# Foreign Exchange Symmetries

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1 Foreign Exchange Symmetries Uwe Wystup MathFinance AG Waldems, Germany 8 September 2008

2 Contents 1 Foreign Exchange Symmetries Motivation Geometric Brownian Motion Model for the Spot Vanilla Options Value A Note on the Forward Greeks Identities Put-Call Parity Homogeneity based Relationships Space Homogeneity Time Homogeneity Put-Call Symmetry Rates Symmetry Foreign-Domestic Symmetry Exotic Options Digital Options Touch Options Barrier Options Quotation Quotation of the Underlying Exchange Rate Quotation of Option Prices Delta and Premium Convention Example Greeks in Terms of Deltas Interpretation of Dual Delta Vega in Terms of Delta Foreign Exchange Symmetries 1.1 Motivation The symmetries of the foreign exchange market are the key feature that distinguishes this market from all others. With a EUR-USD exchange rate of USD per EUR there is an equivalent USD-EUR exchange rate of EUR per USD, which is just the reciprocal. Any model S t for an exchange rate at time t should guarantee that 1 S t is within the same model class. This is satisfied by the Black-Scholes model or local volatility models, but not for many stochastic volatility models.

3 Foreign Exchange Symmetries 3 The further symmetry is that both currencies pay interest, which we can assume to be continuously paid. The FX market is the only market where this really works. In the EUR-USD market any EUR call is equivalent to a USD put. Any premium in USD can be also paid in EUR, any delta hedge can be specified in the amount of USD to sell or alternatively the amount of EUR to buy. Furthermore, if S 1 is a model for EUR-USD and S 2 is a model for USD-JPY, then S 3 = S 1 S 2 should be a model for EUR-JPY. Therefore, besides the reciprocals, also products of two modeling quantities should remain within the same model class. Finally, the smile of a foreign exchange options market is summarized by Risk Reversals and Butterflies, the skew-symmetric part and the symmetric part of a smile curve. In no other market symmetries are so prominent and so heavily used as in FX. It is this special feature that makes it hard for many newcomers to capture the way FX options market participants think. 1.2 Geometric Brownian Motion Model for the Spot We consider the model geometric Brownian motion ds t = (r d r f )S t dt + σs t dw t (1) for the underlying exchange rate quoted in FOR-DOM (foreign-domestic), which means that one unit of the foreign currency costs FOR-DOM units of the domestic currency. In case of EUR-USD with a spot of , this means that the price of one EUR is USD. The notions of foreign and domestic do not refer to the location of the trading entity, but only to this quotation convention. We denote the (continuous) foreign interest rate by r f and the (continuous) domestic interest rate by r d. In an equity scenario, r f would represent a continuous dividend rate. The volatility is denoted by σ, and W t is a standard Brownian motion. We consider this standard model, not because it reflects the statistical properties of the exchange rate (in fact, it does not), but because it is widely used in practice and front office systems and mainly serves as a tool to communicate prices in FX options. These prices are generally quoted in terms of volatility in the sense of this model. 1.3 Vanilla Options The payoff for a vanilla option (European put or call) is given by F = [φ(s T K)] +, (2) where the contractual parameters are the strike K, the expiration time T and the type φ, a binary variable which takes the value +1 in the case of a call and 1 in the case of a put. The symbol x + denotes the positive part of x, i.e., x + = max(0, x) = 0 x.

4 4 Wystup Value In the Black-Scholes model the value of the payoff F at time t if the spot is at S is denoted by v(t, S). The result is the Black-Scholes formula We abbreviate v(s, K, T, t, σ, r d, r f, φ) = φe r dτ [fn (φd + ) KN (φd )]. (3) ˆ S: current price of the underlying ˆ τ = T t: time to maturity ˆ f = IE[S T S t = S] = Se (r d r f )τ : forward price of the underlying ˆ θ ± = r d r f σ ± σ 2 ˆ d ± = ln S K +σθ ±τ σ τ = ln f K ± σ 2 2 τ σ τ ˆ n(t) = 1 2π e 1 2 t2 = n( t) ˆ N (x) = x n(t) dt = 1 N ( x) A Note on the Forward The forward price f is the strike which makes the time zero value of the forward contract F = S T f (4) equal to zero. It follows that f = IE[S T ] = Se (r d r f ) T, i.e. the forward price is the expected price of the underlying at time T in a risk-neutral setup (drift of the geometric Brownian motion is equal to cost of carry r d r f ) Greeks Greeks are derivatives of the value function with respect to model and contract parameters. They are an important information for traders and have become standard information provided by front-office systems. More details on Greeks and the relations among Greeks are presented in Hakala and Wystup [5] or Reiss and Wystup [6]. For vanilla options we list some of them now. (Spot) Delta. The spot delta shows how many units of FOR in the spot must be traded to hedge an option with 1 unit of FOR notional. v S = φe r f τ N (φd + ) (5)

