Foreign Exchange Implied Volatility Surface Copyright Changwei Xiong 2011-2017 January 19, 2011 last update: October 1, 2017 TABLE OF CONTENTS Table of Contents...1 1. Trading Strategies of Vanilla Options... 1.1. Single Call and Put... 1.2. Call Spread and Put Spread... 1.. Risk Reversal...4 1.4. Straddle and Strangle...4 1.5. Butterfly...4 2. FX Option Price Conversions...5. Risk Sensitivities...6.1. Delta...6.1.1. Pips Spot Delta... 7.1.2. Percentage Spot Delta... 7.1.. Pips Forward Delta... 9.1.4. Percentage Forward Delta... 9.1.5. Strike from Delta Conversion... 10.2. Other Risk Sensitivities... 11.2.1. Theta... 11.2.2. Gamma... 11.2.. Vega... 12.2.4. Vanna... 12.2.5. Volga... 12 4. FX Volatility Convention...1
4.1. At-The-Money Volatility...1 4.1.1. ATM Forward... 1 4.1.2. Delta Neutral Straddle... 1 4.2. Risk Reversal Volatility...14 4.. Strangle Volatility...15 4..1. Market Strangle... 15 4..2. Smile Strangle... 15 5. Volatility Surface Construction...17 5.1. Smile Interpolation...17 5.2. Temporal Interpolation...18 5.. Volatility Surface by Standard Conventions...18 6. The Vanna-Volga Method...19 6.1. Vanna-Volga Pricing...19 6.2. Smile Interpolation...20 References...24 2
This note firstly introduces the basic option trading strategies and the Greek letters of the Black-Scholes model. It further discusses various market quoting conventions for the at-the-money and delta styles, and then summarizes the definition of the market quoted at-the-money, risk reversal and strangle volatilities. A volatility surface can be constructed from these volatilities which provides a way to interpolate an implied volatility at any strike and maturity from the surface. At last, the vanna-volga pricing method [1] is presented which is often used for pricing first-generation FX exotic products. An simple application of the method is to build a volatility smile that is consistent with the market quoted volatilities and allows us to derive implied volatility at any strike, in particular for those outside the basic range set by the market quotes. 1. TRADING STRATEGIES OF VANILLA OPTIONS In the following, we will introduce a few simple trading strategies based on vanilla options. These products are traded liquidly in FX markets. 1.1. Single Call and Put The figures below depict the payoff functions of vanilla options. PL PL K S T K S T Short a Put Long a Call PL PL K S T K S T Long a Put Short a Call 1.2. Call Spread and Put Spread
A call spread is a combination of a long call and a short call option with different strikes K 1 < K 2. A put spread is a combination of a long put and a short put option with different strikes. The figure below shows the payoff functions of a call spread and a put spread. PL PL K 1 K 2 S T K 1 K 2 S T CallSpread = CሺK 1 ሻ CሺK 2 ሻ PutSpread = PሺK 2 ሻ PሺK 1 ሻ 1.. Risk Reversal A risk reversal (RR) is a combination of a long call and a short put with different strikes K 1 < K 2. This is a zero-cost product as one can finance a call option by short selling a put option. The figure below shows the payoff function of a risk reversal. PL K 1 K 2 S T RiskReversal = CሺK 1 ሻ PሺK 2 ሻ 1.4. Straddle and Strangle A straddle is a combination of a call and a put option with the same strike K. A strangle is a combination of an out-of-money call and an out-of-money put option with two different strikes K 1 < K ATM < K 2. The figure below shows the payoff functions of a straddle and a strangle PL PL K S T K 1 K 2 S T Straddle = CሺKሻ + PሺKሻ Strangle = CሺK 1 ሻ + PሺK 2 ሻ 1.5. Butterfly 4
A butterfly (BF) is combinations of a long strangle and a short straddle. The figure below shows the payoff function of a butterfly PL K 1 K K 2 S T Butterfly = StrangleሺK 1, K 2 ሻ StraddleሺKሻ K 1 < K, K ATM < K 2 2. FX OPTION PRICE CONVERSIONS Currency pairs are commonly quoted using ISO codes in the format FORDOM, where FOR and DOM denote foreign and domestic currency respectively. For example in EURUSD, the EUR denotes the foreign currency or currency1 and USD the domestic currency or currency2. The rate of EURUSD tells the price of one euro in USD. Under risk neutral measure, Black-Scholes model assumes the FX spot rate follows a geometric Brownian motion characterized by the volatility σ ds t S t = μdt + σdw t (1) where the drift term μ = r d r f and the r d and r f are the domestic and foreign risk free rate respectively. With the assumption of deterministic interest rates, the price of an option on the spot rate can be expressed in Black-Scholes model in terms of an implied volatility σ Bሺω, K, σ, τሻ = ωd f SΦሺωd + ሻ ωd d KΦሺωd ሻ = ωd d (FΦሺωd + ሻ KΦሺωd ሻ) (2) where we define ω = 1 or 1 for call or put respectively, τ = T t for term to maturity, function Φ for standard normal cumulative density function, and d + and d as follows d + = 1 σ τ ln F K + σ τ 2 and d = d + σ τ () In (2), we also have the FX forward F that can be computed using the deterministic rates as T D f F t,t = S t = S D t exp ( (r d r f )ds) (4) d t 5
