Skew Guide

I This has been prepared solely for informational purposes. It is not an offer, recommendation or solicitation to buy or sell, nor is it an official confirmation of terms. It is based on information generally available to the public from sources believed to be reliable. No representation is made that it is accurate or complete or that any returns indicated will be achieved. Changes to assumptions may have a material impact on any returns detailed. Past performance is not indicative of future returns. Price and availability are subject to change without notice. Additional information is available upon request.

DB Guide to Risk Reversals19

th October 2006

Inside: Skew Analysis Risk reversal valuation methodology G-10 risk reversals Analysis of model Valuation Summary Table

Apurv Jain Mark Stafford 212-250-8060

How to value risk reversal contracts and identify

relative value trading opportunities

• We link skewness of the FX spot return distribution to the risk reversal contract and propose a methodology to value risk reversals with respect to implied and realized volatility to find relative value trading opportunities

• In most cases realized skewness of the distribution is less on an absolute value basis than the implied skew

• We observe that a sharp movement in spot is usually

followed by a risk reversal “overvaluation” as “risk premium” increases and implied skew in the following periods is higher than realized skewness of the distribution. Additionally, the back end risk reversals tend to exhibit more overvaluation

• We also discuss applications of conditional variance swaps as a means of taking advantage of the “risk premium”

ON

1m3m1y

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Deutsche Bank

• We link skewness of the FX spot return

distribution to the risk reversal contract and

propose a methodology to value risk reversal with

respect to implied and realized volatility to find

relative value trading opportunities

• In most cases realized skew is less on an absolute

value basis than the implied skew

• We also observe that a sharp movement in spot is

usually followed by a risk reversal “overvaluation”

as “risk premium” increases and implied skew in

the following periods is higher than realized skew.

Additionally, the back end risk reversals tend to

exhibit more overvaluation

• We also discuss applications of conditional

variance swaps as a means of taking advantage of

the “risk premium”

Introduction

The famous Black Scholes model for option pricing has a questionable assumption of constant volatility of the return distribution, which doesn’t hold up to empirical examination. In fact there is well documented correlation between spot levels and the implied volatility levels. This correlation which results in the well known smile or a smirk effect in options on various asset classes is also included in the new and more complex models of option pricing. Risk reversals are liquid instruments that allow exposure to this correlation between volatility and spot levels. In this article, we propose a simple methodology to value risk reversals and propose trade ideas based on those valuations. We take USDJPY as an example to walk through the analysis. Skew- an intuitive explanation Skew means that the return distribution will be asymmetric and the left or the right tail is “heavier.” The absence of skew means that a distribution, for example the normal distribution, will be symmetric with both tails having the same weight. As an illustration of the concept, notice in fig. 1, the lighter distribution has a left tail that is heavier than the darker colored normal distribution. The lighter distribution is referred to as a negatively skewed distribution. In options markets, a thicker left tail of the return distribution means that OTM put options have a higher probability of finishing in the money as compared to an OTM call with the same delta. The delta of an option, which is an approximation of the probability of the option finishing in the money, is computed under Black Scholes by assuming that the return distribution is normal. This phenomenon of skew has been observed in other markets as well

- wherever there is correlation between implied volatility and spot we will see a skew. In the case of equity indices, falls in equity prices are accompanied by sharp increases in volatility. Even if the spot rate is kept constant, an increase in volatility makes an option more expensive since the price of a European or American style option always increases as volatility increases. So naturally, if volatility increases as the spot level falls, the value of the put option increases even more than would be expected with a mere decrease in spot levels. The mechanism described above gives rise to the “skew” in the implied distribution. The implied distribution is what the market expects and the realized distribution is what really happened.

To the extent that implied and realized distributions diverge systematically and consistently, we conclude there is a “risk premium”. The presence of a risk premium is usually accompanied by a trading opportunity. Now imagine the case where on average the “expected” (risk neutral) correlation between implied volatility and spot levels implied by the distribution is on average similar to the correlation observed in reality, over time we would say there is no “risk premium.” However, if we see that the market consistently prices in a different correlation

Fig 1. The light distribution has a negative skew

and the dark one is normally distributed

Fig 2. Out of money puts and deep in the money

calls in equities are more expensive

S&P 500 Dec 2006 Option Strike vs. Implied Vol

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Source: Bloomberg, DB FX Research

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3

between volatility and spot levels than is observed then there is a “skew risk premium” This skew risk premium is in addition to any “insurance premium” the writer of the put option might charge.

Does USDJPY have a skew? When we examine the scatter plot of implied volatility and spot level we see that there is a negative correlation between spot level and implied volatility which is statistically significant. This results in the left tail of the returns being thicker as or a negative skew as explained above.

