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Instantaneous change in the risk-free portfolio

J

Jackie_You

Member
Hi,
When we are using PED approach to derive the Black-Scholes option pricing formula, we have a risk-free portfolio with minus one derivative, and plus delta shares. Why does it say that this portfolio strategy is not self-financing? What's the value of dV(t,St) if it is not -df(t,St) + delta*dSt?

Thanks
 
I think it's because the dV also includes the change in the number of shares i.e. (d(delta)/dt)*s(t) which is not a part of the instantaneous investment gain because you would need a cashflow in/out of the portfolio to change the number of shares you hold.
 
That's right. If you worked out the value of delta at a particular point in time and then fixed the strategy to be holding that fixed number of shares, it would be self-financing, but not if we want the number of shares to be the varying value of delta as it changes over time.
 
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