Managing Gamma and Vega

At financial institutions that hold option positions, the people in charge of hedging and limit management use gamma and vega to see why a daily profit or loss remains after delta hedging, split into gamma and theta.

Profit or loss after delta hedging (gamma and theta)

Even when you trade the underlying asset to match the delta and bring the delta of an option position to zero, its profit or loss does not become zero. A large move in the underlying price produces a gain or loss from gamma, and the passage of time reduces the value by the theta. The steps of delta hedging and how the hedge amount is adjusted are covered on the page Delta Hedging.

For a delta-hedged long option position, the profit or loss over a short period $\Delta t$ can be approximated by the following formula¹.

$$\text{profit or loss} \approx \frac{1}{2}\,\Gamma S^2 \left[\left(\frac{\Delta S}{S}\right)^2 - \sigma^2 \Delta t\right]$$

$S$ is the underlying price, $\Delta S$ is its move over the period, $\Gamma$ is gamma, and $\sigma$ is the implied volatility used to price the option. The first term in the brackets is the square of the return that actually occurred, and it represents the gain from gamma. The second term is the squared return that the implied volatility expects, and it represents the part of theta's loss of value that remains after removing the part matched by interest on the delta hedge's cash. The rest of theta, the interest part, cancels out against the interest on that cash, so the profit or loss is the difference between these two terms².

On this page, a call option (call) with an underlying price of 100, a strike price of 100, a time to expiry of 1 year, an interest rate of 1%, and a volatility of 20% is called the example call. Of the example call's one-day theta of −0.0121, −0.0108 is the loss of value that corresponds to the second term³.

If the return equals $\sigma\sqrt{\Delta t}$, the two terms cancel out and the profit or loss is about zero. This page calls this move the break-even move. With an implied volatility of 20% and one day (with 365 days in a year), it is $0.2 \times \sqrt{1/365} = 1.0468\%$, or a move of about 1.05 when the underlying price is 100.

Deep diveThe gamma part and the interest part of theta

From the Black-Scholes partial differential equation, the price $V$, delta $\delta$, gamma $\Gamma$, and theta $\Theta$ (per year) of an option on an underlying with no dividends satisfy the following relationship².

$$\Theta + rS\delta + \frac{1}{2}\sigma^2 S^2 \Gamma = rV$$

If you buy 1 unit of the option and sell $\delta$ units of the underlying, the cash on hand is $\delta S - V$, and it grows at the interest rate $r$. The profit or loss over the period $\Delta t$ is the sum of the rise in the option's value ($\Theta\Delta t + \delta\Delta S + \frac{1}{2}\Gamma(\Delta S)^2$), the profit or loss on the short underlying ($-\delta\Delta S$), and the interest on the cash ($r(\delta S - V)\Delta t$). Replacing $\Theta$ using the relationship above cancels the delta terms and the interest terms, leaving the formula in the main text.

The −0.0108 in the main text is the gamma part ($\frac{1}{2}\Gamma\sigma^2 S^2 \div 365$), and the remaining −0.0013 of the example call's theta per day is the interest part. The interest part exactly cancels out against the interest on the delta hedge's cash³.

Buy 1 unit of the example call, delta hedge it, and let the underlying price move after one day. The table compares the profit or loss from the formula's approximation with the result of repricing with the Black-Scholes model. To reprice, value the option one day later with 364 days to expiry, then add the short underlying and one day of interest on the cash. The example call's gamma is 0.0197³.

A move of 1 falls short of the break-even move (1.05) and gives a small loss. Moves of 2 and 3 give a gain whether the price goes up or down. The formula's approximation gives the same value for a rise and a fall. Its gap from the repriced profit or loss widens as the move grows, but even for a move of 3 it is about 0.002.

Long and short gamma

A bought option has positive gamma, so once it is delta hedged, it gains on days with large moves and loses on days with small moves. This position is called long gamma. The price of being long gamma is the premium paid when buying the option. In a delta-hedged position, that price shows up bit by bit as theta losses on days with small moves⁴.

A sold option has negative gamma, and its profit or loss goes the other way. A delta-hedged short option gains on days when the underlying moves little, but loses on a large move up or down. That loss is smaller, however, than if the option had been sold and left unhedged⁵. Flip the signs in the table in the previous section. A position that sells 1 unit of the example call and delta hedges it then has a one-day gain of 0.0108 on a day with no move and a loss of 0.0280 to 0.0293 on a move of 2.

