---
title: "Options Market Making in Practice"
book: "Derivatives and Volatility"
subject: quant
language: en
chapter: 26
exercises: 8
source: https://one-course.com/books/quant/5/en/chapter/26-options-market-making-in-practice
---

# Chapter 26 — Options Market Making in Practice

On the eve of an ex-dividend date, two market makers trade 33 000 contracts of a deep in-the-money call with each other, in both directions. Neither takes a position. Both exercise every call they bought. The next morning each finds that some of the calls it sold were never assigned: their holders, among the public, failed to exercise. Each unassigned short call is worth the dividend less a little time value. In this chapter’s model, with 30% of 10 000 public contracts left unexercised, the pair clears $108 553 after fees. The [dividend play](#def-dv-options-market-making-in-practice-play) is a small corner of options market making, but it shows the trade’s character: many small, well-defined edges, each engineered, each with its own costs and risks. This chapter builds the rest of them: a live surface and [theoretical values](#def-dv-options-market-making-in-practice-theo), quote widths, delta-hedging policies, dividends and pin risk, and the limits that keep a book’s risk in buckets.

## 26.1 Fitting a live surface

A market maker quotes thousands of series. It cannot price each from scratch as the underlying moves, so it keeps a surface: a small set of parameters per expiry (chapter 8’s SVI, or a skew model), refitted to the market’s quotes and trades at intervals, and moved with the spot in between by a marking rule (chapter 7).

**Definition 26.1 (Theoretical value).**

The *theoretical value* of an option, for a market maker, is its price on the market maker’s current surface, forward and discount curve: the reference around which it sets its bid and ask, updated with every move of the underlying and every refit.

Between refits the marking rule does the work. If the market’s smile moves with the spot ([sticky delta](https://one-course.com/books/quant/5/en/chapter/7-implied-volatility-and-its-surface#def-dv-implied-volatility-and-its-surface-sticky)) but the theo holds each strike’s volatility ([sticky strike](https://one-course.com/books/quant/5/en/chapter/7-implied-volatility-and-its-surface#def-dv-implied-volatility-and-its-surface-sticky)), the theo drifts off by the skew times the log move of the spot since the fit, divided by $\sqrt\tau$. With a skew of $-0.10$ and a spot at 20% volatility, the mean error in volatility points is:

| refit interval | 1 second | 10 seconds | 1 minute | 5 minutes |
| --- | --- | --- | --- | --- |
| one-month options | 0.0023 | 0.0072 | 0.018 | 0.039 |
| three-month options | 0.0013 | 0.0042 | 0.010 | 0.023 |

A minute’s staleness costs a fiftieth of a volatility point on a one-month option, well inside any width. What makes surfaces go wrong is not the interval but the jumps: an earnings release, a large trade or an index move shifts the whole smile at once. The quoting engine therefore refits on events as well as on a clock, and pulls its quotes while it refits.

## 26.2 Theoretical value and quote widths

Around the theo the market maker quotes a width. Too narrow, and informed traders pick it off whenever its theo is wrong; too wide, and the uninformed flow that pays for everything goes elsewhere. Book 2, chapter 30 treated the width in a game; the build turns it into a formula per series.

**Example 26.2 (The width model).**

Measure everything in volatility points. The theo’s error is normal with a standard deviation of 0.5 point. Uninformed orders arrive at 20 an hour when the half-width is zero, falling as $e^{-w/0.3}$ as it widens; each pays the half-width $w$. Informed orders arrive at $\lambda$ an hour and trade only when the error exceeds $w$, taking the excess. The expected profit per hour, per unit of [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks), is

$$
20\,e^{-w/0.3}\,w-2\lambda\bigl(0.5\,\varphi(w/0.5)-w\,(1-\Phi(w/0.5))\bigr).
$$

Without informed flow the best half-width is 0.30 point and earns 2.21 an hour. With 2, 5, 10 and 20 informed orders an hour it widens to 0.35, 0.42, 0.55 and 0.78, and the profit falls to 1.89, 1.51, 1.07 and 0.65 ([Figure 26.1](#fig-dv-options-market-making-in-practice-width)).

