FX Option Market Conventions

If you receive a currency option quote and work out strikes and the volatility smile from it, or manage the quote data, misreading the type of delta or ATM makes the numbers disagree.

Quoting by volatility and delta

In the over-the-counter currency option market, option prices are quoted not as amounts of yen or US dollars but as implied volatility. Implied volatility is the volatility you get by putting an option's premium into a pricing formula and solving it backward. For currency options, the pricing formula is the Garman-Kohlhagen model (the Black-Scholes model applied to exchange rates). Market participants do not believe the model is correct. They use it as a common yardstick for communicating prices¹. Once the volatility is set, the formula converts it into a premium. The naming of currency options, how premiums are stated, and the Garman-Kohlhagen formula are covered in FX Options.

The market also quotes volatility by delta, not by strike price¹. Delta measures how much the premium moves when the price of the underlying asset moves a little. It lies between 0 and 1 for a call option (call) and between −1 and 0 for a put option (put). In the market, the option to trade is specified by delta instead of strike, as in "a 25-delta call." The strike is then backed out from the volatility and the delta with a formula. Because strikes are set so that the call and the put have deltas of the same absolute value, the upper and lower strikes are not necessarily the same distance from the current rate².

Three kinds of quotes are made for each expiry: the ATM volatility, the risk reversal, and the butterfly (strangle). Risk reversals and butterflies are usually quoted for 25-delta and 10-delta³. All three are stated as volatilities (%), and together they are used to build the volatility smile for each expiry (the shape formed by lining up volatilities by delta or strike).

Types of delta

Currency option delta comes as spot delta and forward delta, and each of these comes in a form that ignores the premium and a premium-adjusted form⁴. The same option has different delta values under different types, so the strike that "25-delta" points to also changes. Which type of delta is the standard depends on the currency pair⁵.

Spot delta is the sensitivity of the option's value to the spot exchange rate. For an option on USD/JPY, it is the amount of foreign currency (dollars) you buy to reproduce the option's price moves under the Garman-Kohlhagen model. Forward delta is the sensitivity to the forward exchange rate and assumes hedging with an FX forward. With $T$ as the number of years to expiry and $r_f$ as the foreign-currency interest rate (continuously compounded), spot delta $\Delta_S$ and forward delta $\Delta_F$ are related as follows⁴.

$$\Delta_S = e^{-r_f T} \Delta_F$$

$e^{-r_f T}$ is the foreign-currency discount factor, and forward delta does not include it. Consider a hypothetical example in which the dollar interest rate is 4.5% and the expiry is three months. Then $e^{-0.045 \times 0.25} = 0.988813$, so a call with a forward delta of 0.25 has a spot delta of 0.2472⁶. In practice, options with long expiries are often quoted in forward delta and options with short expiries in spot delta⁴.

Premium-adjusted delta takes into account that the premium was paid in the foreign currency. If you buy a call and pay the premium in the foreign currency, the option gives you a long position in the foreign currency, while the premium paid gives you a short position in the foreign currency of that amount. Your overall delta therefore stays long, but it is smaller than if you had paid the premium in the domestic currency⁷. Delta that ignores the premium is usually used for currency pairs whose premium is paid in the domestic currency (such as EUR/USD options paid in US dollars)⁴.

ATM conventions

Currency options have several conventions for what counts as ATM. For stock options, an option is usually called ATM when its strike price equals the current price of the underlying asset. Currency options also use the following conventions⁸.

Like the type of delta, the ATM convention depends on the currency pair⁸. For example, the EUR/USD option quotes analyzed in the study by Muck (2022) state the ATM volatility on a delta-neutral ATM basis⁹. The delta-neutral ATM is the strike at which a straddle, buying one call and one put with the same strike, has an overall delta of 0. For non-premium-adjusted delta, the strike $K_{ATM}$ of the delta-neutral ATM is found from the forward exchange rate $F$, the ATM volatility $\sigma_{ATM}$, and the number of years to expiry $T$ with the following formula¹⁰.

$$K_{ATM} = F e^{\sigma_{ATM}^2 T / 2}$$

Consider a hypothetical example in which the forward exchange rate is 140 yen, the ATM volatility is 10%, and the expiry is three months. Then $K_{ATM} = 140 \times e^{0.01 \times 0.25 / 2} = 140.18$ yen. This is slightly above 140 yen, the strike of the ATM forward⁶. The formula is the same for spot delta and forward delta¹⁰.

