Part VII · Theory · Chapter 7A

Why hedging works

The Black-Scholes-Merton equation in five steps

Up to this point the book has described what the dealer does. This chapter answers a deeper question: why the dealer's delta-hedging strategy is the unique correct strategy, and what mathematical structure its hedging P&L actually has. It is the most technical chapter in the book and also the one that justifies every claim that follows.

If mathematics is not your native language, read slowly and accept the steps one at a time. Each formula is followed by what it says in plain words, while the heaviest algebra sits in boxes you can open or skip. The payoff for that patience is large. By the end of Part VII, one idea organizes every options strategy: every delta-hedged option position is, fundamentally, a bet on realized volatility against implied volatility, and nothing else.

1 · Five steps from a hedged book to an equation

The derivation runs in five steps, each building on the last and each doing one specific piece of intellectual work:

1. A model for prices: assume the underlying follows geometric Brownian motion (GBM).
2. Itô's lemma: derive how the option's value changes when the price moves at random.
3. The hedged book: build the delta-hedged portfolio and see that it has no random part.
4. No arbitrage: a portfolio with no risk over the next instant must earn the risk-free rate.
5. The Black-Scholes-Merton equation: write down what that demands of the option's value, a consistency condition.

After the five steps comes the gamma-theta P&L decomposition that Chapter 5C previewed, which drives everything in the dealer-mechanics framework. The chapter ends by connecting option replication to variance swaps and to the way the VIX is built, the bridge to Part IX.

2 · A model for prices

Black and Scholes (1973) and, independently, Merton (1973) assumed that the underlying's price follows a geometric Brownian motion. That phrase rests on two older ideas: the random walk, and the Brownian motion it becomes when its steps grow very small.

From coin flips to Brownian motion

A random walk (Chapter 3B) is a path built from independent steps. The plainest version flips a fair coin at each step, moving up by $h$ on heads and down by $h$ on tails. Each step has mean zero and variance $h^2$. Because the steps are independent, their variances add, so after $n$ steps the position has mean zero and variance $n h^2$. Its standard deviation is $h\sqrt{n}$, the square-root-of-time rule of Chapter 1A.

Now let each step take $\Delta t$ years, so that $n$ steps cover $t = n\,\Delta t$ years. To give the walk a volatility $\sigma$, its variance after one year must be $\sigma^2$ however finely the year is sliced, which fixes the size of each step:

With this step, the variance after $t$ years is $n\sigma^2\Delta t = \sigma^2 t$, whatever $\Delta t$ is. The step must shrink like the square root of the time it takes, not like the time itself. That one scaling rule is the source of everything unusual in this chapter.

As $\Delta t$ shrinks toward zero with the right scaling, the coin-flip walk converges to a Brownian motion, written $W_t$. The sum of many small independent steps is close to normal, whatever each step looks like, so the coin flips disappear into a bell curve. A Brownian motion has four defining properties:

• It starts at zero: $W_0 = 0$.
• Its change over any interval of length $\Delta t$ is normally distributed, with mean zero and variance $\Delta t$.
• Its changes over intervals that do not overlap are independent of each other.
• Its path is continuous, with no jumps.

The path is continuous but so jagged that it has no slope at any point: zoom in on any stretch and it looks as rough as the whole. The notation $dW$ means its change over a very short time $dt$. That change is normal, with mean zero and variance $dt$, so its typical size is $\sqrt{dt}$. For short times $\sqrt{dt}$ is far larger than $dt$: one trading day is $dt = 1/252 = 0.0040$ of a year, but $\sqrt{dt} = 0.063$, almost 16 times larger.

Adding a steady trend gives Brownian motion with drift, $X_t = a\,t + b\,W_t$: a line of slope $a$ with noise of volatility $b$ wrapped around it. Chapter 1D used it for the logarithm of an account's wealth, which drifts at the growth rate $G$ with volatility $\sigma$.

Geometric Brownian motion

Geometric Brownian motion applies the noise to the price in proportion to the price:

In words: in a small instant of time $dt$, the price $S$ changes by a deterministic amount, $\mu S\,dt$, called the drift, and a random amount, $\sigma S\,dW$, called the diffusion. Here $\mu$ is the expected return per year and $\sigma$ the volatility. By construction, $dW$ is normally distributed with mean zero and variance $dt$.

