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Derivatives

Modelling Fair Value Of Implied FX Vol - Part 1

A fair-value measure of implied volatility would be an invaluable tool in the fx options market.

The One-Month Tenor

A fair-value measure of implied volatility would be an invaluable tool in the fx options market. It would enable traders to spot opportunities in the market and take advantage of them, or help a fund manager use the asset class as a source of alpha. Therefore, there have been many attempts to come up with such a measure, ranging from simple historic volatility calculations through to GARCH and beyond.

As implied volatility is a prediction of future volatility it is impossible to directly measure it and there are many influences on the exact value seen in the markets. These influences include:

* Historic volatility--both recent and long term;

* Political environment;

* Overall market confidence, including markets outside of fx itself;

* Future and past events and associated risks; and

* Trading positions held in the market.

To try and build a fair value measure for implied volatility we start by establishing an approach and looking at the one-month maturity. This will subsequently be extended to longer maturities.

 

Approach

To establish a fair-value measure we use regression to model the implied volatilities. For the one month a three-factor model was used and tested over historic data. On each day, the regression was fitted over a fixed-size window of historic data, including the day being tested. This then led to a fair value measure for that day that could be compared against the market value.

To assess the effectiveness of this measure it was then traded. To do this, a trading threshold level was fixed. Once the difference between the fair value and market value breached this level an appropriate trade, either long or short, was put on. This trade was then subsequently closed out when the signal disappeared and the difference between the market volatility at the start and at the end was taken as the profit/loss in volatility points on that trade. These were then totalled over the whole test period.

Two approaches could be taken for defining when a signal disappears. In the first the signal is considered to have disappeared when the fair value and market values cross over. In the second the signal is considered to have disappeared when the difference between fair value and market value drops below the trading threshold level. The second approach is more conservative and considered more applicable when the data series being modelled is less volatile.

 

Example: To start, market vol = 11, fair value = 11.4. Using a trading threshold of 0.3 points or vols, this trading style will buy at 11.

First Approach: If the market vol increased to 11.4 and fair value stayed constant it would sell at 11.4, making a profit of 0.4 vols. If the fair value dropped, however, to 11.2 vols and the market vol increased to 11.2 vols the position would sell at 11.2 giving a profit of 0.2 vols.

Second Approach: If the market vol increased to 11.2 and fair value stayed constant it would then sell at 11.2, making a profit of 0.2 vols. If the fair value dropped, however, to 11.3 vols and the market vol increased to 11.1 vols the position would sell at 11.1 giving a profit of 0.1 vols.

 

Note that this P&L measure ignores time decay, transaction costs and the passing of time. Therefore it is only a valid measure when the signal is used to contribute to a large trading book position. If traded in isolation the other factors will need to be quantified.

 

Modelling One-Month Implied Vol

It is known that financial time series exhibit volatility clustering, i.e. the market exhibits periods of high volatility followed by periods of low volatility. Therefore, different measures of historic volatility should be an important type of indicator for future volatility. For example, the GARCH models for volatility are based on volatility clustering. Naturally recent historical volatility has larger implications for the shorter tenors than for the longer tenors.

It is therefore expected that the one-month vol is heavily dependent on the spot behaviour seen over recent history. For example, spot levels going to an historically very low level, even though historical vols are low or moderate, often may come in conjunction with increased implied vol levels. Therefore the inputs to this model were set to be:

* A measure of how the current spot level compares with a short term historical average;

* The average size of moves seen, weighted to give recent moves more significance; and

* A short term historic volatility measure.

These parameters are then fitted using simple regression on a rolling daily basis so each day the contribution of each parameter could change.

 

Trading The Signal

For the one-month tenor the trading threshold was set to be 0.75 vols, but the position was not closed out again until the implied and fair value measures crossed over, matching the first approach described above.

Figures 1 and 2 show the results from trading the model for EURUSD and USDJPY from the middle of 2003 to the end of 2005.

As can be seen EUR/USD performed best pre 2005 while USD/JPY performed better in 2005. The Royal Bank Of Scotland's internal use of this model reports the overall P&L on a portfolio of 12 currency pairs. The cumulative P&L from June 2003 to end 2005 for this portfolio is shown in figure 3.

It can be seen that this portfolio approach produces a generally increasing P&L.

 

What About Other Maturities?

This combination of spot-based parameters produces an effective model for trading the one-month volatility. Although the model does not work for all currency pairs all of the time, on a portfolio basis it is very effective. This, however, is only for the one-month maturity and it is generally assumed that longer-dated maturities follow very different rules.

Testing this model against the one-year tenor with the same model inputs and thresholds produces very little trading. Reducing the trading threshold to 0.3 produces a reasonable number of trades, but also leads to a net loss of 4.83 vols.

It might be expected that the second approach to deciding the close-out level i.e. closing out the position as soon as the threshold is no longer breached, would be more suitable for the less volatile one-year at-the-money volatility. Moving to this approach improved the P&L to 1.93 vols which, while an improvement, was still not great.

Finally moving the half lives and windows used for the model inputs to a longer period, in line with the one-year maturity, made EUR/USD more profitable but made USD/JPY less profitable.

This suggests that a model based purely on measures derived from spot does not work for the one-year tenor and therefore there must be other factors influencing this tenor. The second part of this Learning Curve will look into which of the many possible factors are important.

 

Conclusion

We have a good model for calculating a fair value of the one-month volatility based purely on inputs derived from the spot history. In practice this is a difficult model to trade outside of a large volatility trading book due to time decay, transaction costs and the passing of time. Longer dated maturities may be easier to trade, but are not well described by this model, and are therefore investigated further in the next Learning Curve.

 

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

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