Bloomberg. Variations on the Vanna-Volga Adjustment. Travis Fisher. Quantitative Research and Development, FX Team. January 26, Version 1.

Transcription

1 Bloomberg Variations on the Vanna-Volga Adjustment Travis Fisher Quantitative Research and Development, FX Team January 26, 27 Version 1.1 c 27 Bloomberg L.P. All rights reserved.

2 Bloomberg FINANCIAL MARKETS COMMODITIES NEWS 2 1 Theory 1.1 Introduction The Vanna-Volga method has been popularized as a way of pricing both vanilla and exotic options given a very limited amount of market data. The best-justified application of the method is to vanilla options, though of course the real interest in the approach lies in its effectiveness at producing reasonable estimates of the market prices of first generation exotics. In this article we address the question of how the Vanna-Volga method should be extended from a method to price vanilla options to a method to price exotics. We explore different viewpoints, which provide alternate explanations for the method s effectiveness. Eventually we describe the approach that has been taken to provide pricing through Bloomberg s OVML function. This is a tweaked version, differing in the treatment of exotics from the standard or modified approaches described in [1]. 1.2 Background We take as a starting point the modified Vanna-Volga adjustment, as described in [1] or [3]. This provides an adjustment to option prices taking into account market price information from three instruments: delta-neutral straddles (known as at-the-monies or ATM), 25-delta risk-reversals (RR), and 25-delta butterflies (BF). The adjustment uses weights w ATM, w RR, and w BF so that a portfolio of ATM, RR, and BF instruments with these weights will have the same vega, vanna, and volga as the option X. The weights are calculated by solving a matrix equation AT M vega RR vega BF vega AT M vanna RR vanna BF vanna AT M volga RR volga BF volga w ATM w RR w BF = X vega X vanna X volga. (1) The price adjustment to the Black-Scholes price of X (with at-the-money implied volatility) is derived from the additional market price of this portfolio over the Black-Scholes price (again, with at-the-money implied volatility). For barrier and touch options, this adjustment is tempered by a weight p, often suggested as the domestic risk-neutral probability of option survival to expiry. Thus the price of X is calculated as X mkt = X BS + p ( w ATM (AT M mkt ATM BS )+ w RR (RR mkt RR BS )+w BF (BF mkt BF BS ) ). (2) Here we ve included the ATM term, which will on closer inspection vanish because by definition the market and Black-Scholes prices of the ATM are both the Black-Scholes price with the at-the-money implied volatility. Document history: Version 1. produced July 11, 26 with Bloomberg internal circulation only. Thanks to Peter Carr, Bruno Dupire, Apollo Hogan, Harvey Stein, and Arun Verma for helpful discussions. All mistakes are mine.

3 Bloomberg FINANCIAL MARKETS COMMODITIES NEWS 3 The additional price relative to the Black-Scholes price in this approach is viewed as a hedging cost of constructing the overhedge needed to zero the vega, vanna, and volga of the option. Note that the cost calculation does not count the entire cost of the overhedge portfolio, but only the additional market price of the portfolio in excess of the Black-Scholes price. 1.3 Reference Volatility The inclusion of the AT M term in Equation 2 hints at one variation of the above adjustment. Instead of using the at-the-money implied volatility for calculating the Black-Scholes prices above, one could use some other volatility. We will refer to this as a reference volatility. In this case, the ATM term no longer vanishes: the Black-Scholes price is now with respect to the reference volatility while the market price is (by definition) with respect to the at-the-money volatility. In this approach, all of the vega, vanna, and volga sensitivities are calculated with respect to the reference volatility. This approach has been studied by Castagna and Mercurio [2], who demonstrate some nice properties of the method. We will remark further on their results in Section Shifting View from Instruments to Greeks There is an equivalent way of writing Equation 1 and Equation 2 which leads to a slightly different interpretation of the adjustment. For y = Equation 1 can be written A = ATM vega RR vega BF vega ATM vanna RR vanna BF vanna ATM volga RR volga BF volga w = x = w ATM w RR w BF X vega X vanna X volga ATM mkt ATM BS RR mkt RR BS BF mkt BF BS = Y ATM Y RR Y BF Aw = x (3) and Equation 2 can be written X mkt = X BS + pw y. (4)