Multiplying the noise by $\sigma S$ scales it by the current price and the volatility. That makes volatility a percentage per unit of time rather than a number of dollars per unit of time. It matches what you see in markets, where volatility scales with the price level: a $1,000 stock has bigger dollar moves than a $10 stock at the same volatility. Dividing both sides by $S$ gives the form Chapter 1A used, $dS/S = \mu\,dt + \sigma\,dW$. The simple return over the instant is a drift plus normal noise.

This is the central modeling assumption of the entire Black-Scholes framework. Prices wiggle in proportion to themselves, which makes log returns normally distributed, or Gaussian, in continuous time. The model is wrong: real returns have fat tails, as Chapter 1B established. But it is wrong in ways that can be corrected at the margins without rebuilding the whole framework. For establishing why dealers hedge the way they do, geometric Brownian motion is fine.

One subtlety needs care: the parameter $\mu$ is the instantaneous expected return in simple-return space, not in log-return space. The expected log return is $\mu - \tfrac12\sigma^2$, the variance drag of Chapter 1A, which Step 2 derives. The two parameters live in different return conventions, both valid as long as you know which one you are using.

3 · Itô's lemma

The option's value depends on the price and on time, $V(S, t)$, so it moves whenever $S$ moves. Ordinary calculus has a rule for this, the chain rule, which would give $dV = (\partial V/\partial S)\,dS$ for the price's part. That is wrong here, because $S$ is random. The correct rule is Itô's lemma, the chain rule for Brownian motion. It rests on one surprising fact: a squared move does not vanish.

Why the square of a move does not vanish

For a smooth path, the change over a short time $dt$ is proportional to $dt$, so its square is proportional to $dt^2$. Over a year cut into $n$ steps, those squares add up to $n \times (1/n)^2 = 1/n$, which vanishes as the steps shrink. That is why ordinary calculus drops every squared change.

A Brownian motion behaves differently, because each squared step $(\Delta W)^2$ has mean $\Delta t$, the variance of the step. The squared steps over a year add up to a total whose mean is 1, a full year, however finely you slice. Its spread shrinks as the slicing gets finer, because the squared steps are independent: for normal steps, the total's standard deviation is $\sqrt{2/n}$.

The running sum of a path's squared changes is its quadratic variation, which for Brownian motion grows exactly like time. That result is usually written as three rules for products of small changes:

Applied to (7A.2), the rules keep only the noise's contribution to the squared move:

The drift's square and the cross term vanish, as the worked example suggested. Over one day the drift squared was 1,600 times smaller than the noise squared, a gap that widens as the interval shrinks.

The chain rule with one more term

Chapter 5C approximated a change in value by delta times the move plus half of gamma times the move squared. Written for a function of both price and time, that second-order Taylor expansion is

plus terms in $dt^2$, $dt\,dS$ and higher powers, all of which vanish by the rules above. In ordinary calculus the $(dS)^2$ term vanishes too, but here, by (7A.3), it is $\sigma^2S^2\,dt$, of the same order as $dt$. Substituting it gives Itô's lemma, the chain rule for a random price:

The extra term is there because $(dS)^2$ is not zero. For ordinary smooth functions, $(dx)^2$ is of order $dt^2$, which you drop in the limit. For Brownian motion, the variance of $dS$ over the interval $dt$ is $\sigma^2S^2\,dt$, of the same order as $dt$ itself, which you must keep. The term $\tfrac12(\partial^2V/\partial S^2)\,\sigma^2S^2\,dt$ is gamma showing up explicitly. It is the quadratic variation contribution that makes options different from linear instruments.

This is the central technical insight of stochastic calculus. The second derivative matters when the underlying is random, and not when it is deterministic. Every "interesting" property of options derives from this single mathematical fact.

