Foreign Exchange Options Market Conventions
FX Spot Market
Currency pairs are commonly quoted using ISO codes in the format FORDOM, where FOR is the foreign currency and DOM is the domestic currency. For example, in the EURUSD pair, EUR is the foreign currency and USD is the domestic currency. The FX spot rate, $x$, for the currency pair FORDOM represents the number of units of DOM is needed to buy one unit of FOR. The domestic currency is also referred to as the numeraire currency.
FX Forward Contract
A forward contract allows an investor to exchange currencies at a time, $T$, at a pre-specified outright forward rate, $f$. The outright forward rate is related to the spot rate by interest-rate parity, $$f=\mathbb{E}[S_T]=xe^{(r_d-r_f)\tau},$$ where $r_d$ is the domestic interest-rate, $r_f$ is the foreign interest-rate and $\tau:=(T-t)$ is the time to maturity.
At inception, an outright forward contract has a value of zero. Thereafter, when markets move, the value of the forward contract is no longer zero but is worth the discounted difference between the expected exchange rate, $f$, and pre-specified outright forward rate, $K$. $$e^{-r_d\tau}(f-K)=xe^{-r_f\tau}-Ke^{-r_d\tau}.$$
It is also worth mentioning that FX forward contracts are oftened quoted as FX forward points. This is the difference between the spot exchange rate and forward outright point as basis points. $$\tilde{f}=10000\times(f-x).$$
FX Vanilla Option Contracts
An option contract is simply (not so simple to be honest) an extension of a forward contract. It provides the investor the right, but not the obligation, to exchange currencies at a time $T$ at a pre-specified rate known as the strike, $K$. The key part here is the additional optionality as in the name. In FX markets, these contracts are usually physically settled. This means that at expiry, the owner of a vanilla FORDOM call (put) option receives $N$ units of FOR and pays $NK$ DOM. Intuitively, the additional optionality comes at a cost. This cost is what traders and market makers want to be able to gauge accurately and is the ultimate motivation for everything we discuss in option theory.
The value of a vanilla option can be computed with the standard Black-Scholes model. $$V(x,K,T,t,\sigma,r_d,r_f,\phi)=\phi e^{-r_d\tau}[f\mathcal{N}(\phi d_+)-K\mathcal{N}(\phi d_-)].$$ The following terms have been abbreviated for simplicity:
- $\tau:=T-t$.
- $f:=\mathbb{E}[S_T]=xe^{(r_d-r_f)\tau}$.
- $\theta_\pm:=\frac{r_d-r_f}{\sigma}\pm\frac{\sigma}{2}.$
- $d_\pm:=\frac{\log(\frac{x}{K})+\sigma\theta_\pm\tau}{\sigma\sqrt{\tau}}=\frac{\log(\frac{f}{k})\pm\frac{\sigma^2}{2}\tau}{\sigma\sqrt{\tau}}.$
- $n(x):=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}=n(-x).$
- $\mathcal{N}(x):=\int_{-\infty}^{x}n(t)\text{d}t=1-\mathcal{N}(-x).$
Delta Conventions
The delta of an option is the percentage of the FOR notional one must buy when selling the option to hold a hedged position. For instance, a delta of 0.35 (35%) indicates buying 35% of the FOR notional to delta-hedge a short option.
In FX markets, we distinguish spot delta, used for a hedge in the spot market, from forward delta, used for a hedge in the forward market. Furthermore, the standard delta is a quantity in percent of foreign currency. Suppose the value of an option with a notional of 1,000,000 FOR was calculated as 73,669 DOM. Assuming a short position with a delta of 40% means, that buying 400,000 FOR is necessary to hedge. However the final hedge quantity will be 326,331 FOR which is the delta quantity reduced by the received premium in FOR. Consequently, the premium-adjusted delta would be 32.63%.