Quoting risk reversals and butterflies

A risk reversal is the difference between the volatilities of a call and a put with deltas of the same absolute value. The 25-delta risk reversal $RR_{25}$ is the volatility of the 25-delta call $\sigma_{25C}$ minus the volatility of the 25-delta put (delta −0.25) $\sigma_{25P}$¹¹.

$$RR_{25} = \sigma_{25C} - \sigma_{25P}$$

As a trade, it combines buying a call and selling a put with deltas of the same absolute value. The call has a strike above the current rate and the put a strike below it, so both are out-of-the-money options¹². By market convention, the price is quoted as the difference in volatilities, not the difference in the premiums of the call and the put. Volatility and premium correspond one to one, so a difference in volatilities corresponds to a difference in premiums¹³. The risk reversal shows the slope (the lopsidedness) of the volatility smile¹¹.

A butterfly is the average of the volatilities of a call and a put with deltas of the same absolute value, minus the ATM volatility¹¹.

$$BF_{25} = \frac{\sigma_{25C} + \sigma_{25P}}{2} - \sigma_{ATM}$$

The market calls this quote a butterfly, but as a trade it refers to a strangle, buying both a 25-delta call and a 25-delta put, set up so that its overall delta is 0¹⁴. The butterfly shows the curvature of the volatility smile (how much volatility rises as you move away from ATM)¹¹.

Finding 25-delta volatilities

The three quotes, ATM, risk reversal, and butterfly, determine the volatilities of the 25-delta call and put. Solving the two formulas above for $\sigma_{25C}$ and $\sigma_{25P}$ gives the following¹⁴.

$$\sigma_{25C} = \sigma_{ATM} + BF_{25} + \frac{RR_{25}}{2}$$ $$\sigma_{25P} = \sigma_{ATM} + BF_{25} - \frac{RR_{25}}{2}$$

Consider a hypothetical example in which a three-month USD/JPY option has the same conditions plus a 25-delta risk reversal of −1.0% and a 25-delta butterfly of 0.3%. To keep the calculation simple, delta is forward delta (without premium adjustment).

Because the risk reversal is negative, the volatility of the put (the USD put) is 1.0 percentage point higher than that of the call (the USD call). If the risk reversal were +1.0% instead, the two would swap, with the call at 10.8% and the put at 9.8%. Because the butterfly is 0.3%, the average of the two, 10.3%, is 0.3 percentage points above the ATM volatility of 10.0%.

Next, the strike is found from each volatility and delta. It is backed out from the forward delta $\Delta$ and the volatility $\sigma$ with the following formula, where $\omega$ is 1 for a call and −1 for a put, and $N^{-1}$ is the inverse of the cumulative distribution function of the standard normal distribution¹⁵.

$$K = F \exp\left(-\omega \sigma \sqrt{T}\, N^{-1}(\omega \Delta) + \frac{\sigma^2 T}{2}\right)$$

Using $N^{-1}(0.25) = -0.674490$, the three strikes are as follows⁶.

The strike of the 25-delta put is 4.99 yen below ATM and the strike of the call is 4.70 yen above it, so the distances from ATM on the upside and downside differ.

Figure description

RR stands for risk reversal and BF for butterfly. From the quotes of ATM 10.0%, RR −1.0%, and BF 0.3%, the volatility is found as 10.0 + 0.3 − 0.5 = 9.8% for the call and 10.0 + 0.3 + 0.5 = 10.8% for the put. From each volatility and the delta, the strike is then backed out: 144.88 yen for the call and 135.19 yen for the put. The quotes are stated as volatility differences and averages, and the strikes are determined afterward.

Deep diveSmile strangles and broker strangles

The formulas above assume that the call and put in the butterfly are the same 25-delta call and put as in the risk reversal. A strangle of this form is called a "smile strangle." In the form called a "broker strangle" (market strangle), by contrast, the strikes of the strangle's call and put do not match the strikes of the risk reversal's call and put. In the EUR/USD quotes Muck (2022) analyzed, the butterfly was the price of a broker strangle¹⁶.

For that reason, 25-delta volatilities found by putting a broker-strangle butterfly directly into the formulas above are approximations. If the volatility smile is not heavily lopsided, the error is small¹⁶. In practice, the strikes are found numerically from the quoted volatilities and the specified deltas. A curve is then fitted to the resulting points of strike and volatility to build the whole volatility surface (the volatility for each expiry and strike)¹⁷¹⁸.

Reading the volatility smile

Lining up the volatilities of the 25-delta put, ATM, and the 25-delta call in order of delta gives the skeleton of the volatility smile. In the example above, from left to right, the put is at 10.8%, ATM at 10.0%, and the call at 9.8%. The negative risk reversal tilts it downward from the put side to the call side, and the positive butterfly adds an upward bend that grows as you move away from ATM. If 10-delta risk reversals and butterflies are also quoted, the same calculation adds the 10-delta points, and interpolating between the points gives the volatility for other deltas and strikes¹¹.

If the Garman-Kohlhagen model held as assumed, volatility would be the same at every delta, and the volatility smile would be flat. In actual markets it is not flat, and its shape shows that the model's assumptions do not hold¹. In the FX market, the usual assumption is that when the rate moves, the smile, viewed as a function of delta, stays the same (sticky-by-delta)¹⁹.