4 Bloomberg FINANCIAL MARKETS COMMODITIES NEWS 4 Combining equations 3 4 we get X mkt = X BS + p(a 1 x) y = X BS + px (A ) 1 y = X BS + px v. Here the vector v = (A ) 1 y is interpreted as a vector of market prices of vega, vanna, and volga. Its entries correspond to the premium that must be attached to these greeks in order to adjust the Black-Scholes price of the ATM, RR, and BF instruments to the market price of those instruments. As a full matrix equation, this becomes ATM vega ATM vanna AT M volga RR vega RR vanna RR volga BF vega BF vanna BF volga 1.5 A Market Price for Greeks? v vega v vanna v volga = Y ATM Y RR Y BF There are several related justifications for why value might directly be attached to an option s volatility greeks. One approach is to view the vega, vanna, and volga as proxies for certain kinds of risk associated to the fluctuation of volatility. Trades which bring additional risk should bring additional compensation for taking the risk. On the other side, if a trade allows you to offload risk to another party you should have to compensate them. We propose that the market may directly attach value to some greeks. In particular, market participants are aware that volatility does fluctuate, and hedging against this fluctuation is standard practice. At the first level of this hedging, traders look to balance their portfolio to have zero vega. In actual market circumstances, this can create a demand for vega, leading to a market price of vega. For example, many banks tend to be writers of vanilla options, and the sale of these will accumulate negative vega in their portfolio. To balance this out, the bank will be willing to pay a premium for options with positive vega, in order to hedge their risk. One might think that since an option which gives positive vega to the buyer s portfolio gives an equal amount of negative vega to the sellers s portfolio, this symmetry should lead to a zero market price for vega. This is not so, however, when there is a correlation between market participants for whom hedging is most important and those whose portfolio naturally tends to be long (or short) vega. As long as participants carry out portfolio management where the vega risk is produced as a simple sum of vegas of all instruments, then vega is implicitly treated as being fungible. This is, as we understand, a widespread practice. Similarly one can imagine there may be direct demand for vanna or volga. There is something problematic here though: it is clear at least for vanna that in a market with barrier options vanna alone cannot command a nonzero price. Consider a knock-out barrier option, with no rebate, where the spot rate is infinitesimally close to the barrier. This option will knock out with probability one (assuming any diffusion component to the process giving the.

5 Bloomberg FINANCIAL MARKETS COMMODITIES NEWS 5 spot rate). Thus the value of the option is zero. This includes any value that would come from the greeks of the option. But the option will have nonzero vanna! This shows that vanna from one source cannot necessarily be used to hedge vanna in the remainder of a portfolio: vanna is not fungible. This difficulty does not present itself in such an obvious way for vega or volga. These both vanish at a knock-out barrier: since the value of the option at the barrier is zero independent of the volatility level, all derivatives of this value with respect to volatility are zero. One way that one can look at the market treatment of vega, vanna, and volga is to consider the market prices implied for these greeks by the vanna-volga adjustment from the viewpoint of Section 1.5. We carried out this exercise with the data from the EUR/USD and JPY/USD vanilla option quotes from June 3, 26. Using a fixed reference volatility of 1%, we calculated the implied market prices of vega, vanna, and volga for vanilla options expiring between two weeks and five years. We used the implied volatility quotes provided to Bloomberg by Bank of America for ATM, RR, and BF at 5, 1, 15, 25, and 35 percent delta. The results of this study are shown in Figure 1 and Figure 2. The first thing to notice is how well the lines for the prices calculated from different delta options agree. This agreement is indicative of how well the vanna-volga method will do in reproducing the vanilla smile. When there is good agreement between, say, the 25 delta implied prices and the 1 delta implied prices of the greeks, there will be good agreement between the price of the 1 delta vanilla options adjusted according to the 25 delta adjustment. Another thing to notice is the general trend of the curves. For vanna, there is a clear term structure. Options with short expiries give an implied price for vanna near zero, while options with longer expiries give higher implied prices for vanna. For vega and volga, there is no such clear term structure. For EUR/USD the price of vega and volga seems fairly constant over time. For USD/JPY there is a clear long term violation of this past about 3 years to expiry, but still constant price for vega and volga seems a better fit then a linear or square root function of time to expiry. If the market actually does treat vega and volga as fungible assets which carry value as hedging tools independent of the option to which they are attached, then this has implications for how the Vanna-Volga adjustment should be weighted for exotic options. Consider a version of Equation 2 which allows different weights for the portion of the adjustment coming from the vega, vanna, and volga terms: X mkt = X BS + p vega X vega v vega + p vanna X vanna v vanna + p volga X volga v volga. (5) If vega and volga are treated equivalently, independent of what sort of option they are attached to, this suggests that the weights p vega and p volga should be equal to 1 for any exotic option. Vanna, on the other hand, most definitely is not fungible, so the weight p vanna should depend on the exotic option. In particular, p vanna must go to zero for options which are