In the notation of Chapter 5C, where $\partial V/\partial S = \Delta$, $\partial^2V/\partial S^2 = \Gamma$ and $\partial V/\partial t = \Theta$, Itô's lemma becomes

Read it this way: over a small interval, the option's value changes by a time-decay piece, $\Theta\,dt$, a directional piece, $\Delta\,dS$, which moves with the underlying, and a curvature piece, $\tfrac12\Gamma\sigma^2S^2\,dt$. The curvature piece is always positive for long-option positions and reflects the convexity benefit.

4 · The hedged book

Now build the dealer's book. Suppose the dealer has bought an option from a customer and hedges it by selling $\Delta$ shares of the underlying, the delta hedging of Chapter 6C. The book is worth

where $\Delta = \partial V/\partial S$ is set at the start of the instant and held fixed through it. Over the instant the short shares lose $\Delta\,dS$, so the book changes by $d\Pi = dV - \Delta\,dS$. When (7A.6) is substituted for $dV$, its $\Delta\,dS$ meets the $-\Delta\,dS$ of the shares and cancels it:

Look at what just happened: the $dS$ terms canceled. The book's change has no random component, so over the instant it is risk-free. That is what delta hedging accomplishes: it removes the $dW$ term and leaves only the deterministic $dt$ terms.

This is the mathematical statement of what Chapter 6C said informally: a delta-hedged book is, at the instant of the hedge, directionless. The $dW$ noise from the underlying has been hedged away. The remaining motion of the book is entirely deterministic, driven by theta and gamma alone, at first order.

Two remarks keep this honest. The first is that "at the instant" matters: delta changes as soon as $S$ moves, so the hedge must be reset continually, which is gamma's chase in Chapter 6D. The second is that a dealer who sold the option holds $-\Pi$, long $\Delta$ shares against a short option. Every sign flips, while the argument that follows is unchanged.

5 · No free money

A book that is risk-free over the next instant must earn the risk-free rate $r$, otherwise there is free money. This is the no-arbitrage condition, the most consequential single principle in derivatives pricing. If the hedged book earned more than $r$, a trader could borrow at $r$, buy the book and keep the difference with no risk, an arbitrage. If the book earned less, the trader would short it and lend the proceeds at $r$. Either trade pushes prices until the gap closes, so

The book earns the risk-free rate on its current value. Combining this with (7A.9), and writing $\Pi = V - \Delta S$:

Cancel $dt$ from both sides and rearrange:

6 · The Black-Scholes-Merton equation

Write the Greeks back out as derivatives and move $r\Delta S$ to the left:

This is the celebrated Black-Scholes-Merton partial differential equation (PDE). A partial differential equation is a relation among a function's partial derivatives. Its solution is not a number but a whole function, here the option's value at every price and time, fixed once you say what it must equal at expiry. Robert Merton received the 1997 Nobel Memorial Prize in Economics for this work, jointly with Myron Scholes. Fischer Black, who would have shared the prize, had died in 1995.

With a dividend yield $q$, the short shares in the book pay out $q\,\Delta S\,dt$ over each instant. Carrying that cost through Steps 3 and 4 turns the term $rS\,\partial V/\partial S$ into $(r - q)S\,\partial V/\partial S$. That is the version whose solution carries the factor $e^{-qT}$ of Chapter 5C's formula.

A consistency condition, not a formula

The Black-Scholes-Merton PDE is not a pricing formula but a consistency condition: any option price function $V(S, t)$ that does not satisfy it creates an arbitrage. Solve it with the condition that at expiry, with no time left, a call is worth $\max(S - K, 0)$. The solution is the Black-Scholes call formula of Chapter 5C, with $d_1$ and $d_2$ as defined there:

More profoundly, the PDE encodes the recipe by which a dealer can replicate the option using only the underlying and cash. The dealer is not "pricing the option" in some metaphysical sense but manufacturing it through dynamic hedging. The shares and cash, adjusted as the price moves, form a replicating portfolio that ends with exactly the option's payoff.

This is the deepest insight in the entire framework. An option's payoff looks as if it needs unusual cleverness to construct, because it is asymmetric and convex. In fact it can be built from a simple time-varying position in the underlying plus cash. The cost of building it is exactly the Black-Scholes price, no more and no less, under the model's assumptions. Any deviation from that price creates an arbitrage.

