Delta Hedging

When trading desks manage an option position, delta hedging sets the basis for how much of the underlying to buy or sell.

What delta hedging offsets

The delta of an option is the ratio of the change in the option price to a small change in the price of the underlying asset. If you hold an amount of the underlying equal to the delta, in the opposite direction, the option's profit or loss from small moves in the underlying price is roughly canceled by the profit or loss on the underlying. This kind of hedging, which uses delta to limit the risk of option price moves, is called delta hedging¹. The definitions and signs of delta and the other Greeks are covered on the page The Greeks.

Consider a hypothetical example in which you sell 1 unit of a call option (call) with an underlying price of 100, a strike price of 100, 0.5 years to expiry, and an interest rate of 1% (continuously compounded). The implied volatility is 20%, and the call sells at 5.88, its price under the Black-Scholes model. The call's delta is 0.5422, so the seller buys 0.5422 units of the underlying. If the underlying price rises from 100 to 101, the call price rises by 0.5562, from 5.8760 to 6.4322, which is a loss for the seller. The underlying held rises in value by 0.5422, which is a gain for the seller. What remains uncanceled is the difference of 0.0140. When the underlying price falls, the loss and the gain are reversed².

A position in which the combined delta of the options and the underlying is zero is called delta-neutral³. When you hold many units of an option, multiply the delta of one unit by the quantity to find the amount of the underlying to hedge with¹. For example, with JGB futures options, whose underlying is JGB futures, the options' delta is converted into a number of JGB futures contracts, and JGB futures are bought or sold to make the position delta-neutral³.

However, delta changes when the underlying price moves or time passes, so a delta-neutral position does not stay neutral once set. The degree to which delta changes with the underlying price is gamma. In theory, the delta is readjusted continuously, but in practice you cannot trade continuously, so it is readjusted at set times⁴. This is called rehedging.

Figure description

The curve shows the delta of a call with a strike of 100, 0.5 years to expiry, an interest rate of 1%, and a volatility of 20%, as the underlying price moves from 70 to 130. The delta is close to 0 when the underlying price is low, rises as the price rises, and approaches 1 when the price is high. At the strike of 100, it is about 0.54.

Hedging steps and a worked example of profit or loss

Delta hedging by the seller of an option proceeds in the following order⁵⁶. At time 0, the seller sells the option, receives the premium, and buys the underlying in the amount of the delta. Any shortfall in paying for it is borrowed, and interest accrues on the cash balance. At each set time, the seller recalculates the delta and trades the difference from the amount held. At expiry, the seller pays the option's payoff and closes out by selling the underlying held. The cash left at the end is the hedged seller's profit or loss.

Let the rehedging interval be $\Delta t$, the interest rate $r$, the underlying price at time $k$ be $S_k$, the delta found at that time $\delta_k$, and the number of hedges up to expiry $n$ (at times 0 through $n-1$). The cash balance $B_k$ after trading at time $k$ is then given by the following formula. Interest is added to the previous balance, and the seller pays for the underlying bought because the delta rose (when the delta falls, the seller receives the proceeds of the sale).

$$B_k = B_{k-1}\, e^{r \Delta t} - (\delta_k - \delta_{k-1})\, S_k$$

The balance at time 0 is the premium received minus $\delta_0 S_0$. At expiry $T$, interest is added to the balance, the $\delta_{n-1}$ units of the underlying held are sold at $S_T$, and the payoff of the call with strike price $K$, $\max(S_T - K, 0)$, is paid⁵.

In the following example, the seller sells the call from the previous section and trades the underlying 4 times, at 0, 0.125, 0.25, and 0.375 years. The underlying price moves from 100 to 104, 98, and 101, and is 103 at expiry. The delta is found at the implied volatility of 20%, and amounts of the underlying are rounded to four decimal places. Interest for 0.125 years is added by multiplying the balance by 1.00125078 ($e^{0.01 \times 0.125}$).

The balance at time 0 is −48.3440: the premium of 5.8760 minus 54.2200 paid for the underlying. At 0.125 years, interest takes the balance to −48.4045, and paying 12.2408 for 0.1177 units of the underlying brings it to −60.6453. At expiry, interest takes the balance to −53.0758, the seller adds 59.4104 from selling 0.5768 units of the underlying at 103, and pays the call's payoff of 3. The remaining cash of 3.3346 is the seller's profit or loss. Discounted to its value at time 0, it is a gain of 3.3180⁷.

