We study Vanna-Volga methods which are used to price first generation exotic options in the Foreign Exchange market. They are based on a rescaling of the correction to the Black-Scholes price through the so-called `probability of survival' and the `expected first exit time'. Since the methods rely heavily on the appropriate treatment of market data we also provide a summary of the relevant conventions. We offer a justification of the core technique for the case of vanilla options and show how to adapt it to the pricing of exotic options. Our results are compared to a large collection of indicative market prices and to more sophisticated models. Finally we propose a simple calibration method based on one-touch prices that allows the Vanna-Volga results to be in line with our pool of market data.
Deep Dive into Vanna-Volga methods applied to FX derivatives : from theory to market practice.
We study Vanna-Volga methods which are used to price first generation exotic options in the Foreign Exchange market. They are based on a rescaling of the correction to the Black-Scholes price through the so-called probability of survival' and the expected first exit time’. Since the methods rely heavily on the appropriate treatment of market data we also provide a summary of the relevant conventions. We offer a justification of the core technique for the case of vanilla options and show how to adapt it to the pricing of exotic options. Our results are compared to a large collection of indicative market prices and to more sophisticated models. Finally we propose a simple calibration method based on one-touch prices that allows the Vanna-Volga results to be in line with our pool of market data.
The Foreign Exchange (FX) option's market is the largest and most liquid market of options in the world. Currently, the various traded products range from simple vanilla options to first-generation exotics (touch-like options and vanillas with barriers), second-generation exotics (options with a fixingdate structure or options with no available closed form value) and third-generation exotics (hybrid products between different asset classes). Of all the above the first-generation products receive the lion's share of the traded volume. This makes it imperative for any pricing system to provide a fast and accurate mark-to-market for this family of products. Although using the Black-Scholes model [3,18] it is possible to derive analytical prices for barrier-and touch -options, this model is unfortunately based on several unrealistic assumptions that render the price inaccurate. In particular, the Black-Scholes model assumes that the foreign/domestic interest rates and the FX-spot volatility remain constant throughout the lifetime of the option. This is clearly wrong as these quantities change continuously, reflecting the traders' view on the future of the market. Today the Black-Scholes theoretical value (BS TV) is used only as a reference quotation, to ensure that the involved counterparties are speaking of the same option.
More realistic models should assume that the foreign/domestic interest rates and the FX spot volatility follow stochastic processes that are coupled to the one of the spot. The choice of the stochastic process depends, among other factors, on empirical observations. For example, for longdated options the effect of the interest rate volatility can become as significant as that of the FX spot volatility. On the other hand, for short-dated options (typically less than 1 year), assuming constant interest rates does not normally lead to significant mispricing. In this article we will assume constant interest rates throughout.
Stochastic volatility models are unfortunately computationally demanding and in most cases require a delicate calibration procedure in order to find the value of parameters that allow the model reproduce the market dynamics. This has led to alternative ‘ad-hoc’ pricing techniques that give fast results and are simpler to implement, although they often miss the rigor of their stochastic siblings. One such approach is the ‘Vanna-Volga’ (VV) method that, in a nutshell, consists in adding an analytically derived correction to the Black-Scholes price of the instrument. To do that, the method uses a small number of market quotes for liquid instruments (typically At-The-Money options, Risk Reversal and Butterfly strategies) and constructs an hedging portfolio which zeros out the Black-Scholes Vega, Vanna and Volga of the option. The choice of this set of Greeks is linked to the fact that they all offer a measure of the option’s sensitivity with respect to the volatility, and therefore the constructed hedging portfolio aims to take the ‘smile’ effect into account.
The Vanna-Volga method seems to have first appeared in the literature in [15] where the recipe of adjusting the Black-Scholes value by the hedging portfolio is applied to double-no-touch options and in [27] where it is applied to the pricing of one-touch options in foreign exchange markets. In [15], the authors point out its advantages but also the various pricing inconsistencies that arise from the non-rigorous nature of the technique. The method was discussed more thoroughly in [5] where it is shown that it can be used as a smile interpolation tool to obtain a value of volatility for a given strike while reproducing exactly the market quoted volatilities. It has been further analyzed in [16] where a number of corrections are suggested to handle the pricing inconsistencies. Finally a more rigorous and theoretical justification is given by [17] where, among other directions, the method is extended to include interest-rate risk.
A crucial ingredient to the Vanna-Volga method, that is often overlooked in the literature, is the correct handling of the market data. In FX markets the precise meaning of the broker quotes depends on the details of the contract. This can often lead to treading on thin ice. For instance, there are at least four different definitions for at-the-money strike (resp., ‘spot’, ‘forward’, ‘delta neutral’, ‘50 delta call’). Using the wrong definition can lead to significant errors in the construction of the smile surface. Therefore, before we begin to explore the effectiveness of the Vanna-Volga technique we will briefly present some of the relevant FX conventions.
The aim of this paper is twofold, namely (i) to describe the Vanna-Volga method and provide an intuitive justification and (ii) to compare its resulting prices against prices provided by renowned FX market makers, and against more sophisticated stochastic models. We attempt to cover a broad range of market conditions by extending our comp
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