# Vanna-Volga Method for Foreign Exchange Implied Volatility Smile. Copyright Changwei Xiong January last update: Nov 27, 2013

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2 This note firstly introduces the basic option trading strategies and the Greek Letters of the Black-Scholes model, and then summarizes the Vanna-Volga method [1] which can be used to construct the implied volatility surface of FX options. In the markets, there are typically three volatility quotes for FX options available for a given market maturity: the delta-neutral straddle, referred to as at-the-money (ATM); the risk reversal for 5 delta call and put; and the vega-weighted butterfly with 5 delta wings. The application of Vanna-Volga pricing method allows us to derive implied volatilities for any option strikes, in particular for those outside the basic range set by the 5 delta put and call quotes. 1. Trading Strategies of Vanilla Options 1.1. Single Call and Put The figures below show the payoff functions of vanilla options. Short a Put Long a Call Long a Put Short a Call 1.. Call Spread and Put Spread

3 A call spread is a combination of a long call and a short call option with different strikes 1 <. 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. CallSpread = C 1 C PutSpread = P P 1 1 Call Spread Put Spread 1.. Risk Reversal A risk reversal (RR) is a combination of a long call and a short put with different strikes 1 <. 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. RR = C 1 P 1 Risk Reversal 1.4. Straddle and Strangle

4 A straddle is a combination of a call and a put option with the same strike. A strangle is a combination of an out-of-money call and an out-of-money put option with two different strikes 1 < ATM <. The figure below shows the payoff functions of a straddle and a strangle. Straddle = C + P Strangle = C 1 + P 1 Straddle Strangle 1.5. Butterfly A butterfly (BF) is a combination of a long strangle and a short straddle. The figure below shows the payoff function of a butterfly. BF = C 1 + P C P, 1 <, ATM < 1 Butterfly. Greek Letters Under a risk neutral measure, the FX spot process is assumed to follow a geometric Brownian motion in Black-Scholes model 4

5 ds S = μdt + dw (1) where the drift term μ = r d r f and r d, r f are the domestic and foreign risk free rate respectively, is the volatility. Then the Black-Scholes vanilla call and put option prices are given as C = D f SN d+ D d N d and P t = D d N d D f SN d+ () where D d = e r dτ and D f = e r fτ are the domestic and foreign discount factor respectively, τ = T t is the term to maturity, function N denotes the standard normal cumulative density function, d + and d are defined as follows d + = ln F + τ, d = d + τ and F = Se μτ τ () The Greek Letters are defined as the sensitivities of the option price to the change in the value of either a state variable or a model parameter. The rest of this section will show the derivation of the Greeks in Black-Scholes model..1. Delta Delta Δ t is the first derivative of the option price with respect to the underlying initial spot S t. Let s first summarize the following relationships N x = 1 N x N d+ = φ d d+ = 1 d + + π e, S = d S = 1 S τ N d = φ d d = 1 d π e = F φ d + (4) where φ is the normal probability density function. Given the above relationships, we can easily derive the call and put delta sensitivities. Δ C = C S = D fn d+ + D f Sφ d+ S D dφ d d S = D fn d+ (5) 5

6 .. Theta Δ P = P S = D dφ d d S D fn d+ + D f Sφ d+ S = D f N d+ 1 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 = C/ t = C/ τ. Let s first derive the following equations: μ τ = + τ + 1 τ ln S t τ = μ τ + d τ = d + τ = μ τ τ 4 τ 1 τ ln S 4 τ 1 τ ln S (6) Therefore, we have Θ C = C t = r fd f SN d+ D f Sφ d+ τ r d dd d N d + D d φ d τ = r f D f SN d+ D f Sφ d+ μ τ + 4 τ 1 τ ln S r dd d N d Se μτ + D d φ d+ μ τ 4 τ 1 τ ln S = r f D f SN d+ D f Sφ d+ τ r dd d N d Θ P = P t = r d dd d N d + D d φ d τ r fd f SN d+ D f Sφ d+ τ (7).. Gamma = r d D d N d + D d φ d μ τ 4 τ 1 τ ln S r fd f SN d+ D f Sφ d+ μ τ + 4 τ 1 τ ln S = r d D d N d r f D f SN d+ D f Sφ d+ τ = Θ C 6