Adding up the daily profit or loss until expiry, long gamma makes a profit on average if the size of the moves that actually occur up to expiry (annualized), called realized volatility, is above the implied volatility, and a loss on average if it is below⁶. You can check this in the next simulator by changing the position and the two volatilities.

Under the same terms as the example call (time to expiry 1 year, implied volatility 20%), buy the call and delta hedge it 250 times over the year (including time 0). Over 2,000 paths with a realized volatility of 25%, the mean profit or loss is 1.99. Ninety percent of the outcomes fall between 0.60 and 3.82⁷. With a realized volatility of 15%, the mean profit or loss becomes −1.97, and switching the position to short flips the sign under the same terms.

Profit or loss depends on where the move happens

Even for a move of the same size, the profit or loss is weighted by the gamma at that time. In the formula in the first section, the $\frac{1}{2}\Gamma S^2$ that multiplies the squared return is large when the underlying price is near the strike price and approaches zero as it moves away from the strike price⁸. So even with the same realized volatility until expiry, the profit or loss differs depending on whether large moves happened near the strike price or far from it.

The next figure shows how gamma changes with the underlying price, for calls with 3 months and 1 year to expiry.

Figure description

This figure plots the gamma of a call with a strike price of 100, an interest rate of 1%, and a volatility of 20% as two curves, with the underlying price on the horizontal axis. Both form a high peak near the strike price of 100. The 3-month curve approaches zero away from the strike price, while the 1-year curve has a gentler peak and does not fall to zero even at the two ends of the plotted range. At an underlying price of 100, the height is 0.0398 for 3 months to expiry and 0.0197 for 1 year, so the shorter the time to expiry, the higher and narrower the peak. The profit or loss from a large move is larger near the strike price because gamma has this shape.

Buy 1 unit of a call with a time to expiry of 3 months and a strike price of 100, delta hedge it, and let the underlying price move 2% in one day. The table compares the results by the underlying price before the move. The other terms are the same as for the example call³.

The gain of 0.0578 on a day with a 2% move at 100, the same as the strike price, is about five times the 0.0115 on a day with a 2% move at 120. For short gamma, a large move near the strike price becomes a large loss.

For example, suppose Bank D (fictional) sells 100 units of this call (3 months, strike price 100), delta hedges it, and the underlying price moves 2% in one day, starting from an underlying price of 100 or 120. The loss on a day with a 2% move is the profit or loss in the table above multiplied by 100, with the sign reversed because this is a sale. The gain on a day with no move is found by multiplying $\frac{1}{2}\Gamma S^2$ in the table by $\sigma^2 \Delta t = 0.2^2 \div 365 = 0.0001096$ and by 100.

At both prices, the loss on a day with a 2% move is more than twice the gain on a day with no move.

Deep diveThe profit or loss formula for continuous hedging

When the delta hedge is adjusted continuously, the profit or loss of a long option position until expiry is given by the following formula⁸.

$$\text{profit or loss} = \int_0^T \frac{1}{2}\,\Gamma_t S_t^2 \left(\sigma_t^2 - \sigma^2\right) dt$$

$\sigma_t$ is the actual (instantaneous) volatility at time $t$, $\sigma$ is the implied volatility, and $\Gamma_t$ and $S_t$ are the gamma and the underlying price at time $t$. The formula multiplies the gap between the actual volatility and the implied volatility by $\frac{1}{2}\Gamma_t S_t^2$ at each time and adds these up, which shows that the profit or loss depends on the path.

Vega and vega by expiry

Vega is the change in an option's price when implied volatility rises by 1 percentage point. Gamma represents the profit or loss from moves in the underlying, while vega represents the profit or loss that arises when the volatility the market uses for pricing changes, even if the underlying price does not move⁵.

Vega is larger for options with a longer time to expiry⁹. Under the same terms as the example call, changing only the time to expiry gives the following values³.

The shorter the time to expiry, the larger gamma and theta are, and the longer it is, the larger vega is. The ratio of vega to gamma (vega ÷ gamma) runs from 5 (3 months) to 60 (3 years), and this ratio is proportional to the time to expiry. The ratio is used later in the worked example to choose the options for the hedge.

Implied volatility is priced in the market for each expiry, and options with different times to expiry are not necessarily priced with the same volatility (Volatility Smiles and Surfaces). So the vega of a position is measured separately for each option expiry. For example, the market risk standards that the Basel Committee on Banking Supervision (BCBS) published in January 2016 define the vega risk factors of the standardized approach as the implied volatilities of options bucketed by maturity. The buckets are 0.5, 1, 3, 5, and 10 years¹⁰. Vega broken down by expiry is called vega by expiry.

Worked example: offsetting gamma and vega with options

The underlying has zero gamma and zero vega, so trading the underlying changes only delta and leaves gamma and vega unchanged. To reduce gamma and vega, you trade options. As an example, suppose a financial institution has sold 100 units of the example call (time to expiry 1 year, strike price 100) to a customer. The position's gamma is −1.9724 and its vega is −39.4479. For the hedge, it buys a call with a time to expiry of 3 months and a strike price of 100 (A) and a call with a time to expiry of 2 years and a strike price of 110 (B). The other terms are the same as for the example call. The values per unit are as follows³.

Offsetting gamma with A alone means buying 49.58 units of A and leaves a vega of −29.59. Offsetting vega with B alone means buying 70.47 units of B and leaves a gamma of −0.9862. With one option, only one of gamma and vega can be brought to zero.

To bring both to zero at once, solve two equations for the quantity $n_A$ of A and the quantity $n_B$ of B, in which total gamma and total vega each balance the amounts sold.

$$0.0398\,n_A + 0.0140\,n_B = 1.9724$$ $$0.1989\,n_A + 0.5598\,n_B = 39.4479$$

The solution is $n_A = 28.33$ and $n_B = 60.40$ (solved with unrounded coefficients). Buying 28.3 units of A and 60.4 units of B gives the following position³.

The remaining delta of −13.766 is offset by buying 13.766 units of the underlying. With gamma now close to zero, theta has also shrunk from 1.211 to 0.032, because most of theta is the gamma part (the first section).

Options with different expiries are used because, for options with the same expiry and the same volatility, the ratio of vega to gamma is the same whatever the strike price. Under the terms of this example, vega (per 1%) is 20 × time to expiry (years) times gamma: 5 times for A, 20 times for the sold 1-year call, and 40 times for B⁹. Two options with the same ratio cannot solve the two equations, so a short expiry with a large share of gamma is combined with a long expiry with a large share of vega.

Total vega is now zero, but vega by expiry remains: 5.629 at 3 months, −39.448 at 1 year, and 33.813 at 2 years. If the implied volatilities of all three expiries rise by 1 percentage point together, the profit or loss is about zero. If only the 1-year volatility rises by 1 percentage point, however, the loss is about 39.45, and if it falls by 1 percentage point, the gain is about 39.45. This is why the previous section measures vega by expiry.

Deep diveWhy the ratio of gamma to vega is set by expiry when the volatility is the same

The gamma and vega (per 1.00 of volatility) of an option on an underlying with no dividends are given by the following formulas⁹. $\phi$ is the density function of the standard normal distribution, and $d_1$ is the same as in the Black-Scholes formula.

$$\Gamma = \frac{\phi(d_1)}{\sigma S\sqrt{T}}$$ $$\nu = S\sqrt{T}\,\phi(d_1)$$

Taking the ratio of the two formulas cancels $\phi(d_1)$ and gives $\nu = \Gamma\,\sigma S^2 T$. The strike price appears only inside $d_1$, so if the volatility $\sigma$ is the same, the ratio does not depend on the strike price. If there is a volatility smile and $\sigma$ differs by strike price, the ratio changes too. In this example, $\sigma S^2 = 0.2 \times 100^2 = 2{,}000$, and dividing by 100 to express vega per 1% gives a ratio of $20T$.

Managing at the portfolio level

Many financial institutions that trade options in liquid markets such as interest rates and currencies do not look for an offsetting trade for each trade. Instead, they measure the risk that remains in the whole position and offset it with futures, swaps, listed options, the underlying, and other instruments. This is called hedging on a portfolio basis. It makes it easier for the institution to take on the trades customers ask for, and it costs less than hedging each trade¹¹. The gamma and vega of the whole position are the sum of each trade's gamma and vega multiplied by its quantity.

