Part VI · Dealers · Chapter 6B

What the dealer inherits

The Greeks of a customer's trade

1 · A bundle of sensitivities

When the dealer sold you the call, it inherited a bundle of sensitivities. The option's value changes when the price of the underlying moves, when implied volatility (IV) moves, when time passes and when interest rates change. Each of these sensitivities is a Greek (Chapter 5C), and each must eventually be managed.

The inheritance is a sign flip: the dealer holds the other side of your contract, so each of its Greeks is the negative of yours.

Vega is more expensive to hedge dynamically than delta. Most dealers manage vega in aggregate across their book rather than hedging each position.

2 · Delta, the Greek hedged first

Delta, the most important of them, is the partial derivative of the option's value with respect to the price of the underlying. If the underlying rises by $1, the option's value rises by about delta dollars. A call's delta lies between 0 and 1, and a put's between −1 and 0. Deep in-the-money (ITM) options behave almost like the underlying itself, with delta near ±1. Deep out-of-the-money (OTM) options barely react, with delta near 0, and at-the-money (ATM) options sit near ±0.5.

The Black-Scholes delta of a call uses the symbols of Chapter 5C: $S$ for the price of the underlying, $K$ for the strike and $T$ for the time to expiry in years. Implied volatility is $\sigma$, the risk-free rate $r$ and the dividend yield $q$. In that notation, the call's delta is

The same $d_1$ appears in every Greek of Chapter 5C, so it pays to read it piece by piece.

Reading $d_1$ piece by piece

• The function $N(\cdot)$ is the distribution function of the standard normal distribution. It takes any number on the real line and returns a probability between 0 and 1: the area under the bell curve to the left of that number.
• The log term $\ln(S/K)$ is the log-moneyness, the log return that would carry the strike to the spot (Chapter 1A). It is positive when spot is above the strike, which is in the money for a call, negative when spot is below, and zero exactly at the money.
• The term $(r - q + \tfrac12\sigma^2)\,T$ adjusts for three things, beginning with the risk-free rate, which raises the forward price above spot, and the dividend yield, which lowers it. The term $\tfrac12\sigma^2$ is the convexity correction: it accounts for the mean of the lognormal distribution sitting above its median in log space.
• The denominator $\sigma\sqrt{T}$ is the standard deviation of log returns over the $T$ years left. Dividing by it puts the log-moneyness on a common scale.

So $d_1$ is a z-score: it says, in standard deviations, how strongly the option is placed to finish in the money. Feeding it to $N$ gives a probability-like number. Multiplying by $e^{-qT}$ adjusts for the dividends you forgo by holding the option rather than the stock itself.

3 · A hedge ratio, and only roughly a probability

Delta, the result of all this, is the dividend-discounted hedge ratio: the number of shares per contract, divided by the multiplier of 100, that neutralizes the option's first-order exposure to spot. The Black-Scholes call price, with $d_2 = d_1 - \sigma\sqrt{T}$, is

Its two $N$ terms are different probabilities under different weightings of the outcomes. $N(d_2)$ is the standard risk-neutral probability of finishing in the money. The call's delta, $e^{-qT}N(d_1)$, is the share-equivalent hedge ratio under the share-numeraire weighting of Chapter 5C. It is close to the chance of finishing in the money but not equal to it, and the gap grows as $\sigma\sqrt{T}$ grows.

A 0.30-delta call behaves as 30 shares of the underlying per 100-share contract. That is its operative meaning, and it is what dealer hedging is built on. A share-equivalent is this unit of exposure: one share's worth of sensitivity to small moves in the underlying.

The informal reading, "a 30% chance of finishing in the money", is roughly right for short-dated options near the money and technically wrong. The chance of finishing in the money is $N(d_2)$, smaller than $N(d_1)$ by about $\varphi(d_1)\,\sigma\sqrt{T}$, where $\varphi$ is the height of the standard normal curve (Chapter 5C).

When you hear a trader say they are "selling 25-delta puts", read it as a hedge ratio of 25 share-equivalents per contract, not as a 25% probability of assignment. For dealer mechanics the share-equivalent reading is the operative one. A dealer short 1,000 contracts of a 0.30-delta call is short $1{,}000 \times 100 \times 0.30 = 30{,}000$ share-equivalents. It behaves as if it were short 30,000 shares of the underlying, and it neutralizes that by buying 30,000 shares, as Chapter 6C describes.

4 · The geometry of $d_1$

Because $d_1$ is a z-score, delta has a picture. The area under the standard normal density to the left of $d_1$ is $N(d_1)$, which is the call's delta up to the dividend factor.

The figure below draws that density with $d_1$ marked on it, at a 4% rate and no dividend. Its sliders set spot and strike (each from 70 to 130), days to expiry (1 to 180) and implied volatility (5% to 80%). A button switches between a call and a put. The readouts give the three pieces of (6B.2): $\ln(S/K)$, the drift term $(r - q + \tfrac12\sigma^2)T$ and $d_1$ itself, with $N(d_1)$ as the call's delta. It opens at the money, with spot and strike at 100, 30 days and 20% IV, where the readouts show 0.000, 0.0049, $d_1 = 0.086$ and a delta of 0.534. Set spot to 105 to reproduce the worked example, then switch to the put.

spot S

$100.0

strike K

$100

days to expiry

30d

implied vol σ

20%

option call put

ln(S/K) = 0.000

(r−q+½σ²)·T = 0.0049

d₁ = 0.086

N(d₁) = call Δ = 0.534

At spot 105 the readouts match the worked example: $d_1 = 0.937$ and a call delta of 0.826. For the put the shaded area moves to the right of $d_1$, and $1 - N(d_1) = 0.174$ is the size of the put's delta. Now keep spot at 105 and drag the days to expiry. At 1 day $d_1$ reaches 4.68, past the chart's right edge, where its marker stops, and delta reaches 1.000: the call is almost certain to finish in the money. At 180 days $d_1$ falls to 0.558 and delta to 0.712. Raising IV to 80% at 30 days has a similar effect, pulling $d_1$ back to 0.342 and delta to 0.634. More uncertainty makes the in-the-money finish less certain, so delta moves toward the middle.

5 · Why an at-the-money call has a delta above one half

With spot equal to the strike the log-moneyness is zero, and (6B.2) becomes

The second term is always positive, and the first is not negative when $r \ge q$, so an at-the-money call then has $N(d_1)$ above 0.5. The second term, from the convexity correction $\tfrac12\sigma^2$, grows with volatility and time. It is what makes high-volatility options drift higher in delta even at the money.

The reason is the shape of the lognormal distribution. Suppose a stock at 100 ends a period with a log return of either +0.2 or −0.2, with equal chance. It ends at $100e^{0.2} = 122.14$ or at $100e^{-0.2} = 81.87$: a gain of $22.14 against a loss of $18.13. The two log returns average zero, which puts the middle of the distribution, in log terms, at a price of 100. The average price, though, is $(122.14 + 81.87)/2 = 102.01$. Even with no expected move in log terms, high volatility means more upside than downside in dollars, because the lognormal distribution is right-skewed by construction.

6 · The asymmetry that drives the rest of this part

The dealer can hedge delta easily and cheaply, by buying or selling shares of the underlying. It cannot hedge gamma cheaply, except by acquiring offsetting options positions. So most dealer books are not gamma-hedged in the literal sense. They carry net gamma exposure and manage it through dynamic delta re-hedging, which is the setup for Chapter 6D onward.

7 · Check yourself