Where to go next, and a summary

OpeningYou can have an edge and still lose

Suppose you know for certain that a $100 stock will realize 30% volatility over the next 30 days, while 30-day ATM options carry an implied volatility of only 25%. You buy 10 ATM calls ($3.02 per share), hedge Delta back to zero with stock at every close, and hold to expiration.

As Chapter 4 showed, once you hedge, only two terms are left in the P&L: Gamma and Theta. Which one is bigger depends on whether realized or implied volatility is higher. This option's Vega is 0.114 per share, so 5 vol points are worth 0.114 × 5 × 1,000 = $570. That is your theoretical edge.

The expectation is positive, and the simulated average is almost exactly the theoretical value. But on any single trade, the spread of outcomes is as large as the expectation itself, and roughly one trade in eight loses money. And all of this assumes you “know realized volatility for certain.” In reality you can only estimate it, and the error is often more than 5 points.

Most of the advanced directions in this chapter look like this: a small edge on each trade and a lot of noise, so they only pay off with heavy repetition, cheap execution and strict position sizing. Sections 16.1–16.6 look at what each of these directions requires. Sections 16.7–16.9 pull the whole course together into one master table, four tools and a 90-day plan, and end with the most important sentence in the course.

16.1Volatility arbitrage: trading the gap between two volatilities

Volatility arbitrage earns the difference between implied volatility and the volatility that is then realized. Direction is just noise to be hedged away.

This formula extends Chapter 4's “breakeven move” to the whole holding period. Every day you are asking the same question: once you square today's move in the stock, is it bigger than the share that implied volatility allowed for?

The two rows are identical because an ATM option's price is almost linear in volatility, so “Vega × vol difference” is very accurate near the money. OTM options are different: their Vega itself changes with volatility (Vomma), so when volatility jumps, the approximation understates the P&L.

The money depends on where the move happens

The formula contains ΓS², so a move of a given size is only worth money where Gamma is large. With 15 days left, 10 contracts of the 100 call earn $157 on the Gamma part from a single 2% daily move with the stock at 100; the same 2% move with the stock at 115 earns only $3.4, because Gamma there is just 0.0013, versus 0.0786 at the money. The two paths work out to the same realized volatility, yet the money differs by more than 40 times.

This is called path dependence. What you buy is not “volatility over the whole period” but “volatility near the strike, close to expiration.” If the stock moves far from the strike early on, the big moves that come later have nothing to do with you.

Two hedging conventions

• Delta at implied volatility (the most common): the daily mark-to-market P&L is smooth, but the total P&L depends on the path. The $157 vs. $3.4 difference above comes from this convention.
• Delta at your forecast of realized volatility: if the forecast is right, the total P&L is in theory locked in as the difference between the option's value at the two volatilities. The price is large mark-to-market swings along the way, and if the forecast is wrong, nothing is locked in.

Both conventions have the same expectation; they differ in where the risk lands. For an individual trader, a third thing matters more: where your “forecast of realized volatility” comes from. Common sources are recent realized volatility, the event calendar (Chapter 10) and comparisons with similar underlyings, and all three have errors. When the edge is only 2 points and the forecast error is 4 points, you don't actually know the sign of the trade.

16.2Discrete hedging: frequency and error

The less often you hedge, the wider the P&L distribution. The standard deviation of the error is roughly inversely proportional to the square root of the number of hedges: hedge four times as often and the error halves.

Continuous hedging exists only in textbooks. In the real world you rebalance only every so often, and every move in the stock between two rebalances goes unhedged, leaving an error. The error averages zero, but it can turn an otherwise steady trade into something close to a coin flip.

Where the approximation comes from: the error in each interval is ½ΓS²(r² − σ²Δt), and the random part of r² has a standard deviation of about √2 σ²Δt. Adding up N independent errors makes the variance grow with N, while the interval length Δt = T/N shrinks each interval's variance by 1/N². Together, the standard deviation falls as 1/√N. The constant √(π/4) comes from averaging Γ² over the possible stock prices and the remaining time. The further you are from these conditions, the worse the approximation: with only 6 hedges, it gives $1,031, while the simulation gives $904.