What is a risk reversal? A risk reversal is a contract which is long 1 unit of a call option (typically 25 delta call option) and short 1 unit of a same delta put option (25 delta put option. Taking the example of USDJPY, a one month 25 delta risk-reversal would be long a one month 118.34 strike USDJPY call and short a 114.77 strike USDJPY put. The implied volatility of the call option (mid market) is 7.43% vs. the put option volatility of 8.03% (spot ref 117.09, ATM volatility 7.6). The difference between the implied volatilities of options with different strikes is called the volatility smirk or smile. This shows that the market price of dollar puts with roughly the same probability of finishing in the money is higher than the price of dollar calls with the same probability of finishing in the money. The explanation for the price difference between the calls and the puts follows directly from the explanation given above about the correlation between spot level and volatility. Essentially a risk reversal contract helps take positions on the spot–vol correlation in addition to a directional bet. Naturally, if the portfolio is delta hedged continuously then the trade only expresses a spot-vol correlation view. It can also be thought about as expressing a dvega/dspot view which means a small change in the vega of the option portfolio given a small change in spot level, holding everything else constant.

How can we see if USDJPY risk reversal is fairly

valued?

Naturally, this leads us to the question if this is justified. There are two aspects to the question – first, if the risk reversal is fairly valued with respect to implied volatility and the second, if the risk reversal is fairly valued with respect to realized volatility. Essentially, these questions mean that if the market participant hedged out the contract such that only exposure to volatility remained then what kind of returns would their portfolios show. In one instance (implied volatility) is how the portfolio would be marked to market and hence no systematic return would be made unless there was a correlation between the spot return and the implied volatility. In another instance, if the portfolio was held to maturity then no systematic return would be made less there

were a correlation with realized volatility. In other words, in the first instance we are testing an “implied skew” behavior and in the second we test the “realized skew” behavior.

Fig 3. USDJPY implied volatility smirk means dollar

puts/yen calls are more expensive

USDJPY skew

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Fig 4. USDJPY spot and implied volatility exhibit a

negative correlation in the past sample (1997-2001)

y = -0.1459x + 0.0632R2 = 0.0795

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Fig 5. spot vs. implied volatility negative correlation

continues in the present sample (Jan 2002-Sep 2006)

y = -0.1562x - 0.0509R2 = 0.1756

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Is the risk reversal fairly valued with respect to

implied volatility? (USDJPY as an example) To answer this question, we start with a broad assumption that a risk reversal is fairly valued if the

implied volatility of a particular strike option remains constant. Continuing with the example above, say the USDJPY spot level falls from 117.09 to 114.77, and the at the money forward volatility increases to 8.03% from 7.6% and similarly, when spot rallies to 118.34 if the at the money volatility goes to 7.43%, then we consider the risk reversal to be fairly valued. Naturally this is a rather simple assumption (also known as sticky by strike). However, given that we look at the “local” changes in spot every day and the related changes in implied volatility accompanying it, this seems to be reasonable. We record the data sample of spot and implied volatility moves into two samples - spot upmoves and spot down-moves.

We take the up-moves sample and measure the change in spot and the related changes in the implied volatility. We perform a regression to get the coefficient βupmove which tells us how much implied volatility moves per unit of spot movement up. Naturally, in case of USDJPY this coefficient is negative since as spot rallies the implied volatility usually sells off. We perform a similar analysis on the down move sample and get the βdownmove which is the coefficient of the change in implied volatility when spot sells off. Note, that this coefficient will also be negative since a spot sell off usually results in an implied volatility rally. To compute the β coefficients we use the data for the last 3 months. We tried different time windows for computing the β coefficients and found that the 3 month gave the best results (as measured by the least amount of squared pricing errors). We then compute how far out of the money the call and the put options are (125 pips for the USDJPY call and 232 pips for the USDJPY put in the above example). And then we calculate the value of the risk reversal with the following formula We look at data from January 2000 to August 2006 and find that the risk reversal seems reasonably correctly valued. There are times when the model value and the market value of the risk reversal diverge and those might be trading opportunities. For example through a large part of 2004 the front end risk reversals were overvalued with respect to implied volatility. Any trading strategy based on that would take advantage of this model valuation to generate positive returns. As can be seen from fig 6,

our model captures the various movements in the price of the risk reversal, which is a true test of how good this simple model is.

Fig 7. The 10 delta risk reversal is overvalued with

respect to implied volatility


Fig 6.The 25 delta risk reversal in the front end is

overvalued

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Value of risk reversal = βupmove*

Out of moneynessof call - β

downmove *Out of moneyness of put

Deutsche Bank@

5

Is the risk reversal fairly valued with respect to

realized volatility?