5 Foreign Exchange Symmetries 5 Forward Delta. The forward delta shows how many units of FOR of a forward contract must be traded to hedge an option with 1 unit of FOR notional. φn (φd + ) = IP [φs T φ f ] 2 (6) K It is also equal to the risk-neutral exercise probability or in-the-money probability of the symmetric put (see Section 1.4.3). Gamma. 2 v S 2 = e r f τ n(d +) Sσ τ (7) Vega. Dual Delta. v σ = Se r f τ τn(d + ) (8) v K = φe r dτ N (φd ) (9) The forward dual delta can also be viewed as the risk-neutral exercise probability. N (φd ) = IP [φs T φk] (10) Dual Gamma Identities 2 v = e r dτ n(d ) K 2 Kσ τ (11) Any computations with vanilla options often rely on the symmetry identities d ± = d σ σ, (12) d ± τ = r d σ, (13) d ± τ = r f σ, (14) Se r f τ n(d + ) = Ke rdτ n(d ). (15)

6 6 Wystup Put-Call Parity The put-call-parity is the relationship v(s, K, T, t, σ, r d, r f, +1) v(s, K, T, t, σ, r d, r f, 1) = Se r f τ Ke r dτ, (16) which is just a more complicated way to write the trivial equation x = x + x. Taking the strike K to be equal to the forward price f, we see that the call and put have the same value. The forward is the center of symmetry for vanilla call and put values. However, this does not imply that the deltas are also symmetric about the forward price. The put-call delta parity is v(s, K, T, t, σ, r d, r f, +1) S v(s, K, T, t, σ, r d, r f, 1) S = e r f τ. (17) In particular, we learn that the absolute value of a put delta and a call delta are not exactly adding up to one, but only to a positive number e r f τ. They add up to one approximately if either the time to expiration τ is short or if the foreign interest rate r f is close to zero. For this reason, traders often prefer to work with forward deltas, because these are symmetric in the sense that a 25-delta call is a 75-delta put. Whereas the choice K = f produces identical values for call and put, we seek the deltasymmetric strike Ǩ which produces absolutely identical deltas (spot, forward or driftless). This condition implies d + = 0 and thus Ǩ = fe σ2 2 τ, (18) in which case the absolute delta is e r f τ /2. In particular, we learn, that always Ǩ > f, i.e., there cannot be a put and a call with identical values and deltas. This is natural as the payoffs of calls and puts are not symmetric to start with: the call has unlimited up-side potential, whereas the put payoff is always bounded by the strike. Note that the strike Ǩ is usually chosen as the middle strike when trading a straddle or a butterfly. Similarly the dual-delta-symmetric strike ˆK = fe σ2 2 τ can be derived from the condition d = 0. Note that the delta-symmetric strike Ǩ also maximizes gamma and vega of a vanilla option and is thus often considered as a center of symmetry. 1.4 Homogeneity based Relationships Space Homogeneity We may wish to measure the value of the underlying in a different unit. This will obviously effect the option pricing formula as follows: av(s, K, T, t, σ, r d, r f, φ) = v(as, ak, T, t, σ, r d, r f, φ) for all a > 0. (19)

