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## Editorial

FX Options Delta

Today let me illustrate the different types of Delta commonly used in the FX Options context. In FX, Delta, and many other Greeks, can be confusing, because they depend on the quotation of the currency pair as well as the currency in which they are calculated. Furthermore, premium can be included or excluded, smile-effect can be included or not included, and numerical approximations may further add to the confusion.

Example Market EUR-USD spot reference S =1.1000, forward rate F =1.1102, USD money market interest rate 1.009%, EUR money market interest rate -0.810%, volatility σ(K) =9.73%, ATM volatility=10.493%, 25-delta-Risk-Reversal RR=-2.466%, 25-Delta-Butterfly BF=0.360%.

Example Product EUR call USD put, strike K =1.1500, maturity 6 months = 182 days between delivery and spot date, EUR nominal (call currency amount) N = 1,000,000, USD nominal (put currency amount)
NK =1,150,000, horizon 5 Dec 2016, spot date 7 Dec 2016, expiry date  5 Jun 2017, delivery date 7 Jun 2017.

Delta generally represents the change of the value of a contract based on the change of the underlying spot. In the case of FX derivatives delta represents the change of the value of a derivative based on the change of the underlying exchange rate. The exchange rate can be quoted in two directions: EUR-USD denotes the price of one EUR in units of USD; USD-EUR denotes the price of one USD in units of EUR. Additional difficulties arise from the possibility to quote a price of an option in both EUR and USD, and that a delta can refer to both a EUR notional and a USD notional. Moreover, a change in the exchange rate can mean either a change in EUR-USD or a change in USD-EUR. Apart from that, delta can be calculated analytically as a derivative of the pricing formula (if there is one) or numerically by a ratio of finite differences. For the finite differences, there are one-sided and two-sided difference quotients. And for the change of the underlying spot one can assume an absolute change or a relative change. Overall, even in a simple Black-Scholes model, this produces a whole variety of deltas (and other sensitivities). Finally, if we include smile, i.e. a dependence of the volatility on moneyness, the ratio of spot and strike price, we will consider smiled delta and more generally smiled Greeks. Using a pricing model beyond Black-Scholes, such as a stochastic volatility model or jump-diffusion model, will give rise to model delta. And here, we need to make assumptions, whether we re-calibrate a model after a spot shift, or assume model parameters to be constant. Let us denote the value of an FX option by v(S) in USD, with S denoting the current price of one EUR in USD.

Spot-Delta and Forward-Delta
Besides assuming a change of the FX spot price one can also assume a change in the value of a forward contract. This is because, one can delta-hedge the spot change risk of an option in the spot market, for which spot delta is required, or in the forward market, for which a forward delta is required. Forward delta differs from the spot delta only by omitting the discount factor in the foreign currency (EUR), more precisely by the EUR discount factor that ensures the interest-rate-forward parity to hold.

Discrete Delta for a Relative Spot Change
It is common practice in risk management to calculate delta numerically via a quotient of finite symmetric differences. This is mainly because delta is calculated on the portfolio level rather than for single transactions. For delta we commonly assume a relative/percentage change of the FX spot by a = 1%, which means we consider the ratio v(S + a/2S) −v(S −a/2S), which is approximately equal to a· S· vs.Therefore, the risk managers’ delta assuming a relative change of the FX spot, and assuming risk is measured in EUR, agrees exactly with the premium-adjusted or premium-included delta of an FX options trader, in the case the option premium is paid in EUR, (up to the factor a).

Relation of Delta to Notional Amounts
Spot-delta N·vs denotes how many EUR to buy when delta-hedging a short option. Thus, N·S·vs denotes the amount of USD to sell. Since the USD notional is N·K, we obtain the USD amount to sell per USD notional of the option as S·vs /K. Equivalently we need to buy − S·vs /K USD per USD notional. Similar results can be derived for premium-adjusted delta.

Smile-Delta
When including the volatility smile, but still using the Black-Scholes formula for valuation, as it is common market practice, we obtain smile-adjusted or smiled Greeks. On the one hand the value of an option v depends directly on the spot S, but on the other hand it also depends on the volatility, which is itself a function of the spot and strike. Therefore, delta must be calculated as vs + VanillaVega · σS(S,K), where the second term is the well-known windmill-adjustment. Using homogeneity σ (aS,aK) = σ (S,K) we know that σS(S,K) = – σK(S,K)·K/S. The slope of the implied volatility on the strike space leads to smile-adjusted valuation of digital options, and similarly to smile-adjusted delta of standard options. The slope is not a consequence of the Black-Scholes model; in fact, it contradicts the Black-Scholes model. It depends on the method chose to interpolate and extrapolate the smile. Consequently, smile-delta is not uniquely determined, but depends on the smile construction procedure in place. Note that all the other variants of delta discussed so far can also be smiled.

Delta Variants Example
As a summary we list the various delta variants in the following table. Option Value is 150 USD pips or 1.36% EUR. Smile-Delta Calculated via parabolic interpolation on the forward-delta space. The interest r denotes the continuously compounded EUR rate.

 Formula Value Description vS +31.73% EUR spot delta: EUR to buy vS·erT +31.60% EUR forward delta – vS·S -34.91% USD per EUR spot delta: USD to buy per EUR notional – vS·S/K -30.35% USD spot delta: USD to buy per USD notional vS-v/S +30.37% EUR premium-adjusted spot delta (vS-v/S)·erT +30.25% EUR premium-adjusted forward delta vS·S-v -33.41% USD pro EUR p.-a. spot delta: USD to buy per EUR notional (vS·S-v)/K -29.05% USD p.-a. spot delta: USD to buy per USD notional vS – VanillaVega ·σK(S,K)·K/S +35,63% EUR parabolically smiled spot delta: EUR to buy

I would particularly like to point out the difference of a spot delta, which is calculated with the smile volatility for the given strike, and the smiled spot delta, which also takes into account the slope of the smile at the strike. If the interpolator is the assumed to be the right choice, then this is the delta hedge to actually perform. In our example, missing out the windmill effect, the delta hedge would be wrong by about 4% of the EUR notional. However, the market quotes from common providers typically use forward delta (with volatility at strike, but without the slope). This is because windmill-adjustment depends on the choice of the interpolator, and there is no market consensus in place for such a choice. The windmill-adjustment can be calculated analytically for the parabolic smile interpolation, which is useful to illustrate the effect, but by no means the market standard, just to make that clear.

The list clearly shows that any discussion about Greeks will take a long time, and the longer you think about it, the worse it gets. Actually, you shouldn’t have read this.

Uwe Wystup, Managing Director of MathFinance

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