rachet options + floaters
Post on 10-Apr-2018
285 Views
Preview:
TRANSCRIPT
-
8/8/2019 Rachet Options + Floaters
1/21
January 2008
RenteDykTM The new boy in class
8. January 2008 Claus Madsen
-
8/8/2019 Rachet Options + Floaters
2/21
-
8/8/2019 Rachet Options + Floaters
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....
-
8/8/2019 Rachet Options + Floaters
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 =
-
8/8/2019 Rachet Options + Floaters
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
-
8/8/2019 Rachet Options + Floaters
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
-
8/8/2019 Rachet Options + Floaters
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
-
8/8/2019 Rachet Options + Floaters
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
-
8/8/2019 Rachet Options + Floaters
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
-
8/8/2019 Rachet Options + Floaters
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
-
8/8/2019 Rachet Options + Floaters
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
+
+
-
8/8/2019 Rachet Options + Floaters
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
-
8/8/2019 Rachet Options + Floaters
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....
-
8/8/2019 Rachet Options + Floaters
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
-
8/8/2019 Rachet Options + Floaters
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...
-
8/8/2019 Rachet Options + Floaters
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
-
8/8/2019 Rachet Options + Floaters
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
-
8/8/2019 Rachet Options + Floaters
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
-
8/8/2019 Rachet Options + Floaters
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
-
8/8/2019 Rachet Options + Floaters
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!
-
8/8/2019 Rachet Options + Floaters
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...
top related