Instructors Corner Risk Reversals

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Outline:

Risk Reversals

February 2006

Global Financial Markets Institute Inc.+1 [email protected]

www.gfmi.com

Author: Kenneth Kapner



© Global Financial Markets Institute Inc. Page 2 of 7

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Risk Reversals

Risk reversal is a commonly used term in the FX markets. Specifically, a risk reversal is:

1. An option strategy combining the simultaneous purchase of out-of-the-

money calls (puts) with the sale of out-of-the money puts (calls). Theoptions will have the same expiration date and similar deltas. (for thosereaders familiar with options strategies, this type of transaction can alsobe categorized as a collar, cylinder or range forward. However, a risk reversal has specific meaning regarding the skewness which is explainedbelow)

2. A market view on both the underlying currency and implied volatility.

This article:• Describes the origins of the risk reversal• Defines the volatility smile and skew

• Examines specific example of quotes• Investigates a hedging example

Origins of the Risk Reversal

The risk reversal has it underpinnings from the limitations of the Black ScholesMerton (BSM) option pricing model. One of the limitations is that the BSMassumes a lognormal distribution. Anyone involved in the spot FX markets willtell you that price ranges are anything but normal or lognormal! In practice, thecurrency markets follow what is known statistically as leptokurtosis. In laymen’sterms, this means that the underlying will trade at extreme prices from the

current spot market more frequently than a normal distribution would suggest.The price action at these extreme levels is referred to as “fat tails”.

Figure 1: Normal Distribution vs. Leptokurtotic Distribution

Normal Distribution

Leptokurtotic Distribution




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Volatility Smile

If more price action occurs at these extreme levels than suggested by the model,the option trader will mark volatility higher to account for the increased

probability of the underlying trading in the “fat tails”. The end result is thatimplied volatility will be higher for out-of-money (OTM) and in-the-money (ITM)options then at-the-money (ATM) options. If there is no bias in marketexpectations of the underlying price, then the picture of volatility is symmetricalaround the at-the-money volatility. This is commonly referred to as the “volatilitysmile”. Notice in Figure 2 how the volatilities are symmetrical around the ATMforward.

Implied Volatility Smile

1 month ATM forward price = 1.1847

7.80

8.00

8.20

8.40

8.60

8.80

9.00

9.20

1.1329 1.1457 1.1537 1.1599 1.165 1.1696 1.1847 1.2006 1.2052 1.2105 1.2168 1.2251 1.238

5 10 15 20 25 30 50 30 25 20 15 10 5

Strike Price/Delta

V o l a t i l i t y

Figure 2: Theoretical Volatil ity Sm ile – USD/ CAD

Volatility Skew

In practice there are supply and demand considerations along with marketexpectations built into option prices. This pushes up the volatility for ITM (OTM)calls (puts) relative to those of the puts (calls). This is referred to as the “skew”.The risk reversal expresses the difference in volatility. In theory, if a currency is

expected to appreciate, calls would be favored over puts and the purchaser of the call would pay a higher volatility relative to puts.




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Implied Volatility Skew

1 month ATM forward price = 1.1847

7.60

7.80

8.00

8.20

8.40

8.60

8.80

9.00

9.20

9.40

1.1329 1.1457 1.1537 1.1599 1.165 1.1696 1.1847 1.2006 1.2052 1.2105 1.2168 1.2251 1.238

5 10 15 20 25 30 50 30 25 20 15 10 5

Strike Price/Delta

V o l a t i l i t y

Figure 3: Volatil ity Skew - USD / CAD Source: Superderivatives

Looking at Figure 3, notice how the volatilities are higher as the CAD isappreciating. The interpretation is that the market expects the CAD to appreciateand/or implied volatility to move higher.

Example

The market has established a 25 (0.25) delta benchmark for risk reversal quotes.

For example, assume the market expects the CAD to appreciate against the USD. A trader quotes a 1-month 25 delta USD/CAD risk reversal of .15 -.28% whereCAD calls are favored over CAD puts. Based on this information, the trader wouldbe willing to do each of the following, assuming volatility is 8.50%:

1. Buy the 25 delta USD put/CAD call at 8.65% and sell the USD call/CADput at 8.50%. The trader pays away the .15%

or2. Sell the 25 delta USD put/ CAD call at 8.78% and buy the USD

call/CAD put at 8.50%. The trader earns the .28 spread

In essence, the dealer is willing to buy the USD put/CAD call at 8.65% and sellthe USD put/CAD call at 8.78%. The dealer is giving a “choice” quote to eithersell the USD Call/CAD put or buy the USD call/CAD put at 8.50%.