7 Foreign Exchange Symmetries 7 Differentiating both sides with respect to a and then setting a = 1 yields v = Sv S + Kv K. (20) Comparing the coefficients of S and K in Equations (3) and (20) leads to suggestive results for the delta v S and dual delta v K. This space-homogeneity is the reason behind the simplicity of the delta formulas, whose tedious computation can be saved this way Time Homogeneity We can perform a similar computation for the time-affected parameters and obtain the obvious equation v(s, K, T, t, σ, r d, r f, φ) = v(s, K, T a, t a, aσ, ar d, ar f, φ) for all a > 0. (21) Differentiating both sides with respect to a and then setting a = 1 yields 0 = τv t σv σ + r d v rd + r f v rf. (22) Of course, this can also be verified by direct computation. The overall use of such equations is to generate double checking benchmarks when computing Greeks. These homogeneity methods can easily be extended to other more complex options Put-Call Symmetry By put-call symmetry we understand the relationship (see [1], [2],[3] and [4]) v(s, K, T, t, σ, r d, r f, +1) = K f v(s, f 2 The geometric mean of the strike of the put f 2 f 2 K K K, T, t, σ, r d, r f, 1). (23) and the strike of the call K is equal to K = f, the outright forward rate. Therefore, the outright forward rate can be interpreted as a geometric mirror reflecting a call into a certain number of puts. Note that for at-the-money (forward) options (K = f) the put-call symmetry coincides with the special case of the put-call parity where the call and the put have the same value Rates Symmetry Direct computation shows that the rates symmetry v + v = τv (24) r d r f holds for vanilla options. This relationship, in fact, holds for all European options and a wide class of path-dependent options as shown in [6].

8 8 Wystup Foreign-Domestic Symmetry One can directly verify the foreign-domestic symmetry 1 S v(s, K, T, t, σ, r d, r f, φ) = Kv( 1 S, 1 K, T, t, σ, r f, r d, φ). (25) This equality can be viewed as one of the faces of put-call symmetry. The reason is that the value of an option can be computed both in a domestic as well as in a foreign scenario. We consider the example of S t modeling the exchange rate of EUR/USD. In New York, the call option (S T K) + costs v(s, K, T, t, σ, r usd, r eur, 1) USD and hence v(s, K, T, t, σ, r usd, r eur, 1)/S ( ) 1 +. EUR. This EUR-call option can also be viewed as a USD-put option with payoff K 1 K S T This option costs Kv( 1, 1, T, t, σ, r S K eur, r usd, 1) EUR in Frankfurt, because S t and 1 S t have the same volatility. Of course, the New York value and the Frankfurt value must agree, which leads to (25). This can also be seen as a change of measure to the foreign discount bond as Numeraire, see, e.g., Shreve s Stochastic Calculus for Finance [7]. 1.5 Exotic Options In FX markets one can use many symmetry relationships for exotic options Digital Options For example, let us define the payoff of digital options by DOM paying: 1 {φst φk}, (26) FOR paying: S T 1 {φst φk}, (27) where the contractual parameters are the strike K, the expiration time T and the type φ, a binary variable which takes the value +1 in the case of a call and 1 in the case of a put. Then we observe that a DOM-paying digital call in the currency pair FOR-DOM with a value of v d units of domestic currency must be worth the same as a FOR-paying digital put in the currency pair DOM-FOR with a value of v f units of foreign currency. And since we are looking at the same product we conclude that v d = v f S, where S is the initial spot of FOR-DOM Touch Options This key idea generalizes from these path-independent digitals to touch products. Consider the value function for one-touch in EUR-USD paying 1 USD. If we want to find the value function of a one-touch in EUR-USD paying 1 EUR, we can price the one-touch in USD- EUR paying 1 EUR using the known value function with rates r d and r f exchanged, volatility

9 Foreign Exchange Symmetries 9 unchanged, using the formula for a one-touch in EUR-USD paying 1 USD. We also note that an upper one-touch in EUR-USD becomes a lower one-touch in USD-EUR. The result we get is in domestic currency, which is EUR in USD-EUR notation. To convert it into a USD price, we just multiply by the EUR-USD spot S Barrier Options For a standard knock-out barrier option we let the value function be v(s, r d, r f, σ, K, B, T, t, φ, η), (28) where B denotes the barrier and the variable η takes the value +1 for a lower barrier and 1 for an upper barrier. With this notation at hand, we can state our FOR-DOM symmetry as v(s, r d, r f, σ, K, B, T, t, φ, η) = v(1/s, r f, r d, σ, 1/K, 1/B, T, t, φ, η) S K. (29) Note that the rates r d and r f have been purposely interchanged. This implies that if we know how to price barrier contracts with upper barriers, we can derive the formulas for lower barriers. 1.6 Quotation Quotation of the Underlying Exchange Rate Equation (1) is a model for the exchange rate. The quotation is a permanently confusing issue, so let us clarify this here. The exchange rate means how much of the domestic currency are needed to buy one unit of foreign currency. For example, if we take EUR/USD as an exchange rate, then the default quotation is EUR-USD, where USD is the domestic currency and EUR is the foreign currency. The term domestic is in no way related to the location of the trader or any country. It merely means the numeraire currency. The terms domestic, numeraire or base currency are synonyms as are foreign and underlying. Commonly, we denote with the slash (/) the currency pair and with a dash (-) the quotation. The slash (/) does not mean a division. For instance, EUR/USD can also be quoted in either EUR-USD, which then means how many USD are needed to buy one EUR, or in USD-EUR, which then means how many EUR are needed to buy one USD. There are certain market standard quotations listed in Table Quotation of Option Prices Values and prices of vanilla options may be quoted in the six ways explained in Table 2. The Black-Scholes formula quotes an option value in units of domestic currency per unit of foreign notional. Since this is usually a small number, it is often multiplied with 10, 000 and