![Expected profit per hour of quoting a half-width around the theo, per unit of vega, for several rates of informed orders; dots mark the best half-width. Informed flow pushes the width out and the profit down. Data: the tutorial.](https://one-course.com/images/onecourse/chapters/quant-5/dv-options-market-making-in-practice/fig-0dfe3679cadc.svg)

***Figure 26.1.** Expected profit per hour of quoting a half-width around the theo, per unit of [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks), for several rates of informed orders; dots mark the best half-width. Informed flow pushes the width out and the profit down. Data: the tutorial.*

Measured in volatility points, the best width does not depend on the option. In price it is that width times the option’s [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks). A one-month at-the-money option on a share at 100 has a [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) of 0.115 a point and a three-month option 0.199, so at 0.42 point the half-widths are 4.8 and 8.4 cents. The inputs are measured, not guessed: the theo’s error from mark-outs (Book 2, chapter 15) of past trades against later theos, and the informed share from mark-outs by counterparty.

## 26.3 Delta-hedging policies

A market maker’s book accumulates gamma from customers’ trades, and its delta moves with every tick. Hedging each move costs the bid–ask spread of the underlying; hedging seldom leaves risk. Time-based hedging, every so often, ignores where the delta is. A band hedges only when it must.

**Definition 26.3 (Hedging band).**

A *hedging band* is a range around the target delta within which the hedge is left alone; when the delta leaves it, the hedge trades back to the band’s nearest edge, not to its centre, so that trades are only as large as the risk requires.

The band’s width follows from a balance. The delta diffuses at a rate set by the gamma, $\Gamma S\sigma$ per unit of $\sqrt t$, so it leaves a band of half-width $H$ after a time of order $H^2/(\Gamma S\sigma)^2$ and costs about $\lambda SH$ in spread each time, with $\lambda$ the proportional cost. The variance of the unhedged part grows as $H^2S^2\sigma^2$. Minimising the cost rate plus a risk-aversion $\gamma$ times the variance rate gives $H^3\propto\lambda S\Gamma^2/\gamma$: a band half-width that scales as the cube root of cost times gamma squared. Whalley and Wilmott derived the optimal band for small costs from a utility-maximising model as an expression in the option’s gamma. The build uses $H=c\,(\lambda S\Gamma^2)^{1/3}$ and treats $c$ as the desk’s risk dial.

**Example 26.4 (Bands against the clock).**

A short one-month at-the-money straddle on a share at 100 (20% volatility) is hedged over its life, checking 13 times a day, with a cost of 5 basis points of the traded notional. Hedging at every check costs 0.525 on average and leaves a P&L standard deviation of 0.321. Once a day costs 0.142 with a standard deviation of 0.855. A band with $c=1$ costs 0.140, the same as daily hedging, with a standard deviation of 0.505, 41% less. With $c=0.25$ it costs 0.313 with 0.313, and every-third-check hedging costs about the same (0.302) with 0.434 ([Figure 26.2](#fig-dv-options-market-making-in-practice-bands)).

![Hedging cost against the remaining risk for a short one-month straddle: time-based hedging at intervals from one to 26 checks (13 checks a day), and bands of increasing width. The band’s frontier lies below the clock’s. Data: the tutorial.](https://one-course.com/images/onecourse/chapters/quant-5/dv-options-market-making-in-practice/fig-2b3ebb16c2ba.svg)

***Figure 26.2.** Hedging cost against the remaining risk for a short one-month straddle: time-based hedging at intervals from one to 26 checks (13 checks a day), and bands of increasing width. The band’s frontier lies below the clock’s. Data: the tutorial.*

## 26.4 Pin risk, dividends and early exercise

Some of a market maker’s risks are not in any Greek. Book 1, chapter 23 described pin risk. A desk short 1 000 calls struck at 100 with the share closing at 100.02 on expiry day does not know how many will be exercised, anywhere from none to all, so it does not know whether it will open Monday short 100 000 shares or flat. It closes such positions before the bell or holds the share position the expected assignment implies and accepts the rest.

