In Chapter 13, we derived the BSM PDE. Now we need to solve it. The call option price is not a single number — it is a surface, a landscape c(t, x) that depends on two coordinates: how much time remains and where the stock price currently sits. As time passes and the stock moves, you travel across this surface.
The option price surface: rising with stock price (Delta), sinking with time (Theta), curved upward (Gamma). The red dashed line is the expiry payoff.
The Itô-Doeblin formula tells us how this surface vibrates as S(t) moves. From Chapter 13, once we set Δ(t) = cₓ and cancel the dW terms, the surface is constrained by the BSM PDE. Now we solve that PDE to get the explicit formula for c(t, x).
The most important operational concept in options trading is delta hedging. When you sell an option, you take on risk. Delta hedging is the continuous act of buying or selling shares to neutralise that risk, moment by moment.
Whether the stock moves up or down, the delta-neutral portfolio has zero net change — the dW term is eliminated entirely.
Delta changes as the stock price moves. A deep in-the-money option has Delta ≈ 1 (moves dollar-for-dollar with the stock). An at-the-money option has Delta ≈ 0.5. A deep out-of-the-money option has Delta ≈ 0. A delta-hedger must rebalance continuously — every time Delta changes, the share position must adjust. This continuous rebalancing is exactly the Itô integral from Chapter 10.
After the dW terms cancel through delta hedging, the no-arbitrage argument forces the remaining deterministic terms to equal the risk-free rate. The result is the famous PDE that the option price c(t, x) must satisfy at every point on the surface.
The PDE is a balance sheet equation. The left side is the total return from holding the option (time decay + price gain + curvature gain). The right side is the return from a risk-free investment. No-arbitrage forces them to be equal. This is the Gamma-Theta trade-off: options buyers pay Theta daily but collect Gamma profits from volatility. Options sellers collect Theta but bleed Gamma.
The Gamma term ½σ²x²cₓₓ is the mathematical reason options have intrinsic value beyond their payoff. Because the option payoff is convex (curved upward), volatility is always beneficial to the holder — every up-down wiggle earns a tiny profit from the curvature. This is Jensen's Inequality applied to finance: the average of a convex function exceeds the function of the average.
The BSM PDE is a partial differential equation with boundary condition c(T, x) = max(x − K, 0). Solving it — using the heat equation transformation — produces the closed-form Black-Scholes formula. This is the explicit answer: plug in five numbers and get the fair price of a European call option.
The formula is: (what you expect to receive from the stock) minus (what you expect to pay for the strike). Both weighted by the probability of the option being in the money.
The BSM formula is a function of five inputs. The Greeks measure how the option price changes when each input changes by a small amount. They are the dials on every options trader's dashboard — and they map directly onto the geometry of the option price surface.
| Greek | Symbol | Measures | Surface view | Practical meaning |
|---|---|---|---|---|
| Δ | Delta | ∂c/∂x — price sensitivity | Slope in price direction | Shares needed to hedge. Ranges 0 to 1. |
| Γ | Gamma | ∂²c/∂x² — delta's rate of change | Curvature of surface | How fast you must rebalance. Highest at-the-money. |
| Θ | Theta | ∂c/∂t — time sensitivity | How surface sinks over time | Daily P&L erosion from time decay. Usually negative for buyers. |
| ν | Vega | ∂c/∂σ — volatility sensitivity | Surface inflation from σ | How much option value rises when market gets nervous (σ increases). |
| ρ | Rho | ∂c/∂r — rate sensitivity | Surface tilt from interest rates | Impact of interest rate changes on option value. Smallest for short-dated options. |
Notice that Vega (∂c/∂σ) does not appear in the BSM PDE itself — because the PDE is derived assuming σ is constant. Vega is the option's sensitivity to a parameter that the model assumes doesn't change. This is why the real world of options trading involves implied volatility — backing out the σ the market is implying from observed option prices — rather than using a fixed input.
There is one relationship in options pricing that holds with the force of a physical law. It requires no model, no assumptions about volatility, and no calculus. It follows purely from no-arbitrage. Violate it and free money exists. Markets eliminate violations within milliseconds.
Both portfolios produce identical payoffs at every possible stock price. Identical payoffs must have identical prices today — or arbitrage exists.
Look carefully at the BSM formula: c = x·N(d₁) − K·e⁻ʳᵀ·N(d₂). Something is conspicuously absent. The stock's expected return α — the drift, the forecast, the analyst's conviction — appears nowhere in the formula. It vanished when we cancelled the dW terms in Chapter 13.
This is the deepest insight in all of options theory. A stock everyone expects to go up dramatically has the same option price as a stock everyone expects to crash — as long as they have the same volatility. This is why options traders care obsessively about implied volatility and pay little attention to analyst price targets. The target is irrelevant. The wiggle is everything.
Enter the five inputs and compute the exact BSM call and put prices, along with all five Greeks. Change α (the expected return) by any amount and watch — the option price does not move. Only σ matters.
Adjust any input and watch the Greeks update. Change Alpha (α) — the call price doesn't move. Only σ drives the option price.
Chapter 14 — the complete BSM picture: