Quant from Scratch by QuantWright

Options and Derivatives · Lesson 8

Capstone: hedge an option and explain the P&L

Sell a call, hedge it with shares every day, and say afterwards where the year's profit or loss came from: the volatility you charged against the one the share moved with, how often you adjusted, what adjusting cost, and a crash the hedge could not follow.

Know the basics? Read the lesson notes

You sell the call

This capstone takes one trade through the options course. You make nine decisions, and the page keeps the score: a point for each one right on the first try (practice isn't scored). Every number is made up, and every count comes from simulated years in made-up worlds, which can't show how real prices or real option quotes behave.

You sell the call from the earlier lessons: the right, but not the duty, to buy a made-up share for €105 (the strike) on one day a year from now. The share is at €100 and interest is 5% a year. You charge the Black–Scholes formula's price (the lesson "Black–Scholes, from the tree"), €6.94, at a volatility (how widely the share's yearly returns typically spread) of 17.42%. That volatility is the price's implied volatility, σimp: the one at which the formula gives the price.

If the share ends at €125 you owe €20; if it ends below €105, nothing. The call's delta, how much its price moves per euro of the share's move, is 0.5347 (the lesson "The Greeks"). To cancel your risk for small moves today, how many shares do you buy?

With no shares, every euro the share rises raises the call you owe by about €0.53, and a big rise costs far more than €6.94. Holding the call's delta, 0.5347 shares, cancels small moves.
The call moves 0.5347 per euro of the share, so 0.5347 shares move the same amount and cancel small moves either way.
Per euro the share moves 1 and the call only 0.5347, so a whole share overshoots: you would gain on rises and lose on falls, a bet on the share. 0.5347 shares cancel small moves.

The shares cost 0.5347 × 100 = €53.47: the €6.94 you received plus €46.53 borrowed at 5%. Holding delta shares against an option is delta hedging.

Your score so far–

The next day

Delta changes as the share moves; its change per euro is gamma, 0.0228 today. The next day the share is at €101, and with 364 days left the formula's delta is 0.5570 (up 0.0223, not quite gamma's 0.0228: gamma itself changes along the move, and a day has passed). What do you do?

0.5570 − 0.5347 = 0.0223. Holding the new delta cancels small moves from here.
That would leave 0.5124 shares while the call now moves 0.5570 per euro. The share rose, so delta rose: you buy 0.5570 − 0.5347 = 0.0223.
Yesterday's 0.5347 shares no longer match the call's 0.5570 per euro: each further euro up would cost you about 0.0223. You buy 0.0223 shares.

Moving the holding to the new delta is rebalancing. After a rise you buy, after a fall you sell: a little buying high and selling low every day. The lesson "The Greeks" showed what pays for it: the hedged seller gains a little on quiet days (theta, the price's fall as time passes) and loses on big moves (gamma). Shares and loan together are the copy of the lesson "Black–Scholes, from the tree": a holding that pays what the call pays. Your P&L (profit and loss) for a year is the cash left on the last day, after selling your shares and paying what the call owes.

One simulated year, rebalanced daily (made-up share, the model's own world)

Top: the share's price; dashed, the €105 strike. Bottom: the shares you hold, from 0 to 1.

Model's world, year 1 of 4
Share on the last day–
The call pays–
Your P&L–

Look at a few years. When the share ends above €105 you end up holding about one share, which you sell to pay what the call owes; when it ends below, about none. Either way the year ends within a few tens of cents of €0.

Your score so far–

In the model's own world

Now 10,000 simulated years, drawn the way the Black–Scholes model says: volatility 17.42% all year, no jumps, and the share grows on average at the 5% interest rate. You rebalance daily, and for now trading is free.

What will the 10,000 years' P&Ls look like?

The €6.94 pays for the copy: the daily buying high and selling low costs about that much over the year. What is left is near €0.
Unhedged you might, but here the shares gain when what you owe grows. Hedged, years land near €0.
The copy costs the formula's price, which you charged, so what you owe and what the copy pays nearly cancel. Rebalancing once a day, not without pause, leaves a small spread.

The year's P&L, 10,000 simulated years (made-up share, the model's own world, before costs)

Sold at 17.42%; the share moves with 17.42%; rebalanced daily. Each bar counts years.

