 Solution

ID: past-exam-of-the-mathematical-models-of-financial-derivatives-course-of-the-university-of-oxford/2015/1/b/solution

Δ

shares and a current cash amount

b

in the bank account. Matching the two next values gives

Δ S_{t + 1}^{u} + R b = V_{t + 1}^{u}, Δ S_{t + 1}^{d} + R b = V_{t + 1}^{d} .

Solving this linear system,

Δ = \frac{V _{t + 1}^{u} - V _{t + 1}^{d}}{S _{t + 1}^{u} - S _{t + 1}^{d}}, b = \frac{S _{t + 1}^{u} V _{t + 1}^{d} - S _{t + 1}^{d} V _{t + 1}^{u}}{R ( S _{t + 1}^{u} - S _{t + 1}^{d} )} .

By no-arbitrage, the claim costs

V_{t} = Δ S_{t} + b

V_{t} = R^{- 1} [q V_{t + 1}^{u} + (1 - q) V_{t + 1}^{d}], q = \frac{R S _{t} - S _{t + 1}^{d}}{S _{t + 1}^{u} - S _{t + 1}^{d}} .

The physical probability

p

does not appear: replication fixes the price of this contingent claim from traded prices.

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