# Modelling Fair Value Of Implied FX Vol: Part 2

In last week's Learning Curve, we showed how measures derived from historic spot could be used via regression to produce a fair-value measure for the one-month volatility.

Beyond The One-Month Tenor

In last week's Learning Curve, we showed how measures derived from historic spot could be used via regression to produce a fair-value measure for the one-month volatility. This measure could then be profitably traded across a number of currency pairs. It was also shown, however, that these same factors do not work for the one-year tenor.

Because of the longer time frame, the one-year implied volatility could be influenced by a far wider set of factors. This could include items such as the interest rate differential, macro-economic data and perceived event risk. This means that a much more detailed investigation was required to obtain the optimal inputs.

Many combinations of different sets of market data were tested using the approach detailed in the previous article. The cumulative P&L for each set was looked at to establish how effective that combination was. The best and worst combinations were analyzed together with how each individual set contributed. This then allowed conclusions to be drawn as to what was important.

The risk of data mining will always be present when we calibrate our model to maximize return. We may end up with a result that is not intuitive and is in reality an artefact of the data sets used. Therefore at each stage care has been taken to firstly ensure the approach makes intuitive sense, and then in addition achieves high returns.

Finding Basic Data Sets

Initial investigations focused on which data series may or may not contribute to the fair value of the one-year implied vol. The trading threshold was set to be 0.3, which tended to give a reasonable number of trades. Positions were closed out as soon as the difference between the implied volatility and fair value became less than the threshold, the second trading approach described in the previous article.

*Data Set 1: Fundamentals*

To get an initial idea of the important data series, 21 series were tested for EUR/USD covering historic vols--both the usual window calculation and an exponential measure--spot levels, risk reversals, strangles, interest rates and interest rate differentials across various maturities. From this set it was concluded that risk reversals, strangles and measures of the current spot level were not useful measures, and longer-dated maturities tended to work best. This is probably due to them being smoother series.

The most important inputs were the historic vol--both the usual calculation and an exponentially smoothed version--and the interest-rate differential. The best-performing combinations all involved a long-term historic volatility, a short-term exponential historic vol and an interest-rate differential.

*Data Set 2: Market Confidence Measures*

This data set introduced two non-fx measures as possible input series: an equity risk premium measure, which is often taken as a confidence measure, and credit spreads, often taken as a risk appetite measure. Both were smoothed using an exponentially weighted moving average. The full set of measures tested were:

* Historic volatility--both calculation styles, various window sizes/half lives;

* Interest-rate differential--various maturities;

* Equity risk premium--various half lives; and

* Credit spreads--various half lives.

The analysis was applied to both EUR/USD and USD/JPY. Key aspects of the results were:

* The long-term historic vol again appeared as an important factor;

* A short-term historic vol measure was an important factor for USD/JPY, whether the normal historic vol or an exponentially weighted measure;

* The equity risk premium made a positive contribution, appearing in many of the top combinations;

* The interest rate differential dropped out of the top combinations; and

* The credit spreads were disliked and appeared in many of the worst, and few of the best, combinations.

Averaging over all possible combinations, this set of data produced better results than data set 1, suggesting a move in the right direction. It is intuitive that an overall market confidence measure should be useful in measuring the fair value of the 1 year volatility, alongside a measure of how volatile the fx market has been.

*Data Set 3: Equity Volatility*

As a final change to the core data set a smoothed version of the VIX, a measure of equity volatility, was included alongside the other measures. This found:

* The VIX made a positive contribution appearing in a number of top combinations; and

* The VIX was noticeably more important for USDJPY than EURUSD.

Final Data Sets To Take Forward

From the above analysis the final set of data series that could contribute to the regression model was fixed at:

* The usual historic volatility measure based on a fixed window with all points equally weighted;

* An historic vol measure using exponential weightings with a varying half life;

* The equity risk premium smoothed using an exponential weighting.

The VIX again smoothed using an exponential weighting.

Four window sizes or half lives corresponding to very short-, short-, medium- and long-term periods were used to give a total of 16 possible series to be chosen from for the regression.

This final data also makes sense intuitively: the normal historic vol measure is a number that an options trader will regularly see and be influenced by. An exponentially weighted volatility can be considered a measure of a trader's memory of spot movements. Recent movements will count more than moves a long time ago, but all will be remembered a bit. You would expect overall confidence in the financial markets to affect the expected future volatility and this is what the equity risk premium measures. Volatility in the equity markets will naturally feed through to the fx markets, particularly in times of high stress.

Optimal Input Tenors

Given the possible data series that could be useful, the optimal tenor for each series was investigated for EUR/USD, GBP/USD, AUD/USD and USD/JPY. During this we were also looking to see if we could apply a single combination across all currency pairs and hence allow a portfolio approach similar to that applied to the one-month tenor.

The optimal half lives for each factor and the P&L achieved is shown in table 1. The overall result is the best performing combination for a portfolio of all four currency pairs.

P&L | Equity Risk | VIX | Historic Vol | Exp. Historic Vol | |

EURUSD | 3.82 | Medium | Very short | Long | Long |

GBPUSD | 2.5 | Very short | Long | Short | Short |

AUDUSD | 3.58 | Short | Very short | Very short | Very short |

USDJPY | 5.93 | Medium | Short | Long | Long |

Overall | 10.07 | Long | Long | Long | Short |

Table 1: Optimal Half Lives/Windows to Maximise P&L |

The P&L of the best performing combination is 71% of the total of each individual model. This overall best model made 1.97 in EUR/USD (vs 3.82), 1.73 in GBP/USD (vs 2.50), 2.01 in AUD/USD (vs 3.58) AND 4.36 in USD/JPY (vs 5.93).

Figure 1 shows the results from trading the best overall inputs for EUR/USD.

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Conclusion

We have found a good model for the one-year at-the-money volatility. The following inputs can be applied across a portfolio and traded profitably:

* A long-term historic vol measure;

* A short-term, exponentially weighted historic vol measure;

* A long-term confidence measure, the equity risk premium; and

* A long term equity volatility measure, the VIX.

In conjunction with the one-month model from the previous Learning Curve, we now have good models for the short and longer dated tenors. Unfortunately they use quite different sets of inputs. The next stage for this work is to try and converge these models and obtain fair value measures for the 2M, 3M and 6M maturities. There are a number of possible approaches for this, with the application of stepwise regression the first path to investigate.

*This week's Learning Curve was written by**Duncan Farnsworth**, head of risk analytics in the currency structuring group at**The Royal Bank Of Scotland**in London.*