Dividends create the opposite kind of opportunity. On the eve of an ex-date, holders of deep in-the-money American calls should exercise when the dividend exceeds the time value they give up, the value of the put with the same strike and expiry (chapter 6). Many do not. Pool, Stoll and Whaley found that in 1996–2006 more than half of the long positions that should have been exercised were not, at a cost to holders of over $491 million, and that market makers captured most of it.

**Definition 26.5 (Dividend play).**

A *dividend play* is a trade on the eve of an ex-dividend date in which market makers buy and sell large, offsetting quantities of a deep in-the-money call among themselves and exercise all their long calls; because assignment falls pro rata on all short positions, their short calls absorb most of the non-assignment left by holders who fail to exercise, and each unassigned short call earns the dividend less the time value.

The arithmetic is short. With $N$ public long contracts of which a share $f$ fails to exercise, and $q$ contracts traded twice between two market makers, $2q+(1-f)N$ calls are exercised against $N+2q$ short positions. The market makers’ unassigned short calls number $2q\,fN/(N+2q)$, which tends to $fN$ as $q$ grows. Fees grow linearly in $q$, so there is a best size, $q^*=\bigl(\sqrt{2fNg\,N/c}-N\bigr)/2$, with $g$ the gain per unassigned call and $c$ the fees per $q$.

**Example 26.6 (One call series).**

A dividend of 50 cents ($50 a contract), a put worth 2 cents ($2), so a gain of $48 per unassigned contract; 10 000 public contracts; fees (an assumption) of $0.10 a contract per trade and $0.05 per exercise, so $0.50 per $q$ for four trades and two exercises. If 30% of holders fail to exercise, the best trade is 32 947 contracts. It leaves 2 605 short calls unassigned, 87% of the 3 000 that were not exercised, for a gross gain of $125 026, fees of $16 474 and a net of $108 553. With 10% failing the best trade is 16 909 contracts and nets $28 591; with 50%, 43 990 contracts and $193 510 ([Figure 26.3](#fig-dv-options-market-making-in-practice-play)).

![Expected net profit of a dividend play on one call series (10 000 public contracts, $48 gain per unassigned call, $0.50 of fees per contract traded each way) against the size of the trade, for three shares of holders failing to exercise; dots mark the best size. Data: the tutorial.](https://one-course.com/images/onecourse/chapters/quant-5/dv-options-market-making-in-practice/fig-3cd605da9f4c.svg)

***Figure 26.3.** Expected net profit of a [dividend play](#def-dv-options-market-making-in-practice-play) on one call series (10 000 public contracts, $48 gain per unassigned call, $0.50 of fees per contract traded each way) against the size of the trade, for three shares of holders failing to exercise; dots mark the best size. Data: the tutorial.*

The expected profit is not the realised one: assignment is random, and a desk that is assigned more than its share keeps less. The trade exists because fees and assignment rules make it cheap, and it disappears where they do not.

**As of September 2026 — Strategy fee caps.**

Cboe Options’ fees schedule of 15 September 2026 caps market-maker and other non-customer transaction fees at $0.00 for merger, short-stock-interest, reversal, conversion and jelly-roll strategies executed in open outcry on the same day in the same class. Dividend strategies are not on that list, so a [dividend play](#def-dv-options-market-making-in-practice-play) there pays ordinary fees.

## 26.5 Bucketed limits

A market maker’s [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) is not one number. Long one-month and short one-year [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) of the same size is flat in total and fully exposed to a change in the term structure. Limits are set by bucket.

**Definition 26.7 (Vega bucket).**

A *vega bucket* is a range of expiries (and, in finer schemes, of moneyness) over which a book’s [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) is summed and limited; a book’s [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) profile is its [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) by bucket, and limits apply to each bucket separately.

When a bucket approaches its limit, the quoting engine does not stop at once. It shades both quotes in the direction that attracts offsetting trades: a desk short [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) raises its bid and ask so that customers sell it options. It withdraws the side that would breach the limit only when the limit is reached.