In this simulation the average was €0.005, with an error bar (a range that misses the long-run average about 1 time in 20) of −€0.002 to €0.011: it contains 0. The typical distance of a year from the average (its standard deviation) was €0.33; 8,729 of 10,000 years ended within €0.50 of 0, before costs.

That spread is the hedging error: what is left because the copy is rebalanced once a day. As in the Greeks lesson's simulation, each hedged day gains or loses a little, and a year adds up 365 of them.

Your score so far–

The share moves more than you charged for

Same sale at €6.94, same daily hedge. But in this world the share's returns spread by 20% a year, not 17.42%. The volatility a share actually moves with is its realised volatility, σreal. What happens to you, on average?

Bigger moves pay the call's owner. As the seller you pay ½ × gamma × move² on each move (the Greeks lesson), so bigger moves cost you more than the €6.94 covered.
The shares cancel delta's part of each move, not gamma's: each day you still pay ½ × gamma × move², and bigger moves make it bigger.
Copying a call on a share that moves with 20% costs the formula's price at 20%, €7.97, more than the €6.94 you charged.

The year's P&L by realised volatility, 10,000 simulated years each (made-up, before costs)

Sold at 17.42%, rebalanced daily. Each bar counts years; dashed, the exact average. Years beyond the axis are drawn at its ends.

Average–
1.05 × (6.94 − price at σreal)–
Years with a loss–

At 20%, the average was −€1.07 (error bar −€1.08 to −€1.06), and you lost in 9,981 of 10,000 years, before costs.

The reason, step by step. You collected €6.94, which grows like cash to 1.05 × 6.94 by the last day. On average the call costs you 1.05 × 7.97 by then: the share grows 5% like cash, so a payoff priced at 20% (the formula's price with 20% put in, €7.97) averages that much (the binomial tree lesson: a price is the average payoff ÷ 1.05 when the share grows 5%). Your shares add nothing on average, since they grow like the cash borrowed to buy them. So the average is 1.05 × (6.94 − 7.97) = −€1.08. In the model's own world, that is the exact average loss before costs, however often you rebalance; the simulation's error bar contains it. Try the other volatilities: below 17.42% you gain on average, above it you lose.

Your score so far–
Theory

Gamma P&L: where a hedged option's P&L comes from

Each day your hedged position earns ½ × Γ × S² × σimp² × Δt from decay (the call you owe loses value as a day passes; you keep that, less the day's interest on what you borrowed) and pays ½ × Γ × m² for the share's move m, with Γ the gamma (0.0228 at the start, when the decay part is the Greeks lesson's €0.0095 a day) and Δt one day in years. A typical day's m² is S² × σreal² × Δt. Each day's difference is that day's gamma P&L; the year's sum is the formula part, and it explains the hedged P&L. Averaged over many years, it comes to what was charged minus what copying the call really costs.

P&L ≈ Σdays ½ Γt St² (σimp² − σreal²) Δt
average P&L = (1 + r)T × [C(σimp) − C(σreal)]
Σdays
the sum over the year's days
Γt, St
gamma (worked out at σimp) and the share's price on day t
σimp
the volatility the price was charged at
σreal
the volatility the share moves with
Δt
one day in years, 1 ÷ 365
interest
the simulation also grows each day's part with interest to the last day, at most × 1.05; the first line leaves this out
C(σ)
the Black–Scholes price of the call at volatility σ
r, T
the interest rate a year and the years to the last day
P&L
the seller's cash on the last day, hedged with delta at σimp, before costs
Worked example: the 20% world

Day one: ½ × Γ × S² = ½ × 0.02282 × 100² = 114.1. σimp² − σreal² = 0.1742² − 0.20² = 0.03035 − 0.04000 = −0.00965. Day one's part: 114.1 × (−0.00965) ÷ 365 = −€0.0030.

If gamma and the share's price stayed put, 365 such days would make 114.1 × (−0.00965) = −€1.10, the idealised estimate. They don't, so the simulation sums each year's own days: the sums averaged −€1.08. The second line: 1.05 × (6.94 − 7.97) = −€1.08.