10 10 Wystup currency pair default quotation sample quote GBP/USD GPB-USD GBP/CHF GBP-CHF EUR/USD EUR-USD EUR/GBP EUR-GBP EUR/JPY EUR-JPY EUR/CHF EUR-CHF USD/JPY USD-JPY USD/CHF USD-CHF Table 1: Standard market quotation of major currency pairs with sample spot prices name symbol value in units of example domestic cash d DOM 29,148 USD foreign cash f FOR 24,290 EUR % domestic % d DOM per unit of DOM % USD % foreign % f FOR per unit of FOR % EUR domestic pips d pips DOM per unit of FOR USD pips per EUR foreign pips f pips FOR per unit of DOM EUR pips per USD Table 2: Standard market quotation types for option values. In the example we take FOR=EUR, DOM=USD, S = , r d = 3.0%, r f = 2.5%, σ = 10%, K = , T = 1 year, φ = +1 (call), notional = 1, 000, 000 EUR = 1, 250, 000 USD. For the pips, the quotation USD pips per EUR is also sometimes stated as % USD per 1 EUR. Similarly, the EUR pips per USD can also be quoted as % EUR per 1 USD. quoted as domestic pips per foreign, in short, d pips. The others can be computed using the following instruction. d pips 1 S %f S K %d 1 S f pips SK d pips (30)

12 12 Wystup This standard inter bank notion must be adapted to the real delta-risk of the bank for an automated trading system. For currency pairs where the risk free currency of the bank is the domestic or base currency, it is clear that the delta is the raw delta of the option, and for risky premium this premium must be included. In the opposite case the risky premium and the market value must be taken into account for the base currency premium, such that these offset each other. And for premium in underlying currency of the contract the market-value needs to be taken into account. In that way the delta hedge is invariant with respect to the risky currency notion of the bank, e.g. the delta is the same for a USD-based bank and a EUR-based bank Example We consider two examples in Table 3 and 4 to compare the various versions of deltas that are used in practice. delta ccy prem ccy Fenics formula delta % EUR EUR lhs δ raw P % EUR USD rhs δ raw % USD EUR rhs [flip F4] (δ raw P )S/K % USD USD lhs [flip F4] (δ raw )S/K Table 3: 1y EUR call USD put strike K = for a EUR based bank. Market data: spot S = , volatility σ = 12%, EUR rate r f = 3.96%, USD rate r d = 3.57%. The raw delta is 49.15%EUR and the value is 4.427%EUR. delta ccy prem ccy Fenics formula delta % EUR EUR lhs δ raw P % EUR USD rhs δ raw % USD EUR rhs [flip F4] (δ raw P )S/K % USD USD lhs [flip F4] δ raw S/K Table 4: 1y call EUR call USD put strike K = for a EUR based bank. Market data: spot S = , volatility σ = 12%, EUR rate r f = 3.96%, USD rate r d = 3.57%. The raw delta is 94.82%EUR and the value is 21.88%EUR.