**Example 26.8 (A month of put demand).**

A toy market maker quotes 15 series (one, three and six months; strikes 90 to 110) for 20 days, 13 checks a day. Orders of 10 contracts arrive three times a check; uninformed customers buy puts two times in three and calls half the time; one order in five is informed. The theo errs by 0.5 point, the half-width is 0.4 point, and the delta is hedged at every check at 5 basis points. Limits are $3 000, $5 000 and $6 000 of [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) per point. With shading, the book’s six-month [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) peaks at $-\$3\,369$ and the P&L is $11 141: edge $36 280, selection $-\$8\,720$, hedging $-\$9\,815$, inventory $-\$6\,605$. Without it, the same flow drives the six-month [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) to $-\$5\,033$, the inventory loses $15 240, and the month ends at $-\$607$ ([Figure 26.4](#fig-dv-options-market-making-in-practice-vega)). One path proves nothing about averages. It shows where the risk goes when nothing leans against the flow.

![Vega by bucket of a toy options market maker facing a month of put demand, with quotes shaded as each bucket approaches its limit (dashed) and without. Data: the tutorial.](https://one-course.com/images/onecourse/chapters/quant-5/dv-options-market-making-in-practice/fig-cfe3c8607676.svg)

***Figure 26.4.** [Vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) by bucket of a toy options market maker facing a month of put demand, with quotes shaded as each bucket approaches its limit (dashed) and without. Data: the tutorial.*

Gamma gets the same treatment by expiry, and the largest single names get their own buckets. The limits are only as good as the [Greeks](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) behind them, which is why the desk’s [Greeks](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) come from the same surface as its quotes.

## 26.6 Tutorial: a toy options market maker

**Goal.** Compute the width model’s best half-widths, compare [hedging bands](#def-dv-options-market-making-in-practice-band) with hedging on a clock, size a [dividend play](#def-dv-options-market-making-in-practice-play), and run a toy market maker with bucketed limits. **End state:** the four figures, the staleness table and the numbers of the weekend problem.