7 Gamma Γ is the first derivative of the delta Δ with respect to the underlying initial spot S t, or the second derivative of the option price with respect to the S t Γ C = C S = Δ C S = D fn d+ = D S f φ d+ S = D 1 fφ d+ S τ Γ P = P S = Δ P S = D fn d+ D f = Γ S C (8).4. Vega Vega V is the first derivative of the option price with respect to the volatility. Let s first derive the following equations: 1 = τ ln F + τ = 1 τ ln F + τ = d + + τ = d (9) d = d + τ = τ = d + Therefore, we have V C = C = D fsφ d+ D d dφ d = D d + d fsφ d+ = D f Sφ d+ τ V P = P = D dφ d d + D fsφ d+ = V C (10).5. Vanna Vanna is the cross derivative of the option price with respect to the initial spot S t and the volatility. The Vanna can be derived as Vanna C = C S = Δ C = D fφ d+ = D fφ d+ τ d + = d S τ V C Vanna P = P S = Δ P = D fφ d+ = Vanna C (11).6. Volga 7

8 Volga is the second derivative of the option price with respect to the volatility. Let s first derive the following equation: d + 1 φ d+ π e = = φ d d+ d + + (1) Therefore, we have Volga C = C = V C = D fs τ φ d + = D d + d fs τφ d+ = V Cd + d Volga P = P = V P = V C = Volga C (1). Construction of Implied Volatility Surface In the FX option market, the volatility surface is built according to the sticky delta rule. The underlying assumption is that options are priced depending on their delta, so that when the underlying asset price moves and the delta of an option changes accordingly, a different implied volatility has to be plugged into the pricing formula. In the market, three quotes are commonly traded, including the at-themoney (ATM) volatility, the risk reversal (RR) for 5Δ call and put, and the vega-weighted butterfly (BF) with 5Δ wings. The notation 5Δ denotes 5% level of the delta, thus a 5Δ call is a call option whose delta is positive 0.5 and a 5Δ put is a put option whose delta is negative 0.5. From these data, one can easily infer three basic implied volatilities, from which on can further build up the entire implied volatility surface for a given market maturity. The method is summarized as follows..1. FX Market Quotes The ATM volatility atm quoted in the FX markets is associated with a 0Δ straddle (Δ C + Δ P = 0). According to (5), the ATM strike atm corresponding to the atm can be derived as Δ C + Δ P = N d+ + N d+ 1 = 0 N d+ = 0.5 d + = 0 (14) 8

9 τ ln F + atm atm atm τ = 0 atm = Fe 1 atmτ The Risk Reversal is a typical structure of a long call and a short put with a symmetric 5Δ. The RR price RR is quoted as the difference between the two implied volatilities RR = ΔC ΔP (15) The BF is structured by a long 5Δ strangle and a short ATM straddle. The BF price BF in terms of volatility is defined as BF = ΔC + ΔP It is evident to derive the following equations atm (16) ΔC = ATM + BF + RR and ΔP = ATM + BF RR (17) The strikes corresponding to ΔC and ΔP can then be derived from (5) Δ C = D f N + dδc = δ and Δ, P = D f N + dδp = δ ΔC, ΔP (18) where δ = 5% for 5Δ. Let s define α = N 1 1 δdf, where N 1 is the inverse of the standard normal cumulative density function. One would have the following strikes ΔC = Fe α ΔC τ+ 1 ΔC τ and ΔP = Fe α ΔP τ+ 1 ΔP τ (19) For typical market parameters and for τ up to two years, α > 0 and ΔP < atm < ΔC. To ease the notation, we denote the three strikes by i, i = 1,,, where 1 = ΔP, = atm and = ΔC. Similarly we have i, i = 1,,, where 1 = ΔP, = atm and = ΔC for the three basic implied volatilities. Based on the strikes and the implied volatilities, one can infer three option market prices as C mkt i, i, i = 1,,. The table below summarizes the results described above 9

