Rachet Options + Floaters

8/8/2019 Rachet Options + Floaters

1/21

January 2008

RenteDykTM The new boy in class

8. January 2008 Claus Madsen


2/21


3/21

22

Definition

The Components:

Coupon is a function of the 10Y CMS (Constant Maturity Swap) rate + a

fixed spread A sold Ratchet cap on the 10Y CMS rate

A sold Bermuda call option with a strike equal to 105

The future coupon can only fall never rise! Structures with similar

characteristics as RenteDykTM are usually referred to as One-Way-Floaters

Bermudan options are a well known product but what is a Ratchet? see

next slide....


4/21

33

What is a Ratchet cap?

A Ratchet option (sometimes called a Reset option or Cliquet option) is a series of

consecutive forward starting options. The first is active immediately, the second

becomes active when the first one expires etc. Each option is struck at-the-money

when it becomes active that is the future strikes are stochastic!

Ratchet features can be incorporated into other structures like for example

RenteDykTM. (They are frequently encountered in structured equity products)

The payoff from a Ratchet Cap can be expressed as:

This implies that C(T) can be expressed as (assuming a zero spread):

( ) [ ( ) ( ), 0]TPayoff T t Max R T C t =

( ) ( ) [ ( ), ( )]C T T t Min R T C t =


5/21

44

Valuation of Ratchet Options - I

For the valuation of Cliquet Options (the terminology normally used in

the equity market) we have the following methods available:

(European Style Cliquet Option) Closed form solution due to Rubenstein

(1991) - price as a series of forward-starting options. More sophisticated

techniques where the future strike only change if at the reset date the

stock price is below/above (put/call) can be handled by the method ofGray and Whaley (1999)

A more general approach is the binomial pricing method from Huag and

Haug (2001), which relies on the CRR binomial model

Monte Carlo simulation is however the most natural approach for the

pricing of Cliquet Options, due to the path-dependency embedded in the

determination of the future strike levels


6/21

55

Valuation of Ratchet Options - II

For Rachet Options (now we are back in the interest-rate world!) the

complexity is somewhat greater when it comes to valuation - analytical

formulas are hard to come by even in simple cases!). Therefore, forRatchet Options we would in general need some kind of numerical

method and the natural pricing methodology to choose is Monte Carlo

Simulation

However a finite difference method can also be used by adding an extra

state-variable


7/21

66

Visualizing the stochastic property ofthe future strike levels

10-Year DKK Swap-Rate

(Init Coupon 4.725%, Fixing 2 times a year 30.12 and 30.6)

3.00%

3.50%

4.00%

4.50%

5.00%

5.50%

6.00%

6.50%

7.00%

31-12-

1998

31-12-

1999

31-12-

2000

31-12-

2001

31-12-

2002

31-12-

2003

31-12-

2004

31-12-

2005

31-12-

2006

Dates

Swap-Rate/Coupon


8/21

77

Valuation of Ratchet Options - III

For this particular type of Ratchet Cap embedded in RenteDykTM we

have the following additional issues:

We need a 2-factor model, due to the fact that the short-rate and the

10Y CMS Rate are not 100% correlated see next slide....

We need to take into account the volatility for At-The-Money Swaptions

and over time volatility for Out-Of-The-Money Swaptions due to the

fact that we expect to have an increased exposure to lower and lower

strike levels as time goes - Volatility Smile dependency!

These 2 factors clearly add complexity to the pricing of RenteDykTM


9/21

88

Correlation betweenThe Short-Rate and The 10-CMS Rate (avg Corr: 0.79)

Correlation Structure

(180 days rolling Correlation)

1.50%

2.50%

3.50%

4.50%

5.50%

6.50%

7.50%

31-12-

1998

31-12-

1999

31-12-

2000

31-12-

2001

31-12-

2002

31-12-

2003

31-12-

2004

31-12-

2005

31-12-

2006

Dates

Swap-Rate

-1.0000-0.8000-0.6000-0.4000

-0.20000.00000.20000.40000.6000

0.80001.0000

Correlation

Short-Rate 10-Year Swap Correlation


10/21

99

The Pricing of RenteDykTM

What Interest Rate Model to choose?

In this case it is natural to use the Libor Market Model as it is straightforward to

extend to multiple factors more precisely the Extended Libor Market Model.

Extended, as we need to include the Volatility Smile. Candidates are for example:

The CEV version of Andersen and Andreasen (2000) Alternatively the

Displaced Diffusion version (Rebonato (2001))

Lognormal-Mixture Model of Brigo and Mercurio (2000)

The stochastic-Volatility model of Andersen and Brotherton-Ratcliffe (2001) The SABR model of Hagan, Kumar, Lesniewski and Woodward (2002)

A short-rate model is not the obvious choice let us however still pursue that

possibility for the following practical reasons: Speed of calculation, comparison

with other mortgage bonds which normally are valued using a 1-factor short-rate model - simplicity and therefore transparency


11/21

1010

The Pricing of RenteDykTM Short-Rate Models - I

The most popular short-rate model in the market is the Hull-White model:

Our Hull-White implementation for RenteDykTM is a Monte-Carlo based pricing

method the standard seems to be a lattice implementation, this (interesting)

discussion will however not be pursued here....

A few interesting issues for the Hull-White models implication for the volatility

sensitivity and for volatility smiles will be given here....