1. **The hedger**, band or clock, vectorised over paths: `def hedge_paths (paths: np.ndarray, book: Sequence[Option], vol: float , dt: float , cost: float , band: float | None = None , every: int = 1 ) -> dict [str , np.ndarray]: """Delta-hedge a book along simulated paths (rows) at flat volatility, paying `cost` per unit of traded notional. With `band` = c the hedge trades back to the edge of the band c (cost S Gamma^2)^(1/3) when the delta leaves it; otherwise it re-hedges fully every `every` steps. Returns per-path P&L (after costs), costs and trade counts.""" n, steps = paths.shape steps -= 1 def book_at (s, t): v, d, g = np.zeros_like(s), np.zeros_like(s), np.zeros_like(s) for o in book: pv, pd, pg = _bs_vec(s, o.strike, o.expiry - t, vol, o.right) v, d, g = v + o.qty * pv, d + o.qty * pd, g + o.qty * pg return v, d, g hedge, cash, costs, trades = np.zeros(n), np.zeros(n), np.zeros(n), np.zeros(n) value0 = book_at(paths[:, 0 ], 0.0 )[0 ] for i in range (steps): s = paths[:, i] _, delta, gamma = book_at(s, i * dt) if band is not None : h = band * (cost * s * gamma * gamma) ** (1.0 / 3.0 ) new = np.clip(hedge, -delta - h, -delta + h) else : new = -delta if i % every == 0 else hedge trade = new - hedge cash -= trade * s costs += cost * np.abs(trade) * s trades += np.abs(trade) > 1e-12 hedge = new value_end = book_at(paths[:, -1 ], steps * dt)[0 ] pnl = value_end - value0 + cash + hedge * paths[:, -1 ] - costs return {" pnl " : pnl, " costs " : costs, " trades " : trades}` **Listing 26.1.** Delta hedging on a clock or in a band. code/firm/optmm/firm_optmm.py
2. **The [dividend play](#def-dv-options-market-making-in-practice-play)** and its best size: `def dividend_play (public: float , fail: float , q: float , gain: float , trade_fee: float , exercise_fee: float ) -> dict : """Two market makers trade q contracts twice with each other (each ends long q and short q) and exercise all their longs. Public holders own `public` contracts and a share `fail` of them does not exercise. Assignment falls pro rata on all short open interest (public + 2 q). Each short call left unassigned gains `gain` (the dividend less the time value); costs are four trade fees and two exercise fees per q. Returns the expected combined profit.""" exercised = 2 * q + (1 - fail) * public shorts = public + 2 * q unassigned = 2 * q * (1 - exercised / shorts) cost = q * (4 * trade_fee + 2 * exercise_fee) return {" unassigned " : unassigned, " gross " : gain * unassigned, " cost " : cost, " net " : gain * unassigned - cost, " captured " : unassigned / (fail * public) if fail > 0 else 0.0 } def best_play (public: float , fail: float , gain: float , trade_fee: float , exercise_fee: float ) -> tuple [float , dict ]: """The trade size q maximising the expected net profit: with a = 2 fail public gain and c the cost per q, the net is a q / (public + 2 q) - c q, maximal at q = (sqrt(a public / c) - public) / 2 (zero if negative).""" a = 2 * fail * public * gain c = 4 * trade_fee + 2 * exercise_fee q = max ((math.sqrt(a * public / c) - public) / 2 , 0.0 ) if c > 0 else math.inf return q, dividend_play(public, fail, q, gain, trade_fee, exercise_fee)` **Listing 26.2.** The dividend play’s expected profit. code/firm/optmm/firm_optmm.py
3. **The quote**, shaded by the bucket’s use of its limit: `def quote (o: Option, spot: float , t: float , surf: SkewSurface, half_width_vol: float , bucket_vega: float , limit: float , skew_at_limit: float = 1.0 ) -> tuple [float , float ]: """Bid and ask around the theo, half-width in volatility points times vega; as the bucket's vega approaches its limit the quotes are shifted (in volatility points, up to skew_at_limit) to discourage trades that add to it, and the side that would breach the limit is withdrawn (NaN).""" value, vega = theo(o, spot, t, surf) use = bucket_vega / limit shift = skew_at_limit * max (min (use, 1.0 ), -1.0 ) bid = value - (half_width_vol + shift) * vega ask = value + (half_width_vol - shift) * vega if use >= 1.0 : bid = math.nan if use <= -1.0 : ask = math.nan return bid, ask` **Listing 26.3.** Quotes around the theo with limit shading. code/firm/optmm/firm_optmm.py
4. **Run** `dv_optmm.stale_table()` , `width_study()` , `hedge_study()` , `dividend_study()` , `toy_mm()` and `fig_optmm.py` .

**What to change next.** Make the half-width depend on the informed share by series; hedge the toy market maker with a band; draw the [dividend play](#def-dv-options-market-making-in-practice-play)’s assignment at random and plot the distribution of its profit.

## 26.7 Build: the options quoting engine

**Purpose.** The miniature firm’s options quoting engine: theos from a live surface, widths from a model, a band hedger, the [dividend play](#def-dv-options-market-making-in-practice-play)’s arithmetic, and bucketed limits that shade quotes.

**Interface.** `theo(option, spot, t, surface)` $\to$ (value, [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) per point); `width_profit`, `optimal_width(err, lam_u, w_scale, lam_i)`; `band_half_width`, `hedge_paths(paths, book, vol, dt, cost, band, every)`; `should_exercise`, `dividend_play(public, fail, q, gain, trade_fee, exercise_fee)`, `best_play`; `Bucket(name, lo, hi, limit)`, `vega_by_bucket`, `quote(option, spot, t, surface, half_width_vol, bucket_vega, limit)`.

**Rules.** Widths are set in volatility points and converted by [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks); hedges trade to the band’s edge; a limit shades before it withdraws; every trade’s mark-out is recorded for the width model.

