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QUESTION

Calculate the delta values for the followind two exact solutions
of the Black-Scholes equation:

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\item[(a)]
$V(S,t)=AS$

\item[(b)]
$V(S,t)=A\exp(rt)$

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Comment on the associated trading strategies.


ANSWER

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\item[(a)]
$V(S,t)=AS,\ A=$const. (Asset only portfolio)

This solves Black-Scholes since:

$\frac{\partial v}{\partial t}=0,\ \frac{\partial V}{\partial
S}=A,\ \frac{\partial^2V}{\partial S^2}=0$

Therefore in Black-Scholes: $\frac{\partial V}{\partial
t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2V}{\partial
S^2}+rS\frac{\partial v}{\partial S}-rV=0$

LHS=$0+\frac{1}{2}\sigma^2S^2\times 0+rSA-rAS=0=$RHS

$\Delta=\frac{\partial V}{\partial S}=A\equiv$ amount of
underlying asset at each point in time in portfolio: obvious from
value $V$.

\item[(b)]
$V(S,t)=Ae^{rt},\ A=$const. (Risk free solution).

$\frac{\partial V}{\partial t}=Are^{rt},\ \frac{\partial
v}{\partial S}=0,\ \frac{\partial^2v}{\partial S^2}=0$

Therefore in Black-scholes

LHS$=Are^{rt}+\frac{1}{2}\sigma^2S^2\times0+rS0-rAe^{rt}=0=$RHS

$\Delta=\frac{\partial V}{\partial S}=0\rightarrow$ as risk free
solution, no risky assets held.

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