T t T t where D d = exp ( r d ds) and D f = exp ( r f ds) are the domestic and foreign discount factor respectively. The option price quote convention may vary [2]. Options can be quoted in one of four relative quote styles: domestic per foreign (fd), percentage foreign (%f), percentage domestic (%d) and foreign per domestic (df). The call and put price defined in (2) are actually expressed in domestic per foreign style (also known as the domestic pips price), denoted by V fd. With the notional amount N f expressed in foreign currency, we have V fd = N f Bሺω, K, σ, τሻ. The other price quote styles have the following conversions with respect to V fd V %f = V fd S, V %d = V fd K, V df = V fd SK (5) It is very important to note that this technique of constructing all these different quote styles only works where there are two notionals given by strike K = N d N f, in foreign and domestic currencies, and there is a fixed relationship between them, which is known from the start. This is true for European and American style vanilla options, even in the presence of barriers and accrual features, but is most definitely not true for digital options. Suppose one has a cash-or-nothing digital which pays one USD if the EURUSD FX rate fixes at time T above a particular level (sometimes called strike, which actually leads to the confusion). The digital clearly has a USD notional (= $1, the domestic notional) so we can obtain percentage domestic (%USD) and foreign per domestic (EUR/USD) prices. However, there is no EUR notional (the foreign notional) at all so the other two quote styles are meaningless [].. RISK SENSITIVITIES Risk sensitivity of an option is the sensitivity of the price to a change in underlying state variables or model parameters. We will present some basic types of risk sensitivities in the context of Black-Scholes model..1. Delta 6
Delta is the ratio of change in option value to the change in spot or forward. There are several definitions of delta, such as spot/forward delta, pips/percentage delta, etc. Since FX volatility smiles are commonly quoted as a function of delta rather than as a function of strike, it is important to use a delta style consistent with the market convention..1.1. Pips Spot Delta The pips spot delta is defined in Black-Scholes model as the first derivative of the present value with respect to the spot, both in domestic per foreign terms, corresponding to risk exposures in FOR. This style of delta implies that the premium currency is DOM and notional currency is FOR. It is commonly adopted by currency pairs with USD as DOM (or currency2), e.g. EURUSD, GBPUSD and AUDUSD, etc. By assuming N f = 1 and hence V fd = Bሺω, K, σ, τሻ, the pips spot delta is equivalent to the standard Black-Scholes delta Δ S = V fd S = ωd fφሺωd + ሻ + ωd f Sφሺd + ሻω d + S ωd dkφሺd ሻω d S = ωd fφሺωd + ሻ (6) where we have used the following identities Φሺωd + ሻ = ωφሺd d + ሻ = ω + 2π exp ( d 2 + 2 ), Φሺωd ሻ = ωφሺd d ሻ = ω F K φሺd +ሻ Φሺ xሻ = 1 Φሺxሻ, d + S = d S = 1 Sσ τ (7) with φ the normal probability density function. To understand pips spot delta, assuming DOM is the numeraire, if one wants to hedge a short option position of N f notional with a premium of N f V fd in DOM, he must long N f Δ S amount of the spot S t by taking a long in N f Δ S units of FOR and a short in N f Δ S S t units of DOM..1.2. Percentage Spot Delta The percentage spot delta (also known as premium adjusted pips spot delta) is defined as a derivative of the present value with respect to the spot, both in percentage foreign terms, corresponding to risk exposures in DOM. This style of delta implies that the premium currency and notional currency 7