10 atm Market Quotes Implied Volatilities Strikes 1 = ΔP = atm + BF RR 1 = ΔP = Fe α ΔP τ+ 1 ΔP τ RR = ΔC ΔP = atm = atm = Fe 1 atmτ BF = ΔC + ΔP atm = ΔC = atm + BF + RR = ΔC = Fe α ΔC τ+ 1 ΔC τ.. Replicating Portfolio The Vanna-Volga method serves the purpose of defining an implied volatility surface that is consistent with the three basic implied volatilities. The Black-Scholes model assumes a flat-smile volatility that is constant over time. In real financial markets, however, volatility is stochastic and traders hedge the associated risk by constructing portfolios that are vega-neutral in the Black-Scholes world (flat-smile). Suppose there exists a portfolio X consisting of a long position in a call C with a strike, a short position in amount of the underlying spot S, and three short positions in ω i amount of calls C i with strike i, respectively X = C S ω i C i (0) where the option prices are assumed to be given by the Black-Scholes formula. The dynamics of the portfolio X depends on the movements of both S and, which can be written as dx = dc ds ω i dc i (1) By Ito s lemma, we have 10

11 dx = C t ω C i i dt + C t S ω C i i ds + C S ω C i i d Theta Delta Vega + 1 C S ω C i i S dsds + 1 C ω C i i dd Gamma Volga () + C S ω C i i dsd S Vanna Choosing the and the weights ω i so as to zero out the coefficients of ds, d, dd and dsd, the replicating portfolio is then locally risk free at time t (given that the gamma and other higher order risks are ignored) and must have a return at risk free rate dx = r d Xdt () Therefore, when the flat volatility is stochastic and the options are valued with Black-Scholes formula, we can still have a locally perfect hedge. The weights ω i can be determined by making the replicating portfolio is vega-, volga- and vanna-neutral, i.e. it is fully hedged with respect to the stochastic flat volatility risk. Given a flat volatility (usually choose = = atm ), we therefore have Vega: C = ω i C i Volga: C = ω i C i (4) Vanna: C S = ω C i i S From previous derivation (10) (11) and (1), we have for the strike 11

12 Vega: Volga: Vanna: C = V C = Vd +d C S = Vd S τ (5) The (4) can then be written in matrix form as V 1 V V ω 1 1 V 1 d + 1 d 1 V d + d V d + d ω = V d + d (6) V 1 d 1 V d V d ω d Therefore, there exists a unique solution to the system for the strike ω 1 = V ln ln V 1 ln ln, ω = 1 1 V ln ln 1 V ln 1 ln, ω = V ln ln 1 V ln 1 ln (7) A smile-consistent price ς for the call with the strike is obtained by adding to the Black- Scholes price the cost of implementing the above hedging strategy at prevailing market prices C mkt,ς = C, + ω i C mkt i, i C i, (8) The new option price is thus defined by adding to the flat smile Black-Scholes price the cost difference of the hedging portfolio induced by the market implied volatilities with respect to the flat volatility... Approximation for Implied Volatilities A market implied volatility curve can then be constructed by inverting (8), for each considered, through the Black-Scholes formula. By taking the first order expansion of the (8) in (usually choose = = atm ), we have 1

13 C mkt,ς C, + ω i V i ( i ) (9) Substituting ω i with the results in (7) and considering that V = ω i V i in (4), we have C mkt,ς C, + V y i i V C, + V (ς ) (0) where ς is the first order approximation of the implied volatility for strike, and the coefficients y i are given by y 1 = ln ln ln 1 ln 1, y = ln 1 ln ln 1 ln, y = Simplifying (0) gives the first order approximation ln ln 1 ln ln (1) 1 ς y i i () Which tells that the implied volatility ς can be approximated by a linear combination of the three basic volatilities i. A more accurate second order approximation, which is asymptotically constant at extreme strikes, is obtained by expanding the (8) at second order in C mkt,ς C, + V (ς ) + 1 V d + d (ς ) C, + ω i V i ( i ) + 1 V i d + i d i ( i ) V (ς ) + 1 V d + d (ς ) V y i i V + 1 V y id i + d i ( i ) () d + d (ς ) + (ς ) P + Q 0 where 1

14 P = ς and Q = y i d i + d i ( i ) Solving the quadratic equation in () gives the second order approximation ς d + d (P + Q) d + d (4) 14

15 REFERENCES 1. Mercurio, F. and Castagna, A., The vanna-volga method for implied volatilities, Risk Magazine: Cutting Edge Option pricing, p , March 007. Online resource: 15

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