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# Using the Black Scholes Equation for Prediction

I am finding it hard to do these question, attached as picture. I do not get how to start the questions.

#### Solution Preview

The Black-Scholes Equation needs to be satisfied as shown in (1)

dv/dt + ½*(SIGMA)^2*S^2*d2V/dS2 + rS*dV/dS - rV = 0 (1)

We are given a form of solution as

V(S,t) = a*S^n*exp{mt} (2)

Find: dV/dt = ma*S^n *exp{mt} (3)

Find: dV/dS = na*S^(n -1) *exp{mt} (4)

Hence: d2V/dS2 = n(n-1)a*S^(n -2)*exp{mt} (5)

Putting (2), (3), (4) and (5) into (1) we get

ma*S^n*exp{mt} + 0.5*n(n-1)(sigma)^2*S^2*a*S^(n -2)*exp{mt} + nrSa*S^(n -1)*exp{mt} - raS^n *exp{mt} = 0 (6)

Taking a common factor of aS^n*exp{mt }out of (6) we get

{aS^n*exp(mt) }*{m + 0.5*n(n-1)&#963;^2 + nr - r} = 0

So

m + 0.5*n(n-1)**sigma)^2 + nr - r = 0

Therefore

m = - r(n-1) - 0.5*n(n-1)*sigma^2

m = (1 - n)*{r + 0.5*(sigma)^2 }

Boundary conditions for European Call option

Denote a call option by C(S,t)

- Boundary conditions are:

S = 0, C(0,t) = 0 for all t

S tends to infinity, C( S,t) tends to S

For t = T C(S,T) = max(S - E, 0)

- Explore the boundary conditions and ...

#### Solution Summary

Using the Black Scholes Equation and proving solution with boundary conditions

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