Holds when
  • the share moves as the model says, with no jumps, at a steady σreal;
  • daily moves are small (½Γm² is a small-move estimate); the "≈" is the hedging error;
  • before costs; the second line also needs the share to grow on average at the interest rate, as in the model's own world here.
Breaks when

The hedge is rebalanced less often, costs are charged, or the share jumps: the next three screens.

The situationA made-up share is at €50, and a call on it has gamma 0.02. You sold the call at σimp = 20%; the share moves with σreal = 22%.

The idealised estimate: if gamma and the share's price stayed put through a year of daily hedging, how much would you lose, in euros?

Hint: ½ × 0.02 × 50² × (0.20² − 0.22²), with 365 days × 1 ÷ 365 = 1 year.

½ × 0.02 × 2,500 = 25, and 0.04 − 0.0484 = −0.0084, so 25 × (−0.0084) = −€0.21: a loss of €0.21.
½ × Γ × S² = ½ × 0.02 × 2,500 = 25. σimp² − σreal² = 0.04 − 0.0484 = −0.0084. A year: 25 × (−0.0084) = −€0.21, a loss of €0.21.

Check it year by year, then hedge less often

The formula part gives each simulated year its own number. Below, each dot is one year of the 20% world, rebalanced daily: the formula part across, the actual P&L up. The dots sit close to the line where the two are equal. Their spread around it, the standard deviation of actual minus formula part over all 10,000 years, is the hedging error: €0.42.

Formula part against the actual P&L, 1,000 of the 10,000 simulated years (made-up, 20% world, before costs)

Each dot: one year. Dashed: actual = formula part. The same years at every frequency; only the hedge changes. Years beyond the axis are drawn at its edge; the two axes differ in scale.

Spread around the line–
Times daily's–
Average P&L–

Now rebalance weekly instead: 7 times fewer adjustments. By how much will the spread around the line grow?

A week's move is about √7 times a day's, and the error grows with the move squared: 7 times a day's. But there are 7 times fewer weeks, and random errors add up like the square root of their count: √7 times less. 7 ÷ √7 = √7 = 2.65.
Each week's error is about 7 times a day's (the move is √7 times bigger, squared), but there are 7 times fewer of them, and random errors add up like the square root of their count. Net: √7 = 2.65 times.
A week without adjusting lets the hedge drift further from delta, so each week's error is bigger. Net: about √7 = 2.65 times the daily spread.

Press the buttons. In this simulation, weekly: €1.10, 2.63 times daily's. Every four weeks: €2.11, 5.06 times, against √28 = 5.29.

The average stayed at −€1.07 for all three, as the formula's second line says: in the model's own world, rebalancing less often adds spread, not an average loss, before costs. With k days between adjustments, the spread grows about √k times: each gap's error is its move² minus the move² you'd expect, about k times a day's in size, but there are k times fewer gaps, and random errors add up like √(count).

Your score so far–

What adjusting costs

Now every share bought or sold costs 0.1% of its value, a stand-in for fees and the gap between buying and selling prices. In the 20% world, daily rebalancing spent on average €0.68 a year on adjustments, plus €0.11 on the first purchase and the last sale, which every frequency pays.

Weekly makes 7 times fewer adjustments. About what will its adjustments cost?

Each weekly trade is bigger than a daily one: a week's change in delta is about √7 times a day's, since ups and downs in the week partly cancel. 7 times fewer trades, each √7 times larger: √7 = 2.65 times less in all.
A week's change in delta is about √7 times a day's (ups and downs in the week partly cancel), so 7 times fewer trades, each about √7 times larger, cost √7 = 2.65 times less.
Within a week, ups and downs in delta partly cancel, so weekly trades buy and sell less in all: about √7 = 2.65 times less.

Spread against costs, by how often you rebalance (made-up, 20% world, 10,000 simulated years)

Dark: the hedging error's spread, before costs. Orange: average costs a year, all trades.

In this simulation, weekly adjustments cost €0.26, 2.63 times less than daily; every four weeks, €0.13, 5.10 times less (√28 = 5.29).

A week's change in delta is about √7 times a day's, because ups and downs within the week partly cancel: 7 times fewer trades, each about √7 times larger. Rebalancing more often: a smaller spread and larger costs. Less often: the reverse. No frequency removes both; which matters more depends on the cost per trade and on the spread a seller can carry, and this simulation shows only the trade between them.