Now we try to answer the question of does the portfolio of long one unit 25 delta call and short one unit 25 delta put pay off if held to maturity. Notice, in the earlier case, we evaluated the question of if spot moved instantaneously and one were to sell the portfolio (having hedged all other risks but volatility risk) then would the return on the portfolio be positive. In the earlier case the only volatility one cared about was the implied volatility since we assume that only a very small interval of time passes before the portfolio is revalued and hence the exposure to realized volatility is minimal. Now we examine if the realized volatility shows any correlation with the spot levels. In other words is the realized skew equal to the implied skew of the distribution. For this we use some of the well known results in financial literature from Breeden and Litzenberger (1978) to infer the risk neutral distribution of the returns from the option prices. The formula derived for the relationship between the implied skew and the risk reversal is as follows:

Here t is the time to maturity and n the evaluation time. This formula enables us to compute the risk neutral implied skew from the risk reversal prices. Then we take the actual log returns of the USDJPY spot and compute the skew of that distribution. If the returns were indeed normally distributed, then the skew should be zero- in other words the distribution should be symmetric and there should be no correlation between spot level and volatility. The results in fig. 8 show that the realized skew is always higher than the implied skew or that in this small sample, the portfolio produces positive returns. In other words selling dollar puts and buying dollar calls produces positive returns for the market maker when the investor buys “insurance."

Why does being long USD calls / short USD puts

in USDJPY produce a positive return? Implied volatility has a risk premium built in it in addition to the future expected realized volatility. The time period of the data (Apr 03 – July06) is a “small sample” in that there is no event when the yen appreciated with a shock. Essentially, the strategy that systematically sells dollar puts provides insurance against the possibility of a big sell off in spot. A deeper question remains- is the risk premium justified? I.e. is the volatility risk premium merely a convenience cost (i.e.

Fig 10. And naturally the realized risk reversal has a

higher value than the implied

USDJPY Realized vs. Implied Risk reversal

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Fig 8. Realized volatility and spot had a similar

negative relationship from 1997-2001

y = -0.3364x + 0.0625R2 = 0.0488

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Fig 9. However, since 2002 this relationship has

broken down

y = 0.0252x + 0.0106R2 = 0.0006

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Skewt,n

= 4.4478* RiskReversalt,n

(25)/At The

Money Volatilityt,n

the amount of money you pay to not have to synthetically manage your portfolio to create a call option) or is there a covariance with a risk factor involved. An interesting insight is that yen rallies, carry trade unwinds and bouts of risk aversion are all correlated. Naturally, when the market is more risk averse, it is willing to pay a higher “insurance premium” for the same risk. However, the central question remains- What is the real macroeconomic risk? – an example could be a global recession risk due to a US current account deficit.

How can we use variance swaps to take

advantage of the “risk premium?” The data makes us think that a smart way to trade yen volatility would be to be long only implied volatility and skew and sell the realized volatility and skew. A simple variance swap pay off is just the difference between the implied and realized variance. Now structures like the conditional variance swap utilize the spot and volatility correlation to improve the volatility break even and the contract is structured as follows:

Mathematically the pay off can be expressed as: Here if i-1’th spot fixing is below the level H then

HSi <−11 = 0 and if it is above then = 1

How do the suggested trades take advantage of

the “risk premium?” The short conditional variance swap leg enables selling the realized volatility when spot sells off, which is usually lower than the market implied volatility and has little relation to spot level. Essentially, the conditional variance swap collects the risk premium when spot sells off, which is results in

positive returns. In a low volatility environment a conditional variance swap is an ideal way to leverage a core view that volatility will pick up over the coming months. Any increase in the “uncertainty” or increase in risk premium as spot sells off or due to some macro economic event will also favor this trade since typically higher the risk aversion, the higher the premium we collect and hence the better the strike improvement. The BoJ’s stance on interest rates also helps us determine the feasibility of the trade. As the BoJ looks unlikely to raise rates soon, we are confident of low realized volatility in the near future. Not unnaturally, the carry trade is back.

In the rest of this piece we examine risk reversal

value for other G10 currency pairs.

22Payoff StrikeRealised σσ −=

∑= −

<

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i i

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SN i

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log25211

1Payoff1

σ

• If spot remains above 114 the variance swap works in the usual way

• If spot moves below 114, fixings below 114 do not contribute to the variance calculation