Some stylized facts 1: The Hull-White model has that property that there is a negative correlation

between (caplet) volatility and interest rates (a desirable feature) i.e. the

(caplet) volatility increases when rates are falling, the reason being that we

have the following formula:

This effect is shown on the next slide....

1 11

(0, , )T Caplet

ev

F T T

+

+


12/21

1111

Implied Vol Sensitivity for the Hull-White Model

This figure illustrates this effect which is obviously more pronounced for shorter

maturity Caplets than for longer maturity Caplets

Hint: Black Caplet Vols are given as decimals

1

3

5

7

9

0

0,1

0,2

0,3

0,4

0,5

BlackCapletVol

YC-Shifts

Black Caplet Vol implied from HW Model


13/21

1212

The Pricing of RenteDykTM Short-Rate Models - II

Some stylized facts 2:

Embedded in the Hull-White model is a volatility smile in the implied Black

Caplet volatility

This smile is however not changeable with a different parameterization of theHull-White model as there is no direct control over it it is embedded in the

model

In general the slope of this embedded Black volatility smile is small compared

to the market

There can exist cases where it is not possible to imply a Black CapletVolatility given the price obtained from the Hull-White Model the HW price

is unattainable! (this happens for high values of in the Hull-White Model)

We also have that smirks are not possible in the Hull-White Model

This feature is shown on the next slide....


14/21

13

Implied Volatility Surface for the Hull-White Model

1

3

5

7

9

0,1

0,15

0,2

0,25

0,3

0,35

-300-250-200-150 -100 -500 50 100 150 200 250 300

Black

CapletVol

Change in Strike-Level

Black Caplet Implied Vol-Surface from HW

Model


15/21

1414

The Pricing of RenteDykTM Short-Rate Models - III

We can now list the following issues connected with using a 1-Factor model moreprecisely the Hull-White Model:

Even though the model embeds the (desirable) feature that (Black) volatilityrises when rates falls we have limited control over it, the relative slope isnearly constant for all parameterizations. A slightly higher relative slope canthough be observed for short maturities and for mean-reversion => 0

It is not possible to fit the ATM humped Cap Volatility curve as only smallhumps are normally observable. A large hump is only possible in case the YC isdecreasing which is not the norm...

The embedded volatility smile in the Hull-White model has a too small slope the relative slope is approximately constant across different parameterizations

There can exist cases where it is not possible to Imply a Black Caplet Volatility given the price obtained from the Hull-White Model the HW price isunattainable! (happens for high values of in the Hull-White Model)

Smirks are not possible in the Hull-White Model

As it is a one-factor model the 10-year rate will be a deterministic function ofthe short-rate

Hint: When using a simpler model than theoretically is required knowing the

limitations is necessary...


16/21

1515

Risk-Measures....

We will here look at 3 issues:

The distribution of Ratchet prices across the maturity spectrum

On slide 16 it can be seen that the most of the value of the Ratchet

Option embedded in RenteDykTM is located around the 10-year

maturity point...

The delta-vectors

On slide 17 the distribution of sensitivies across the maturity

spectrum Key-Rate sensitivities are shown

The stability of duration measures for different increments

On slide 18 it is obvious that duration is a function of the

increment, that is duration is scale dependent!

Besides that we will from a theoretically point discuss the stability of duration

measures over time


17/21

1616

Ratchet Price Distribution

Distribution of Ratchet Value

0

2040

60

80

100

120

0.00

2.00

4.00

6.00

7.99

9.99

12.00

14.01

15.99

18.00

20.01

22.00

23.99

26.01

28.00

30.00

Dates

0

0.5

1

1.5

2

2.5

3

3.5


18/21

1717

Delta Vectors Distribution of Sensitivities

DV for RenteDyk

-2

-1

0

1

2

3

0.01

1.00

2.00

5.00

8.00

10.00

13.00

15.00

20.00

30.00

40.00

Maturity

Absolute

Sensitivities


19/21

1818

Duration as a function of the Increment. Unit = 100 Bp

Duration Sensitivity

0

0.5

1

1.5

2

2.5

100 90 80 70 60 50 40 30 20 10

Increment

Du

ration


20/21

1919

Risk-Measures The nature of Ratchet Options

Wilmott (2002) highlights the danger of volatility risk in Rathets, as it can be

shown that the gamma of a Cliquet option is the sum of gamma values for regular

options because the gamma of a forward-starting option is zero before starting

time. This may create the impression that risk-management is easy in this case.However, for this type of option, hedging can be quite complex because delta,

vega and gamma have discontinuities around reset times!

The above entails the following:

Even under a stable interest-rate environment, risk-measures will behave erratic

as we are getting nearer and nearer to the fixing-date and risk-measures will

be discontinous as we pass the fixing-date!


21/21

2020

Some closing remarks on Risk-Measures for RenteDykTM

1-order Risk-Measures are scale dependent, i.e. it is a function of the selected

increment

Risk-Measures (duration and convexity) will even under a stable interest rateenvironment behave erratic as we are getting nearer and nearer to the fixing-date

and risk-measures will be discontinous as we pass the fixing-date

Risk-Measures are discontinous around the fixing-dates

Vega is remarkably higher than for Capped-Floaters

Delta-Vectors in general it can be said that the sum of the Delta-Vectors is not

equal to OAS-Price Risk. That is DVs for RenteDykTM (only) indicate/highlight the

relative importance of a shift at different parts of the curve

So lets be careful out there...

Rachet Options + Floaters

Documents

Transcript of Rachet Options + Floaters