The examination manual of the U.S. Federal Reserve Board says that option positions should have limits on each of delta, gamma, vega, theta, and rho¹². Many trading units explain their daily profit or loss by the risks they manage, such as delta, gamma, vega, theta, and rho. The manual says that if the unexplained profit or loss is large, the control function should investigate the cause and, where necessary, revise the valuation models or inputs (such as volatility)¹³.

A short gamma position depends on market liquidity. Its hedge assumes that, during large moves, the underlying can be traded quickly at a reasonable cost. If liquidity dries up and the hedging trades cannot be done, the actual risk is larger than expected. The institutions hit hardest when liquidity dries up are those that have sold options and are short volatility and gamma¹⁴.

Assumptions and limitations

The gamma, vega, and theta on this page are values from the Black-Scholes model with constant volatility. The relationship in which the gain from gamma balances the loss from theta assumes that the option is priced with the right volatility. Because the model does not match reality in some respects, the relationship is meant for building intuition². Implied volatility in the market differs by strike price and expiry, and volatility can move when the underlying price moves, so the actual profit or loss also includes a part that comes through vega.

The gamma approximation holds for small moves. The larger the move, the wider the approximation error (about 0.002 for ±3 in the table in the first section), and if the underlying price jumps a lot in one day, a profit or loss that the approximation cannot capture arises before the hedge can be adjusted. Because of gamma, delta hedging works less well against large changes in the underlying price⁵.

Notes

1. Martin Haugh, "Black-Scholes and the Volatility Surface" (Columbia University, 2013; hereafter Haugh), pages 11–12. The formula in the main text removes the delta term of the source's formula through the hedge, sets $\Delta\sigma = 0$, and adds the theta term from equation (23). ↩

2. Haugh, page 11, "The Relationship between Delta, Theta and Gamma," equations (23) and (24). It says that for a delta-hedged position the gain from gamma is offset by the loss from theta, that this assumes the correct implied volatility, and that it should be used to help intuition. ↩ ↩² ↩³

3. Calculated with this site's Black-Scholes calculation, with an interest rate of 1% (continuously compounded), no dividends, and a volatility of 20%. Vega is per 1%, and theta is per day (365 days in a year). Values are rounded to four decimal places (three in the position table of the worked example), and totals and ratios are computed from unrounded values. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷

4. Haugh, page 9. A long gamma position can make money by rebalancing to stay delta-neutral, but the premium for that long gamma is paid up front. ↩

5. Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual" (hereafter the FRB manual), Section 4330.1 (page dated February 1998, page 6; page 615 of the PDF). ↩ ↩² ↩³

6. Haugh, page 13, "Some Answers to Delta-Hedging Questions." ↩

7. Calculated with this site's delta hedging calculation, with 2,000 paths, random seed 1, and delta computed at implied volatility. The profit or loss is per unit, discounted to time 0, and rounded to two decimal places. The 90% range runs from the 5th to the 95th percentile. ↩

8. Haugh, page 13, equation (27) and the paragraph after it. The source's formula is for the option seller, and the main text flips the sign to show the buyer's side. The source also says the dollar gamma approaches zero when the option is deep in the money or deep out of the money. ↩ ↩²

9. Haugh, the gamma formula on page 8, and the definition and formula of vega on pages 9–10, with the explanation that vega grows with the time to expiry. The source's formulas include the dividend yield $q$, and the main text sets $q = 0$. ↩ ↩² ↩³

10. 小立敬, "トレーディング勘定の抜本的改定（FRTB）に関するバーゼル委員会の最終規則の概要" (Overview of the Basel Committee's final rules on the fundamental review of the trading book (FRTB)), 野村資本市場クォータリー, Spring 2016, page 90, Figure 7 (page 15 of the PDF; in Japanese). ↩

11. FRB manual, Section 4330.1 "Options," "Hedging on a Portfolio Basis" (page dated April 2003, page 7; page 616 of the PDF). ↩

12. FRB manual, Section 2010.1 "Market Risk," "Limits on options positions" (page dated February 1998, page 7; page 51 of the PDF). ↩

13. FRB manual, Section 2100.1 "Financial Performance," around "Unexplained Profits and Losses" (page dated April 2003, page 9; page 150 of the PDF). ↩ ↩²

14. FRB manual, Section 2030.1 "Market Liquidity Risk of Trading Activities" (page dated September 2006, page 2; page 85 of the PDF). ↩