Your score so far–

A crash the hedge can't follow

Back to free trading, and to the crash world of the lesson "Implied volatility and the smile". In 1 year in 10, on one day, the price is multiplied by 0.78 (a 22% fall) on top of its usual move: a jump, a move too large and sudden for a daily hedge to follow. The usual moves are calmer, 15.74%, so the year's log returns still spread by 17.42%, the volatility you charged, and the share still grows 5% a year on average.

You sell at €6.94 and rebalance daily. With σreal = σimp over the year, the formula says each year should end near €0. How will the 10,000 years look, before costs?

On usual days the share moves at 15.74%, less than the 17.42% you charged, so their formula part is a small gain. The crash is one 22% move, and ½ × gamma × move² describes small moves only.
The year's spread matches, but it arrives differently: calmer usual days, which you gain on, and one 22% fall, which a daily hedge can't follow.
It is the other way round. Usual days are calmer than 17.42%, which you gain on; on the crash day your shares fall 22% at once.

The year's P&L in the crash world, 10,000 simulated years (made-up, daily, before costs)

Top: years without a crash. Bottom: crash years. Each panel has its own height scale. The small hump near €0 among crash years: 131 ended above −€1, and in 114 of them you held under 0.1 shares when the crash came, with the share well below €105.

The 8,953 years without a crash averaged +€0.70 (error bar €0.69 to €0.71); the 1,047 crash years, −€5.58 (−€5.79 to −€5.37), and the worst lost €18.27. You gained in 8,934 of 10,000 years; the average of all years was €0.05 (€0.00 to €0.09). These counts are this simulation's, before costs.

Why one day costs so much: say the share falls 22% on the first day, from €100 to €78. Your 0.5347 shares lose 0.5347 × 22 = €11.76. The call you owe falls from €6.94 to €0.53, €6.41 in your favour. Net, about −€5.35 in one day. Your shares fell all the way at the old delta; rebalancing the next day comes too late. The year's volatility matched the price, but part of it came in one jump, which the formula's hedge misses.

Your score so far–

Explain a year's P&L

Splitting a result into the parts that made it is P&L attribution. For a crash-world year, rebalanced daily, with the 0.1% costs back on, there are four parts: the formula part (usual days, with σreal = 15.74%), the crash day's own result, the costs, and the rest, the hedging error. They add up to the year's P&L.

Year 15 of the simulation crashed on day 288 and ended at −€9.28 after costs. Its formula part was +€0.64, its crash day −€8.94, its costs −€0.69. How much did the hedging error lose, in euros?

Hint: the four parts add up to −9.28, so the rest is −9.28 − 0.64 + 8.94 + 0.69.

−9.28 − 0.64 + 8.94 + 0.69 = −0.29: the hedging error lost €0.29.
The rest is the total minus the other three: −9.28 − (+0.64) − (−8.94) − (−0.69) = −0.29, a loss of €0.29.

Three simulated years, explained part by part (made-up, crash world, daily, after costs)

The first year without a crash and the first two with one. The parts add up to the total.

Crash-world year 1 had no crash: quiet days paid +€1.00, costs took €1.02 (more than the crash world's average €0.70: a busy year, its price near the strike on most days, where delta moves most, and its last sale of about a whole share), the hedging error added €0.16, total +€0.14. In year 4 the crash on day 121 made −€7.37 of its −€7.44. The formula part is collected a little at a time; the crash is paid in one day.

Your score so far–

What the simulation can't show

Every count here comes from made-up worlds built to the rules stated on each screen. They can't show real prices, which can jump in ways no list of worlds covers. They hold the implied volatility fixed for the year, while real option prices move: a call sold at 17.42% and valued again later at a higher implied volatility shows a loss before the last day. They leave out liquidity, whether you can trade as much as you need near the price shown, and margin, cash a broker asks a seller to put up when the share moves against them, which can force a sale at the worst moment. Dividends and exercising the option early are left out too.

On the crash day here, your rebalance happened at the day's closing price, at the usual cost. What real problem does that leave out?