both are FOR. It is used by currency pairs like USDJPY, EURGBP, etc. In Black-Scholes model, the percentage spot delta has the form Δ %S = V %f S S = S S (V fd S ) = V fd S V fd S = Δ K S V %f = ωd f F Φሺωd ሻ (8) which shows that the percentage spot delta is the pips spot delta premium-adjusted by percentage foreign option value. This can be explained by assuming FOR is the numeraire. If one wants to hedge a short position with a premium of N f V fd /S t in FOR, the delta sensitivity with respect to 1/S t must be V fd S 1 S = 1 S V fd V fd S 2 S 1 S 2 S = V fd Δ S S t (9) To hedge the delta risk, one must long N f (V fd Δ S S t ) amount of the spot 1/S t by taking a long in N f (V fd Δ S S t ) units of DOM and a short in N f (V fd /S t Δ S ) units of FOR. This is equivalent to taking a short in N f (Δ S S t V fd ) units of DOM and a long in N f (Δ S V fd /S t ) units of FOR, which translates into the percentage spot delta Δ S V %f. Whether pips or percentage delta is quoted in markets depends on which currency in the currency pair, FORDOM, is the premium currency, and the definition of premium currency itself is a market convention. If the premium currency is DOM, then no premium adjustment is applied and the pips delta is used, whereas if the premium currency is FOR then the percentage delta is used. Despite the fact that market convention involves different delta quotation styles, they are mutually equivalent to one another (referring to [4] for more details). The difference between pips delta and percentage delta comes naturally from the change of measure between domestic and foreign risk-neutral measures. Consider the case of a call option on FORDOM, or to be more thorough, a FOR call/dom put. If the two counterparties to such a trade are FOR based and DOM based respectively, then they will agree on the price. However, the price will be expressed and actually exchanged in one of two currencies: FOR or DOM. From a domestic investor s point of view, if the premium currency is DOM, the premium itself is 8
riskless and the hedging of the option can be done by simply taking Δ S amount of FORDOM spot. If however the premium currency is FOR, there will be two sources of currency risk: 1) the change in intrinsic option value due to the move in underlying spot. 2) the change in premium value converted from FOR to DOM due to the move in FX rate. Apparently to hedge the two risks, one must take Δ S and V fd /S t amount of spot position respectively..1.. Pips Forward Delta The pips forward delta is the ratio of the change in forward value (in contrast to present value!) of the option to the change in the relevant forward, both in domestic per foreign terms Δ F = V F;fd F = ωφሺωd +ሻ = Δ S D f (10) by the following facts V F;fd = V fd D d = ωfφሺωd + ሻ ωkφሺωd ሻ,.1.4. Percentage Forward Delta d + F = d F = 1 Fσ τ (11) The percentage forward delta is defined as the ratio of the change in forward value to the change in the forward, both in percentage foreign terms Δ %F = V F;%f F F = F F (V F;fd F ) = V F;fd F V F;fd F = Δ F V F;%f = ω K F Φሺωd ሻ (12) Again, the percentage forward delta is the pips forward delta premium-adjusted by forward percentage foreign option value. The choice between spot delta and forward delta depends on the currency pair as well as the option maturity. Spot delta is mainly used for tenors less than or equal to 1Y and for the currency pair with both currencies from the more developed economies. Otherwise, the use of forward delta dominates. It is obvious that the spot delta and forward delta differ only by a foreign discount factor D f. Since the credit crunch of 2008 and the associated low levels of liquidity in short-term interest rate 9
products, it became unfeasible for banks to agree on spot deltas (which include discount factors) and, as a result, market practice has largely shifted to using forward deltas exclusively in the construction of the FX smile, which do not include any discounting [5]..1.5. Strike from Delta Conversion It is straightforward to compute strikes from pips deltas. However, since explicit strike expressions in percentage deltas are not available, we must solve for the strikes numerically. It can be seen that the percentage deltas are monotonic in strike on put side, but this is not the case on call side. Using percentage forward delta as an example, the expression of a call delta is Δ %F = K F Φ ( 1 σ τ ln F K σ τ 2 ) (1) Obviously, the delta has two sources of dependence on strike and the function is not always monotonic. This may result in two different solutions of strike. To avoid the undesired solution, the numerical search can be performed