Lecture 3 : The Ito formula and SDE

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Lecture 3 The Ito formula and SDE
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Remember
This is only the most elementary form of the Ito
formula. A more general form will follow.
However, we first have to extend the Ito integral
to n-dimensions.
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The first equality above is the self-financing
condition.
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This is the Black-Scholes partial differential
equation. The replication principle implies that
the price v(t,x) of the option must satisfy
For the case of a European Call one can derive an
explicit solution of this partial differential
equation.
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Note Here B(x,t) denotes a function and not
Brownian motion !
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Under the conditions above, as solution to SDE
has the Markov property.
Relationship between SDE and PDE
i.e. the function p satisfying
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Why is the KBE important for us ?
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From this we derive the following important
result
This is left as an exercise.