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.