6 Bloomberg FINANCIAL MARKETS COMMODITIES NEWS d 1d 15d 25d 35d Implied Price of Vega $/vega years d 1d 15d 25d 35d Implied Price of Vanna $/vanna years d 1d 15d 25d 35d Implied Price of Volga $/volga years Figure 1: Implied EUR/USD prices for vega, vanna, and volga from June 3, 26 vanilla option prices.

7 Bloomberg FINANCIAL MARKETS COMMODITIES NEWS d 1d 15d 25d 35d Implied Price of Vega yen/vega years Implied Price of Vanna yen/vanna d 1d 15d 25d 35d years.2 Implied Price of Volga.15.1 yen/volga d 1d 15d 25d 35d years Figure 2: Implied USD/JPY prices for vega, vanna, and volga from June 3, 26 vanilla option prices.

8 Bloomberg FINANCIAL MARKETS COMMODITIES NEWS 8 certain to immediately knock out. The standard suggestion of weighting by the probability of survival to expiry without knocking out satisfies this requirement. 1.6 Approximation to Stochastic Volatility Another approach to justify the Vanna-Volga adjustment is to begin with a model in which volatility is stochastic and work out how this additional risk is related to the option s greeks. The basic start in this direction is the Itô expansion developed in [1], which can be interpreted as a robustness result [2]. Related investigations into approximations of stochastic volatility models are contained in [5], [4], and [7]. We will discuss this circle of ideas here, with an eye towards understanding what the adjustment coefficients might look like for exotic options. Consider a stochastic volatility model driven by volatility following a general diffusion: ds = µdt+ σdw 1 S dσ = f(σ,t)dt + g(σ,t)dw 2, where W 1 and W 2 are possibly correlated Brownian motions. Applying Itô s Lemma gives the value X T at time T of an exotic claim as X T = X T T T T T T X t dt X S ds 2 X S 2 (ds)2 X σ dσ 2 X σ 2 (dσ)2 2 X S σ dsdσ. When σ is constant, the last three terms of this equation vanish and the first three give the Black Scholes price of the option. For the stochastic volatility case, the last three terms give an adjustment to the Black-Scholes price. Note that the integrands of the Itô integrals for the vega, vanna, and volga terms vanish once the option knocks out. This suggests that for the adjustment written as in Equation 5, the weights p vega, p vanna, and p volga should be chosen so that the corresponding terms of the adjustment approximate the estimated values of the stochastic integrals. The standard choice of the probability of survival to expiry can be seen as one way of estimating this; another obvious suggestion would be the expected time to knock out divided by T. In any case, it is clear that this line of thought will have all of the weights decrease to zero for options which are at the barrier level.