The formula gives delta at any price, €78 included. The problem is trading on it: in a crash the price you can sell at, and how much you can sell, may be much worse than the closing price.
That is liquidity: when everyone sells at once, the gap between buying and selling prices widens and large trades move the price. The simulation charged 0.1% as on any day.
Nothing on this page stops interest. The gap is liquidity: in a crash, selling may only be possible at worse prices than the close, at higher costs.
Your score so far–
Practice 1 of 3

A friend's sale

The situationA friend sells the same made-up €105 call at σimp = 30%, for €11.92, and hedges it in a model world where the share moves with 25% and grows on average at the 5% interest rate. At 25% the formula's price is €9.95.

Before costs, what is your friend's average P&L on the last day, in euros?

Hint: (1 + r)T × [C(σimp) − C(σreal)], with T = 1.

1.05 × (11.92 − 9.95) = 1.05 × 1.97 = €2.07: the copy costs €9.95 and your friend charged €11.92.
The copy costs the formula's price at the volatility the share moves with, €9.95; your friend charged €11.92. The €1.97 difference grows at 5%: 1.05 × 1.97 = €2.07, whatever the rebalancing, before costs.
Practice 2 of 3

Adjusting less often

The situationIn a made-up model world, rebalancing a hedge every day spends on average €0.70 a year on adjustments.

About how much would rebalancing every 4 days spend on adjustments, in euros?

Hint: with k days between adjustments, adjusting costs shrink about √k times.

√4 = 2, so 0.70 ÷ 2 = €0.35: 4 times fewer trades, each about twice as large.
Not 0.70 ÷ 4: a 4-day change in delta is about √4 = 2 times a day's, since ups and downs partly cancel. 4 times fewer trades, each twice as large: 0.70 ÷ 2 = €0.35.
Practice 3 of 3

Nine years in ten

The situationA made-up course seller writes: "Sell options and hedge them every day. In our simulation it gained in 9 years out of 10. Low risk."

What should you ask first?

A share of winning years says little: in this lesson's crash world the hedged seller gained in 8,934 of 10,000 years before costs, and crash years averaged −€5.58.
Delta hedging is what made 9 in 10 possible; it doesn't make the tenth year small. In the crash world, crash years averaged −€5.58 before costs.
In this lesson's crash world, gains came in 8,934 of 10,000 years, before costs, but crash years averaged −€5.58 and the worst lost €18.27. Whether the seller's simulation has jumps at all decides how the tenth year looks.
✓

Lesson done

A delta-hedged seller's year, before costs, is close to the sum over its days of ½ × gamma × S² × (σimp² − σreal²) × 1/365, when the share moves without jumps. In this lesson's simulated model world, charging the volatility the share moved with averaged about €0, with a spread from daily rebalancing; charging 17.42% while it moved with 20% lost €1.07 on average, against the exact 1.05 × (6.94 − 7.97) = −€1.08. In these simulations, rebalancing every k days widened the spread about √k times and cut adjusting costs about √k times. In a simulated world with crashes, the same yearly volatility arrived partly in one jump, which the daily hedge could not follow: most years gained a little, crash years lost a lot. These counts come from made-up worlds; real prices, real quotes, liquidity and margin are outside them.

Your scoreAnswer the steps above to see it.

Key ideas

Delta hedging
holding delta shares against an option, so that small moves of the share cancel
Rebalancing
moving the holding to the new delta, at chosen intervals
Hedging error
what a delta hedge leaves because it is rebalanced at intervals, not without pause; in the Black–Scholes model's world its spread grows about √k times with k days between adjustments
Realised volatility
the volatility a share actually moves with, as against the implied volatility a price was charged at
Gamma P&L
a delta-hedged option's result from one day: about ½ × gamma × S² × (σimp² − σreal²) × the day in years, for the seller, when moves are small
P&L attribution
splitting a result into the parts that made it: here the formula part, a jump, costs and the hedging error
Jump
a move too large and sudden for the hedge to follow, outside what ½ × gamma × move² describes

P&L ≈ Σdays ½ Γt St² (σimp² − σreal²) Δt, no jumps, before costs

average P&L = (1 + r)T × [C(σimp) − C(σreal)], in the model's world, before costs

Lesson notes

A seller who hedges a call with delta shares, rebalanced daily, ends the year with a result close to the sum over its days of ½ × gamma × S² × (σimp² − σreal²) × 1/365, when the share moves without jumps and before costs: what matters is the volatility charged against the volatility the share moved with. Rebalancing at intervals leaves a hedging error, costs grow with the rebalancing, and a jump falls outside the formula. Every number is made up or simulated, and every count belongs to its simulation, not to real markets.