Those two columns are the trade-off. The error falls as 1/√N, while slippage rises roughly as √N: the number of shares per rebalance shrinks as 1/√N, and the number of rebalances grows as N. Going from once a day to 8 times a day cuts the error from $455 to $150 but raises slippage from $33 to $96, almost three times as much. The model splits each day into equal intervals and doesn't distinguish trading hours from overnight. In real markets you can't rebalance overnight, so hedging several times a day helps less than this suggests.

Three hedging rules used in practice

• Fixed time: once a day, at the close. Simple, but it leaves a full day of exposure between hedges.
• Delta threshold: rebalance only when Delta drifts beyond a set amount (such as ±50 shares, or ±10% of portfolio Delta). Rebalancing happens when the stock has actually moved, so the error is smaller for the same cost.
• Price grid: rebalance each time the stock moves a fixed amount (such as half a daily standard deviation). It works about as well as the threshold rule and is easier to set up in advance with conditional orders.

No rule can handle a gap. If the stock gaps 5% overnight, you have no chance to rebalance in between: positive Gamma makes extra money in a gap, negative Gamma loses extra, and no hedging frequency changes that.

16.3Relative value on the volatility surface

Surface trades don't ask whether an option is expensive in absolute terms, only which part of the same underlying's surface is expensive relative to another.

Chapter 6 broke implied volatility into three dimensions: level (ATM IV), skew (across strikes) and term structure (across expirations). Plot them together and you get the volatility surface. A relative-value trade buys a relatively cheap part of the surface, sells a relatively expensive part, and hedges away what the two have in common.

Two common skew measures: the 25-Delta risk reversal = 25Δ call IV − 25Δ put IV = 22.5 − 29 = −6.5 points, which measures “how much more the left side costs than the right”; and the 25-Delta butterfly = average of the wings − ATM = (29 + 22.5)/2 − 25 = 0.75 points, which measures “how much more the wings cost than the middle.” In the table, the 95 put and the 105 call have Deltas of almost exactly ±0.25.

Skew trades

If −6.5 sits at the steepest end of this stock's skew over the past year, you might conclude that downside protection has been bid up too far. There are three ways to express that view:

• Risk reversal: sell the 95 put and buy the 105 call. With this skew you take in a net $0.309/share; if the surface were flat, the same combination would cost $0.271. The catch is a Delta of about +49 shares per set, which you have to hedge with stock before it counts as a “pure skew” trade.
• Put spread: buy the 100 put and sell the 95 put. With this skew it costs $1.450, versus $1.797 on a flat surface. When you want protection but it looks expensive, use a spread to sell the expensive leg back (Chapter 11).
• Put ratio spread: buy 1 ATM put and sell 2 OTM puts. You sell more skew, but the downside becomes a naked short, so it has to come with a tail-hedge budget.

Steep skew usually has a reason: the market is pricing some kind of decline. A relative-value trade assumes you can explain why that pricing has gone too far. If you can't, you are just selling insurance.

Term-structure trades

IVs at different expirations are tied together by one hard constraint: total variance σ²T must increase with time to expiration. From this you can work out the forward volatility between two expirations:

Example: 30-day IV is 40% (with an earnings report inside that window) and 60-day IV is 30%. Forward volatility = √((0.09 × 60 − 0.16 × 30)/30) = 14.1%. The market is saying that over the 30 days after earnings, volatility will be only about 14%. If the stock normally realizes around 25%, the far month looks cheap, and a calendar spread that sells the near month and buys the far month (Chapter 9) is a bet on exactly that view.