The practical implication is that when you buy an option, you are buying the right to a payoff that the dealer constructs for you through their hedging. The premium you pay is the cost of the construction kit, plus the dealer's margin, plus compensation for residual risks the model does not capture: jump risk, transaction costs and fat tails. The variance risk premium of Chapter 5E is the cumulative spread between what dealers charge and what the model says is fair. It is the structural compensation dealers require to be in this business.

Where the expected return went

Look again at (7A.13): the expected return $\mu$ appears nowhere in it. It entered through the price model, (7A.2), and reached Itô's lemma through $dS$. It disappeared in Step 3, when the hedge canceled the $dS$ terms, the only route by which the drift could reach the book. So the option's price does not depend on how fast anyone expects the underlying to grow.

That settles the question Chapter 4B left open: which probabilities turn an expected payoff into a price. Because $\mu$ does not matter, you may price as if the underlying grew at the risk-free rate, then discount the expected payoff at that rate. Probabilities chosen that way are the risk-neutral probabilities of Chapter 5C.

7 · The hedging P&L: gamma against theta

Now the PDE gives the most important formula in this book. Suppose the dealer delta-hedges continuously while the realized volatility $\sigma_R$ differs from the volatility used to price the option, the implied volatility at entry $\sigma_I$. Over each instant, the book's gain above what the same money would earn as cash is approximately

where $\Gamma$ is computed at the implied volatility.

Where the hedging P&L comes from

Step 1: expand using the actual move. The dealer values and hedges the option at $\sigma_I$, so $\Theta$, $\Delta$ and $\Gamma$ are the Black-Scholes values at $\sigma_I$. Over the instant, the Taylor expansion gives $dV \approx \Theta\,dt + \Delta\,dS + \tfrac12\Gamma(dS)^2$. Here $(dS)^2$ is the square of the move that actually happened, not its model average.

Step 2: add the hedge and the financing. The short shares cancel $\Delta\,dS$, as in Step 3. Subtracting what the book would have earned as cash, $r(V - \Delta S)\,dt$, leaves

$$\begin{aligned} d\Pi - r\Pi\,dt \approx{}& \tfrac12\Gamma(dS)^2 \\ &+ (\Theta - rV + r\Delta S)\,dt \end{aligned}$$ (7A.16)

Step 3: use the PDE at the implied volatility. The option's price satisfies (7A.12) with $\sigma_I$ in place of $\sigma$, which, rearranged, says $\Theta - rV + r\Delta S = -\tfrac12\Gamma\sigma_I^2S^2$.

Step 4: substitute and factor. The second term of Step 2 becomes $-\tfrac12\Gamma\sigma_I^2S^2\,dt$, so that factoring out $\tfrac12\Gamma S^2$ gives (7A.15). The only change from Step 3 of the derivation is that the squared move is now the realized one, not $\sigma_I^2S^2\,dt$.

Read (7A.15) this way: over each interval, the delta-hedged dealer earns gamma times the difference between the realized squared return and the implied variance. This is the entire economic content of a delta-hedged option position. If the squared returns average the implied variance, the bracket averages zero and the book earns nothing beyond cash. If they do not, the book earns or pays the difference, scaled by $\tfrac12\Gamma S^2$, with these signs:

• If you are long the option (long gamma) and realized volatility exceeds implied, you make money. Every wiggle in spot lets you scalp gamma, buying low and selling high in your hedge, at a rate that in aggregate exceeds the theta you are paying.
• If you are long the option and realized falls short of implied, you lose, because you bought too much volatility.
• If you are short the option (short gamma, the dealer's typical position), the signs flip. You earn the implied volatility, as the theta you collect, and pay the realized volatility, as the rebalancing losses.

A short-gamma dealer wants quiet markets, while a long-gamma dealer wants moves.

This is the punchline of the gamma-theta decomposition. Three choices set up the bet: strike selection affects the size of gamma, expiration selection affects its speed, and the implied volatility at entry sets the implied side. To first order, the resulting P&L is a function of whether realized exceeded implied.

The figure runs (7A.15) one day at a time. Its top panel draws a simulated spot path whose volatility is the realized-vol setting. Its bottom panel accumulates the hedged P&L of one contract of an at-the-money option, strike 100 with 60 trading days to expiry, with gamma computed at the implied vol.

