(a) The arbitrage bounds for the interest rate are 0.5 (1 + 1.5. r r) ≤ ≤ (b) Both the price of stock and the price of the option have to satisfy the risk-neutral valuation principle. This gives us the following set of equations Solving these equations we find that = 5% (and q = 0.55). r The Black-Scholes PDE is given by: or i.e. (a) See Lecture Notes (b) The value process is given by: V(t) = + V(t) hB(t)B(t) hS(t)S(t) To be a self-financing portfolio strategy, the process must satisfy V = + dV(t) hB(t)dB(t) hS(t)dS(t) In order for the relative portfolios, the value process must satisfy V where To solve the above SDE we use = and define = ln(V) and use Itô: V0 v0 Z Integrating gives and We also know that The portfolio is now easily found to be (B = e-rt): and
(a) The arbitrage bounds for the interest rate r are 0.5 ≤ (1 + r) ≤ 1.5. (b) Both the price of stock and the price of the option have to satisfy the risk-neutral valuation principle. This gives us the following set of equations Solving these equations we find that r = 5% (and q = 0.55). The Black-Scholes PDE is given by: or i.e. (a) See Lecture Notes (b) The value process V(t) is given by: V(t) = hB(t)B(t) + hS(t)S(t) To be a self-financing portfolio strategy, the process V must satisfy dV(t) = hB(t)dB(t) + hS(t)dS(t) In order for the relative portfolios, the value process V must satisfy where To solve the above SDE we use V0 = v0 and define Z = ln(V) and use Itô: Integrating gives and We also know that The portfolio is now easily found to be (B = e-rt): and
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