**Acceptance tests.** `code/firm/optmm/tests/`: without informed flow the best width is the demand’s scale, and informed flow widens it; an infinite [dividend play](#def-dv-options-market-making-in-practice-play) captures every failure to exercise and the best size beats its neighbours; quotes are symmetric at zero use, shaded when long [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks), and one side is withdrawn at the limit; bands cost less than hedging at every check and hold more risk than tight bands.

**Stretch.** Random assignment in the [dividend play](#def-dv-options-market-making-in-practice-play); widths by series from recorded mark-outs; gamma limits by expiry.

Sources and further reading

- M. Pool, H. R. Stoll and R. E. Whaley, “Failure to exercise call options: an anomaly and a trading game”, *Journal of Financial Markets* 11(1) (2008) 1–35.
- A. E. Whalley and P. Wilmott, “An asymptotic analysis of an optimal hedging model for option pricing with transaction costs”, *Mathematical Finance* 7(3) (1997) 307–324.
- Cboe Exchange, Inc., fees schedule, 15 September 2026, footnote 13.

## 26.8 Exercises

**Exercise 26.1 ★.**

Why does the best half-width equal the demand’s scale (0.3 point) when there is no informed flow?

**Solution of Exercise 26.1.**

With no informed flow the profit is $20\,e^{-w/0.3}w$. Its derivative $20\,e^{-w/0.3}(1-w/0.3)$ vanishes at $w=0.3$, where the gain per trade and the loss of trades balance at the margin.

**Exercise 26.2 ★.**

Convert a half-width of 0.42 volatility point into cents for one-month and three-month at-the-money options on a share at 100.

**Solution of Exercise 26.2.**

$0.42\times0.115=0.048$ and $0.42\times0.199=0.084$: 4.8 and 8.4 cents a share, $4.83 and $8.37 a contract.

**Exercise 26.3 ★.**

Check the [dividend play](#def-dv-options-market-making-in-practice-play)’s count of unassigned calls for $N=10\,000$, $f=0.3$, $q=32\,947$.

**Solution of Exercise 26.3.**

$2q=65\,894$, $fN=3\,000$, $N+2q=75\,894$: $65\,894\times3\,000/75\,894=2\,605$.

**Exercise 26.4 ★★.**

Derive the best trade size $q^*$ of the [dividend play](#def-dv-options-market-making-in-practice-play).

**Solution of Exercise 26.4.**

The net is $g\,2q\,fN/(N+2q)-cq=aq/(N+2q)-cq$ with $a=2fNg$. Its derivative is $aN/(N+2q)^2-c$, zero at $(N+2q)^2=aN/c$, so $q^*=(\sqrt{aN/c}-N)/2$; the second derivative is negative. With $a=288\,000$, $N=10\,000$, $c=0.5$: $q^*=(75\,895-10\,000)/2=32\,947$.

**Exercise 26.5 ★★.**

Why does a [hedging band](#def-dv-options-market-making-in-practice-band) trade to its edge rather than to its centre?

**Solution of Exercise 26.5.**

Trading to the edge is the smallest trade that restores an acceptable risk. Trading to the centre pays for a larger trade and resets the position to where the next random move is as likely to need another trade as not. Inside the band the risk is accepted by design, so paying to reduce it further is waste.

**Exercise 26.6 ★★.**

Why does the stale-surface error grow as the square root of the refit interval and fall with the option’s expiry?

**Solution of Exercise 26.6.**

The error is the skew times the log move since the fit, divided by $\sqrt\tau$. The log move’s standard deviation grows as the square root of the interval, and the skew’s effect on a strike’s volatility is larger for short expiries, where the smile is steeper in [log-moneyness](https://one-course.com/books/quant/5/en/chapter/7-implied-volatility-and-its-surface#def-dv-implied-volatility-and-its-surface-surface): hence $1/\sqrt\tau$.

**Exercise 26.7 ★★★.**

*Coding.* Draw the [dividend play](#def-dv-options-market-making-in-practice-play)’s assignments at random (each short call assigned with the same probability) for $f=0.3$ and the best size, and give the standard deviation of the net profit.

