Editorial
Vanna-Volga and the Greeks
Many people ask us what’s so special about calculating Greeks in the vanna-volga approach. Isn’t it like calculating bumped values and ratios of differences?
On the other hand, did you ever wonder, why for instance a professional system like SuperDerivatives doesn’t provide vanna-volga Greeks along with its “market price”, which is known to be based on vanna-volga and other adjustments?
Figure 1: Difference of vanna-volga based KO call option value and its corresponding vanilla option value, strike on the x-axis
For exotics, we like to see prices consistent with vanilla options. A simple example is a barrier option, say a down-and-out call in USD-JPY with barrier 102, spot 109.24. When we move the barrier away, we want to observe a monotone convergence of the barrier option price to the vanilla option price. In particular, the barrier option price should never exceed the vanilla option price. Figure 1 shows the difference of the vanilla and the regular knock-out option value for varying strikes. In a Black-Scholes model with constant volatility the barrier option value is always below the vanilla option value.
Surprisingly we observe that in the vanna-volga approach the knockout option value can be slightly higher than the vanilla option value, so the difference of a vanilla minus knockout option value can become negative, see the blue curve in Figure 1. Analyzing this, it turns out that we are in a situation where for the vanilla option value the vanna-volga hedging costs are negative or, more generally, where the sum of the vanna and the volga term in the vanna-volga formula is negative and the vanna-volga value is below the flat volatility Black-Scholes value.
For knockout options the vanna and the volga term are usually multiplied with some factor smaller than one (often a function of the no-touch probability of the barrier) and the correction term to the Black-Scholes value is negative, but not negative enough anymore to cause a knock-out option value above the vanilla option value.
To force minimum consistency one can at the very least cap the barrier option value at the vanilla option value or floor their difference at zero, see Figure 2.
Figure 2: Difference of vanna-volga based KO call option value and its corresponding vanilla option value, floored at zero, strike on the x-axis
Implementing such a consistency rule is quite easy, but we now lose smoothness of the value function. And as we know this may cause jumps and spikes in the Greeks, especially when we compute derivatives by finite differences, i.e. bumping market data.
Figure 3 shows the vega profile for varying strikes for the above example computed by bumping volatility by an absolute value of 0.1%.
Figure 3: Vega on the strike space of a regular knock-out call, comparing vanna-volga approach with and without consistency rule (cap)
Figure 4 shows the gamma profile for varying strikes for the above example computed by bumping spot by an absolute value of 0.01.
Figure 4: Gamma on the strike space of a regular knock-out call, comparing vanna-volga approach with and without consistency rule (cap)
In this example the kinks and jumps we see in the graphs are unpleasant, but not dramatic. The problem is that the kinks occur at parameter levels that are not easy to predict – in contrast to non-smooth behavior at a barrier level, which is known in advance and allows us to compute one-sided finite differences or shift the barrier.
Non-smooth value functions at non-predictable locations of kinks in combination with parameter bumping can now really lead to exploding Greeks, as can be seen in Figure 5. Just by selecting other strikes we hit a region where gamma takes an arbitrary value.
Figure 5: Exploding gamma on a different strike grid of a regular knock-out call, cause by a vanna-volga approach with consistency rule
Of course, one can try to model smoother transition rules between the curves for each consistency rule, but this will lead to an endless patchwork.
The situation becomes even worse, if we value vanilla options from an already constructed smile surface. Then the vanilla options are valued correctly, but one would need to add many more rules to avoid inconsistencies caused by valuation based on the smile and valuation based on the vanna-volga approach[1].
Valuing first generation exotics with the vanna-volga approach approximated their prices well in past and still yields very fast and simple rough indicative prices for touch contracts in FX markets. However, computing vanna-volga Greeks was and remains complicated. This is caused by having to impose many consistency rules to a seemingly simple vanna-volga formula.
Today we discussed only some of the technical difficulties and traps but didn’t even explore if the Greeks computed in vanna-volga are useful and correct quantities for risk management.
Overall, we learn that if we impose conditions to make pricing of exotics in vanna-volga approaches consistent with vanilla options, we obtain instable Greeks. Without consistency conditions, the Greeks might be smoother, but the values may lead to arbitrage. It will require a lot of patchwork to cure all the headaches at one go.
Consequently, we consider it rather worth investing the time to compute stable Greeks within consistent models, knowing that there are still enough numerical challenges left. And with the model class of SLV and its variants we have models at hand, that are flexible enough to fit the prices in many FX option markets.
[1] Recall MathFinance Newsletter 352 of 28 May 2019, https://www.mathfinance.com/newsletter-352/
Andreas Weber and Uwe Wystup
MathFinance AG
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Overview
Traditionally, Quantitative Finance has revolved around the development of parsimonious models that yield some economic understanding of financial markets. In recent years, there has been a change in this paradigm by embracing data-driven methods from AI and ML. Here are some reasons that explain this shift: greater amounts of financial data are available that require fast processing; financial analysis and computations are supplemented by non-financial data, such as textual data, in order to create new insights; data-driven methods allow to detect trends and market changes that would not be observed with a rigid model. At the same time, platform technology has taken over trading of spot and derivatives in financial markets. Pricing models, Greeks and risk calculation have to be faster and more accurate than ever before. The special issue welcomes contributions that explore innovative uses of AI / ML methods and platform technology in Quantitative Finance. These can involve economic, quantitative, computational and technological aspects.
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