Thursday, March 3, 2016

black scholes - Dynamic Delta Hedging And a Self Financing Portfolio


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

with Δ1t=CSBtCtCSS,Δ2t=BtCtCSS
It is emphasized that the strategy (CS,1) is not self financing. I have no doubt about the derivation. Rather, I'm interested to know, in the context of dynamic delta hedging, is it actually valid to set Δt (the amount of the underlying to buy or sell) to CS instead of Δ1t as shown above? For example, in this paper on page 4 the authors investigate delta hedging strategies by setting Δ1t to CS (e.g. N(d1) for a European call option) and explicitly claim the self financing property. So how would one actutally compute the correct Δt for a delta hedging strategy in a BS world?



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=Ctdt+CSμSdt+CSσSdW+122CS2σ2S2dt


We differentiate the weight wS:


dwS=wStdt+wSSdS+122wSS2dS2=wStdt+wSSμSdt+wSSσSdW+122wSS2σ2S2dt=(2CStdt+2CS2μSdt+2CS2σSdW+123CS3σ2S2dt)


Hence:


SdwS=(2CStSdt+2CS2μS2dt+2CS2σS2dW+123CS3σ2S3dt)wSdS=(CSμSdt+CSσSdW)dwSdS=2CS2σ2S2dt



Terms cancel and we obtain:


dX=Ctdt2CStSdt2CS2μS2dt2CS2σS2dW123CS3σ2S3dt122CS2σ2S2dt


Hence we conclude that the self-financing condition dX=dC+wSdS is not verified: indeed the term in 2C/St would not appear if it was self-financing, in which case it would read:


dX=(Ctdt+CSμSdt+CSσSdW+122CS2σ2S2dt)CS(μSdt+σSdW)=Ctdt+122CS2σ2S2dt


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=wCCS


Recall that after having cancelled the random terms in dXt through the choice of (wC,wS), we get:


dXt=wC(Ct+12σ2S22CS2)dt


From the risk-free return constraint, we then obtain dropping time subscripts:



dX=rXdtwC(Ct+12σ2S22CS2)dt=r(wSS+wCC)dtwC(Ct+12σ2S22CS2)=rwC(CSS+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 portfolioCt+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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