The Greeks: Delta, Gamma, Vega, and How Traders Manage Risk

Chapter 7 is one of the meatiest chapters in the first part of Wilmott’s book. It does two big things: first, it derives the actual Black-Scholes formulas for calls, puts, and binary options step by step. Second, it introduces the Greeks, which are the sensitivity measures that traders live and die by every single day. Wilmott makes an interesting argument early on: getting the hedging right is more important than getting the price right. Let me explain why.

From Equation to Formula

In the previous chapter we saw the Black-Scholes PDE. Now Wilmott shows how to actually solve it for specific contracts. The process involves a series of variable changes that transform the complicated-looking equation into the simple diffusion equation.

The key steps: change to future value terms, flip time so we solve forward instead of backward, switch from stock price S to log-return x = log(S), and then shift coordinates. After all these substitutions, the Black-Scholes equation becomes:

dW/dt = (1/2) d2W/dx2

This is just the basic diffusion equation, which has a known fundamental solution: the Gaussian bell curve that starts as a spike and spreads out over time.

Using the superposition principle (linearity of the equation), you can write the solution for any payoff as an integral of this fundamental solution weighted by the payoff function. For a call option with payoff max(S - E, 0), working through the integral gives the famous formula:

Call = S * e^(-D(T-t)) * N(d1) - E * e^(-r(T-t)) * N(d2)

Where N is the cumulative normal distribution function, and:

d1 = [log(S/E) + (r - D + sigma^2/2)(T-t)] / [sigma * sqrt(T-t)]

d2 = d1 - sigma * sqrt(T-t)

Here S is the stock price, E is the strike, r is the risk-free rate, D is the dividend yield, sigma is volatility, and T-t is time to expiry.

For a put, you can either repeat the derivation or use put-call parity:

Put = E * e^(-r(T-t)) * N(-d2) - S * e^(-D(T-t)) * N(-d1)

There is a nice approximation for at-the-money forward options (when S equals the forward price): the call value is roughly 0.4 * S * sigma * sqrt(T-t). Quick mental math for traders.

Binary options (which pay $1 if the stock is above the strike at expiry, nothing otherwise) have an even simpler formula:

Binary Call = e^(-r(T-t)) * N(d2)

Now, the Greeks

Here is where the chapter gets really practical. The Greeks are partial derivatives of the option price with respect to various inputs. They tell you how sensitive your position is to changes in the market, and they determine your hedging strategy.

Delta (the big one)

Delta is the rate of change of option value with respect to the stock price:

Delta = dV/dS

For a call, delta ranges from 0 (way out of the money) to 1 (deep in the money). For a put, it ranges from -1 to 0. Delta tells you how many shares you need to hold to hedge one option.

Delta hedging is the core of options trading. You hold the option and short delta shares of the underlying. This creates a delta-neutral position where small moves in the stock do not affect your portfolio value. But delta changes as S changes, so you have to continuously adjust. This is called dynamic hedging or rebalancing.

Wilmott points out that the financial world splits into speculators (who do not hedge) and hedgers. Among hedgers, delta hedgers believe they have a better price for the option and try to lock in profit by eliminating all directional risk. If your hedging is accurate, the profit or loss is set the moment you trade. If your hedging is sloppy, no initial edge will save you.

Gamma (the curvature)

Gamma is the second derivative of the option price with respect to the stock:

Gamma = d2V/dS2

Gamma measures how fast delta changes. High gamma means you need to rehedge frequently; low gamma means your hedge is stable.

Here is a fascinating fact Wilmott mentions: in a long gamma position, you make money 32% of the time on individual rehedges and lose money 68% of the time. But when you win, you win bigger. The net effect averages out to the risk-free rate. (This becomes clearer in Chapter 47.)

Gamma hedging goes beyond delta hedging. Since you cannot add gamma to your portfolio using just the underlying stock (its second derivative is zero), you need to trade additional options. A minimum gamma-neutral portfolio needs two different option types plus the underlying.

For calls and puts, gamma is always positive and peaks near the strike. For binary options, gamma changes sign, which creates interesting complications.

Theta (time decay)

Theta is the rate of change of the option price with time:

Theta = dV/dt

Theta, gamma, and the option value are connected through the Black-Scholes equation itself. In a delta-hedged portfolio, theta contributes to earning the risk-free rate in a completely deterministic way, unlike gamma which contributes the right amount only on average.

For at-the-money options, theta is typically negative and largest in magnitude. Your option loses value every day just from the passage of time. This is the “time decay” that options sellers collect and options buyers pay for.

Speed

Speed is the third derivative of the option value with respect to the stock price:

Speed = d3V/dS3

Traders use gamma to estimate how much they will need to rehedge if the stock moves by $1. But that is only an approximation. Speed gives the next order correction, telling you how gamma itself changes with the stock price. It matters most for large moves or when the option is near the strike close to expiration.

Vega (the tricky one)

Vega is the sensitivity of the option price to volatility:

Vega = dV/d(sigma)

Wilmott flags something important: vega is fundamentally different from the other Greeks because volatility is a parameter, not a variable. The stock price and time are variables that change in the model. Volatility is an input that we assume we know.

In reality, volatility is uncertain. If you plug 20% into the formula, vega tells you how much your price changes if the true volatility is actually 21% or 19%.

You can vega hedge by trading options against each other (since the underlying has zero vega). This reduces your dependence on volatility, which is a big deal because volatility is arguably the least well-known input in the whole model.

But there is a catch. Vega is only meaningful for options with single-signed gamma (like plain calls and puts). For binary options, increasing volatility makes the price go up in some regions and down in others. The vega might be near zero even though the option is hugely exposed to volatility changes.

Rho

Rho is the sensitivity to the interest rate:

Rho = dV/dr

It is usually the least important Greek for short-dated equity options, since interest rates do not move much day to day. But for longer-dated options or in volatile rate environments, it matters.

Implied Volatility: Turning the Formula Inside Out

The Black-Scholes formula takes volatility as input and gives price as output. But what if you already know the market price? You can reverse the formula to find what volatility is “implied” by the market price. This is implied volatility.

Since there is no closed-form inverse, you use numerical methods like Newton-Raphson iteration. Start with a guess (say 20%), compute the model price, see how far off you are, adjust by the ratio of the error to the vega, and repeat until convergence. Wilmott even provides the VBA code.

The big discovery when you compute implied volatility across many strikes and expiries: volatility is not constant. You get the famous volatility smile (or skew, or frown). For equity indices like the S&P 500, implied volatility typically slopes downward with strike (lower strikes have higher implied vol). This is fundamentally inconsistent with the constant-volatility Black-Scholes model, and it is one of the central puzzles explored later in the book.

In equity markets, traders assume the smile is “sticky” with respect to strike. In FX markets, traders assume it is sticky with respect to delta. Wilmott suggests this difference might be partly because of the different charts each group stares at all day, not necessarily any deep financial reason.

Why Hedging Beats Pricing

Wilmott’s key insight: if your hedging is accurate, you have locked in your profit the moment you trade. Future uncertainty does not matter because it has been hedged away. But if your hedge is wrong, the initial price advantage is meaningless because random fluctuations will dominate your P&L.

This is why traders obsess over the Greeks. They are not just theoretical curiosities. They are the actual quantities you use to decide how many shares to hold, how often to rebalance, and how exposed you are to the things you cannot control.

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