Deriving the Black-Scholes Equation

When we have tools like Ito’s Lemma and the Geometric Brownian Motion (GBM) model, we can now start proving the Black-Scholes equation.

1. Start by defining the variables.

Assume that:

• S(t): Price of the underlying asset
• V(t,S): Value of the contract (e.g., European call option)
• T: The expiration time of the option
• r: risk-free rate
• μ: Asset growth rate
• σ: Asset volatility

Where the asset price follows a GBM:

dS=μSdt+σSdB

And we want to evaluate V(t,S)

2. Use Ito’s Lemma with V(t,S)

Since V is a function of both t and S, we use Ito's Lemma:

dV = ∂t∂V dt + ∂S∂V dS + 1/2 ∂S²∂²V (dS)²2

Substituting dS from GBM:

dV=∂t∂V​dt+∂S∂V​(μSdt+σSdB)+21​∂S2∂2V​σ2S2dt

Rearranging:

dV=(∂t∂V​+μS∂S∂V​+21​σ2S2∂S2∂2V​)dt+σS∂S∂V​dB

3. Create a Delta-Hedged Portfolio

Create a portfolio:

Π=V−ΔS

The change in the portfolio:

dΠ=dV−ΔdS

Substituting the values from above:

dΠ=(∂t∂V​+μS∂S∂V​+21​σ2S2∂S2∂2V​)dt+σS∂S∂V​dB−Δ(μSdt+σSdB)

Rearranging:

dΠ=(∂t∂V​+μS∂S∂V​−ΔμS+21​σ2S2∂S2∂2V​)dt+(σS∂S∂V​−ΔσS)dB

To eliminate risk (hedge randomness):

Set Δ=∂S/∂V

It will be:

dΠ=(∂t∂V​+21​σ2S2∂S2∂2V​)dt

4. Use the No-Arbitrage principle

If the portfolio is risk-free, it should grow at a rate of r:

dΠ=rΠdt=r(V−ΔS)dt=r(V−S∂S∂V​)dt

Pair the dt terms on both sides:

∂t∂V​+21​σ2S2∂S2∂2V​=r(V−S∂S∂V​)

Move to the other side:

∂t∂V​+rS∂S∂V​+21​σ2S2∂S2∂2V​−rV=0

This is the Black-Scholes equation:

∂t∂V​+rS∂S∂V​+21​σ2S2∂S2∂2V​−rV=0

5. Payoff Function for Call Option

To solve this PDE, boundary conditions are required, such as:

European Call Option with strike K

Payoff at t=T: V(T,S)=max⁡(S−K,0)V(T, S) = \max(S - K, 0)V(T,S)=max(S−K,0)

From here, we can use the method of transforming the PDE and solving it analytically to obtain the closed-form Black-Scholes formula.

Reference: Deriving the Black-Scholes Equation

From https://www.quantstart.com/articles/Deriving-the-Black-Scholes-Equation/