9 Bloomberg FINANCIAL MARKETS COMMODITIES NEWS 9 Another approach which may be enlightening is to make more specific assumptions about the stochastic volatility terms that appear, and try to approximate the resulting model. Fouque, Papanicolaou, and Sircar carried out such an approach in their book [5], where they work with a stochastic volatility model with a very high rate of mean reversion. This has fast stochastic volatility term which reverts to a mean level on a shorter time scale than the options in which one is interested. Forming a perturbative approximation, the authors find an expansion with base term the Black-Scholes price under historical volatility. The adjustment relative to this base term is written in terms of a partial differential equation with source terms that are functions of the option s Black-Scholes greeks. In [4] this approach is revised to work with a fast/slow stochastic volatility model, having in addition a slow stochastic volatility term which reverts to a mean level on a longer time scale than the one in which one is interested. They also move to an implied rather than historical reference volatility. A remarkable feature of their approach is that they justify a method of estimating a particular reference volatility σ and constants v, v 1, and v 3 from an implied volatility surface. From these the price P of a vanilla option on underlying S traded at time t with expiration at time T is approximated as P mkt = P BS +(T t)v P vega +((T t)v 1 +(v 3 /σ ))S(t)P vanna. (6) Here the Black-Scholes price and greeks are all calculated with respect to the volatility σ. If we view t, T, and S(t) as constants this is recognizable as a truncated version of the vanna-volga adjustment. For exotic options, the approach gives a partial differential equation for the adjustment X adj = X mkt X BS ( L BS X adj = 2v X vega + 2v 1 SX vanna + v 3 S ) S (S2 X gamma ), (7) wherel BS is the usual Black-Scholes partial differential operator L BS = t + 1 ( 2 (σ ) 2 S 2 2 S 2 + r S ) S. In the expansion that Fouque, Papanicolaou, and Sircar use, the leading order terms are from the option s vega, vanna, and the first three derivatives with respect to spot (including delta and gamma). From their point of view, the volga term is higher order. In contrast, Stoikov [7] carries out an expansion approximating a stochastic volatility model with only a slow volatility term. He works within within a general stochastic volatility framework ds = µsdu+ σ(u)sdw 1 u dσ = b(σ,u)du+a(σ,u)(ρdw 1 u + 1 ρ 2 dw 2 u ) for W 1 and W 2 uncorrelated Brownian motions.

10 Bloomberg FINANCIAL MARKETS COMMODITIES NEWS 1 He studies the problem of pricing the additional value an option will provide given an already held portfolio and certain risk preferences. Ignoring (setting zero) the portfolio term, his first order adjustment term is [ T X adj = E t T t T t 1 2 a2 X volga (σ,u) du+ ρaσsx vanna (σ,u) du+ ( b ρaµ σ (T t)a2 µ 2 σ 3 ) ] X vega (σ,u) du. (8) The vanna-volga adjustment could be viewed as a way of trying to approximate solutions to Equation 7 or to Equation 8 by a linear combination of the option s vega, vanna, and volga. This makes the method a cheap approximation to a first order approximation to a stochastic volatility model. 1.7 Control Variates A final attempt to describe the effectiveness of the method portrays the correction as an adjustment to the Black-Scholes price using the observable prices of the at-the-money straddles, risk-reversals, and butterflies as control variates. This line of thought was first suggested in [6]. Here, one considers the Black-Scholes market prices of instruments as easily estimated values with a high correlation to the correct market price. This immediately leads to a weighted adjustment X mkt = X BS + β ATM (ATM mkt ATM BS ) + β RR (RR mkt RR BS ) + β BF (BF mkt BF BS ). (9) This leaves open the question of what the optimal choice of the control variate coefficients β ATM, β RR, and β BF should be. We could choose a sample of models that differ from the Black-Scholes model with reference volatility and minimize the variance of the resulting estimated values of X mkt. For a sample of Black-Scholes models with volatility differing only slightly from the reference volatility, minimizing the variance requires weights which would give a portfolio of ATM, RR, and BF with vega matching that of the exotic option. This is just another way of expressing the goal of being hedged against volatility changes. If we consider a delta-hedged option (assume X is delta neutral), then for a sample of Black-Scholes models with volatility and spot differing slightly from the reference volatility and original spot, minimizing the variance will require matching of vega, vanna, volga. This leads to the beta weights being the same as the weights w from Equation 1. By considering other samples, one could justify other choices for the control variate coefficients. It is not clear what a canonical choice should be.