within a range ሺK min, K max ሻ that encloses the proper strike solution. We can choose the strike by pips delta as the upper bound K max (because a pips delta maps to a strike that is always larger than that of a percentage delta) and the lower bound K min can be found numerically as a solution to the equation below (where K min maximizes the Δ %F ) [6] Δ %F K = Φሺd ሻ F 1 Fσ τ φሺd ሻ = 0 Φሺd ሻσ τ = φሺd ሻ (14) However, the function below fሺkሻ = Φሺd ሻσ τ φሺd ሻ (15) is also not monotonic. It has a maximum σ τ when K 0 and a minimum when K = F exp ( 1 2 σ2 τ), which can be used to find the K min. The table below summarizes the delta and strike conversion of the 4 delta conventions. Table 1. Deltas and delta neutral straddle strikes Delta Convention Delta from Strike Strike from Delta 10
pips spot Δ S ሺKሻ = ωd f Φሺωd + ሻ Kሺδ Δ S ሻ = F exp ( σ2 τ 2 ωσ τφ 1 ( ωδ D f )) pips forward Δ F ሺKሻ = ωφሺωd + ሻ Kሺδ Δ F ሻ = F exp ( σ2 τ 2 ωσ τφ 1 ሺωδሻ) percentage spot Δ %S ሺKሻ = ωd f K F Φሺωd ሻ Kሺδ Δ %S ሻ (K min, Kሺδ Δ S ሻ) for ω = 1 percentage forward Δ %F ሺKሻ = ω K F Φሺωd ሻ Kሺδ Δ %F ሻ (K min, Kሺδ Δ F ሻ) for ω = 1.2. Other Risk Sensitivities In the following context, we will only express the risk sensitivities in domestic per foreign terms for simplicity. Assuming the value of option is given in the Black-Scholes model, e.g. V fd = Bሺω, K, σ, τሻ, the risk sensitivities can be derived as follows..2.1. Theta Theta θ is the first derivative of the option price with respect to the initial time t. Converting from t to τ, we have θ = B/ t = B/ τ. Let s first derive the partial derivatives d + τ = (( μ σ + σ 2 ) τ + 1 σ τ ln S t K ) τ = μ 2σ τ + σ 4 τ 1 2σ τ ln S K (16) d τ = (d + σ τ) = μ τ 2σ τ σ 4 τ 1 2σ τ ln S K The theta can then be derived using identity D f Sφሺd + ሻ = D d Kφሺd ሻ, that is θ = B t = ωr fd f SΦሺωd + ሻ D f Sφሺd + ሻ d + τ ωr dd d KΦሺωd ሻ + D d Kφሺd ሻ d τ = ωr f D f SΦሺωd + ሻ ωr d D d KΦሺωd ሻ D f Sφሺd + ሻ σ 2 τ (17).2.2. Gamma 11
Spot (forward) Gamma Γ is the first derivative of the spot (forward) delta Δ with respect to the underlying spot S t (forward F t,t ), or equivalently the second derivative of the present (forward) value of the option with respect to the spot (forward) Γ S = 2 B S 2 = Δ S S = D fφሺd + ሻ Sσ τ, Γ F = 2 B F 2 = Δ F F = φሺd +ሻ Fσ τ (18) The call and the put option with an equal strike have the same gamma sensitivity..2.. Vega Vega V is the first derivative of the option price with respect to the volatility σ. Let s first derive ( 1 d + σ = σ τ ln F K + σ τ 2 ) σ = 1 σ 2 τ ln F K + τ 2 = d + σ + τ = d σ (19) d σ = (d + σ τ) = d + σ σ τ = d + σ Therefore, we have V = B σ = D fsφሺd + ሻ d + σ D dkφሺd ሻ d σ = D fsφሺd + ሻ d + d = D σ f S τφሺd + ሻ (20) The call and the put option with an equal strike have the same vega sensitivity..2.4. Vanna Vanna V S is the cross derivative of the option price with respect to the initial spot S t and the volatility σ. The Vanna can be derived as V S = 2 B S σ = Δ S σ = D fφሺd + ሻ d + σ = D fφሺd + ሻd = Vd σ Sσ τ (21) The call and the put option with an equal strike have the same vanna sensitivity..2.5. Volga Volga V σ is the second derivative of the option price with respect to the volatility σ V σ = 2 B σ 2 = V σ = D fs τ φሺd +ሻ d + d + σ = D fs τφሺd + ሻ d +d σ = Vd +d σ (22) 12
using the fact that φሺd + ሻ ( 1 = 2π exp ( d 2 + 2 )) = φሺd d + d + ሻd + + (2) The call and the put option with an equal strike have the same volga sensitivity. 4. FX VOLATILITY CONVENTION In liquid FX markets, Straddle, Risk Reversal and Butterfly are some of the most traded option strategies. It is convention that the markets usually quote volatilities instead of the direct prices of these instruments, and typically express these volatilities as functions of delta, e.g. δ = 0.25 or 0.1, which are commonly referred to as the 25-Delta or the 10-Delta. Let s define a general form of delta function Δሺω, K, σሻ, whick can be any of the pips spot Δ S, pips forward Δ F, percentage spot Δ %S or percentage forward Δ %F. The δ in Black-Scholes model can be computed by the delta function Δሺω, K, σሻ from a strike K and a volatility σ. Providing a market consistent volatility smile σሺkሻ at a maturity, there is a 1- to-1 mapping from δ to K such that δ = Δ(ω, K, σሺkሻ). 4.1. At-The-Money Volatility FX markets quote the at-the-money volatility σ atm against a conventionally defined at-themoney strike K atm. There are mainly two types of at-the-money definitions: ATM forward and ATM delta-neutral straddle. A market consistent volatility smile σሺkሻ must admit the fact that σሺk atm ሻ = σ atm. 4.1.1. ATM Forward In this definition, the at-the-money strike is set to the FX forward F t,t K atm = F t,t (24) This convention is used for currency pairs including a Latin American emerging market currency, e.g. MXN, BRL, etc. It may also apply to options with maturities longer than 10Y. 4.1.2. Delta Neutral Straddle 1