The sale and the hedge

The seller sells a made-up European call (the right, not the duty, to buy a share for €105, the strike, on one day a year away) on a share at €100, with interest 5% a year. The price, €6.94, is the Black–Scholes formula's at an implied volatility σimp = 17.42% (the volatility, how widely the share's yearly log returns typically spread, at which the formula gives the price). To cancel small moves the seller buys the call's delta, 0.5347 shares, for €53.47: the €6.94 plus €46.53 borrowed. That is delta hedging. Delta changes by gamma, 0.0228 per euro, as the share moves: at €101 a day later it is 0.5570, so the seller buys 0.0223 more. That is rebalancing.

The model's own world

In 10,000 simulated years in which the share moves as the Black–Scholes model says (volatility 17.42%, no jumps, growing on average at 5%), with daily rebalancing and free trading, the seller's P&L (the cash left on the last day) averaged €0.005, error bar (two standard errors either side) −€0.002 to €0.011, with a standard deviation of €0.33; 8,729 years ended within €0.50 of 0. The shares and loan copy the call, and the copy costs the price charged. The spread is the hedging error, left because the copy is rebalanced once a day.

Realised volatility different from the price

When the share moves with a realised volatility σreal of 20%, the copy costs the formula's price at 20%, €7.97. The seller charged €6.94, so on average loses 1.05 × (7.97 − 6.94) = €1.08 by the last day. The simulation's average was −€1.07 (error bar −€1.08 to −€1.06), and the seller lost in 9,981 of 10,000 years, before costs. Below 17.42% the seller gains on average.

Hedging less often, and costs

Each simulated year's formula part, the sum over its days of its own gamma P&L, sat close to its actual P&L: in the 20% world the spread of actual minus formula part was €0.42 with daily rebalancing, €1.10 weekly (2.63 times) and €2.11 every four weeks (5.06 times), close to √7 = 2.65 and √28 = 5.29. A longer gap means a bigger error each time (the move squared) but fewer of them, which add up like the square root of their count. The average stayed at −€1.07, before costs.

With a cost of 0.1% of every share bought or sold, daily adjustments cost on average €0.68 a year in the 20% world, weekly €0.26 (2.63 times less) and every four weeks €0.13 (5.10 times less), plus about €0.11 for the first purchase and last sale. A week's change in delta is about √7 times a day's, because ups and downs partly cancel. More frequent rebalancing trades a smaller spread for larger costs; the simulation shows the trade, not which side to take.

A crash the hedge can't follow

In a made-up crash world (1 year in 10 has one day on which the price is multiplied by 0.78, a 22% fall; usual moves at 15.74%, so the year's log returns still spread by 17.42%), the same sale and daily hedge gave, before costs, an average of +€0.70 in the 8,953 years without a crash and −€5.58 in the 1,047 crash years; the worst lost €18.27, and the average of all years was €0.05. A jump is too sudden for the hedge: a 22% fall on the first day costs the 0.5347 shares €11.76 while the call falls only €6.41, about −€5.35 in a day.

P&L attribution splits a year into its parts. Simulated year 15, after costs: formula part (usual days at 15.74%) +€0.64, crash day −€8.94, costs −€0.69, hedging error −€0.29, total −€9.28.

What the simulation can't show

Real prices, which can jump in ways no list of made-up worlds covers; real option prices, whose implied volatility moves during the year; liquidity, whether a trade of the size needed happens near the price shown, which is worst in a crash; margin, cash a broker asks a seller to put up when the share moves against them; dividends and exercising the option early.

Formula: where a hedged option's P&L comes from

For a seller hedged with delta at σimp, on a share moving without jumps at a steady σreal, before costs: each day the position earns ½ΓS²σimp²Δt from decay (the call losing value as a day passes, less the day's interest on the loan) and pays ½Γm² for the move m, whose typical square is S²σreal²Δt. Each day's difference is its gamma P&L; the year's sum is the formula part.

