South Korean Won to U.S. Dollar 3-Month At-the-Money Option-Implied Volatility

The at-the-money option-implied volatility is the exchange-rate volatility at which the forward-based option pricing formula reproduces traded won-dollar option premiums, evaluated at a 3-month maturity and a strike equal to the forward. The formulas of Black (1976) and Garman and Kohlhagen (1983) treat volatility as the one unobserved scale parameter of the option price, so a traded premium inverts to a single implied volatility, and currency option markets quote their volatility surfaces around an at-the-money point (Reiswich and Wystup 2010). The series is the denominator of the carry-to-risk ratio, the 3-month at-the-money option-implied volatility in the definition of Galati, Heath, and McGuire (2007) and Aquilina et al. (2024), published as a series of its own in annualized percent.

Here $\sigma_i$ is the implied volatility of trade $i$ executed in week $w$, with strike $K_i$, forward $F_i$ at its execution time and maturity $\tau_i$ in years, so $c_0$ is the fitted volatility at a 3-month maturity and a strike equal to the forward. The smile is quadratic rather than a full arbitrage-free parameterization such as that of Gatheral and Jacquier (2014), which a week of trades cannot pin down, and each value is a weekly surface estimate rather than an end-of-day reading, in effect an average over the week's trades whose week-to-week changes carry the autocorrelation that averaging induces (Working 1960).

The options trade alongside the offshore non-deliverable forward market, whose implied yields can diverge from onshore rates (Ma, Ho, and McCauley 2004), and the inputs are public trade records of offshore won-dollar options subject to U.S. reporting requirements, a subset of the offshore market. The series is an option-implied volatility, not a realized or model-filtered one, and it differs from the global volatility that Menkhoff et al. (2012) build from realized exchange-rate moves. Because option premiums include compensation for crash risk (Jurek 2014), its level describes priced risk rather than the volatility that follows, and non-overlapping weeks avoid the correlation of overlapping windows (Hansen and Hodrick 1980).

The volatility is estimated weekly from option trades in four steps, using transaction prices from public trade records rather than dealer mid quotes, and targets the 3-month at-the-money option-implied volatility of the definition of Galati, Heath, and McGuire (2007) and Aquilina et al. (2024).

(1) Trade selection. Vanilla offshore non-deliverable won-dollar options are kept, excluding digital, barrier, volatility and variance products, deliverable options and trades whose notional is disclosed only as a capped value, and duplicate reports, re-reports and block allocations are merged into one trade. A merged trade counts as a package trade if any of its reports flags it as one, and package trades are excluded because a premium quoted for a strategy is not the price of its individual legs (Reiswich and Wystup 2010).

(2) Trade-time forward and type screening. The forward for each trade is the level of offshore non-deliverable forwards (Ma, Ho, and McCauley 2004) within five minutes of its execution, widening to fifteen and then thirty minutes when trades are sparse, carried to the option's maturity by that day's forward curve and checked against the forward implied by put-call parity on call and put pairs (Garman and Kohlhagen 1983). A trade whose two call-or-put indications disagree and whose price cannot settle the type is excluded, and near-simultaneous call and put trades of equal strike, maturity and notional are merged into one straddle, whose implied volatility does not depend on those indications.

(3) Inversion. Each premium is inverted for $\sigma$ from the forward-based formula of Black (1976), which coincides with Garman and Kohlhagen (1983) for currency options.

The formula is shown for a dollar call with premium $P$ per dollar notional $N$ and dollar discount factor $\mathrm{DF}$, with puts following by put-call parity and discounting at a flat 3-month U.S. government bill yield.

(4) Weekly surface and revisions. Options with maturities of exactly 3 months trade thinly, so all observations executed in a Monday–Friday week are pooled and the 3-month point is interpolated from a quadratic smile with a linear maturity term.

