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Deriving Black–Scholes From a Self-Financing Replicating Portfolio

Original question

Can you help me derive the Black-Scholes partial differential equation using a self-financing delta-hedged portfolio?
Hold one option, short its delta in shares, and finance any rebalancing through a cash account. The stock diffusion cancels instantaneously. Equate the resulting hedge gain, including dividends on the short position, to the interest on the hedged position. For continuous dividend yield qq, the result is Vt+12σ2S2VSS+(r−q)SVS−rV=0V_t+\tfrac12\sigma^2S^2V_{SS}+(r-q)SV_S-rV=0.

Write the gains equation for a self-financing hedge

Let BtB_t denote the value of the cash leg and define total portfolio value Ht=Vtprice−ΔtSt+BtH_t=V_t^{\mathrm{price}}-\Delta_t S_t+B_t. Here Vtprice=V(t,St)V_t^{\mathrm{price}}=V(t,S_t) while VtV_t below is a partial derivative. For a continuous dividend yield qq, the self-financing gain is
dH=dV−Δ(dS+qS dt)+rB dt.dH=dV-\Delta(dS+qS\,dt)+rB\,dt.
With dS=μS dt+σS dWdS=\mu S\,dt+\sigma S\,dW, Itô’s lemma gives
dV=Vt dt+VS dS+12σ2S2VSS dt.dV=V_t\,dt+V_S\,dS+\tfrac12\sigma^2S^2V_{SS}\,dt.

Set Δ=VS\Delta=V_S. The combined option and stock gain now contains only a drift, and the locally riskless portfolio earns rH dtrH\,dt. Substitute H=V−ΔS+BH=V-\Delta S+B and cancel rBrB on both sides to obtain the stated PDE. If you expand the product d(ΔS)d(\Delta S) directly, the additional rebalancing terms have opposite signs in the cash-account differential. The self-financing gains equation captures their net cancellation.

Differentiate the option as a function of time and spot, then subtract the gains of the stock holding, including its dividends. The stock-short quantity is chosen so that the Brownian terms cancel. When delta changes, financing that trade transfers value between the stock and cash legs. This is why you use the gains equation for a self-financing strategy instead of simply differentiating V−ΔSV-\Delta S and treating every resulting term as profit. Once the diffusion is cancelled, compare the remaining local return with what the net portfolio value earns at the risk-free rate.

Read the dividend and funding terms in a numerical check

Suppose a claim has S=90S=90, V=8V=8, delta 0.400.40, and gamma 0.0250.025 per dollar. Use annual parameters r=4%r=4\%, q=2%q=2\%, and σ=30%\sigma=30\%. The option and short stock together are worth 8−36=−288-36=-28. Their required local riskless return is 0.04(−28)=−1.120.04(-28)=-1.12 dollars per year. The curvature term is 12(0.30)2(90)2(0.025)=9.1125\tfrac12(0.30)^2(90)^2(0.025)=9.1125, while the short position pays dividends at 0.40(0.02)(90)=0.720.40(0.02)(90)=0.72 per year. Equating the gain gives Vt+9.1125−0.72=−1.12V_t+9.1125-0.72=-1.12, so Vt=−9.5125V_t=-9.5125 per year. Adding any cash balance changes both sides by its interest accrual and leaves this result unchanged.
Use the options pricer to see how spot and volatility affect the option value the hedge is trying to replicate.

Interactive example

Move volatility and expiry in Black–Scholes

This European call uses strike 100, a continuously compounded risk-free rate of 5% and zero dividends. The benchmark spot is 100.

Price at spot 100 is 10.45 · delta 0.637
Call prices across spot prices compared with the expiry payoff01020307085100115130Spot priceValue
  • Call price today
  • Expiry payoff

References and next steps