A delta-neutral straddle (DNS) is a straddle with zero combined call and put delta, such as Δሺ1, K atm, σ atm ሻ + Δሺ 1, K atm, σ atm ሻ = 0 (25) If the Δሺω, K, σሻ is in the form of pips spot delta (6) or pips forward delta (10), the ATM strike K atm corresponding to the ATM volatility σ atm can be derived as Φሺd + ሻ Φሺ d + ሻ = 0 Φሺd + ሻ = 0.5 K atm = F exp ( 1 2 σ atm 2 τ) (26) Alternatively, if the ΔሺK, σ, ωሻ takes the form of percentage spot delta (8) or percentage forward delta (12), the ATM strike K atm can be derived as Φሺd ሻ Φሺ d ሻ = 0 Φሺd ሻ = 0.5 K atm = F exp ( 1 2 σ atm 2 τ) (27) The table below summarizes the ATM forward and ATM DNS strikes with associated delta definitions Table 2. Deltas and delta neutral straddle strikes Delta Convention Delta Formula Delta of ATM Forward ATM DNS Strike ATM DNS Delta pips spot ωd f Φሺωd + ሻ ωd f Φ (ω σ atm τ 2 pips forward ωφሺωd + ሻ ωφ (ω σ atm τ 2 percentage spot percentage forward ω K F Φሺωd ሻ 2 τ ) F exp ( σ atm 2 ) 1 2 ωd f 2 τ ) F exp ( σ atm 2 ) 1 2 ω K ωd f F Φሺωd ሻ ωd f Φ ( ω σ atm τ ) F exp ( σ atm 2 τ 2 2 ) 1 2 ωd f exp ( σ 2 atmτ 2 ) ωφ ( ω σ atm τ 2 2 τ 2 τ ) F exp ( σ atm 2 ) 1 2 ω exp ( σ atm 2 ) It is evident that if the ATM strike is above (below) the forward, the market convention must be that deltas for that currency pair are quoted as pips (percentage) deltas [7]. 4.2. Risk Reversal Volatility FX markets quote the risk reversal volatility σ δrr as a difference between the δ-delta call and put volatilities. Providing a market consistent volatility smile σሺkሻ, it is given by σ δrr = σሺk δc ሻ σሺk δp ሻ (28) where δ-delta smile strikes K δc and K δp can be inverted from the delta function such that 14
Δ(1, K δc, σሺk δc ሻ) = δ, Δ( 1, K δp, σሺk δp ሻ) = δ (29) 4.. Strangle Volatility There are two types of strangle volatilities. 4..1. Market Strangle Market strangle (MS, also known as brokers fly) is quoted as a single volatility σ δms for a delta δ. The δ-delta market strangle strikes K MS,δC and K MS,δP for the call and put are both calculated in Black-Scholes model with a single constant volatility of σ atm + σ δms, such that at these strikes the call and put have deltas of ±δ respectively Δ(1, K MS,δC, σ atm + σ δms ) = δ, Δ( 1, K MS,δP, σ atm + σ δms ) = δ (0) This gives the value of the market strangle in Black-Sholes model as V δms = B(1, K MS,δC, σ atm + σ δms, τ) + B( 1, K MS,δP, σ atm + σ δms, τ) (1) This value must be satisfied by a market consistent volatility smile σሺkሻ, such that the V δms below must be equal to the V δms V δms = B(1, K MS,δC, σ(k MS,δC ), τ) + B( 1, K MS,δP, σ(k MS,δP ), τ) (2) Note that, at these strikes we generally have Δ (1, K MS,δC, σ(k MS,δC )) δ, Δ ( 1, K MS,δP, σ(k MS,δP )) δ () Providing a calibrated volatility smile σሺkሻ that is consistent with the market, it is easy to derive the market strangle volatility from the smile. The procedure takes the following steps 1. Choose an initial guess for σ δms (e.g. σ δms = σ δss ) 2. Compute the market strangle strikes K MS,δC and K MS,δP by (0). Compute the strangle value V δms in (1) and the V δms 4. If V δms 4..2. Smile Strangle in (2) is close to V δms then the V δms is found, otherwise go to step 1 to repeat the iteration 15
Providing a market consistent volatility smile σሺkሻ is available, it is more intuitive to express the strangle volatility σ δss as σ δss = σሺk δcሻ + σሺk δp ሻ 2 σሺk atm ሻ (4) This is called smile strangle volatility, where the smile strikes K δc and K δp are given by (29). Given the market quoted σ atm, σ δrr and σ δms, one can build a volatility smile σሺkሻ that is consistent with the market. The procedure takes the following steps 1. Preparation: Determine the delta convention (e.g. pips or percentage, spot or forward) Determine the at-the-money convention (e.g. ATMF or ATM DNS) and its associated ATM strike K atm Choose a parametric form for the volatility smile σሺkሻ (e.g. Polynomial-in-Delta interpolation) Determine the market strangle strikes K MS,δC and K MS,δP by (0) using σ atm + σ δms Compute the value of market strangle V δms in (1) 2. Choose an initial guess for σ δss (e.g. σ δss = σ δms ). Use σ atm, σ δrr and σ δss to find the best fit of σሺkሻ such that with the smile strikes K δc and K δp given by (29), we have σሺk atm ሻ = σ atm σሺk δc ሻ σሺk δp ሻ = σ δrr (5) σሺk δc ሻ + σሺk δp ሻ σሺk 2 atm ሻ = σ δss 4. Compute the value of the market strangle V δms and K MS,δP using the σሺkሻ fitted in step. in (2) with the market strangle strikes K MS,δC 16
