Editorial
Calendar arbitrage in the FX volatility surface
Today I would like to share an observation with you that my colleagues Alexander and Andreas discovered when analyzing FX volatility market data.
When we construct the FX volatility smile based on the usual FX brokers’ data ATM, 25-delta risk reversal (RR) and butterfly (BF), 10-delta risk reversal and butterfly, there is a decent fan club that uses an SVI (stochastic volatility inspired) parametric function for the total variance on the log-moneyness space.
We consider market data of 07 Jan 2014 for AUDJPY as follows:
| Tenor | 10 RR | 25 RR | ATM | 25 BF | 10 BF |
| 6M | -5.475 | -2.725 | 11.71 | 0.6 | 2.525 |
| 1Y | -7.1 | -3.525 | 12.44 | 0.65 | 2.975 |
| 2Y | -11.625 | -5.9 | 13.30 | 0.6 | 3.225 |
When we generate the volatility smile curves using an SVI parameterization we see the following variances on the log-moneyness space:
Calendar arbitrage occurs if the total variance, which is volatility squared times the time to maturity σ²T, for a given moneyness F/K and a longer maturity is smaller than the total variance for the same moneyness (and possibly a strike K adjusted by the ratio of forwards F of the two maturities respectively) and a shorter maturity. Such a scenario would yield negative forward variances. One can visualize calendar arbitrage by checking if the total variance curves intersect. If they do, then there is a (theoretical) calendar arbitrage opportunity. A real arbitrage opportunity would require including bid-offer spreads and a liquid market of very low delta-options.
In the present scenario, we observe a rather good fit of the SVI-curves to the market volatility quotes. They are non-intersecting in the region of quoted volatilities. However, if we zoom out and plot the total variance curves on a wider log-moneyness range (x-axis), the curves for 2Y (red) and 1Y (black) intersect.
While the phenomenon is outside the usual range of traded options, it can still cause problems when using the volatility surface as a building block of an exotics model, such as a local volatility model. Note that this isn’t a market-based arbitrage, but a model-based arbitrage. It is the SVI parameterization that yields this effect.
Solution: switch on the control-wings button in the MathFinance volatility surface MFVal. This ensures that calendar arbitrage is prevented in the SVI parameterization:
The 2Y curve (red) is now on top of the 1Y curve (black). This input for an exotics model is much more suitable.
Summary
Applying various interpolation techniques to construct the FX volatility surface from market quotations may introduce arbitrage. We showed a calendar arbitrage example with the popular SVI-parameterization and an idea how to fix it. Generally volatility surface construction requires detailed attention and permanent inspection.
Uwe Wystup, Managing Director of MathFinance
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