11 Bloomberg FINANCIAL MARKETS COMMODITIES NEWS Consistency Features In [2], Castagna and Mercurio point out two consistency features of the vanna-volga adjustment when a reference volatility is used. The first consistency feature is for the implied volatility smile implied from the ATM and 25 delta RR and BF. The vanna-volga adjustment gives a price and hence an implied volatility for a vanilla call for any strike. If one chooses three other strikes and repeats the adjustment using those implied volatilities as inputs, from these points the same curve will be recovered. The second consistency feature is that for any European claim which can be replicated as a (possibly infinite) portfolio of vanillas, the price derived from the vanna-volga adjustment for that option will match the vanna-volga price of the portfolio. These are almost obvious once one shifts view as we did in Section 1.4. For any three strikes, the adjusted prices of the call options C(K i ), i = 1,2,3, are given by C mkt (K i ) = C BS (K i )+x v, (1) where as in Section 1.4 x is the vector of vega, vanna, and volga for the option C(K i ) and v is the vector of market prices of vega, vanna, and volga. The first consistency result is pointing out that we can recover the market prices of vega, vanna, and volga from the prices of these options C(K i ). Explicitly, set B = C(K 1 ) vega C(K 2 ) vega C(K 3 ) vega C(K 1 ) vanna C(K 2 ) vanna C(K 3 ) vanna C(K 1 ) volga C(K 2 ) volga C(K 3 ) volga with these sensitivities calculated relative to the reference volatility, and C mkt (K 1 ) C BS (K 1 ) y C = C mkt (K 2 ) C BS (K 2 ). C mkt (K 3 ) C BS (K 3 ) Then Equation 1 can be written in vector form as y C = B v. This is completely analagous to the equation relating v to the market prices of the ATM, RR, and BF. The market prices of the greeks can be recovered as v = (B ) 1 y C, giving the first consistency result. For any European claim which can be formed as a portfolio of vanilla options, by the linearity of summation (or integration) the vega of the portfolio is the summed (or integrated) vega of the vanillas. Similarly for vanna and volga. The adjustment to the price of the option is a linear function of the vega, vanna, and volga of the portfolio, which is the same as the summed (or integrated) adjustment to the vanillas. This gives the second consistency result.

12 Bloomberg FINANCIAL MARKETS COMMODITIES NEWS 12 2 Practice 2.1 A Compromise The most problematic part of the vanna-volga adjustment is the weight used to reduce the magnitude of the correction at the barrier for knock-out and touch options. In the earliest publications describing the method, Lipton and McGhee note that the coefficients are empirically adjusted and Wystup notes that there are different beliefs among market participants about the unwinding cost, both admissions that it is not theoretically clear what to use for the weights. Most recently, Castagna and Mercurio describe the treatment of path-dependent options as an unsolved issue for which ad hoc procedures are usually used. For our variation of the vanna-volga adjustment, we adjust our practice with regards to this weight in two ways. First, instead of taking as a starting point the domestic risk-neutral probability of missing the barrier, we take the average of the domestic and foreign risk-neutral probabilities. This symmetrization helps to preserve the foreign/domestic symmetry inherent in foreign exchange options. More significantly, we carry out the adjustment as an adjustment based on market prices of vega, vanna, and volga, and treat vega and volga differently than vanna. The adjustment we use prices a knock-out option according to the formula X mkt = X BS + (1/2+1/2p sym )X vega v vega + p sym X vanna v vanna + (1/2+1/2p sym )X volga v volga, (11) where p sym is the symmetrized probability of not hitting the barrier and v vega, v vanna, and v volga are the market prices of vega, vanna, and volga as calculated from the ATM and 25 delta RR and BF market prices. This treatment of vega and volga is a compromise between the views of Section 1.5 (which suggests that vega and volga should be completely unweighted) and the view of Section 1.6 (which suggests that vega and volga should be weighted by a function which goes to zero as spot approaches the barrier). This compromise is further justified empirically; this variation of price better matches the market data we have available than either the fully weighted or fully unweighted variations. 2.2 Forward Starting Barriers For options with forward starting barriers, the weights given in Equation 11 are adjusted to account for this forward starting nature. In the limiting case of terminal barrier options, where the barrier is monitored only at the expiry date, the option is a European claim with a truncated payoff function. For such an option, just like for a vanilla option, the full weight of the vanna-volga correction should apply. This is the consistent way to carry out the adjustment, as discussed in Section 1.8. Our compromise method to interpolate between fully