P&L ≈ Σdays ½ Γt St² (σimp² − σreal²) Δt
average P&L = (1 + r)T × [C(σimp) − C(σreal)]
Γt, St
gamma (at σimp) and the share's price on day t
σimp, σreal
the volatility charged; the volatility the share moves with
Δt
1 ÷ 365
interest
the simulation also grows each day's part with interest to the last day, at most × 1.05
C(σ)
the Black–Scholes price at volatility σ
r, T
the interest rate a year; the years to the last day
Worked example

Sold at 17.42%, share moving with 20%: on day one ½ × 0.02282 × 100² = 114.1; 0.1742² − 0.20² = 0.03035 − 0.04 = −0.00965; day one's part 114.1 × (−0.00965) ÷ 365 = −€0.0030. A whole year at fixed gamma and price: −€1.10; the simulated years' own sums averaged −€1.08, the exact average 1.05 × (6.94 − 7.97) = −€1.08.

Holds whenBreaks when

The share jumps (a crash day), costs are charged, or the hedge is rebalanced less often, which widens the "≈".

One more worked example

New made-up numbers

The same €105 call is sold at 25%, for €9.95, and the share moves with 20%, at which the formula's price is €7.97. In the model's world, before costs, the average P&L on the last day is 1.05 × (9.95 − 7.97) = 1.05 × 1.98 = €2.08, whatever the rebalancing.

Common mistakes

Check yourself

  1. A call is sold at an implied volatility at which the formula gives €8.00. The share then moves with a volatility at which the formula gives €8.50; interest is 5% and the call has a year to run. In the model's world, before costs, what is the hedged seller's average P&L on the last day?
    Answer

    1.05 × (8.00 − 8.50) = −€0.525: the copy costs €8.50, €0.50 more than was charged, grown at 5%.

  2. On one day ½ × Γ × S² = 50, the call was sold at 20% and the share moves with 15%. What is that day's gamma P&L for the seller, before interest growth?
    Answer

    50 × (0.04 − 0.0225) ÷ 365 = 50 × 0.0175 ÷ 365 = €0.0024, a gain.

  3. Daily rebalancing leaves a hedging error with a spread of €0.40. About what spread would rebalancing every 9 days leave, in the model's world?
    Answer

    About √9 = 3 times wider: €1.20.

  4. Why can't a daily hedge follow a 22% fall in one day?
    Answer

    The hedge holds the old delta through the whole fall, and rebalancing the next day comes too late. With 0.5347 shares, a fall from €100 to €78 lost €11.76 on the shares while the call fell €6.41: about −€5.35.

Try it yourself

Run the script. Every number in this lesson comes from one Python script with a fixed random seed (below). It hedges the call through 10,000 years in each world, adds up each year two ways (the cash flows, and the days' results grown with interest) and stops if they disagree, and checks each simulated average against 1.05 × (C(σimp) − C(σreal)). Change the cost, the frequency or the crash size and see which numbers move.

Work it out by hand. In a spreadsheet, draw a year of daily prices, work out delta each day with NORM.S.DIST, and keep a running cash column: add the sale, pay for shares bought, receive for shares sold, add a day's interest. On the last day sell the shares and pay the call.

The code behind every number (Python, seed 2026111181)
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For the precise: why the average is exact, and the crash world's average

The average. In the model's world the share grows on average at the interest rate, so any rebalancing's gains on the shares, discounted, average 0, and the payoff's discounted average is the formula's price at σreal. The seller's last-day cash is 1.05 × C(σimp) − payoff + the shares' gains, so its average is 1.05 × (C(σimp) − C(σreal)) for daily, weekly or four-weekly rebalancing alike. The script checks every simulated average against it (within three standard errors) at 12%, 15%, 17.42%, 20% and 25%.

The crash world. Its year-end prices are distributed as in the lesson "Implied volatility and the smile", where this call's price is €6.83. The crash is placed on one day drawn evenly from the year, so day by day the share is not built to grow at the interest rate, and the hedged average, €0.05 before costs, is not a price. After costs, crash years averaged −€6.07, years without a crash −€0.02.

Honest notes

Where this comes from