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Hedging Example

Assume a U.S. company is importing goods from Germany and is invoiced inEuros. The invoice is EUR 1,000,000 and is due in three months. The company is

concerned that the USD will depreciate. Assume current market prices are asfollows:

Spot 1.2073

Forward 1.2146

EUR Call 25 Delta Strike/Vols 1.2519 /9.15%

Premium USD 800

EUR Put Strike/Vols 1.1779 /8.95%

Premium USD 800

To hedge the position, the company buys a EUR call/USD put at 25 delta andsells a corresponding EUR put/USD call.

Figure 4 shows the company’s exposure in three months. Assume the current 3-month forward price is 1.2146. and the company could lock in the forwardexchange rate at this price. Hence, this will be their benchmark to compare thehedge using the risk reversal.

Expiry Payoff Profile

-15,000

-10,000

-5,000

0

5,000

10,000

15,000

1 .

1 1 9 6

1 .

1 2 9 6

1 .

1 3 9 6

1 .

1 4 9 6

1 .

1 5 9 6

1 .

1 6 9 6

1 .

1 7 9 6

1 .

1 8 9 6

1 .

1 9 9 6

1 .

2 0 9 6

1 .

2 1 9 6

1 .

2 2 9 6

1 .

2 3 9 6

1 .

2 4 9 6

1 .

2 5 9 6

1 .

2 6 9 6

1 .

2 7 9 6

1 .

2 8 9 6

1 .

2 9 9 6

1 .

3 0 9 6

U/L Price

P r o f i t / L o s s

Contract 1

Contract 2

Contract 3

Contract 4

Contract 5

Contract 6

Net position

Figure 4: Net Exposure of U.S. Company




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Figure 5 shows the risk reversal. Contract 2 is the EUR call / USD put. Contract3 is the EUR put /USD call. The blue line is the net position of the two contracts


-8,000

-6,000

-4,000

-2,000

0

2,000

4,000

6,000

8,000

1 .

1 1 9 9

1 .

1 2 9 9

1 .

1 3 9 9

1 .

1 4 9 9

1 .

1 5 9 9

1 .

1 6 9 9

1 .

1 7 9 9

1 .

1 8 9 9

1 .

1 9 9 9

1 .

2 0 9 9

1 .

2 1 9 9

1 .

2 2 9 9

1 .

2 3 9 9

1 .

2 4 9 9

1 .

2 5 9 9

1 .

2 6 9 9

1 .

2 7 9 9

1 .

2 8 9 9

1 .

2 9 9 9

1 .

3 0 9 9

U/L Price


Contract 1Contract 2Contract 3

Contract 4

Contract 5Contract 6

Net position

Figure 5: Risk Reversal

Figure 6 combines the two positions resulting in what is commonly referred toas a collared position. Contract 1 is the company’s original exposure to the Euro.Contract 2 is the EUR call / USD put. Contract 3 is the EUR put / USD call. Theblue line is the net position of the three contracts (exposures).


-15,000

-10,000

-5,000

0

5,000

10,000

15,000

1 . 1

1 9 8

1 . 1

2 9 8

1 . 1

3 9 8

1 . 1

4 9 8

1 . 1

5 9 8

1 . 1

6 9 8

1 . 1

7 9 8

1 . 1

8 9 8

1 . 1

9 9 8

1 . 2

0 9 8

1 . 2

1 9 8

1 . 2

2 9 8

1 . 2

3 9 8

1 . 2

4 9 8

1 . 2

5 9 8

1 . 2

6 9 8

1 . 2

7 9 8

1 . 2

8 9 8

1 . 2

9 9 8

1 . 3

0 9 8

U/L Price


Contract 1

Contract 2

Contract 3

Contract 4

Contract 5

Contract 6

Net position

Figure 6: Combined Exposure of U.S. Company w ith Risk Reversal




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It is interesting to note that the company is more concerned with the efficacy of the hedge. The trader, on the other hand, will be more interested in the detailedaspects of the risk reversal including the impact it will have on the option books’ risk measurements, i.e. the Greeks.

This article:• Described the origins of the risk reversal• Defined the volatility smile and skew• Examined specific example of quotes• Investigated a hedging example

Any questions or comments can be addressed to the author [email protected]

I wanted to thank:Superderivatives for supplying the data.ChisholmRoth for using Magellan spreadsheets illustrating the option payoff profiles.

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Instructors Corner Risk Reversals