**Solution of Exercise 26.7.**

The market makers’ assigned calls are hypergeometric: $65\,894$ of $75\,894$ short positions, $72\,894$ exercises. With 200 000 draws the net profit averages $108 552 with a standard deviation of $870 ($871 from the hypergeometric formula); its first percentile is $106 502. Assignment risk is small at this size.

**Exercise 26.8 ★★★.**

*Find the flaw.* “Our toy market maker made $11 141 with limits and lost $607 without: limits add about $11 700 a month.”

**Solution of Exercise 26.8.**

One path. Over 20 seeds of the same toy the difference averages $9 585 with a standard deviation of $11 597 (a standard error of $2 593), and limits did worse in 6 of the 20. Limits reduce inventory risk, and on average in this toy that also helps the P&L, because the flow is one-sided. But a single month is a single draw.

## 26.9 Problem: The Dividend-Play Morning

**Problem 26.1.**

Weekend problem — what the non-exercisers are worth

A share pays a 50-cent dividend tomorrow. Its deep in-the-money call series has 10 000 contracts of public open interest; the put with the same strike is worth 2 cents. Two market makers consider a [dividend play](#def-dv-options-market-making-in-practice-play).

**Part I — Exercise.**

1. Should a holder of the call exercise tonight, and what does failing to cost per contract?
2. Why can early exercise of a call be optimal only just before an ex-date (with zero rates)?
3. What did Pool, Stoll and Whaley find about how often holders fail?
4. Who holds the other side of the public’s calls, and what happens to them when holders fail?
5. Why does assignment pro rata over short positions make the play work?

**Part II — The play.**

6. Write the number of the market makers’ unassigned short calls as a function of $q$ .
7. What is the most they can capture, and at what size?
8. Give the fees per contract traded each way under the chapter’s assumption.
9. Compute the best size and the net profit when 30% fail.
10. Give the net profit when 10% and when 50% fail.

**Part III — Risks.**

11. What happens if the market makers are assigned more than their pro-rata share?
12. What happens if the public exercises more than expected?
13. How does the absence of a fee cap for dividend strategies change the trade?
14. What other risk does the desk carry overnight on these positions?
15. How would you estimate the failure rate for a series in advance?

**Part IV — Judgement.**

16. Who loses from the play, and is it the market makers’ doing?
17. Should the desk do the trade at 10% expected failure?
18. How would a rule change on exercise or fees end the play?
19. State the *named result* : the expected profit of a [dividend play](#def-dv-options-market-making-in-practice-play) on one call series when a given share of holders fails to exercise, net of fees.
20. In one sentence: where does the [dividend play](#def-dv-options-market-making-in-practice-play) ’s profit come from?

**Solution of Problem 26.1.**

**1.** Yes: the dividend ($50 a contract) exceeds the put’s value ($2); failing costs $48 a contract. **2.** With zero rates a call’s early exercise gains nothing except the dividend, which the holder of the share receives and the holder of the call does not. **3.** More than half of the long positions that should have been exercised were not (1996–2006), costing holders over $491 million. **4.** Other short positions, including dealers’. Their shorts are assigned less than they would be if every holder exercised, so they keep the dividend on the unassigned part. **5.** The market makers add $2q$ short positions that share the assignments with the original writers. As $q$ grows, their shorts become almost all of the short interest, so almost all the non-assignment falls to them. **6.** $2q\,fN/(N+2q)$. **7.** $fN=3\,000$ contracts, approached as $q\to\infty$. **8.** $0.50: four trades at $0.10 and two exercises at $0.05. **9.** 32 947 contracts each way; $108 553 net (2 605 unassigned, $125 026 gross, $16 474 fees). **10.** $28 591 (16 909 contracts) and $193 510 (43 990 contracts). **11.** The gain shrinks; the standard deviation from random assignment is about $870 at the best size. **12.** Fewer failures: the gain falls, and at the planned size the fees can exceed it (at 10% failure the best size is half as large). **13.** Fees rise linearly with $q$, so the best size and the profit shrink; with no fees the play would capture all $144 000 at an unlimited size. **14.** The share position of exercised calls and any imbalance of assignment, carried through the ex-date open, and operational risk on exercise instructions. **15.** From the series’ history of exercises on past ex-dates, the share of open interest held by retail accounts, and how deep in the money it is. **16.** The holders who fail to exercise. The market makers do not cause the failure, but they redirect its value from the original writers to themselves. **17.** Only at a size that pays the fees: at 10% failure the best trade nets $28 591 on fees of about $8 500, still worth it on these numbers. **18.** Automatic exercise of calls whose dividend exceeds the time value, or fees on dividend strategies high enough to make $q^*$ zero. **19.** With 30% of 10 000 public contracts unexercised, a $48 gain per unassigned call and $0.50 of fees per contract traded each way: $108 553 at the best size of 32 947 contracts ($28 591 at 10%, $193 510 at 50%). **20.** From option holders who fail to exercise, collected through the pro-rata assignment rule.