With the same call, the result is reversed on a path where the underlying price swings widely to 108, 95, and 107 and ends at 97 at expiry. The call's payoff at expiry is 0, but the seller loses 4.4888 (4.4664 in value at time 0)⁷. In the seller's delta hedge, a rise in the underlying price increases the delta, so the seller buys more of the underlying, and a fall reduces the delta, so the seller sells. When the price moves a lot, the seller buys high and sells low by larger amounts, and a loss results. Conversely, a party that buys the call and delta hedges it has the opposite loss and gain on these two paths. For the buyer of an option, the delta changes in a favorable direction when the price moves. As a result, large price moves produce a gain for the buyer and a loss for the seller⁸.

Consider a hypothetical example in which Bank A sells 100 units of the same call. The amounts bought and the profit or loss in the table all become 100 times larger, and the underlying bought at time 0 is 54.22 units. Before discounting, the profit or loss is a gain of 333.46 on the path through 100, 104, 98, and 101, and a loss of 448.88 on the path through 108, 95, and 107. The same sale can end in a gain or a loss, depending on the path the underlying price takes.

Rehedging frequency and the variability of profit or loss

If the underlying price moves with a volatility equal to the implied volatility, the mean profit or loss is close to zero even when you rehedge at intervals, and quadrupling the number of rehedges roughly halves the variability of profit or loss⁹¹⁰. To check this, take the same short call as in the previous section's example and generate 2,000 random paths of the underlying price (Monte Carlo Simulation). The size of the price moves that actually occur up to expiry (annualized) is called realized volatility. The paths are generated so that realized volatility is 20%, and the underlying price grows on average at the same rate as the interest rate¹¹.

Hedging 50 times up to expiry, including the first hedge at time 0, gives a mean profit or loss of −0.03 (standard error of the mean 0.02), which is close to zero. The standard deviation is 0.68, and 90% of the 2,000 paths fall between −1.16 and 1.08. The standard deviation of profit or loss for different numbers of hedges is as follows.

A single hedge means hedging only at time 0 and making no adjustments afterward. Each time the number of hedges is roughly quadrupled, from 12 to 50 and from 50 to 200, the standard deviation roughly halves.

The size of the variability can be estimated with the following approximation, using the option's vega $\nu$ (the change in price when volatility changes), the volatility $\sigma$, and the number of hedges $N$¹².

$$\sigma_{\text{P\&L}} \approx \sqrt{\frac{\pi}{4}}\, \frac{\nu\, \sigma}{\sqrt{N}}$$

The example call's vega is 28.05 per 1.00 of volatility (0.2805 per 1%), so for $N = 50$ the estimate is $0.886 \times 28.05 \times 0.2 \div \sqrt{50} \approx 0.70$. The right-hand column of the table shows the values from this formula, which are close to the simulated values for 12 or more hedges. With few hedges, such as a single one, the approximation is rough. If you look at the price only at intervals, you measure the size of the moves in between (the volatility) from a small number of observations, so the measured value has an error. The value captured by hedging is off by the same error, so the variability is proportional to vega and inversely proportional to the square root of the number of hedges¹³.

When realized volatility differs from expectations

Option prices are higher when volatility is higher. So if realized volatility is greater than the implied volatility at which the option was sold, a seller who delta hedges loses on average. If realized volatility is smaller, the seller gains on average¹⁴. When the underlying price grows on average at the same rate as the interest rate, the mean profit or loss approaches the difference between the option price at realized volatility and the price at which the option was sold¹⁵.

For the example call (sold at 5.88), the results of hedging 50 times at different realized volatilities are as follows¹¹.

The theoretical value is the price at which the call was sold minus the price at realized volatility. At a realized volatility of 30%, it is 5.8760 − 8.6776 = −2.8016, a loss of about 2.80. At 15%, it is 5.8760 − 4.4730 = 1.4030, a gain of about 1.40. For a party that buys the call and hedges it, the signs of profit or loss in the table are reversed.