The same constraint lets you check whether quotes are consistent: with 30-day IV at 40%, 60-day IV can't be below 40% × √(30/60) = 28.3%. Otherwise total variance would fall with time to expiration, and a calendar spread would contain a riskless opportunity. When you see quotes like this in practice, it is usually bad data, or a factor you haven't accounted for, such as dividends or stock borrow.

Applied to events, it also lets you back out the implied size of the earnings move itself (Chapter 10). If IV is normally 25% and 30-day IV has been pushed up to 40%, the event contributes variance of (0.40² − 0.25²) × 30/365 = 0.0080. That is equivalent to a one-standard-deviation move of about 8.95% on earnings day, or an average absolute move of about 7.1%.

Across underlyings: the index and its components

An index's volatility depends on both its components' volatilities and the correlations between them. With equal weights and equal component volatilities:

With 10 equal-weighted stocks each at 35% IV and the index at 20% IV, the implied correlation backs out to about 0.25. If the average correlation among the components is 0.5, index IV should be close to 26%. Selling index volatility and buying component volatility is called a dispersion trade, and at heart it sells correlation: in normal times, when stocks go their own way, it makes money; in a crisis, correlations spike and the losses come all at once. This is an institutional tool, and it is enough for an individual to know it exists. It also explains why index skew is usually steeper than single-stock skew: when the index falls, the correlations among its components rise together.

16.4The market maker's view

Market makers earn the bid–ask spread and volume; direction and volatility are inventory risks they have to keep under control. Understand their constraints and you will know how to place your orders.

A 30-day ATM call has a theoretical value of $3.02, and a market maker might quote it 2.97 bid, 3.07 ask. On each contract traded, they expect to earn half the spread: $0.05 per share, or $5 per contract. Set that $5 against the risk they carry. In Section 16.2, 10 of the same options hedged once a day had a per-trade error with a standard deviation of about $455, while the spread income on 10 contracts is only $50. For a $50 edge to hold up statistically takes roughly (2 × 455 / 50)² ≈ 331 uncorrelated trades. A market maker can do that many in a day; an individual trader can't in a lifetime.

What sets the width of a quote

• Vega and volatility uncertainty: this option's Vega is 0.114, so misjudging IV by 0.5 points costs $0.057, and the spread has to cover at least that.
• Gamma and time to expiration: near expiration and near the money, Gamma is large, hedging error is large, and spreads widen.
• Events: before earnings and major data releases, information asymmetry grows, and market makers widen their spreads so informed traders don't “pick them off.”
• Inventory: when a market maker has already sold too many puts, they raise both the bid and the ask on puts, to draw in sellers and sell fewer themselves. Part of the skew you see in quotes is market-maker inventory.
• Competition: the more actively an underlying trades, the tighter its quotes.

Five rules for trading against market makers

1. Use limit orders. Start at the mid and give up $0.01–0.05 at a time; don't just cross the spread.
2. Check the spread as a share of price first: on contracts where spread ÷ mid is over 10%, the cost of getting in and out will eat most of your edge.
3. Enter multi-leg structures as a single combo order, so one leg doesn't fill while the price of the other runs away.
4. Quotes are widest in the first few minutes after the open and around major data releases. Keep orders that aren't urgent out of those windows.
5. Work out your exit cost in advance: on illiquid contracts, a stop-loss order can fill at a very bad price.

A worked example: a 30-day iron condor (sell the 95 put and 105 call, buy the 90 put and 110 call, IV 25%) takes in $1.519 in theory. If each leg has a $0.10 bid–ask spread and you cross the spread both to open and to close, the cost is 4 × $0.05 × 2 = $0.40, or 26% of the premium. If your plan is to close once you have made half ($0.76), that $0.40 is 53% of your target profit. Seller strategies, with their “high win rate, small profit per trade,” are most vulnerable to exactly this kind of invisible cost.

16.5Cross-asset options

Indexes, rates, commodities, currencies and even volatility itself all have options. You read the Greeks the same way; what changes is how the underlying moves and how the contract settles.