13 Foreign Exchange Symmetries Greeks in Terms of Deltas In Foreign Exchange markets the moneyness of vanilla options is always expressed in terms of deltas and prices are quoted in terms of volatility. This makes a ten-delta call a financial object as such independent of spot and strike. This method and the quotation in volatility makes objects and prices transparent in a very intelligent and user-friendly way. At this point we list the Greeks in terms of deltas instead of spot and strike. Let us introduce the quantities + = φe r f τ N (φd + ) spot delta, (32) = φe r dτ N (φd ) dual delta, (33) which we assume to be given. From these we can retrieve Interpretation of Dual Delta d + = φn 1 (φe r f τ + ), (34) d = φn 1 ( φe r dτ ). (35) The dual delta introduced in (9) as the sensitivity with respect to strike has another - more practical - interpretation in a foreign exchange setup. It is easy to verify that for vanilla options the domestic value v(s, K, τ, σ, r d, r f, φ) (36) corresponds to a foreign value ( 1 v S, 1 ) K, τ, σ, r f, r d, φ up to an adjustment of the nominal amount by the factor SK. From a foreign point of view the delta is thus given by 1/S ln( φe rdτ 1/K N ( φ ) + (r ) f r d σ2 τ) σ τ ( = φe rdτ N φ ln( S ) + (r ) K d r f 1 2 σ2 τ) σ τ (37) =, (38) which means the dual delta is the delta from the foreign point of view. Now, we list value, delta and vega in terms of S, +,, r d, r f, τ and φ. Value. v(s, +,, r d, r f, τ, φ) = S + + S e r f τ n(d + ) e r dτ n(d ) (39)

14 14 Wystup (Spot) Delta. v S = + (40) Vega. v σ = Se r f τ τn(d + ) (41) Notice that vega does not require knowing the dual delta Vega in Terms of Delta Mat/ 50% 45% 40% 35% 30% 25% 20% 15% 10% 5% 1D W W M M M M M Y Y Y Table 5: Vega in terms of Delta for the standard maturity labels and various deltas. It shows that one can neutralize a vega position of a long 9M 35 delta call with 4 short 1M 20 delta puts. This offsetting, however, is not a static, but only a momentary hedge. The mapping v σ = Se r f τ τn(n 1 (e r f τ )) is important for trading vanilla options. Observe that this function does not depend on r d or σ, just on r f. Quoting vega in % foreign will additionally remove the spot dependence. This means that for a moderately stable foreign term structure curve, traders will be able to use a moderately stable vega matrix. For r f = 3% the vega matrix is presented in Table 5.

15 Foreign Exchange Symmetries 15 The most important result of this paragraph is the fact that vega can be written in terms of delta, which is the main reason why the FX market uses implied volatility quotation based on deltas in the first place. References [1] Bates, D. (1988). Crashes, Options and International Asset Substitutability. Ph. D. Dissertation, Economics Department, Princeton University. [2] Bates, D. (1991). The Crash of 87 Was It Expected? The Evidence from Options Markets. The Journal of Finance. 46:3. July 1991, pp [3] Bowie, J. and Carr, P. (1994). Static Simplicity. Risk. (8) [4] Carr, P. (1994). European Put Call Symmetry. Cornell University Working Paper. [5] Hakala, J. and Wystup, U. (2002). Foreign Exchange Risk. Risk Publications, London. [6] Reiss, O. and Wystup, U. (2001). Efficient Computation of Option Price Sensitivities Using Homogeneity and other Tricks. The Journal of Derivatives. Vol. 9 No. 2. [7] Shreve, S.E. (2004). Stochastic Calculus for Finance II. Springer. [8] Wystup, U. (2006). FX Options and Structured Products. Wiley Finance Series.

16 Index base currency, 9 Black-Scholes formula, 4 butterfly, 6 delta, 4 delta quotation convention, 10 delta-symmetric strike, 6 DOM, 3 domestic, 3, 9 dual delta, 5 dual gamma, 5 underlying, 9 vanilla option, 3 vega, 5 vega in terms of delta, 14 vega matrix, 14 exotic option, 8 FOR, 3 foreign, 3, 9 foreign exchange symmetries, 2 foreign-domestic symmetry, 8 forward delta, 5 gamma, 5 geometric Brownian motion, 3 Greeks, 4 Greeks in terms of deltas, 12 homogeneity, 6 numeraire currency, 9 put-call delta parity, 6 put-call symmetry, 7 put-call-parity, 6 quotation, 9 rates symmetry, 7 space-homogeneity, 7 spot delta, 4 straddle, 6 time-homogeneity, 7 16

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