• Strike improvement of 0.6 vols from 8.85% to 8.25% in 1 year maturity

Deutsche Bank@

7

EURUSD

At the moment, our model says that the EURUSD front end risk reversals which just switched sign from being positive to negative at -0.2 vols for EUR puts are slightly expensive. The back end (=1 year) risk reversal at 0.35 vols is also somewhat over priced with respect to implied volatility although it has been tending towards fair value. The 1m 25 delta strikes are 1.2380 and 1.2720 for vols of 6.8 and 6.55 respectively. We expect the implied to rally if spot goes to 1.2380 and to sell off if spot retraces back to 1.2720 and be lower. However, we believe that the market is overestimating the possibility of a “regime shift” once more after a sharp spot move. So our view is that the front end risk reversals are somewhat overvalued. Our analysis of risk reversal with respect to realized volatility indicates that it is overvalued. The other general finding is that risk reversals rarely pay for themselves if they are higher than 0.5 vols. Unless, we expect a big macro shock, we would prefer to be a seller rather than a buyer at those levels. When we have sharp and persistent divergences from the model in the front or the back end it implies a trading opportunity. When the risk reversal price produced by the model consistently provides a value above the current price, it means that the risk reversals are “cheap.”- For example, at the moment EURUSD 1m risk reversals are cheap. Historically, in the middle of 2002 the risk reversals were cheap with respect to implied volatility. That pattern continued through that year and gradually they returned to fair value in 2003. Looking at EURUSD spot we see three main regimes. Before 2002 EURUSD was in a downward trend and after 2002 it was in an uptrend that lasted until 2005. Since then it has traded sideways and is in what we classify as the third regime. We observe that risk reversals tend to trend with spot values. In the first EUR bearish regime any spot moves lower meant an increase in implied volatility. There was a regime change in 2002, during which the risk reversal changed from favoring EUR downside to EUR upside and in the second regime EURUSD moves up meant an increase in implied volatility. In the third regime of EURUSD trading sideways we see that EURUSD risk reversals are fairly valued and spot increases still mean implied volatility increases.

Fig 11. From 1999-2002 EUR puts were better bid

and if spot fell, implied volatility rallied

y = -9.6956x - 0.0497R2 = 0.0355

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Fig 12. EURUSD spot “regime” change in 2002 led to

EUR calls being more expensive

EURUSD spot rate

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Fig 13. Front End EURUSD risk reversals are slightly

overvalued with respect to implied volatility

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GBPUSD GBP risk reversals have just flipped direction from being bid for sterling calls to being bid for sterling puts. They had been bid for sterling calls since 2001 when spot started trending higher. At the moment, with the 1 month risk reversal at -0.18 favoring downside, we think that the market has overreacted with respect to the dollar rally and the probability of a sharp dollar up move is less than implied by the risk reversal. We agree that any GBPUSD rally is more likely to result in spot being back in the “range” which is bearish volatility, however according to our model the odds are higher that the dollar up move is subdued than is currently implied by the distribution. At the moment we prefer to sell the USD calls and buy USD puts and then wait to buy them back for a small profit when the market prices in a lower chance of a spot break and the risk reversal price comes down. Until recently, the back end risk reversals which were priced at 0.30 vols seemed overvalued with respect to implied volatility. However, with the recent move down to 0.2250 they have come closer to being fairly valued. Historically, GBPUSD implied volatility has rallied with spot rallies, however realized volatility rarely tends to follow. Although, GBPUSD traded sideways and then mostly downward from 1996 to 2002, implied volatility still rallied when spot moved up. This indicates to us that the market might have been overpricing the risk reversals in that period, or attaching a large risk premium to a large dollar move downward. Notice, that the realized volatility and spot levels display no significant relationship during the current sample period of 1999-2005. This led to the realized skew being quite low and hence the realized risk reversal was overvalued for most of the 1999-2005 period. The simplest way to formulate a trading strategy, based on the risk reversal model is to examine periods when the model valuation is significantly and persistently different from the actual valuation. Naturally, if the model values are higher than the risk reversal values, we like buying and if the model values are lower than the current risk reversal values, we like selling. For example in GBPUSD, we note that in Sep 2002 buying risk reversals (buy GBP calls / sell GBP puts) would have generated substantial positive returns, as would have selling them (sell GBP calls / buy GBP puts) in November and December 2003. When spot is in a trending cycle then risk reversals tend to be have higher values, which may be ascribed to overvaluation or possibly a higher risk

premium on a spot move up. In times of sideways movement risk reversals tend to be reasonably correctly valued, perhaps because people are unwilling to buy vanilla options when they see realized volatility being low and the expected realized volatility in the near future is low as well. With little speculative interest and little realized volatility the price of the risk reversal decreases. Interestingly we notice the pattern that the front end risk reversal is close to fair value in these situations. However, notice that the back end of the risk reversal is still slightly overvalued according to the model. This probably reflects a term structure of risk premium which is upward sloping or liquidity and hedging costs. Additionally, for this currency pair also we find that when the risk reversals price gets to more than 0.5 vols they rarely tend to pay for themselves.