Table 16-4 Options on common asset classes: read the specs before you think about strategy

Volatility as the underlying

VIX options are the easiest to misuse. Spot VIX can't be traded directly; VIX options are priced off the VIX future for the matching month. In calm periods, VIX futures usually sit above spot (an upward-sloping term structure), so the intuition that “VIX is only 15, so a 20 call is cheap” is often wrong: the matching month's future may already be near 18. VIX is right-skewed and mean-reverting, so long calls lose Theta most of the time and pay off only in sharp sell-offs. They work better as part of a tail hedge (Chapter 11) than as a directional bet.

16.6Short-option exposure in structured products

The high coupon on snowballs and autocallable notes is what the investor gets for selling a put with a barrier.

Structured products package options as “notes” and sell them to investors. The most common type is the autocallable note; a variant common in China is called the snowball. Below, we take one apart using a hypothetical set of typical terms.

Term

12 months, linked to a stock index, with the starting level set at 100

Knock-out

Observed monthly from month 3; if the index is ≥ 100, the note ends early and pays the annualized coupon for the months held

Knock-in

Observed every trading day; once the index is ≤ 75, the note is marked “knocked in”

Maturity

Never knocked in: the full 12 months of coupon. Knocked in and below 100 at maturity: the principal loses as much as the index fell, with no coupon

The three outcomes correspond to three kinds of options. Early knock-out and reaching maturity without a knock-in are both digital options: “meet the condition, get a fixed payment.” The loss after a knock-in is exactly the seller's payout on a down-and-in put with a strike of 100 and a barrier of 75. Put together, the investor = holds a short-term deposit + sells a knock-in put + gives up the index's upside (a rally only buys an early exit and a few months of coupon).

Where the coupon comes from, in numbers

Price these terms on 2,000 risk-neutral paths (4% rate, no dividends, IV 25%) and solve for the coupon that makes the note's present value exactly equal to the principal: the fair annualized coupon is 18.7%. That looks far higher than the premium on a one-year ATM put (7.9% of principal). But in present-value terms, all the coupons the investor might receive are worth only 6.14% of principal:

• 4.19% pays for the knock-in put they sold, 68% of the coupons' present value;
• 1.96% is interest on the principal while it is tied up.

There are three reasons the 18.7% annualized figure looks high: almost half the paths (48.9%) knock out in month 3 and collect only 3 months of coupon; the average holding period is about 0.50 years; and the 16.1% of paths that knock in and lose get no coupon at all. Only 6.4% of paths neither knock out nor knock in, and collect a full year.

The coupon rises with IV, and so does the probability of a knock-in loss. A high coupon is the price of high risk. Issuers launch higher-coupon products when markets are volatile precisely because that is when the put being sold is worth more, and more likely to be triggered.

What the investor actually sold

• Volatility: with the coupon fixed at 18.7%, a rise in IV from 25% to 30% lowers the note's present value by about 2.3% of principal; a drop in IV to 20% raises it by 2.7%. This is a negative-Vega position.
• The downside tail: losses appear only after a big drop, and the average loss is 27%, the same shape as selling an OTM put.
• The upside: a big rally in the index only gets you an early knock-out and a few months of coupon.
• Liquidity: notes usually can't be sold at fair value whenever you like, so once one knocks in, it is hard to cut your loss.

The issuer's profit is hidden in the coupon. If the same terms are quoted at 15%, that is 3.7 points below fair value. The coupon's present-value factor is 0.328 (every 1 point of annualized coupon is worth 0.328% of principal), so the investor is effectively giving up 3.7 × 0.328 ≈ 1.2% of principal.

16.7Master table: strategies × three exposures

Any structure can be written as three signs: direction, time and volatility. Decide which signs you want first, then pick the structure.

Chapter 14's “three-dimensional view” asks you to answer three questions before you trade: are you bullish or bearish, do you want time working for you or can you live with it working against you, and do you expect volatility to rise or fall? The table below puts the 24 structures from across the course under the same parameters (stock at 100, 30 days, IV 25%; the far months of the calendar and diagonal are in parentheses) and computes Delta, Theta and Vega for each one.