Unadjusted Deltas
- Spot delta: The sensitivity of a vanilla option's premium with respect to the spot rate, $x$, is given as, $$\Delta_s(K,\sigma,\phi):=\frac{\partial V}{\partial x}=\phi e^{-r_f\tau}\mathcal{N}(\phi d_+).$$
- Forward delta: An alternative solution to a spot hedge is a hedge with a forward contract. The sensitivity of a vanilla option's premium with respect to the outright forward rate, $f$, is given as, $$\Delta_f(K,\sigma,\phi):=\frac{\partial V}{\partial f}=\phi\mathcal{N}(\phi d_+).$$
Premium-Adjusted Deltas
- Premium-adjusted spot delta: This takes care of the correction induced by payment of the premium in FOR currency, which is the amount by which the delta hedge in foreign currency has to be corrected. The delta can be represented as, $$\Delta_{s,pa}:=\Delta_s-\frac{V}{x}=\phi e^{-r_f\tau}\frac{K}{f}\mathcal{N}(\phi d_-).$$
- Premum-adjusted forward Delta: As in the case of a spot delta, a premium payment in foreign currency leads to an adjustment of the forward delta. The resulting hedge quantity is given by, $$\Delta_{f,pa}(K,\sigma,\phi):=\Delta_f-\frac{V}{f}=\phi e^{-r_f\tau}\frac{K}{f}\mathcal{N}(\phi d_-).$$
Sadly, which deltas are used in practice varies depending on the currency pair and cannot be attributed to a rule-based system.
At-The-Money Conventions
As you are probably figuring out for yourself, nothing is straightforward in the options market. There tends to be two prominent ways of defining the at-the-money (ATM) strike.
- Forward ATM: This takes into account that the risk-neutral expectation of the future spot is the forward price, which is a natural way of specifying the middle. It is very common for currency pairs with a large interest rate differential (emerging markets) or long maturity (typically 10Y onwards).
- Delta-neutral saddle (DNS): This ensures that a straddle (long call and long put with the same strike) has zero combined delta. The strike that satisfies this is the ATM strike. This convention is considered as the default notion for short-dated FX options (up to 1Y typically).
The table below summarises the different delta conventions as well as ATM conventions.
| Delta convention | Forward ATM | Delta-neutral saddle | ||
|---|---|---|---|---|
| Strike | Delta | Strike | Delta | |
| Spot delta | $fe^{\frac{1}{2}\sigma^2\tau}$ | $f$ | $\frac{1}{2}\phi e^{-r_f\tau}$ | \(\Delta_{\text{spot}}(K_{\text{saddle}})=0\) |
| Forward delta | $fe^{\frac{1}{2}\sigma^2\tau}$ | $f$ | $\frac{1}{2}\phi$ | \(\Delta_{\text{fwd}}(K_{\text{saddle}})=0\) |
| Premium-adjusted spot delta | $fe^{-\frac{1}{2}\sigma^2\tau}$ | $f$ | $\frac{1}{2}\phi e^{-r_f\tau}e^{\frac{1}{2}\sigma^2\tau}$ | \(\Delta_{\text{pa-spot}}(K_{\text{saddle}})=0\) |
| Premium-adjusted forward delta | $fe^{-\frac{1}{2}\sigma^2\tau}$ | $f$ | $\frac{1}{2}e^{\frac{1}{2}\sigma^2\tau}$ | \(\Delta_{\text{pa-fwd}}(K_{\text{saddle}})=0\) |
Delta-Strike Mapping
While it is common in equity markets to quote strike-volatility or strike-price pairs, this is usually not the case in FX markets. It is standard to receive implied volatility-delta pairs from market data providers. The investor is then tasked with transforming implied volatility-delta to strike-price pairs respecting the delta and ATM conventions. This section will outline the algorithms which can be used to that end.