The fit covers maturities of 21–213 days and $|m_i|\le 0.15$, starting from a least-absolute-deviation fit, trimming residuals beyond three times the scaled median absolute deviation and refitting the rest by least squares. The published value is $c_0$, the point where the strike equals the forward, so at the money is defined at the forward, one of several at-the-money conventions in currency options (Reiswich and Wystup 2010), and it may differ from a delta-neutral-straddle quote. A week is published only when enough observations survive trimming, including some near 3 months and near the money, over a wide enough maturity span, and a failing week is left empty and never filled from other weeks. The quadratic smile is preferred to the arbitrage-free SVI of Gatheral and Jacquier (2014) because a week supplies too few trades for its parameters, non-overlapping weeks avoid overlap correlation (Hansen and Hodrick 1980), and weekly pooling costs little for questions of persistence because test power depends on sample span rather than sampling frequency (Shiller and Perron 1985). Trades are processed in a fixed order so that repeated computation reproduces the same value (Demmel and Nguyen 2013), and because corrections, cancellations and late reports can arrive after a week closes, affected weeks are recomputed and past values can be revised (Orphanides and van Norden 2002).

The ratio places the won within the carry-trade literature, where a position borrows in a low-interest currency and lends in a high-interest one. Fama (1984) showed that the forward premium is a biased guide to subsequent spot changes, the forward-premium anomaly, so a forward-implied differential embeds a time-varying risk premium and is not a gauge of the depreciation to come, and Lustig, Roussanov, and Verdelhan (2011) trace the payoffs of currency portfolios sorted by forward discount to a common carry factor tied to global risk. Scaling the differential by option-implied volatility, as Galati, Heath, and McGuire (2007) do, restates carry in units of the risk the option market prices, the sense in which Bhansali (2007) describes a carry position as economically close to a short position in currency volatility.

The scaling matters because carry and volatility move against each other in stress. Brunnermeier, Nagel, and Pedersen (2008) document that carry positions build up gradually and unwind abruptly as risk and funding illiquidity rise, which leaves high-interest currencies exposed to crash risk, and Menkhoff et al. (2012) identify global foreign-exchange volatility as the risk to which high-interest currencies are exposed. A jump in implied volatility therefore moves the ratio toward zero even when the differential is unchanged, the configuration Aquilina et al. (2024) examine around a sharp unwinding of yen-funded carry positions in which the won sits on the long side of the basket, and Daniel, Hodrick, and Lu (2017) document how deep and prolonged carry drawdowns can be.

For the won the ratio condenses two Korea-specific features into one number. Its numerator is an offshore non-deliverable forward-implied differential, and because the offshore market trades apart from the onshore one its implied yields can diverge from onshore rates (Ma, Ho, and McCauley 2004), while the covered-parity basis further separates it from the onshore money-market spread (Du, Tepper, and Verdelhan 2018). Its sign records whether, on forward prices, the won is a target or a funding currency against the dollar, so a negative reading means that a position long the won pays carry. Koijen et al. (2018) stress that carry is what accrues only if prices stay unchanged, and Burnside et al. (2011) relate carry payoffs to peso problems, rare adverse states that a short sample may not contain, so the ratio characterizes current pricing and is not a forecast.

The ratio is an exact identity, the forward-implied differential divided by the at-the-money implied volatility, so every weekly change is fully accounted for by changes in those two companion series, which show which of them moved. The three series are one construction and are not independent evidence about one another.

For currency desks and investors with won exposure, the ratio states on one scale how much forward-implied rate differential a won position carries relative to the exchange-rate volatility that options price, whose global component Menkhoff et al. (2012) identify as the risk priced in carry. A reading near zero means the differential is small relative to priced volatility, and a change of sign marks a switch in which side of a won-dollar forward receives the carry (Koijen et al. 2018).

Carry and option protection are two sides of one trade-off. Burnside et al. (2011) and Jurek (2014) hedge carry positions with currency options and find that removing crash exposure leaves part of the average carry payoff in place, and Bhansali (2007) shows that the carry position itself behaves like a short volatility position, whose exposure is priced by the implied volatility in the denominator. A won-based holder of dollar assets hedges by selling dollars forward, a position long the won, so a negative differential is a hedging cost and a positive one a hedging pickup, and the ratio sets that 3-month forward-hedge carry against the price of hedging with options at the same horizon (Garman and Kohlhagen 1983; Reiswich and Wystup 2010).

The series is a descriptive record and is not a forecast, nor a trading rule. Regressions of asset payoffs on persistent financial ratios suffer the small-sample bias documented by Stambaugh (1999), rules chosen after inspecting a sample invite the data-snooping biases analyzed by Lo and MacKinlay (1990) and White (2000), and carry strategies have suffered deep and prolonged drawdowns (Daniel, Hodrick, and Lu 2017) that a short sample may not contain.