5. If V δms is close to the true market strangle V δms then the σሺkሻ is found, otherwise go to step 2 to repeat the iteration. 5. VOLATILITY SURFACE CONSTRUCTION Table 2 presents an example of ATM, risk reversal and smile strangle volatilities at a series of maturities. Each maturity may associate with different ATM and delta conventions. In previous section, we have shown how to extract the five volatilities, at ±10D ±25D and ATM respectively, from market quotes for each maturity subject to its associated market convention. It is often desired to have a volatility surface, so that an implied volatility at arbitrary delta/strike and maturity can be interpolated from the surface. Table. ATM, risk reversal and smile strangle volatilities with associated conventions Maturity ATM Convention Delta Convention ST10D ST25D ATM RR25D RR10D 1M ATM DNS Spot Percentage 0.7% 0.28% 9.1% -1.1% -2.09% M ATM DNS Spot Percentage 1.01% 0.6% 9.59% -1.4% -2.72% 6M ATM DNS Spot Percentage 1.% 0.44% 10.00% -1.66% -.15% 1Y ATM DNS Spot Percentage 1.67% 0.51% 10.9% -1.88% -.66% Y ATM DNS Forward Percentage 2.4% 0.68% 10.58% -1.90% -.59% 5Y ATM DNS Forward Percentage 2.65% 0.74% 10.86% -2.00% -.64% 7Y ATM DNS Forward Percentage 2.80% 0.7% 11.6% -2.20% -.85% 10Y ATM DNS Forward Percentage 2.75% 0.57% 12.4% -2.6% -4.60% 12Y ATMF Forward Percentage 2.2% 0.64% 12.7% -2.78% -4.44% 15Y ATMF Forward Percentage 2.16% 0.62% 1.0% -.1% -5.07% 20Y ATMF Forward Percentage 2.1% 0.6% 1.0% -.18% -5.08% 5.1. Smile Interpolation There are many ways to perform smile interpolation. Polynomial in delta is one of the simple and widely used methods. It employs a 4 th order polynomial that allows a perfect fit to five volatilities (or a 2 nd order polynomial if just fitting to three volatilities). The parameterization is as follows 4 ln σሺkሻ = a j xሺkሻ j, xሺkሻ = MሺKሻ MሺZሻ (6) j=0 where a j s are the coefficients to be calibrated (exactly) to the market data. The function Mሺ ሻ provides a measure of moneyness that often takes the form 17
MሺKሻ = Φ ( 1 v τ ln F K ) (7) The parameter Z can be set to the forward F or the at-the-money strike K atm such that the xሺkሻ provides a measure of distance from the Z. The parameter v in (7) can simply use v = σ atm. However, to be more adaptive, one may choose v = σሺkሻ, together with which the (6) must be solved iteratively for the σሺkሻ. This interpolation method is named after the fact that the measure of moneyness (7) is similar to the definition of forward delta (10). The calibration of the coefficients a j is straightforward. From previous discussion, we are able to retrieve 5 volatility-strike pairs ሺσ i, K i ሻ for i = 1,,5 at a given maturity from market quotes, i.e. volatilities at strikes corresponding to ±10D, ±25D and ATM subject to proper delta and ATM conventions. Based on the 5 volatilities, we are able to form a full rank linear system from (6), which can then be solved for the coefficients a j s. 5.2. Temporal Interpolation The most commonly used temporal interpolation assumes a flat forward volatility in time. This is equivalent to a linear interpolation in total variance. For example, if we have σ atm ሺpሻ and σ atm ሺqሻ at maturities p and q respectively, subject to the same ATM and delta convention, we may interpolate an ATM volatility at a time t for p < t < q by the formula q t 2 σ atm ሺtሻt = q p σ atm 2 t p ሺpሻp + q p σ atm 2 ሺqሻq (8) The temporal interpolation in ±10D and ±25D volatilities may follow the same manner. 5.. Volatility Surface by Standard Conventions Table 2 shows that the market convention on ATM and delta style may vary from one maturity to another. Such jumps in convention introduce inconsistency in definition of the ATM strikes and δdeltas at different maturities. We must choose a consistent set of smile conventions for marking an ATM strike and δ-deltas strikes at all maturities [8]. A pragmatic choice is to use delta-neutral ATM and 18