Fig. 14. The implied vol and spot correlation have a

positive relationship

y = 5.8963x - 0.0496R2 = 0.0927

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Fig. 15. However, GBPUSD realized volatility has

shown no correlation to realized spot movements

y = -3.3271x + 0.0353R2 = 0.0022

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Deutsche Bank@

9

NZDUSD

With the exception of USDJPY and the yen crosses, NZDUSD exhibits the most extreme risk reversals in the G10 FX world, between 0.5 and 1 vol favoring kiwi puts. The market preference for puts on the kiwi dollar is due to the currency’s position as the highest yielder in the G10. With 1 year rates above 7.5%, long kiwi dollar positions are naturally very popular with FX carry traders. During bouts of risk aversion, these carry trades are unwound dramatically and the currency lurches lower, causing higher actual volatility and higher implied volatility as traders scramble to buy gamma to cover their positions. We find that in our model, front end risk reversals are undervalued with respect to moves in implied volatility. However, as we shift to longer dated risk reversals, we find that the market overestimates the change in implied volatility when spot changes. Additionally, as seen in fig. 17, the realized skew of the NZDUSD distribution is a lot less negative in our sample than implied by the value of the risk reversal. This suggests the back-end Kiwi risk reversals are overvalued both in terms of implied and realized volatility moves. This overvaluation becomes most apparent after the early 2004 drawdown of 2003 carry trade (NZDUSD up 30% in ’03). Since mid 2004, rises in US rates have reduced the interest rate differential and removed the attraction of USD of a funding currency thus reducing the interest in the NZDUSD carry trade. The NZDUSD realized skew exceeded the implied skew for a brief period when the NZD fell sharply but it seems the longer dated risk reversals priced in a “regime change” of falling spot and hence they have not been worth it subsequently as they have remained around 0.75 vols since 2004.

Fig 16. Realized skew is not as negative as implied

skew. Risk reversals rarely pay for themselves if

they exceed 0.5 vols

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Fig 18. NZDUSD front end risk reversal is

undervalued with respect to implied volatility

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Fig 17. Back End NZDUSD risk reversals are

overvalued with respect to implied volatility

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GBPJPY

Like any other carry trades, GBPJPY risk reversals are bid for downside with investors demanding protection for long cash positions. The 1m risk reversal is priced at -0.6 vols and the one year is -0.72 vols. At these levels both the front end and back end risk reversals seem overvalued with respect to implied volatility. Also, on average the risk reversal rarely pays for itself in GBPJPY. We find that while there is a significant negative relationship between implied volatility and spot levels, there is no significant relationship between the spot moves and the changes in the realized volatility (from 2002- today, fig. 21). Naturally, this results in the skew implied by the distribution having a much more negative value than the actual skew in the realized distribution. Like other currency pairs analyzed before, spot trends have definitely driven the risk reversal prices in this currency pair as well. There was a big sell off in GBPJPY spot in the year 2000 when GBPJPY went from around 179 to 148.72 and since then there have been several small sharp sell offs. We think that these sharp “crashes” might cause a market “crash” risk premium. For a certain amount of time, the risk reversal more than pays for itself and after that the “risk premiums” turn more negative and the risk reversal becomes overvalued. This dynamic leads us to suggest conditional variance swaps or premium collecting trades for this currency pair.

Fig 22. GBPJPY realized risk reversal has usually

exhibited a less negative skew than implied

Realized vs. Implied Risk reversal

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Fig 20. Back End GBPJPY risk reversals are over

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25 del Model Value

25 del Price


Fig 21. GBPJPY realized volatility shows no

relationship with spot moves in our sample

y = -0.0528x - 0.0784R2 = 0.0053

-6

-4

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6

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1 m onth change in m ean GBPJPY Spot

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Deutsche Bank@

11

AUDJPY

AUDJPY has consistently been one of the most popular carry pairs due to the high Australian interest rates (6.4%) and the low Japanese interest rates (0.66%). With a history of frequent and sharp sell offs, for example in 2000, AUDJPY went from around 71 to 55.99; this pair exhibits a significant skew- 1 month 25 risk reversal is valued at -0.95 vols and the one year is -1.5 vols. It is also a reasonable argument that this implied skew includes a “crash premium.” At these levels the front end risk reversals are fairly valued with respect to implied volatility but the back end risk reversals are overvalued with respect to the implied volatility. As expected, the realized skew of the return distribution is consistently less negative or less on an absolute value basis than is implied by the value of the risk reversal contract. This means that the risk reversal is overvalued with respect to realized volatility. When we glance at the charts we notice that AUDJPY follows the typical yen cross sell off pattern- sell off happens, short and long end implied risk reversals widen considerably, for a short period of time front end risk reversal pays for itself and then gradually it becomes more expensive, while the back end risk reversal frequently doesn’t pay for itself (esp. if kept to maturity). We like buying 6 month AUDJPY variance swaps at 7.8% while the regular var swap is at 8.15%. In this case the fixings below 87 will not count towards the var swap calculation.

