Let's assume the usual Black Scholes assumptions hold. My question is related to an answer on this question. There, the weights (Δ1t,Δ2t) are derived which form a locally risk free portfolio Xt=Δ1tSt+Δ2tCt
Answer
Main references
As explained in my comments, the correct approach to derive the hedging portfolio would be the one described in Gordon's answers to the following questions:
The hedging portfolio Ct−(∂C/∂S)St is not self-financing
We can check that the hedging portfolio (wC,wS)=(1,−∂C/∂S) is not self-financing. Letting Xt be the portfolio value, we have − dropping time subscripts:
dX=dC+d(wSS)=dC+(SdwS+wSdS+dwSdS)
The differential of the option is:
dC=∂C∂tdt+∂C∂SμSdt+∂C∂SσSdW+12∂2C∂S2σ2S2dt
We differentiate the weight wS:
dwS=∂wS∂tdt+∂wS∂SdS+12∂2wS∂S2dS2=∂wS∂tdt+∂wS∂SμSdt+∂wS∂SσSdW+12∂2wS∂S2σ2S2dt=−(∂2C∂S∂tdt+∂2C∂S2μSdt+∂2C∂S2σSdW+12∂3C∂S3σ2S2dt)
Hence:
SdwS=−(∂2C∂S∂tSdt+∂2C∂S2μS2dt+∂2C∂S2σS2dW+12∂3C∂S3σ2S3dt)wSdS=−(∂C∂SμSdt+∂C∂SσSdW)dwSdS=−∂2C∂S2σ2S2dt
Terms cancel and we obtain:
dX=∂C∂tdt−∂2C∂S∂tSdt−∂2C∂S2μS2dt−∂2C∂S2σS2dW−12∂3C∂S3σ2S3dt−12∂2C∂S2σ2S2dt
Hence we conclude that the self-financing condition dX=dC+wSdS is not verified: indeed the term in ∂2C/∂S∂t would not appear if it was self-financing, in which case it would read:
dX=(∂C∂tdt+∂C∂SμSdt+∂C∂SσSdW+12∂2C∂S2σ2S2dt)−∂C∂S(μSdt+σSdW)=∂C∂tdt+12∂2C∂S2σ2S2dt
Independence of the pricing PDE and the option weight wC
Note that the confusion around the hedging portfolio is in part due to the fact that the pricing PDE does not depend on the weight of the option wC. As stated in your question, the correct stock weight wS is:
wS=−wC∂C∂S
Recall that after having cancelled the random terms in dXt through the choice of (wC,wS), we get:
dXt=wC(∂C∂t+12σ2S2∂2C∂S2)dt
From the risk-free return constraint, we then obtain − dropping time subscripts:
dX=rXdt⇔wC(∂C∂t+12σ2S2∂2C∂S2)dt=r(wSS+wCC)dt⇔wC(∂C∂t+12σ2S2∂2C∂S2)=rwC(−∂C∂SS+C)
Hence the derivative weight wC can be cancelled.
A note on hedging portfolios
Note the difference between our approach and the one described in your paper:
- Here, we hold a portfolio of options and stocks and we require this portfolio to return the risk-free rate;
- In your paper, we hold a portfolio made up on an option combined with stocks and riskless bonds and we require its value to be 0.
wC(t)Ct+wS(t)St=B(t)⏟(1)Our hedging portfolio⟺Ct+wS(t)St+wB(t)B(t)=0⏟(2)Your paper's hedging portfolio
The self-financing condition is different in both cases:
(1):CtdwC(t)+dwC(t)dCt+StdwS(t)+dwS(t)dSt=0(2):StdwS(t)+dwS(t)dSt+BtdwB(t)+dwB(t)dBt=0
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