The profit or loss on each path varies around the mean. This is because the profit or loss from delta hedging depends not only on the size of realized volatility but also on the path the underlying price takes¹⁶. The standard deviation of 1.67 in the 30% row is large partly because the delta is computed at 20%, which differs from realized volatility (see the next section). The formula that splits profit or loss into gamma and the passage of time (theta), and the differences across paths, are covered on the page Managing Gamma and Vega.

How the volatility used for the delta matters

If the expected return on the underlying equals the interest rate, computing the delta at realized volatility or at implied volatility does not change the mean profit or loss, but it does change how the profit or loss varies¹⁵. Suppose you knew realized volatility in advance, computed the delta with it, and hedged continuously. The profit or loss at expiry would then not depend on the path. It would be fixed at the difference between the price at realized volatility and the price at which the option was traded¹⁷. The profit or loss before expiry, however, rises and falls depending on the path¹⁷. If the delta is computed at implied volatility, the profit or loss at expiry depends on the path and is not known in advance¹⁸.

This difference remains when you hedge at intervals. If the volatility used for the delta differs from realized volatility, hedging more often does not reduce the variability of profit or loss much¹⁹. For the example short call with a realized volatility of 30%, the standard deviation of profit or loss is as follows¹¹.

With the delta computed at realized volatility, quadrupling the number of hedges roughly halves the standard deviation. With the delta computed at implied volatility, going from 50 to 200 hedges lowers it only from 1.67 to 1.39. In both cases, the mean profit or loss is around −2.8, close to the theoretical value of −2.80. This is because, if the expected return on the underlying equals the interest rate, the mean profit or loss is the same regardless of how the delta is computed or how many times you hedge¹⁵.

In practice, you cannot know realized volatility up to expiry in advance, and the size of price moves also changes over time. For this reason, Derman's lecture notes say that a trading desk would compute the delta at the current implied volatility at each point. This implied volatility moves together with recent realized volatility, but it is not the same²⁰.

Rehedging frequency and trading costs

Rehedging more often reduces the variability, but each trade in the underlying has a cost, so more is lost to costs. Rehedging less often reduces costs but makes the profit or loss less certain²¹²². The simulator on this page does not include trading costs, so its standard deviation keeps falling as the number of hedges rises.

Instead of rehedging at set times, you can also rehedge only when the delta has changed by at least a set amount²³. In a worked example in Derman's lecture notes, trading costs 0.1% of the value traded, and the option is rehedged each time the delta moves by 0.02. There were 202 rehedges, the loss from costs was 0.63, and the standard deviation of profit or loss was 0.38. In the same example, both the loss from costs and the variability of profit or loss are smaller than when rehedging at every one of 1,000 time steps²³.

To reduce the need to rehedge in the first place, you can combine the option with other options so that the position's gamma is close to zero²². Managing gamma by combining options is covered in Managing Gamma and Vega.

Risks that delta hedging leaves

The calculations so far rest on the assumptions of the Black-Scholes model: the underlying price moves without jumps, volatility is constant up to expiry, and trading has no costs. The error from hedging at intervals remains even if all the other assumptions hold²². In practice, price jumps, changes in future volatility, and trading costs come on top of this error. The range of profit or loss that a party selling and hedging options needs to allow for is therefore even wider²².

The change in an option's price can be approximated by splitting it into a delta term, a gamma term (proportional to the square of the change in the underlying price), and a vega term (proportional to the change in volatility)²⁴. Delta hedging can bring only the delta term to zero. The gamma term when the underlying price moves a lot and the vega term when implied volatility moves remain.

Notes

1. Japan Exchange Group, "デルタ（用語集）" (Glossary: delta, in Japanese). It gives an example of hedging the purchase of 10 units (1 unit = 1,000 shares) of a call with a delta of 0.3 by selling 0.3 × 10 × 1,000 = 3,000 shares. ↩ ↩²

2. Prices and deltas are calculated with the Black-Scholes formula. The price is 5.8760, the delta is 0.5422, and the price when the underlying price is 101 is 6.4322. ↩

3. Japan Exchange Group, "国債先物オプション入門" (Introduction to JGB Futures Options, in Japanese), by Takahiro Hattori, 2022, page 41 (the subheading "ガンマ・トレーディング" (Gamma trading) in Section 7.3). It gives an example in which buying 100 ATM calls has a delta equal to 50 JGB futures contracts and selling 50 JGB futures brings the delta to zero, and it calls a position with zero delta "デルタ・ニュートラル" (delta-neutral). ↩ ↩²