forward pips delta as the standard conventions. We may convert the volatility-strike pairs ሺσ i, K i ሻ for i = 1,,5 at maturity t to (σ i, K i) such that the (σ i, K i) at all maturities follow the same unified standard conventions. The temporal interpolation is then performed on the standardized volatilities, e.g. between σ 25D ሺpሻ and σ 25D ሺqሻ to get a 25D volatility at an interim time t for p < t < q. Following the same manner, five volatilities σ iሺtሻ can be obtained, along with their associated strikes K iሺtሻ (inverted from the δ-delta values given the standard conventions we have chosen). The last step is then to build a smile based on the σ iሺtሻ and K iሺtሻ for strike interpolation. The conversion from ሺσ i, K i ሻ to (σ i, K i) can be simple. We must at first fit a smile σሺkሻ to the ሺσ i, K i ሻ. To be consistent across maturities, it is ideal to choose Z = K atm in (6). This requires to find iteratively the K atm and the σ atm = σ(k atm ) that conform to the standard ATM and delta conventions (e.g. equation (26) must be satisfied). Once the smile σሺkሻ is available, it is trivial to find all the (σ i, K i) at the ±10D and ±25D subject to the standard conventions. 6. THE VANNA-VOLGA METHOD The vanna-volga method is a technique for pricing first-generation FX exotic products (e.g. barriers, digitals and touches, etc.). The main idea of vanna-volga method is to adjust the Black-Scholes theoretical value (TV) of an option by adding the smile cost of a portfolio that hedges three main risks associated to the volatility of the option: the vega, vanna and volga. 6.1. Vanna-Volga Pricing Suppose there exists a portfolio P with a long position in an exotic trade X, a short position in amount of the underlying spot S, and short positions in ω 1 amount of instrument A 1, ω 2 amount of instrument A 2 and ω amount of instrument A. The hedging instruments A i s can be the straddle, risk reversal and butterfly, as they are liquidly traded in FX markets and they carry mainly vega, vanna and volga risks respectively that can be used to hedge the volatility risks of the trade X. By construction, the price of the portfolio and its dynamics must follow 19
P = X S ω i A i, dp = dx ds ω i da i (9) We may estimate the Greeks in Black-Scholes model and further express the price dynamics in terms of the stochastic spot S and flat volatility σ. By Ito s lemma, we have dp = ( X t ω i Theta + ( X σ ω i Vega A i σ A i t ) dt + ( X S ω i Delta A i S ) ds + 1 X 2 ( 2 S 2 ω i Gamma 2 A i S 2 ) dsds ) dσ + 1 X 2 ( 2 σ 2 ω 2 A i i σ 2 ) dσdσ + ( 2 X S σ ω 2 A i i ) dsdσ S σ Volga Vanna (40) Choosing the and the weights ω i so as to zero out the coefficients of ds, dσ, dσdσ and dsdσ, the portfolio is then locally risk free at time t (given that the gamma and other higher order risks can be ignored) and must have a return at risk free rate. Therefore, when the flat volatility is stochastic and the options are valued in Black-Scholes model, we can still have a locally perfect hedge. The perfect hedge in the three volatility risks implies that the following linear system must be satisfied vegaሺxሻ vegaሺa 1 ሻ vegaሺa 2 ሻ vegaሺa ሻ ω 1 ( vannaሺxሻ) = ( vannaሺa 1 ሻ vannaሺa 2 ሻ vannaሺa ሻ) ( ω 2 ) (41) volgaሺxሻ volgaሺa 1 ሻ volgaሺa 2 ሻ volgaሺa ሻ ω This perfect hedging is under an assumption of flat volatility. Due to non-flat nature of the volatility surface, additional cost between A i ሺσ smile ሻ and A i ሺσ flat ሻ must be accounted into the price of the trade X to fulfil the hedging. As a result, the vanna-volga price X VV of the trade X is computed as follows X VV ሺσ smile ሻ = X TV ሺσ flat ሻ + ω i (A i ሺσ smile ሻ A i ሺσ flat ሻ) (42) where X TV ሺσ flat ሻ is the theoretical Black-Scholes value using a flat volatility (e.g. σ flat = σ atm ), A i ሺσ smile ሻ and A i ሺσ flat ሻ are the prices of the hedging instrument valued with a volatility smile and a flat volatility respectively. 6.2. Smile Interpolation 20
