock price is currently $70. It is known that at the end of three months it will be either $72 or $68. The risk-free interest rate is 10% per annum with continuously compounding.

1. What is the value of a three-month European call option with a strike price of $70 using the no-arbitrage argument?

2. What is the value of a three-month European call option with a strike price of $70 using the risk-neutral valuation?

Answer 1

With the no-arbitrage approach, the call option value is calculated using following equation

c = hS + PV(-hS- + c-)

Where, h is the hedge ratio

S is the current stock price

S- is the expected price if goes down

c- is the expected call option value if stock price goes down

c- = Max(0,S--X) = Max(0,68-70) =0

c+ = Max(0,S+-X) = Max(0,72-70) = 2

h = (c+- c-)/(S+-S-) = 2-0 / 72-68 = 0.5

Continously compounded present value factor = 1/ert = 1/e0.10*(3/12) = 0.97531

c = 0.5*70 +0.97531*(-0.5*68 + 0)

c = $1.8395

With the risk neutral valuation, the call option is calculated using the following equation

c = PV[*c+ + (1-)*c-]

Where, is risk neutral probability

= (FV(1)-d)/u-d

Where, d is down factor

u is up factor

d = S-/S = 68/70 = 0.971429

u = S+/S = 72/70 = 1.0285714

FV(1) =1*ert = 1.025315

= (1.025315-0.971429)/(1.0285714-0.971429)

= 0.94301

c = 0.97531[0.94301*2 + (1-0.94301)*0]

= 0.97531*(1.88602)

= $1.8395

It can be seen that the price of option remains same regardless of valuation approach used.