4. Martin Haugh, "Black-Scholes and the Volatility Surface" (Columbia University, 2013), Section 5, "Delta Hedging" (page 12). ↩

5. Haugh, Section 5, "Mechanics of the Excel spreadsheet" (page 13), equations (25) and (26). The source adds interest as $r \Delta t$ (simple interest) and includes a dividend yield $q$. The formula in the text uses continuous compounding and no dividends. ↩ ↩²

6. Michael Kamal, "When You Cannot Hedge Continuously" (Goldman, Sachs & Co., 1998), page 2. ↩

7. Values calculated by finding the Black-Scholes delta and rounding amounts of the underlying and money amounts to four decimal places. Discounting to the value at time 0 multiplies the profit or loss by $e^{-0.01 \times 0.5} = 0.99501248$. ↩ ↩²

8. Japan Exchange Group, "国債先物オプション入門" (Introduction to JGB Futures Options, in Japanese), by Takahiro Hattori, pages 41–43 (the subheading "ガンマ・トレーディング" (Gamma trading) in Section 7.3). ↩

9. Kamal, page 3 (Table 1 and the paragraph after it). On 50,000 paths in which a one-month ATM put is hedged 21 and 84 times, the mean profit or loss is close to zero and the standard deviation is 16.3% and 8.1% of the premium. ↩

10. Emanuel Derman, "Lecture 2: Dynamic Replication" (2008), Section 2.6.1 (page 28). In an example of a one-month ATM option hedged with the delta computed at 20%, equal to realized volatility, with the expected return set equal to the interest rate, going from 21 to 84 hedges cuts the standard deviation from 0.42 to 0.21. ↩

11. Values calculated with this site's simulator with random seed 1 and 2,000 paths. The underlying price follows geometric Brownian motion at realized volatility, with an expected return of 1%, equal to the interest rate. Profit or loss is per unit of the option and discounted to its value at time 0. ↩ ↩² ↩³

12. Kamal, page 4, equations (1)–(3). The source writes vega as $\kappa$ and gives the coefficient $\sqrt{\pi/4}$ as 0.886. The source's $N$ is the number of rehedges. The approximation column in the text uses the number of hedges including the first hedge at time 0 as $N$. ↩

13. Kamal, pages 4–5 ("Understanding the Results Intuitively"). ↩

14. Haugh, Section 5, "Some Answers to Delta-Hedging Questions" (page 13). ↩

15. Derman, "Lecture 3: Transactions Costs," Section 3.1 (page 3). When the expected return on the underlying equals the interest rate, the expected value of the call is the Black-Scholes price, regardless of how the delta is computed and even without hedging. ↩ ↩² ↩³

16. Haugh, Section 5 (page 13), equation (27) and the paragraphs around it. ↩

17. Derman, Lecture 2, Section 2.5.1, "Hedging with Realized (Known) Volatility" (pages 18–21). ↩ ↩²

18. Derman, Lecture 2, Section 2.5.2, "Hedging with Implied Volatility" (pages 22–24). ↩

19. Derman, Lecture 2, Section 2.6.1 (pages 28–29). In an example with the delta computed at an implied volatility of 40% and a realized volatility of 20%, quadrupling the number of hedges does not reduce the standard deviation in the same way. ↩

20. Derman, Lecture 2, Section 2.5.2 (page 24, end of the section). ↩

21. Derman, Lecture 3, Section 3.2.1, "Rehedging at regular intervals" (page 4). ↩

22. Kamal, "Trading Volatility and its Risks" (page 5). ↩ ↩² ↩³ ↩⁴

23. Derman, Lecture 2, Section 2.6.1 (page 28), and Lecture 3, Section 3.2.1 (pages 4–5). In an ATM call example, a delta trigger of 0.02 and a cost of 0.1% give 202 rehedges, a loss of 0.63, and a standard deviation of 0.38. ↩ ↩²

24. Haugh, Section 4, "Delta-Gamma-Vega Approximations to Option Prices" (pages 11–12). ↩