The records cover only offshore trades subject to U.S. reporting requirements, a subset of the offshore market, and recent weeks can be revised as late reports arrive, so their values are provisional in the sense of Orphanides and van Norden (2002). Desks that read the ratio beside the forward-implied differential and the implied volatility should treat the three as one identity and attribute a change in the ratio by checking which leg moved.

KRFXIV3M is the weekly 3-month at-the-money-forward implied volatility of the won against the dollar, a constructed value read off a surface fitted to a week of reported option premiums rather than a price observed in a market, and the object tested is its natural logarithm on the pinned sample of 104 complete weeks from 2024-10-04 to 2026-09-25 recorded for sign-off. Volatility is the positive scale parameter of the option price in Black (1976) and Garman and Kohlhagen (1983), and its innovations scale with its level, so the log is the natural scale, and the deterministic term is pinned to a constant, since a trend in log-volatility would send the volatility to zero or to infinity and is therefore excluded as uninformative rather than tested.

Over this span of about two years the unit-root reading of the log level is ambiguous, and the ambiguity is the kind in which neither null can be rejected. The augmented Dickey and Fuller (1979) test, lag-augmented as in Said and Dickey (1984), does not reject a unit root with a statistic of −2.38215 at p = 0.146839, the Phillips and Perron (1988) test agrees at −2.5539 with p = 0.102903, and the KPSS test does not reject level stationarity either, at 0.203212 with p > 0.10 (Kwiatkowski et al. 1992). With both nulls surviving, the triplet cannot separate a persistent stationary process from a unit-root process, the low-power configuration that a short span produces (Shiller and Perron 1985), so no order of integration is assigned and the series is described neither as integrated nor as stationary. The GLS-detrended test of Elliott, Rothenberg, and Stock (1996), which the matrix reserves for exactly such ambiguous readings as a confirmation of the Dickey-Fuller leg, was not run in this pinned pass, so the ambiguity stands unresolved rather than being settled by preference.

The first-order autocorrelation of the log level is 0.891712, an implied half-life of about 6.05 weeks, a descriptive persistence summary and not an integration claim, read against the near-unity small-sample downward bias of the autoregressive estimate (Andrews 1993). Serial-dependence and break diagnostics are read on the 103 weekly log changes from 2024-10-11 to 2026-09-25, each taken between two consecutive published weeks. The Ljung and Box (1978) portmanteau, refining the Box and Pierce (1970) form, does not reject white noise at lags 13 and 26, with Q = 7.78964 at p = 0.85703 and Q = 19.4397 at p = 0.81734, and the heteroskedasticity-robust automatic portmanteau of Escanciano and Lobato (2009), which guards against the volatility clustering that can distort a plain portmanteau on a volatility series, agrees at 0.467141 with p = 0.494305. The Bai and Perron (1998) procedure, by the Bai and Perron (2003) algorithm, finds no break in the mean of the differenced log series (Perron 1989). The non-rejections are weak evidence rather than proof of white noise, since 103 changes give a portmanteau at these lag lengths limited power, and pooling a week of trades acts like averaging, which induces a positive first-order correlation in changes (Working 1960) that tests at these lags are poorly placed to detect.

The series carries the model-derived caveat of its archetype, since each value is a construction from reported prices through a fitted surface. Each week is computed from that week's trades alone and the sample is never re-estimated, so no smoother rewrites the history, but late trade reports can still revise a past week after first publication. Replaying the archive in file-date order over 104 weeks, the 95th percentile of the absolute gap between each week's first-published and final volatility is 0.0589 volatility points, below the median weekly standard error of the at-the-money estimate of 0.0792 that serves as the gate's tolerance, and no week was empty at first publication (Orphanides and van Norden 2002). The three verdicts of this group are tied by an exact identity, since the carry-to-risk ratio equals the forward-implied differential divided by this volatility in every week, so they are not independent evidence of one another, and the ambiguity here neither strengthens nor weakens the clean integrated readings of the other two.

The series is weekly with no posited seasonal period, so the seasonal-unit-root machinery of Hylleberg et al. (1990), the Canova and Hansen (1995) seasonal-stationarity test and seasonal-dummy regressions are inapplicable and were deliberately not run (Ghysels and Osborn 2001; Beaulieu and Miron 1992).