Two sliders set the implied and the realized vol, from 5% to 60%, starting at 20% and 30%. A third picks the path seed, from 1 to 99, starting at 7. A switch flips the position between long and short gamma. Three readouts give the volatility the path actually realized, the terminal P&L and the sign that realized against implied predicts. Step through a few seeds, then set realized equal to implied, then below it, and flip to short gamma.

implied vol

20%

realized vol

30%

path seed

#7

position long Γ short Γ

realized vol = 30.1%

terminal P&L (long Γ) = +$123

predicted sign = positive

At the starting settings, seed 7's path realizes 30.1% and the long-gamma book ends about $123 up, as predicted. At 30% realized, every seed ends positive, but by amounts from about $35 to over $600: the sign is reliable, while the size depends on the path. With realized set equal to implied, the final P&L scatters around zero, from about −$160 to +$170. At 10% realized every seed loses. The predicted sign compares the path's realized volatility, not the slider's, with implied. Flipping to short gamma turns each curve upside down, because the short book earns what the long book pays.

The honest caveats

The "and nothing else" framing skips several real-world second-order effects that matter in practice:

• Vega risk on IV moves: a delta-hedged position is not vega-neutral. If implied volatility changes between entry and exit, that mark-to-market vega P&L is a separate component on top of the gamma-against-realized term. For long-dated options with large vega, it can dominate.
• Discrete hedging: the decomposition assumes continuous rebalancing, while real hedging happens in steps. The path of spot between rebalances adds variance around the theoretical gamma-scalping P&L. Two traders running the same position with different hedge frequencies realize different P&L from the same gap between realized and implied.
• Skew and surface dynamics: options that sit on a steep part of the volatility surface accrue P&L from vanna and volga as spot and volatility move together. Those terms are not captured by the simple $\Gamma \times (\sigma_R^2 - \sigma_I^2)$ decomposition, and Chapter 7B works them through.
• Transaction costs: hedging is not free, so the gamma-scalping P&L is gross. Net P&L can be meaningfully smaller, especially in less liquid underlyings or on short-DTE options, where rebalancing is frequent.

For most retail-scale, short-to-medium-DTE, large-cap delta-hedged positions, the first-order framing of realized against implied captures the bulk of P&L variability. For longer-dated positions, exotic surface trades or low-liquidity underlyings, the second-order terms matter and should be modeled explicitly.

Hedging in steps, and paying for each step

The second and fourth caveats pull against each other, as Chapter 6E showed. Hedging more often tames the path's noise but pays the spread more often. The next figure puts both into one book. It holds long at-the-money calls on a $600 underlying and delta-hedges them along one simulated path of five-minute bars, 78 to a trading day. The path's volatility equals the implied volatility, so hedging continuously earns nothing on average before costs.

Four schedules hedge the same path: on every bar (the row labeled continuous), five times a day, at each daily close and every three days. On each bar, a schedule's P&L adds up three things. The first is the gamma-theta term of (7A.15). The second is the delta that the stale hedge leaves open, the calls' delta at the start of the bar less the hedge held, times the bar's move. The third is the cost of any rebalance, including the first hedge: the shares traded, times the price, times the transaction cost.

Hedging on every bar leaves no delta open, so that schedule's curve is the pure gamma-theta P&L less its costs. Sliders set the implied vol (5% to 80%, starting at 20%), the days to expiry, labeled DTE (5 to 90, starting at 30), and the number of contracts (1 to 500, starting at 100). Two more set the transaction cost (0 to 20 basis points, starting at 2) and the seed (1 to 500, starting at 7). Below the chart, a table gives each schedule's final P&L. Set the cost to zero and step through seeds, then raise the cost.

IV σ

20%

DTE

30d

contracts

100

transaction cost (bps)

2.0

seed

7

Hedge frequency	Final P&L
Continuous (theoretical)	−$21,529
Intraday (5 hedges/day)	−$15,167
Daily close	+$11,548
Every 3 days	+$4,471

At the starting settings, the 100 calls carry about 5,340 shares of delta. Hedging on every bar rebalances 2,339 times and ends about $21,500 down, of which about $19,400 is transaction cost. Five hedges a day end about $15,200 down, the daily close about $11,500 up and every three days about $4,500 up. All four share the same gamma-theta term, a loss of about $2,100 on this path, so everything else in each result is open delta and cost.