The sign of the risk reversal shows in which direction the options that protect against a sharp move are more expensive. In USD/JPY, a negative risk reversal means USD puts are more expensive. Demand for protection against a sharp move toward a stronger yen and weaker dollar is strong, and market participants are more wary of a move in that direction. The risk reversal is not necessarily an indicator that predicts where the rate will go. Through market participants' perception of risk, however, it is one clue to the background of exchange rate moves²⁰. For example, after the market turmoil of March 2020, even as USD/JPY moved back toward a weaker yen, the three-month 25-delta risk reversal on USD/JPY showed stronger wariness of a stronger yen than in normal times²¹.

Notes

1. Matthias Muck, "Arbitrage-free smile construction on FX option markets using Garman-Kohlhagen deltas and implied volatilities" (Review of Derivatives Research, 2022; hereafter Muck 2022), Section 2.1. ↩ ↩² ↩³

2. Bank of Japan, "リスク・リバーサルからみた為替変動へのリスク認識" (Risk perceptions of exchange rate moves seen through risk reversals, Bank of Japan Review 2012-J-14, August 2012, in Japanese; hereafter BOJ 2012), footnote 7 on page 8. ↩

3. OpenGamma, "Vanilla Forex Options: Garman-Kohlhagen and Risk Reversal/Strangle" (May 2012; hereafter OpenGamma 2012), Section 3 "Smile." ↩

4. Muck 2022, Section 2.1, equations (4) and (5), and note 3. The paper's symbol $B_f(t,T)$ (the price of a foreign-currency discount bond) corresponds to $e^{-r_f T}$ on this page. ↩ ↩² ↩³ ↩⁴ ↩⁵

5. Martin Haugh, "Foreign Exchange, ADR's and Quanto-Securities" (Columbia University, 2013; hereafter Haugh FX), pages 3–4, "Other Deltas." ↩ ↩²

6. $e^{-0.045 \times 0.25} = 0.988813$, and 0.25 × 0.988813 = 0.2472. The strikes and their distances from ATM are rounded to two decimal places. ↩ ↩² ↩³

7. Haugh FX, page 3, "Delta and the Option Premium." This page uses the same term, "premium-adjusted delta." ↩

8. Haugh FX, page 4, "What Does At-the-Money (ATM) Mean?" ↩ ↩²

9. Muck 2022, Section 5, "Data." The ATM volatilities follow the ATM-DNS convention. ↩

10. Muck 2022, Section 2.2 "At-the-money volatilities, risk reversals, and butterflies," equation (6) and the unnumbered equation right after it. The paper's $X_{ATM}$ corresponds to $K_{ATM}$ and $T - t$ to $T$. ↩ ↩²

11. OpenGamma 2012, Section 3 "Smile." ATM is the delta-neutral straddle, the risk reversal measures the skew of the smile, and the strangle measures its curvature. ↩ ↩² ↩³ ↩⁴ ↩⁵

12. BOJ 2012, pages 1–2, "リスク・リバーサルとは" (What is a risk reversal?), and footnote 6 on page 8. ↩

13. BOJ 2012, pages 2–3 and footnote 9 on page 8. The difference in volatilities "is referred to as the price by trading practice" (our translation). ↩

14. Muck 2022, Section 2.2, and equation (21) in Section 5. ↩ ↩²

15. OpenGamma 2012, Section 3, equation (2). The paper's $t$ corresponds to $T$ on this page, and this page writes the formula in expanded form. ↩

16. Muck 2022, Section 2.2 (the distinction between smile strangles and broker strangles), and the paragraph after equation (21) in Section 5. ↩ ↩²

17. Haugh FX, pages 4–5, "Building an FX Volatility Surface." ↩

18. Martin Haugh, "Black-Scholes and the Volatility Surface" (Columbia University, 2013), page 4, Section 3 "The Volatility Surface." The notes describe the volatility surface as a function of strike and time to maturity. ↩

19. Haugh FX, page 3, the paragraph on sticky-by-delta two paragraphs before "Delta and the Option Premium." ↩

20. BOJ 2012, the summary on page 1, page 2, and note 1 to Chart 4 on page 3. The source says the risk reversal "is not necessarily an indicator for predicting future exchange rate moves themselves" (our translation). ↩

21. Bank of Japan, "店頭デリバティブ取引データからみた通貨オプション市場 ―新型コロナウイルス感染症下での取引動向―" (The currency option market seen through OTC derivatives trade data: trading trends under COVID-19, Bank of Japan Review 2022-J-10, June 2022, in Japanese), the main text on page 3 and Chart 4 "ドル／円のリスクリバーサル" (USD/JPY risk reversal) (three-month 25-delta risk reversal, weekly, source: Bloomberg). ↩