It is straightforward to compute strikes from unadjusted deltas. However, since explicit strike expressions in premium-adjusted deltas are not available, we must solve for the strikes numerically. For a given spot delta, $\Delta_s$, and corresponding volatility, $\sigma$, the strike can be retrieved explicitly. $$K=f\exp\left(-\phi\mathcal{N}^{-1}\left(\phi e^{r_f\tau}\Delta_s\right)\sigma\sqrt{\tau}+\frac{1}{2}\sigma^2\tau\right).$$ The equivalent forward delta defintion is given below. $$K=f\exp\left(-\phi\mathcal{N}^{-1}\left(\phi\Delta_f\right)\sigma\sqrt{\tau}+\frac{1}{2}\sigma^2\tau\right).$$
For a premium-adjusted forward delta, we obtain the following equation. $$\Delta_{f,pa}(K,\sigma,\phi)=\phi\frac{K}{f}\mathcal{N}(\phi d_-)=\phi\frac{K}{f}\left(\phi\frac{\log\left(\frac{f}{K}-\frac{1}{2}\sigma^2\tau\right)}{\sigma\sqrt{\tau}}\right).$$ This cannot be solved for strike in closed-form, it must be solved using a numerical procedure. This is straightforward for put delta because the put delta is monotone in strike. This means we can employ a simple procedure such as the bisection or secant method. This is not the case for the call delta. Different schemes can be employed here such as Brent's root searcher, Newton-Raphson method, Nelder-Mead simplex, etc.
Implied Volatility Smiles and Surfaces
To reiterate, in FX markets it is common to use the delta to measure the degree of moneyness. Consequently, volatilities are assigned to deltas (for any delta type), rather than strikes. For example, it is common to quote the volatility for an option which has a premium-adjusted delta of 0.25.
The curve constructed by interpolating through various volatility-delta pairs is called the volatility smile. It is called a smile because this curve exhibits skewness and curvature resembling that of a smile. In FX markets, the volatility smile is given implicitly as a set of restrictions implied by market instruments. This is FX-specific, as other markets quote volatility versus strike directly. A consequence is that one has to employ a calibration procedure to construct a volatility versus delta or volatility versus strike smile.
Finally, the volatility smile is not static. It varies with maturity, leading to a volatility surface. The volatility surface is a function of both moneyness (delta or strike) and time to maturity.
The key part of construction is to build a curve such that it matches the market quotes. The FX market uses three volatility quotes for a given delta (typically 0.25 and 0.10). Simply, we aim to build a curve function $\sigma:A\mapsto B$ that maps strikes to implied volatility such that $\sigma(K_{\text{ATM}})=\sigma_{\text{ATM}}$, $\sigma(K_{25-\text{RR}})=\sigma_{25-\text{RR}}$ and $\sigma(K_{25-\text{BF}})=\sigma_{25-\text{BF}}$.
- ATM volatility, $\sigma_{\text{ATM}}$.
- Risk reversal (RR), $\sigma_{25-\text{RR}}$: A risk reveral is a strategy where you are long an out-of-the-money (OTM) call and short an OTM put with the same expiry and underlying currency pair.
- Butterfly (BF), $\sigma_{25-\text{BF}}$: A call butterfly (call fly) is a strategy where you are long a low-strike call, short two middle-strike calls and long a high-strike call. All options have the same expiry, strikes are spaced evenly and share the same underlying currency pair.
Risk reversals are used to measure the skewness of the smile, the extra volatility added to the 0.25 delta put volatility compared to the 0.25 delta call volatility. Call flies are used to measure the curvature of the smile, the extra volatility added to the 0.25 delta put and call volatility compared to the ATM volatility. Often, a simplified formula is used to easily calculate the 0.25 delta volatilities given these quotes. Let $\sigma_{25-\text{C}}$ be the volatility of a call with delta of 0.25 and $\sigma_{25-\text{P}}$ be the volatility of a put with delta of 0.25. Then, $$\sigma_{25-\text{C}}=\sigma_{\text{ATM}}+\frac{1}{2}\sigma_{25-\text{RR}}+\sigma_{25-\text{BF}},$$ $$\sigma_{25-\text{P}}=\sigma_{\text{ATM}}-\frac{1}{2}\sigma_{25-\text{RR}}+\sigma_{25-\text{BF}}.$$
Volatility Smile Calibration
The last step in this process is now to employ some volatility smile calibration technique. This could be a polynomial interpolation, local volatility model such as Dupire's, stochastic volatility model such as SABR, and many more. We will touch on this in future posts!