13 Bloomberg FINANCIAL MARKETS COMMODITIES NEWS 13 monitored and terminal barriers replaces p sym by p = p sym +(1 p sym ) T start t, (12) T t where as before t is the time of trade, T is the time of option expiry, and T start is the start time for barrier monitoring. If in addition the barrier ends before expiry, no additional adjustment is made. 2.3 Reference Volatility Despite the hints that a different reference volatility might give better results, we have not seen sufficient evidence to change from the de-facto standard of using the ATM mid volatility as the reference volatility. This means that viewed as an overhedge, the ATM term of the vanna-volga adjustment vanishes. Note, however, that when the adjustment is broken down as vega, vanna, and volga terms, none of these terms vanishes. 2.4 Using Knock-In Options and Rebates as Building Blocks Another problematic feature of the vanna-volga adjustment is how to treat knock-in options and options with rebates. The compromise adjustment described in Equation 11 is justified for a knock-out option, with the vanna portion of the adjustment forced to zero in the limit of the option certainly being knocked out. For knock-in options, one could use a similar formula, replacing p sym in Equation 11 with 1 p sym, which is the probability of hitting the barrier. Unfortunately, this will not satisfy the no-arbitrage condition that a knock-out option plus a knock-in option equals a vanilla option. With this in mind, we will instead price knock-in options as the difference between the vanna-volga price of a vanilla option and the vannavolga price of the knock-out option. Rebates on knock-in and knock-out options are respectively no-touch and one-touch options. For each of these, the vanna term of the adjustment should be forced to zero as spot approaches the barrier, because in that limit the option price becomes deterministically determined. For concreteness, we provide here a table showing how all kinds of single, double, and sequential barrier options can be decomposed into a combination of vanilla options and knock-out options.

14 Bloomberg FINANCIAL MARKETS COMMODITIES NEWS 14 Barrier Option up-and-out barrier option down-and-out barrier option up-and-in barrier option up-and-out barrier option double-barrier-knock-out option double-barrier-knock-in option up-and-in down-and-out sequential barrier option up-and-in down-and-in sequential barrier option up-and-out down-and-in sequential barrier option up-and-in up-and-out sequential barrier option down-and-in up-and-out sequential barrier option down-and-in up-and-in sequential barrier option down-and-out up-and-in sequential barrier option down-and-in down-and-out sequential barrier option Decomposition UOx = UOx DOx = DOx UIx = x - UOx DIx = x - DOx DBKOx = DBKOx DBKIx = x - DBKOx UIDOx = UIDOx UIDIx = x - DOx - UIDOx UODIx = UOx - DBKOx UIUOx = UOx(H up ) - UOx(H dn ) DIUOx = DIUOx DIUIx = x - UOx - DIUOx DOUIx = DOx - DBKOx DIDOx = DOx(H dn ) - DOx(H up ) Note that when considering the up-and-in down-and-out sequential barrier options as essentially a knock-out option, the probability of not hitting the barrier is calculated as the probability of not knocking out. This includes all paths which never reach the upper barrier as well as all paths which reach the upper barrier but thereafter do not reach the lower barrier. Similarly for the down-and-in up-and-out sequential barrier options. 2.5 No-Arbitrage Ceilings and Floors Since the vanna-volga adjustment is not made according to any model, there is no reason to suppose that arbitrage is not possible. Indeed, it is not hard to find examples where the adjusted price of a vanilla call as a function of strike is a non-convex function, which implies arbitrage. Such an example was given in [1]. It is not even hard to construct examples where the adjusted price of a vanilla call will be negative. To prevent the most obvious and egregious of these arbitrage violations, we put floors