Fig 23. Naturally, AUDJPY has a negative

relationship between spot and implied volatility

y = -0.2357x + 0.0234R2 = 0.1622

-2

-2

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0

1

1

2

2

-3.00 -2.00 -1.00 0.00 1.00 2.00 3.00

1 m onth change in AUDJPY Spot

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Impl

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Fig 24. Front end AUDJPY risk reversals seem fairly


-5.00

-4.00

-3.00

-2.00

-1.00

-

1.00

2.00

3.00

Jan-00

Jul-00

Jan-01

Jul-01

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Jul-02

Jan-03

Jul-03

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Jan-06

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25 del Model Value

25 del Price


Fig. 26. AUDJPY realized skew is consistently less

negative than the implied skew

Realize d vs. Implied Risk reversal

-3.00

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Mar

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99

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-00

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-04

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-05

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-06

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06



Fig 25. Back End AUDJPY risk reversal has been

consistently overvalued according to the model

-5.00

-4.00

-3.00

-2.00

-1.00

-

1.00

2.00

3.00

Jan-00

Jul-00

Jan-01

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25 del Model Value

25 del Price


AUDUSD

AUDUSD 1m risk reversal is -0.5 vol and the back end is -0.57 vols. At these levels the front end risk reversals are fairly valued with respect to implied volatility while the longer dated risk reversal contracts seem overvalued with respect to implied volatility. The risk reversal is overvalued with respect to realized volatility (fig 30) so we like collecting the risk premium via instruments like conditional variance swaps. We see that spot and implied volatility have a slight negative correlation. The spot trend and risk reversal pricing dynamics work similar to other currency pairs. We find that as AUDUSD rallied steadily from 0.63 to 0.80 from 2002 to 2003 risk reversals went bid for AUD calls, reflecting a spot regime change. Again, we notice that the implied distribution seems to overestimate the skew present in the realized return distribution. Also, in keeping with the big spot move leading to risk reversal overvaluation, we observe that the back end risk reversal became “overvalued” with respect to implied volatility after we saw a massive sell off in AUDUSD spot in end of 2003 to the middle of 2004.

Fig 27. AUDUSD front end risk reversals seem fairly


-3.00

-2.50

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-

0.50

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25 del Model Value

25 del Price


Fig 29. AUDUSD back end risk reversals seem over


-1.40

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

-

0.20

0.40

0.60

Jan-00

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25 del Model Value

25 del Price


Fig. 30. AUDUSD realized risk reversal has been

consistently less negatively skewed than the implied


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Feb-

98

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98

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Fig 28. AUDUSD spot and implied volatility have a

slightly negative correlation

y = -6.7021x - 0.0123R2 = 0.017

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Deutsche Bank@

13

USDCAD USDCAD 1m risk reversal at 0.1 vol and 1y at 0.07 vol is fairly valued with respect to implied and realized volatility. The spot and implied volatility have a slight negative correlation which is not very stable through time which means that the risk reversals flip signs from positive to negative frequently.In general, we observe that whenever spot is trading within a range the front end risk reversals are usually fairly priced, However, when spot breaks out of range then the risk reversal contracts become bid in that direction. For example if USDCAD rallies and breaks the range then the risk reversal is bid for USD calls and if USDCAD sells off and breaks the range then CAD calls are more expensive. Essentially the idea of the market “over reaction” seems to be prevalent. However, for a short period of time the risk reversals do pay for themselves as – for example in May 2002 the risk reversals more than paid for themselves. However, in June 2003 the risk reversals were overvalued in 2003 as USDCAD had a sharp decline and the risk reversal went bid for CAD puts. Like other currency pairs in USDCAD risk reversal also, we find that it is easier to sell into the rallies by using the model carefully to find a persistent overvaluation the market. This “overvaluation” seems to result from an overestimation of the probability of the sharp spot move continuing.

Fig 34. USDCAD spot and implied volatility have a

slightly negative correlation which is not very stable

y = -2.8797x + 0.0068R2 = 0.0127

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Fig 31. USDCAD implied and realized skew are

usually close to each other and fairly valued


-1.50

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Feb-

98

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Fig 32. USDCAD front end risk reversals are faily


-1.50

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25 del Model Value

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Fig 33. USDCAD back end risk reversals are fairly

valued at the moment

-1.00

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25 del Model Value

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EURCHF

EURCHF implied volatility and spot show a slight negative correlation and the 1m risk reversals are priced at -0.05 vols which means that they are almost “flat” (i.e. call and put have almost similar implied volatilities). At this level the front end risk reversals are fairly valued with respect to implied volatility (fig. 35). The flat level implies that the spot has not had a massive trend so it is not a surprise that it is “fair value.” However, the back end risk reversal at -0.125 vol is slightly overvalued (fig 37) and the actual skew is consistently less on an absolute value basis than what is implied by the distribution derived from the risk reversal contract. Again, we see that EURCHF spot shows the classic sharp sell off and slower rallies up, which like the previous currency pairs, means that the implied skew is likely to be more negative and the longer term risk reversals are more likely to be “overvalued” Additionally, since in CHF crosses there is the belief of CHF being a crisis currency, longer dated CHF calls might be seen as providing insurance.

