Risk Reversals & Their Relationship With Spot

  • 10 Mar 2003
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This article focuses on the relationship between spot and risk reversals. Using five reference currency pairs chosen for the liquidity of their out-of-the-money options we investigate potential causal links between spot and risk reversals. While a causal link from risk reversals to spot cannot be entirely ruled out in some instances, the relationship appears to be very weak, whereas there is strong evidence supporting a causal link from spot to risk reversals. Consequently, charts showing spot versus risk reversals should be interpreted with caution, bearing in mind that spot is the leading force.

Out-Of-The-Money Options & Risk Reversals:
A Quick Refresher Course

Out-of-the-money options trade at a premium or discount to the at-the-money options to reflect the set of probabilities assigned by the market to different spot scenarios. Figure 1 shows the current volatility surface for euro/dollar: the cheapest level is for at-the-money options while the wings are trading at a premium. This means market players expect the distribution of spot to have fatter tails than a lognormal distribution: in other words, they assign a relatively high probability to scenarios where spot deviates a lot from the forward.

Out-of-the-money options are usually quoted in terms of delta. The delta of an option represents, among other things, the probability that the option will expire in the money. An at-the-money forward option will have a delta of around 50. On either side of the at-the-money option, the option with the smallest delta (of the call and the put) will be quoted. As such, the scale will rise from zero to 50 and decrease back to zero, as shown on figure 1.

The market convention for quoting out-of-the-money options is via strangles and risk reversals, usually constructed by using 25 and 10 delta options. A 25-delta strangle is obtained by buying--or selling--a 25-delta call and a 25-delta put. As such, it is a symmetric structure, with an aggregated delta of zero.

A 25-delta risk reversal is obtained by the contemporaneous purchase of a 25-delta call and sale of a 25-delta put, or vice-versa. As such, it is an asymmetric structure, with an aggregated delta of 50: There is a 50% chance of ending up between the two strikes and a 25% probability of ending up beyond either of the strikes. A risk reversal will benefit from a directional move in spot. Gamma and vega will actually change sign depending on where spot is trading. Risk reversals are generally quoted as an implied volatility spread between the two strikes. For instance a 25-delta risk reversal in euro/dollars quoted at +1% in favour of the euro calls implies that the euro call trades at a 1% premium in volatility terms to the euro put.

Risk Reversals & Spot

With risk reversals being an aggressive bullish, or bearish, strategy, their evolution looks strikingly similar to that of spot. Consequently charts of spot against risk reversals are a common occurrence in the foreign exchange market, as shown in Figure 2 for dollar/yen. Since it is not clear which series is leading the other, the interpretation of such plots is, however, questionable. Do risk reversals contain some information relevant for spot--in other words, does the volatility market impact the spot market? Or are risk reversals merely responding to the latest spot movements?

We addressed these questions using three years of data in five reference currency pairs: euro/dollar, dollar/yen, euro/yen, cable and aussie/U.S. dollar. We first noted there is always a strong correlation between the spot and risk reversals time series, even though this correlation tends to decrease slightly with the term of the risk reversals considered. In this article we report our results using spot versus one-year 25-delta risk reversals.

Do Risk Reversals Contain Information
Relevant To Spot?

We study the relationship between spot and risk reversals using a methodology called the Granger causality test. This methodology analyses the presence or absence of feedback between variables and infers a possible causal relationship. In this context, the risk reversals will (partially) cause the current level of the spot if past values of the risk reversals can be considered useful information for explaining the current level of spot. Specifically, we let yt be the daily returns on spot and RRt be the risk reversals. A simple way of testing for causality is to set up the following model:

yt = (sum)*t-iyt-i + (sum)ßt-iRRt-i + *t

and test the hypothesis that:

H0:ß1= ß2 =...= ßk = 0

If this null hypothesis is rejected than we can conclude that the past values of the risk reversals have an impact on the current level of the spot.

Using our five reference currency pairs, we first tested whether risk reversals contain some information relevant to spot: our results are presented in Table 1.

The hypothesis that risk reversals cause movement in spot is strongly rejected for all currencies apart from euro/yen. This suggests risk reversals contain little relevant information on future values of spot. For instance, there is no statistical evidence that a higher level of risk reversals should be followed by an increase in spot. Further investigation in the euro/yen case shows that the result that we obtain--we accept that risk reversals can have influence on spot--seems to be due to our observation window: this relationship did not hold prior to 2000. This might be a consequence of the fact that euro/yen has traded in ranges for protracted periods of times since 2000. Indeed, the past three years have been marked first by dollar strength and then by dollar weakness, meaning euro/dollar and dollar/yen were in the spotlight while euro/yen was trading as a cross.

Does Spot Contain Some Information
Relevant To Risk Reversals?

Our next step consisted in testing the reverse relationship: after all, spot being the most liquid instrument in the FX market, it makes intuitive sense that spot might cause risk reversal movement. Table 2 shows the result of the reverse causality test: is spot relevant in explaining risk reversals?. The answer to that test is an emphatic 'yes' for all currency pairs, with an associated p-value of 0.

Consequently, we conclude that spot is causing risk reversals rather than the opposite. This means for example that an increase in spot is likely to be followed by an increase in risk reversals.

This result does not come as a complete surprise: in the foreign exchange market, spot tends to be the most liquid instrument. Therefore, it is logical that it should move first. Furthermore, implied volatility can be interpreted in terms of market makers' expectations, one-month implied volatility for at-the-money options today is the expected realised volatility over the next month. In this context, out-of-the-money options quoted by the market give the expected distribution of spot seen by the market today. In particular, risk reversals, which essentially define the skewness of the probability distribution function of spot, can be interpreted as a measure of bullishness toward a particular currency. For instance, if out-of-the-money call options are better bid than put options, the underlying probability distribution function implied by the option market will be skewed to the upside (Figure 3 shows a slightly exaggerated skewed PDF for euro/dollar). In this context, the fact that risk reversals do not cause spot is symptomatic of the lagging nature of market sentiment measures.


Our analysis has shown that in most cases, the risk reversal market is reacting to spot movements rather than the opposite. Consequently, charts of spot versus risk reversals should be interpreted with caution, always remembering that spot is the leading force.

This week's Learning Curve was written byAnne SanciaumeandAlexei Jiltsov, in the quantitative foreign exchange research group atLehman Brothersin London.

  • 10 Mar 2003

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