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 = E_P [S¹₁] / (1 + r)                                                                                                  (1)

 

Where, E_P [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 π_C is the arbitrage-free price of C at t = 0, then we can define an asset S² by

S²₀ = π_C and S²₁ = C

Let us consider a portfolio ξ, where

ξ = (ξ¹, ξ²) and S₀ = (S¹₀, S²₀) and V(ω) = ξ  . S₀                                                                (3)

 

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, V₁(ω₊) = V₁(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, V₀ = ξ¹ S¹₀ + ξ² S²₀                               (7)

And for risk free portfolio, V₁ = V₀ (1 + r)                                                                 (8)

 

Here, V₁ = p V₁(ω₊) + (1-p) V₁(ω_-) = V₁(ω₊)                           (From (5))

 

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

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

       Þ π_C = 9.5238

 

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

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

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

 

Hence, E_P [C / (1 + r)] = 10/(1 + r) = 10/1.05 = 9.5238 = π_C

Hence prooved.

 

 

(d) When C = E_P [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