**Question**:

**Answer**:

**(a)** Let *r* be the single period interest rate. The models consists of set Ω = {ω+, ω-} with probability measure,

P({ω+}) = *p* = ½, P({ω-}) = 1 – *p* = ½

The stock price can attain two values S11(ω+) = 120 and S11(ω-) = 90 at *t* = 1

For model to be arbitrage free,

S10 = EP [S11] / (1 + *r*) (1)

Where, EP [S11] = *p* * S11(ω+) + (1 – *p*) * S11(ω-)

= ½ * 120 + ½ * 90

= 105 (2)

Therefore, S10 = 100 = 105 / (1 + *r*)

Þ *r* = 5%

**(b)** Let *C* be a European Call option, and πC is the arbitrage-free price of *C* at *t* = 0, then we can define an asset S2 by

S20 = πC and S21 = C

Let us consider a portfolio ** ξ**, where

Here, S21(ω+) = C(ω+) = (S11(ω+) – K)+ = 120 – 100 = 20

S21(ω-) = C(ω-) = (S11(ω-) – K)+ = (90 – 100)+ = 0 (4)

Since the portfolio is risk free, *V*1(ω+) = *V*1(ω-) (5)

Þ 120 ξ1 + 20 ξ2 = 90 ξ1 (Using (3))

Þ ξ1 = - ⅔ ξ2

Keeping ξ2 = -1, we get ξ1 = ⅔

Hence, the portfolio ** ξ** = (⅔ , -1) (6)

To calculate the initial investment, we have, *V*0 = ξ1 S10 + ξ2 S20 (7)

And for risk free portfolio, *V*1 = *V*0 (1 + *r*) (8)

Here, *V*1 = *p* *V*1(ω+) + (1-*p*) *V*1(ω-) = *V*1(ω+) (From (5))

Hence, (⅔ x 100 - πC) (1 + *r*) = ⅔ x 120 – 20 (Using (4), (6), (7) and (8))

Þ (⅔ x 100 - πC) = 60 / 1.05 = 57.1428

Þ πC = 9.5238

**(c)** The expected value of call option *C* at *t* = 1 is given by

EP [*C*] = EP [S21] = *p* * S21(ω+) + (1 – *p*) * S21(ω-)

Þ EP [*C*] = EP [S21] = 10 (Using (4)) (9)

Hence, EP [*C* / (1 + *r*)] = 10/(1 + *r*) = 10/1.05 = 9.5238 = πC

Hence prooved.

**(d)** When *C* = EP [*C* / (1 + *r*)] = 9.5238 under the original measure P, the option is under-priced and arbitrage opportunities exist.

One can borrow an amount equal to 9.5238/ (1 + *r*) = 9.07 at *t* = 0 (so that he pays back 9.5238 at *t* = 1) and buy such an option.

The value of the Option at *t* = 1 is C = 10, leading to a profit of 10 – 9.5238 = 0.4762 at *t*=1

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