One seed proves little, so look across all 500 of the figure's seeds. At zero cost the four schedules each average close to zero, while the spread of final P&L widens as hedging gets rarer. Its standard deviation is about $3,100 hedging every bar, $9,600 at five a day, $20,600 daily and $35,600 every three days. At 2 basis points the average cost runs the other way: about $19,000, $5,400, $2,700 and $1,800.

Both patterns follow square-root rules in the hedging interval $\delta t$. Over one interval the move, and with it the open delta, grows like $\sqrt{\delta t}$, which makes the error the interval adds grow like $\delta t$. Summed over $T/\delta t$ independent intervals, the total error grows like $\sqrt{\delta t}$. Hedging five times less often gives about $\sqrt5 = 2.2$ times the spread, as from $9,600 to $20,600. The shares traded at each rebalance also grow like $\sqrt{\delta t}$. The number of rebalances, though, falls like $1/\delta t$, so the total cost grows like $1/\sqrt{\delta t}$ as the interval shrinks. Hedging about 15 times as often, 2,339 rebalances against 155, should cost about $\sqrt{15} = 3.9$ times as much, and the figure gives 3.5.

No schedule wins on both counts: the spread falls as the interval shrinks while the cost rises. That is the trade-off of Chapter 6E, which finds where the net result is best for an SPX book.

8 · Why the dealer feedback loop is built into the math

This is also the punchline for why aggregate dealer positioning moves the market. Suppose dealers are collectively short gamma. By (7A.15) they are mechanically losing money to volatility. The more volatility there is, the more they lose. To minimize losses they hedge aggressively, which means buying as the market rises and selling as it falls. That hedging creates volatility, which makes them lose more, which makes them hedge more.

The feedback loop is mathematically built into the gamma-theta decomposition just derived. It is not a story or a metaphor: it falls out of the PDE. Chapter 8D formalizes it into the effective-volatility equation, which quantifies how dealer feedback amplifies or dampens the underlying's volatility depending on the sign of aggregate dealer gamma.

9 · The variance swap connection

Add up (7A.15) over the life of the option. Each instant's gain is weighted by $\tfrac12\Gamma S^2$, which changes as spot and time move. Now suppose you held a position whose gamma falls like $1/S^2$ as spot rises, so that $\Gamma S^2$ stays at a constant, $g$. The weight then comes out of the sum. Write $P$ for the book's total gain above cash over a life of $T$ years:

Here $\sigma_R^2 = \tfrac{1}{T}\sum (dS/S)^2$ is the realized variance, annualized. A payoff proportional to realized variance less a fixed number is exactly a variance swap (Chapter 4B). So integrating the hedging P&L over the life of the option and weighting by $1/S^2$ gives, almost exactly, the payoff of a variance swap. This is not coincidence but the foundation of model-free variance replication, and the reason the VIX (Chapter 9A) is constructed as a weighted strip of options with $1/K^2$ weights.

Demeterfi, Derman, Kamal and Zou's 1999 Goldman Sachs paper, "More Than You Ever Wanted to Know About Volatility Swaps", showed how. A portfolio of options weighted by $\Delta K/K^2$ across all strikes, where $\Delta K$ is the spacing between strikes, replicates the payoff of a variance swap when it is dynamically delta-hedged. The intuition is that weighting by $1/K^2$ creates a "gamma flat" portfolio. Its $\Gamma S^2$, which turns a squared return into dollars and is 100 times the per-share dollar gamma of Chapter 6D, is the same wherever spot is. That is exactly the property required to replicate realized variance.

This is the mathematical thread connecting dealer hedging to the VIX. The VIX is, fundamentally, the cost of replicating variance through delta-hedged options. Every option you trade contributes some amount of replicated variance to the dealer's book. Aggregate dealer gamma is, essentially, the aggregate variance-replication exposure of the entire options ecosystem. Chapter 9A returns to this connection in detail.

10 · Check yourself