15 Bloomberg FINANCIAL MARKETS COMMODITIES NEWS 15 and ceilings for the vanna-volga adjusted price of options at the most obvious targets where such a comparison exists with a simpler option. For all options there is a floor at zero. For terminal barrier options, there is a ceiling at the vanilla option value. For knock-out barrier options and double-barrier knock out options, there is a ceiling at the corresponding terminal barrier option value. For sequential barrier options, there is a ceiling at the inner barrier option value. For double one-touch options (the rebates on double-knock-out barrier options) there is a ceiling at each one-touch option value for the two barrier levels. For double no-touch options (the rebates on double-knock-in barrier options) there is a ceiling at each no-touch barrier option value for the two barrier levels. 2.6 Bid/Ask Indicative Spread Generation The market data that we have been able to obtain coincides with Wystup s assessment that the bid/ask spread quoted for barrier options varies with the strike and distance of barrier from spot, while for one-touch and no-touch options the spread quoted is a constant percent of notional (depending possibly on the source of the quote and the currency pair). We have mimicked this split, providing a constant spread for one-touch and no-touch options, though we have not attempted to find values for this spread as a function of currency pair. As a baseline value, we have chosen 3.5% of notional for the fixed spreads. For the variable spreads on barrier options, we generate a spread which depends on the magnitude of the terms of the vanna-volga adjustment. We have been able to achieve reasonable results with a spread calculated as where X spread v spread = (A ) 1 = 1/2v spread vega X vega + 1/2v spread vanna X vanna + 1/2v spread volga X volga, (13) ATM mkt ask ATMBS bid RR mkt ask RRBS bid BFask mkt BFBS bid. (14) For a variety of reasons, we want to restrict the market data input to the mid prices of the ATM, RR, and BF together with the spread at the money. Using this data, we approximate the market asking prices for the RR and BF by assuming that the volatility spread at the 25 delta call and put is the same as the volatility spread at delta. The meaning of RR mkt ask and BFask mkt in Equation 14 is the price of the 25 delta risk-reversal and butterfly calculated with the volatilities for the calls and puts increased by one half of the ATM volatility spread. The meaning of AT Mbid BS, RRBS bid, and BFBS bid in Equation 14 is the price of these instruments calculated with the reference (ATM mid volatility) reduced by one half of the ATM volatility spread. The baseline spread calculated in Equation 13 is capped at an amount of 1% of the option price to prevent (as happens in some cases) overly wide spreads. This is furthermore widened to a minimum of the ATM price spread.

16 Bloomberg FINANCIAL MARKETS COMMODITIES NEWS 16 In most cases, the bid and ask price are indicated symmetrically around the best price, which is the price calculated according to the vanna-volga adjustment using the mid prices as discussed above. If, however, the resulting bid price is negative the bid price is quoted at zero and the ask price is increased to keep the spread as calculated. In such a case the best price is no longer a mid price. References [1] Carr, P., Hogan, A., and Verma, A. Vanna-Volga Method For 1st Generation Exotics in FX, Bloomberg working paper, 26. IDOC [2] Castagna, A. and Mercurio, F. Consistent Pricing of FX Options, working paper, 25. [3] Castagna, A. and Mercurio, F. The vanna-volga method for implied volatilities., Risk, Jan. 27 [4] Fouque, J.-P., Papanicolaou, G., Sircar, R., and Sølna, K. Timing the Smile, Wilmott Magazine, 24. [5] Fouque, J.-P., Papanicolaou, G., and Sircar, R. Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, 2. [6] Lipton, A., and McGhee, W. Universal Barriers, Risk, May 22. [7] Stoikov, S. Pricing Options From the Point of View of a Trader, to appear in IJTAF, 26. [8] Wystup, U. The Market Price of One-Touch Options in Foreign Exchange Markets, Derivatives Week, 12, 13, 23.