Fig 35. EURCHF front end risk reversals are fairly

valued at the moment

-3.00

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25 del Model Value

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Fig 38. EURCHF implied volatility and spot show a

slight negative correlation

y = -7.1506x - 0.01R2 = 0.0314

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Fig 37. EURCHF back end risk reversal is over valued

according to the model

-2.00

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Fig 36. EURCHF realized skew is consistently less

negative than the implied skew


-2.00-1.50-1.00-0.500.000.501.001.502.00Ja

n-99

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Deutsche Bank@

15

EURJPY

EURJPY has similar characteristics to USDJPY and other yen crosses; essentially there is a stable negative relationship between spot and implied volatility. The front end risk reversal is overvalued (fig. 39) with respect to implied volatility. Similarly, the back end risk reversal is overvalued and we notice that the skewness implied by the distribution is more (on an absolute value basis) than what is realized in actual returns. The implied and realized skew become less negative in the end of 2000 and beginning of 2001 as EURJPY rallied sharply resulting in the implied and realized skew changing across all models. Since then, EURJPY has experienced a steady uptrend and occasional sharp draw-downs. In 2003 EURJPY experienced sharp down moves and naturally the back end risk reversal in EURJPY suddenly became overvalued after 2003 due to the “insurance premium” attached to the possibility of another sharp down move. After the sharp down move it is also interesting how even in the front end risk reversal the implied and realized skew diverge due to the addition of the “risk premium”. In our opinion it is these drawdowns that provide a good trading opportunity, either via buying the risk reversal or via conditional variance swaps. These are reflected by our model - we see that the model line is consistently closer to zero as compared to the actual price of the risk reversal almost through the life of the model (refer to chart).

Fig 41. EURJPY realized skew has been consistently

less negative than implied by the risk reversals


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Fig 39. EURJPY front end risk reversal is overvalued

according to the model

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Fig 38. EURJPY has exhibited negative correlation of

spot and implied volatility from 2002- today

y = -0.111x - 0.0125R2 = 0.0954

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Fig 40. Back end EURJPY risk reversal has been

consistently overvalued wrt implied volatility

-2.50

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USDCHF

As a “safe haven” currency CHF, typically rallies in times of crisis. Usually, in times of crisis implied volatility shoots up also. Thus we find that USDCHF spot and implied volatility have a stable negative correlation. The 1m (front end) risk reversal is bid 0.30 for USD calls and we think they are overvalued (fig. 43) Once again, this is a function of the market reasoning that there will be a sharp dollar up move and as spot breaks out of range the volatility will go bid. We agree with the logic presented, but disagree with the probability of the sharp dollar move. We think that the probability of the sharp dollar move up is considerably less than implied by the risk reversal distribution. The back end risk reversals have come in over the last two weeks from -0.45 vols (bid for CHF puts) to -0.23 for CHF puts which is much closer to fair value with respect to implied volatility (fig 44).There is a possible argument that they might have a risk premium built in and hence might seem more “overvalued.” With the implied and realized skew of the front end risk reversal close to zero , this risk reversal also seems fairly valued with respect to realized volatility. However when we examine the back end for implied vs. realized skew, we find that the risk reversal is overvalued with respect to realized volatility. Our preferred trade in USDCHF would be to buy the back end risk reversal in times of a crisis and collect the crisis premium.

Fig 44. USDCHF back end risk reversal is slightly

overvalued

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Fig 43. USDCHF front end risk reversal is overvalued

valued according to the model

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Fig 45. USDCHF realized skew is close to the skew

implied from the risk reversals


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Fig 42. USDCHF spot and implied volatility have a

stable negative correlation

y = -11.684x - 0.1076R2 = 0.2375

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Deutsche Bank@

17

Analysis of the risk reversal model

We observe a few systematic patterns from our skew valuation models. Below we discuss and summarize the observations of valuations with respect to implied and realized volatility and also note possible improvements to the model. Observations about the risk reversal valuation

with respect to implied volatility

• When spot is trading in a range, front end risk reversals tend to be close to fair value with respect to implied volatility as the market seems quite good at pricing the risk reversal with respect to implied volatility during those times. We don’t observe many systematic trading opportunities, however from time to time we find that risk reversals become over or undervalued.