The vanna-volga method may also serve a purpose of interpolating a volatility smile based on the market quoted at-the-money volatility σ atm, the δ-delta risk reversal volatility σ δrr, and lastly the δdelta smile strangle volatility σ δss (converted from market strangle volatility σ δms by the method in section 4..2). From the relationship in (5), we can derive the following quantities Strikes K 1 = K δp Implied Volatilities σ 1 = σሺk δp ሻ = σ atm + σ δss σ δrr 2 K 2 = K atm σ 2 = σሺk atm ሻ = σ atm K = K δc σ = σሺk δc ሻ = σ atm + σ δss + σ δss 2 where the ATM strike K atm is given by the at-the-money convention, and the δ-delta smile strikes K δc and K δp are solved by (29). We will follow a similar analysis as in section 6.1. Suppose we have a perfect hedged portfolio P that consists of a long position in a call option X with an arbitrary strike K, a short position in amount of spot S, and three short positions in ω i amount of call options A i with strikes K 1 = K δp, K 2 = K atm and K = K δc. The perfect hedge in the three volatility risks admits that the following linear system must be satisfied vegaሺxሻ vegaሺa 1 ሻ vegaሺa 2 ሻ vegaሺa ሻ ω 1 ( vannaሺxሻ) = ( vannaሺa 1 ሻ vannaሺa 2 ሻ vannaሺa ሻ) ( ω 2 ) (4) volgaሺxሻ volgaሺa 1 ሻ volgaሺa 2 ሻ volgaሺa ሻ ω where these volatility sensitivities can be estimated in Black-Scholes model assuming a flat volatility flat volatility σ (usually we choose σ = σ atm ). Plugging the closed form Black-Scholes vega, vanna and volga in (20) (21) and (22) respectively, the (4) becomes 1 VሺK 1 ሻ VሺK 2 ሻ VሺK ሻ ω 1 VሺKሻ ( d + d ሺKሻ) = ( Vd + d ሺK 1 ሻ Vd + d ሺK 2 ሻ Vd + d ሺK ሻ) ( ω 2 ) (44) d ሺKሻ Vd ሺK 1 ሻ Vd ሺK 2 ሻ Vd ሺK ሻ ω 21
where Vd + d ሺKሻ is short for VሺKሻd + ሺKሻd ሺKሻ. By inverting the linear system, there is a unique solution of ω for the strike K, such that ω 1 = VሺKሻ VሺK 1 ሻ ln K 2 K ln K K ln K 2 K 1 ln K K 1, ω 2 = VሺKሻ VሺK 2 ሻ ln K K 1 ln K K ln K 2 K 1 ln K K 2, ω = VሺKሻ ln K ln K K 1 K 2 VሺK ሻ ln K K 1 ln K K 2 (45) A smile-consistent volatility v (i.e. a Black Scholes volatility implied from the price by the vanna-volga method) for the call with the strike K is then obtained by adding to the Black-Scholes price the cost of implementing the above hedging at prevailing market prices, that is CሺK, vሻ = CሺK, σሻ + ω i (CሺK i, σ i ሻ CሺK i, σሻ) (46) where the function CሺK, σሻ stands for the Black-Scholes call option price with strike K and flat volatility σ. A market implied volatility curve can then be constructed by inverting (46), for each considered K. Here we introduce an approximation approach. By taking the first order expansion of (46) in σ, that is we approximate CሺK i, σ i ሻ CሺK i, σሻ by ሺσ i σሻvሺk i ሻ, we have CሺK, vሻ CሺK, σሻ + ω i ሺσ i σሻvሺk i ሻ (47) Substituting ω i with the results in (45) and using the fact that VሺKሻ = ω i VሺK i ሻ, we have CሺK, vሻ CሺK, σሻ + VሺKሻ ( y i σ i σ) CሺK, σሻ + VሺKሻሺv σሻ v y i σ i (48) where v is the first order approximation of the implied volatility v for strike K, and the coefficients y i are given by y 1 = ln K 2 K ln K K ln K 2 K 1 ln K K 1, y 2 = ln K K 1 ln K K ln K 2 K 1 ln K K 2, y = ln K K ln K 1 K 2 ln K ln K (49) K 1 K 2 22
This shows that the implied volatility v can be approximated by a linear combination of the three smile volatilities σ i. A more accurate second order approximation, which is asymptotically constant at extreme strikes, can be obtained by expanding the (46) at second order in σ CሺK, vሻ CሺK, σሻ + VሺKሻሺv σሻ + 1 2 V σሺkሻሺv σሻ 2 CሺK, σሻ + ω i (VሺK i ሻሺσ i σሻ + 1 2 V σሺk i ሻሺσ i σሻ 2 ) VሺKሻሺv σሻ + Vd +d ሺKሻ ሺv σሻ 2 2σ (50) d +d ሺKሻ 2σ VሺKሻ y i σ i VሺKሻσ + VሺKሻ 2σ y id + d ሺK i ሻሺσ i σሻ 2 ሺv σሻ2 + ሺv σሻ (v σ + yid + d ሺK ሻሺσ i i σሻ2 ) 0 2σ Solving the quadratic equation in (50) gives the second order approximation v σ + σ + σ 2 + ሺ2σሺv σሻ + y i d + d ሺK i ሻሺσ i σሻ 2 ሻd + d ሺKሻ d + d ሺKሻ (51) where d + d ሺKሻ stands for d + ሺKሻd ሺKሻ that is evaluated with a flat volatility σ. 2
REFERENCES 1. Mercurio, F. and Castagna, A., The vanna-volga method for implied volatilities, Risk Magazine: Cutting Edge Option pricing, p. 106-111, March 2007. Online resource: http://www.risk.net/data/risk/pdf/technical/risk_0107_technical_castagna.pdf 2. Clark, I., Foreign Exchange Option Pricing - A Practitioner s Guide, Wiley Finance, 2011. pp. 41-4.. Clark, I., Foreign Exchange Option Pricing - A Practitioner s Guide, Wiley Finance, 2011. pp. 4. 4. Clark, I., Foreign Exchange Option Pricing - A Practitioner s Guide, Wiley Finance, 2011. Chapter. 5. Clark, I., Foreign Exchange Option Pricing - A Practitioner s Guide, Wiley Finance, 2011. pp. 48-49. 6. Reiswich. D and Wystup, U., FX Volatility Smile Construction, Working Paper, 2010. pp. 9-11. Online: https://www.econstor.eu/bitstream/10419/40186/1/61825101.pdf 7. Clark, I., Foreign Exchange Option Pricing - A Practitioner s Guide, Wiley Finance, 2011. pp. 52 8. Clark, I., Foreign Exchange Option Pricing - A Practitioner s Guide, Wiley Finance, 2011. pp. 69-70 24