Four patterns stand out in this table:

• Theta and Vega almost always have opposite signs: structures that pay for time are buying volatility, and structures that collect time are selling it. This is another way of stating Chapter 4's Gamma–Theta trade-off.
• The exceptions are calendars and diagonals (highlighted in the table): selling the near month and buying the far month makes both Theta and Vega positive. They earn from the near month decaying faster than the far month, and they bet that far-month IV won't fall.
• A vertical spread's signs flip as the stock moves: a bull call spread has positive Vega with the stock at 100 (the long leg is at the money). Once the stock rises past the midpoint between the two strikes (about 102.5), the short leg is closer to the money, and Theta and Vega flip sign together. At 106, Theta becomes +1.32 and Vega becomes −3.72. The master table is only a snapshot of “right now.”
• When the signs match, look at the tail: the cash-secured put, the covered call and the bull put spread are all + / + / −. The difference is on the downside: the first two have almost no protection, while the third has a capped loss. The three signs pick the candidates; the maximum loss makes the final choice. The collar and the risk reversal have small Theta and Vega, and their signs depend on the actual skew.

16.8Four tools and a 90-day plan

The course leaves you with four things you can use every day: the calendar and dashboard tell you when, and whether it is expensive; the matrix tells you what to use; the rule card tells you how big; the journal tells you whether you did it right.

Table 16-7 The four tools: where they are, what they answer, how to use them

The order of the four tools is the skeleton of Chapter 14's nine-step process. Their value comes from repetition: use the same template a hundred times and you will see your real win rate, average P&L and most common mistakes. In the journal, be especially careful to flag trades that “made money but were done wrong.” Losing money forces you to reflect, but making money rewards the wrong behavior. Appendix D has four templates you can copy directly.

The 90-day practice plan

The plan has three stages of four weeks each. Each stage ends with a pass standard; if you don't meet it, repeat the stage instead of moving on. Checkmarks are saved only in your own browser.

Completed 0 / 12 weeks

Why no orders in the first four weeks? Because those weeks are for building a sense of what is “normal”: the range a stock's IV usually sits in, how much it rises before earnings, how wide the spreads are. Without that reference, you can't tell whether any given day's quotes are expensive or cheap. Why only defined risk in the second stage? Because that stage is for practicing execution and record-keeping, and the worst case has to be small enough that it doesn't stop you from continuing to practice.

16.9Course summary: first, learn not to lose big

Options are tools for shaping risk: they let you decide in advance how much you make or lose in each scenario. Using them as tools for leverage is where almost every “way to die” in this course begins.

One view, three ways to express it. Your account is $20,000, and you think a $100 stock will rise 10% within 90 days; its IV is 30%. The 90-day ATM call costs $6.41 per share, and the 100/110 call spread costs $3.68 per spread.

The all-in calls go almost to zero in three of the six scenarios: if the stock doesn't rise, you lose everything. At market prices, the risk-neutral probability that the stock ends above the calls' breakeven of 106.41 is only about 34%. The spread version looks like it makes too little, but its worst case is 1.8%: you can make a judgment of the same quality fifty times, while the all-in version can end everything at once. The third way is what options are really for: first decide the worst outcome you can bear, then use a structure to shape the payoff into what you want.

The math of losses is asymmetric: lose 10% and you need 11.1% to get back to even; lose 30% and you need 42.9%; lose 50% and you need 100%; lose 80% and you need 400%. Chapter 13's rule card and Chapter 15's list of how accounts die say the same thing: control the worst case first, then talk about returns. Case 3 uses a hundred trades to show that even when every trade has a positive expectation, oversized positions make most accounts shrink over time.

Sixteen chapters, one sentence each

CasesThree cases

ToolsChecklist and self-test

Checkmarks are saved only in your own browser.

Self-test: five questions