How to eliminate the uncertainty or the risk

Question

Assignment:Financial Partial Differential Equations : Black-Scholes and Ito's Lemma

If dS=μSdt+ σSdX and we let ζ=1/S, then dζ=adt+bdX  , where  a=(-μ+σ²)ζ and b=σζ . Define V¯(ζ,t)=V(S,t)/S=ζV(1/ζ,t)= . Suppose that the stock pays dividends continuously:

•    D(S,t) => dividend

•    If dividend is paid continuosly:

    * D(S,t)dt = D0Sdt

    * D₀ is a constant dividend rate

Derive the equation for V¯(ζ,t) directly by using Itô’s lemma.

(Hint: Take II=V-ΔS=V¯(ζ,t)/ζ- Δ/ζ as the portfolio during the derivation)

Notes: Derivation of B-S Equation:

 V denotes the value of an option that depends on the value of the underlying asset S and time t, ie, V = V(S,t). In a time step dt, the underlying asset pays out a dividend SD₀dt, where D₀ is a constant known as the dividend yield. S satisfies dS=μSdt+σSdX.

 According to Itô's lemma, the random walk followed by V is given by:

            dV=(∂V/∂S)dS+ [∂V/∂t+(1/2 ) σ²S² (∂²V/∂S² )]dt

V has at least one t derivative and two S derivatives. Construct a portfolio consisting of one option and a number -Δ of the underlying asset. This number is not yet known. The value of this portfolio is:
    II=ΔS

Because the portfolio contains one option and a number -Δ of the underlying asset, and the owner of the portfolio receives SD0dt for every asset held, the earnings for the owner of the portfolio during the time step dt is:
    dII=dV-ΔdS-ΔSD₀dt

Using *, it is apparent that Π follows the random walk:

        dII=(∂V/∂S-Δ)dS+[∂V/∂t+(1/2 ) σ²S² (∂²V/∂S² )]dt

The random component in this random walk can be eliminated by choosing:

    Δ=∂V/∂S    

This results in a portfolio whose increment is wholly deterministic:

        dII=[∂V/∂t+(1/2 ) σ²S² (∂²V/∂S² )-ΔSD₀]dt

Because the return for any risk-free portfolio should be r,

        rIIdt=dII=[∂V/∂t+(1/2 ) σ²S² (∂²V/∂S² )-ΔSD₀]dt

Substituting ** and *** into **** and dividing by dt:

    ∂V/∂t+(1/2 ) σ²S² (∂²V/∂S² )+(r-D₀)S(∂V/∂S)-rV=0

This is the Black-Scholes partial differential equation. The key idea of deriving this equation is to eliminate the uncertainty or the risk. dII is not a differential in the usual sense. It is the earning of the holder of the portfolio during the time step dt. Therefore, ΔSD₀ appears. In the derivation, in order to eliminate any small risk, Δ is chosen before an uncertainty appears and does not depend on the coming risk. Therefore, no differential of Δ is needed.