• A sharp spot movement usually results in the front end risk reversal paying for itself for some time but then the market begins to “overestimate” the volatility changes with respect to spot movement. At that point we find the front end risk reversal contract becoming overvalued with respect to implied volatility. These result in the most systematic trading opportunities of selling the overvalued risk reversal.

• The effect of a sharp spot movement on back end risk reversals is greater. The back end risk reversals seem to have a higher tendency to become “overvalued” with respect to implied volatility and remain that way for longer periods. It seems that the market usually “overestimates” the magnitude and the life of the trend and prices in a “regime shift” The back end of the risk reversal is usually slightly overvalued according to the model. However, the back end overvaluation could also reflect a term structure of “risk premium” which is upward sloping.

• We find the overvaluation effects to be exacerbated in case of carry currencies. Usually, many market participants are invested in the carry trade. These currency pairs experience sharp declines and hence these investors bid up the prices of the lower interest rate currency calls as they all want to buy “protection.” So after the carry currency pair experiences a sharp sell off, the investors attach a higher subjective probability of a sell off happening again.

Observations about risk reversal valuation with

respect to realized volatility

• We observe that the realized skew of the

return distribution is less on an absolute value basis in most currency pairs than implied by the risk reversal contracts.

• We observe that the difference between the realized and implied skew is more substantial in the higher carry currencies. We think a similar explanation of the “crash risk premium” drives the difference in the realized and implied skew.

• In the event of a sharp spot movement the realized skew exceeds the implied skew for a certain period of time and the risk reversal “pays for itself.” However, after the event the implied skew continues to be higher and the market overestimates the possibility of the regime shift continuing and also becomes more willing to pay the insurance premium via the difference between the implied and the realized skew.

• A good (but rough) rule of thumb is that risk reversals rarely pay for themselves when they are above 0.5 vols.

Further improvements to the model

With our model, we are inherently subscribing to some sort of a linear relationship between spot and volatility which is closer to a Heston type stochastic volatility model (although we allow for the correlation to vary across time) than a local volatility model. There is no theoretical reason at all for the spot and volatility relationship to be linear. However, given that we look at daily changes and do separate the upmoves from the down moves we have eliminated the first order issues with assuming the relationship to be linear. To be theoretically more correct we should have a volatility and spot relation model for every currency pair, which would then be calibrated to fit the various market observed option prices and end up as a time varying mix of local and stochastic volatility model and we would simulate the path taken by spot and then estimate the returns of our investment strategies to really capture the risk reversal value with respect to both implied and realized volatility. All this will naturally require considerable calibration and computation. That seems impractical at this stage. Additionally, we still need to uncover the “macroeconomic” or consumption based asset price factor that underpins the purported risk premium- remember that in any standard asset

pricing model such as the CAPM, the ICAPM or APT, we need covariance with the consumption portfolio (proxied by various “factors” usually). The issue of the risk premium in currencies remains a thorny one and as practitioners, we are interested in the “basic risk factors” as long as they either help us in hedging them or in terms of any predictive or return explanatory powers. However, there are other easier and more fruitful improvements at hand. We could correct for constant interest rate assumptions in our model. We could also make some distributional assumptions about the skewed currency pairs, while computing the realized skew. Obviously, any departure from normality costs computation times and increases complexity. Yet another application of the skew analysis could be to use them in conjunction with the carry trade(s) since they appear to be closely related. It would also be interesting to extend this analysis to emerging markets since the “risk factors” there are more apparent. On another front, we could develop a cross sectional model that would enable us to compare risk reversals across currency pairs and identify common characteristics leading to risk reversal portfolio opportunities or trading models. A cross sectional model using a spot range breakout indicator, interest rate differential, country default probability and current account deficits might be an interesting extension.

Currency Pair Risk Reversal*

EURUSD OvervaluedGBPUSD OvervaluedNZDUSD UndervaluedGBPJPY OvervaluedAUDJPY Fairly valuedAUDUSD Fairly valuedUSDCAD Fairly valuedEURCHF Fairly valuedEURJPY OvervaluedUSDCHF OvervaluedUSDJPY Overvalued

Risk Reversal Summary Table

This has been prepared solely for informational purposes. It is not an offer, recommendation or solicitation to buy or sell, nor is it an official confirmation

of terms. It is based on information generally available to the public from sources believed to be reliable. No representation is made that it is accurate

or complete or that any returns indicated will be achieved. Changes to assumptions may have a material impact on any returns detailed. Past performance is not indicative of future returns. Price and availability are

subject to change without notice. Additional information is available upon request.

Dedicated to Mom, Rachel and Manu

*1m risk reversal valued wrt implied vol

Documents

Skew Guide