## 26.10 Interview questions

**Interview question 26.1 ★ trader.**

How do you set the width of an option quote?

**Solution of Interview question 26.1.**

In volatility points, then times [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks): wide enough to cover the theo’s error against informed traders, narrow enough to keep uninformed flow; estimated from the mark-outs of past trades by counterparty type and adjusted for inventory and limits.

*What the interviewer is looking for: error of the theo, adverse selection, [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) scaling, mark-outs.*

**Interview question 26.2 ★★ trader, researcher.**

How often should an options market maker delta-hedge?

**Solution of Interview question 26.2.**

Not on a clock: in a band around the target delta whose width grows with the cost of hedging and shrinks with gamma and risk aversion (half-width proportional to the cube root of cost times gamma squared), trading to the band’s edge. On the chapter’s example a band gives the same cost as daily hedging with 41% less P&L noise.

*What the interviewer is looking for: the band, and why it beats a clock.*

**Interview question 26.3 ★★ trader.**

Explain a [dividend play](#def-dv-options-market-making-in-practice-play). Who loses?

**Solution of Interview question 26.3.**

On the eve of an ex-date, market makers trade large offsetting amounts of a deep in-the-money call and exercise their longs. Pro-rata assignment leaves most of the public’s unexercised calls assigned against someone else, so the market makers keep the dividend on their unassigned shorts. The holders who fail to exercise lose, and the original writers lose what they would have kept.

*What the interviewer is looking for: pro-rata assignment, failure to exercise, who loses.*

**Interview question 26.4 ★★ developer.**

Design the part of a quoting engine that updates theos as the underlying ticks. What must be fast, and what can be slow?

**Solution of Interview question 26.4.**

Fast: the forward and each series’ theo and delta from the current surface parameters, by closed forms and cached per-expiry quantities, with the marking rule between refits; quote updates when the theo moves by more than a threshold. Slow: surface refits (on a clock and on events), limit checks, width parameters from mark-outs. The quotes are pulled while a refit runs.

*What the interviewer is looking for: separation of hot path and refit, event-driven refits.*

**Interview question 26.5 ★★ risk, trader.**

Why limit [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) by bucket rather than in total? What does the desk do as a bucket nears its limit?

**Solution of Interview question 26.5.**

Offsetting [vegas](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) in different expiries are exposed to changes in the term structure, which total [vega](https://one-course.com/books/quant/5/en/chapter/4-greeks-and-the-hedging-p-l#def-dv-greeks-and-the-hedging-pnl-greeks) hides. Near a limit the desk shades its quotes to attract offsetting trades, hedges with listed options in the bucket, and withdraws the side that would breach the limit only at the limit.

*What the interviewer is looking for: term-structure risk, shading before withdrawing.*

**Interview question 26.6 ★★★ trader.**

It is expiry Friday and the share is sitting on your largest short strike. What do you do?

**Solution of Interview question 26.6.**

That is pin risk: uncertain assignment. Reduce the position before the close (buy back the strike or trade the pin), size the weekend share position to the expected assignment, and keep capacity to hedge Monday’s open; know the exercise cut-offs and the automatic-exercise threshold.

*What the interviewer is looking for: assignment uncertainty and acting before the close.*