How to Derive the Black-Scholes PDE in Simple Terms

Goal: We want to determine the fair value of a European call option on a non-dividend-paying stock by constructing a continuous hedging portfolio which prevents any arbitrage and, as a consequence, is risk-free.

 

Step 1:  We construct a hedging portfolio

 

• P = V(S, t) - ΔS   (equation 1)

 

where V is the value of our call option depending on the price of the underlying stock S and time t.

 

Step 2:  We consider the instantaneous change in our hedging portfolio value so that we can evaluate the option price from the volatility of the underlying stock at any time (continuous hedging)

• dP = dV - ΔdS   (equation 2)

Step 3: We define the instantaneous change in the price of the underlying stock

• dS = μS dt + σS dW  (equation 3)

• Drift term: μS dt, deterministic term. The drift term is the expected rate of return of the stock.
• Stochastic / diffusion term: σS dW. The diffusion term captures the random (stochastic) movements in the stock price, and W is the Brownian motion (Wiener process).
• μ and σ are, respectively, the rate of return and the volatility.
• dW captures the random and unpredictable component of price movements.

Step 4: We use Itô's lemma to find the differential of a function of a stochastic process. Itô's lemma is the stochastic analogue of the chain rule in standard differentiation

 

dV = (∂V/∂t) dt + (∂V/∂S) dS + (1/2) (∂²V/∂S²) (dS)²  (equation 4)

 

 ∂V/∂t is called "theta"

 ∂V/∂S is called "delta"

 (1/2) (∂²V/∂S²) is called "gamma"

 

Step 5: We square the (dS) term from equation (3) and get:

 

(dS)² = (μS dt + σS dW)²

       = μ²S² (dt)² + 2μσS² (dt dW) + σ²S² (dW)²

 

Note on Multiplication Rules:

 

 (dt)² = 0

 (dt dW) = 0

 (dW dW) = dt

 

Therefore:

 

(dS)² = 0 + 0 + σ²S² dt = σ²S² dt   (equation 5)

 

Step 6: We substitute the (dS)² term in equation (4) with the (dS)² term from equation (5):

 

dV = (∂V/∂t) dt + (∂V/∂S) dS + (1/2) σ²S² (∂²V/∂S²) dt

 

or simplifying:

 

dV = (∂V/∂t) dt + (∂V/∂S) dS + (1/2) σ²S² (∂²V/∂S²) dt   (equation 6)

 

Step 7: We substitute the dV term in equation (2) with the dV term from equation (6)

 

dP = dV  Δ dS

   = [(∂V/∂t) dt + (∂V/∂S) dS + (1/2) σ²S² (∂²V/∂S²) dt]  Δ dS

   = [(∂V/∂t) + (1/2) σ²S² (∂²V/∂S²)] dt + [(∂V/∂S)  Δ] dS

 

We set Δ = dV/dS so that (dV/dS - Δ)dS = 0 and we finally obtain: 

 

dΠ = (dV/dt + 1/2 * σ^2 * S^2 * d²V/dS²) dt (equation 7)

 

=> Now our continuous hedging portfolio only depends on a deterministic term and is thus insulated from the price change of the underlying stock.

 

Step 8: Since our hedging portfolio is risk-free (equation 1), we can posit:

 

dΠ = r Π dt with r = risk-free rate

 

and we substitute the Π term from equation (1) in equation (8):

 

dΠ = r (V - ΔS) dt

 

=> dΠ = r (V - (dV/dS) S) dt (from step 7)

 

=> dV/dt + 1/2 * σ^2 * S^2 * d²V/dS² = r V - r S dV/dS

 

=> dV/dt + r S dV/dS + 1/2 * σ^2 * S^2 * d²V/dS² - r V = 0 (BS PDE)

 

The Black-Scholes partial differential equation is derived: 

 

dV/dt + r S dV/dS + 1/2 * σ^2 * S^2 * d²V/dS² - r V = 0