CUTTING EDGE. CROSS-CURRENCY EXOTICS
Smiling
hybrids
Vladimir Piterbarg develops a multi-currency
model with foreign exchange skew suitable
for valuation and risk management of forexlinked hybrids, in particular power-reverse
dual-currency (PRDC) swaps. The emphasis
of the article is on model calibration to forex
options across different maturities and strikes
Power-reverse
dual-currency (PRDC) swaps
are by far the most widely
traded and liquid cross-currency exotics. Continuing interest in these
structures, driven mainly by Japanese investors looking for higher
yields, has resulted in large books of PRDC swaps being accumulated
by most exotics dealers in the market. Large, concentrated exposures
require sophisticated and robust models to risk-manage them properly.
PRDC swaps are essentially long-dated (30 years being quite typical)
swaps with coupons that are options on the forex rate (most commonly a dollar/yen rate). They are often Bermuda-style callable, or
have knock-out provisions. PRDC swaps are exposed to moves in the
foreign exchange rate and the interest rates in both currencies. By now,
most dealers have standardised on a three-factor modelling framework
(see, for example, Sippel & Ohkoshi, 2002), with the forex rate following a lognormal process, and the interest rates in the two currencies
driven by one-factor Gaussian models. Keeping the number of factors
to the minimum allows us to use partial differential equation (PDE)based methods for valuation, an essential requirement for large books.
The choice of Gaussian assumptions for the interest rates and lognormality for the forex rate has allowed for a very efficient, essentially
closed-form, calibration to at-the-money options on the forex rate.
Forex options exhibit a significant volatility skew (induced in longerdated options, ironically, by dealers trying to hedge their PRDC swap
positions). Moreover, the structure of PRDC swaps, epitomised in the
typical choice of strikes for the coupons, as well as the callability/knockout features, makes them particularly sensitive to the forex volatility
skew. Incorporating a skew in a model is by no means trivial. While
various mechanisms for including a skew in the model are well known
(local volatility, stochastic volatility and/or jumps), their applicability
in the context of PRDC swap modelling is hindered by the problem of
66
piterbarg.indd 66
calibration, exacerbated by the presence of stochastic interest rates.
In this article, we build a cross-currency model that incorporates
forex volatility smiles. We keep one-factor assumptions for the interest rates, but use the local volatility (constant elasticity of variance
(CEV)-type) process for the forex dynamics. This avoids introducing any more stochastic factors, keeping the speed and accuracy of
valuation the same as in the ‘standard’, lognormal model. Most of
the focus is on the problem of calibrating forex options for various
maturities and strikes simultaneously. The main aim of this article is
a calibration procedure that can be performed essentially instantaneously. The procedure is obtained by combining our recently developed techniques of skew averaging (see Piterbarg, 2005c and 2005d)
with derived Markovian representation of the dynamics of the forward forex rate that is exact for European-style options.
In addition, the effect of forex volatility skew on PRDC swaps
(cancellable and knock-out varieties) is studied, and is found to be
significant. In particular, by analysing the shapes of the payouts of
the securities, it is determined that the slope of the forex volatility
smile is a major factor affecting the values, thus endorsing our focus
on introducing skews into PRDC swap modelling.
Limitations and alternatives
The subject of PRDC swap modelling, despite its importance in practice, has received scant attention in the literature. Particularly disappointing is the lack of published research on long-dated forex smile
modelling (the subject of this article) except for some interesting results
presented at conferences (see Balland, 2005). The model we develop is
not intended as the final answer to all forex smile-related problems. Single-factor interest rate assumptions, while sufficient for PRDC swaps,
make the model unsuitable for some of the more complicated hybrids,
such as constant maturity swap-spread linked ones. For such derivatives, a model similar to that of Schlogl (2002), in which Libor market
models are used for interest rates, yet with sufficient control over forex
smiles and a good forex option calibration algorithm (something that
Schlogl does not address) will probably need to be developed. In addition, such a model would serve as a good platform to incorporate interest rate volatility smiles by using, for example, skew-enhanced Libor
market models as in Andersen & Andreasen (2000).
We note that the model we develop does not attempt to exactly
match implied forex volatilities for all strikes. In fact, since the CEV
specification for local volatility is used, this would be impossible in
most markets exhibiting convex smiles. Our focus is on developing a
model in which the slope of the volatility skew can be controlled, thus
allowing us to assess the skew exposure of PRDC swaps (which, as
mentioned above, is very significant).
The CEV specification is a typical choice for introducing volatility
skews in many contexts (see, for example, Andersen & Andreasen,
2000, where it is used for Libor market models). Theoretical limitations (having non-zero probability of absorption at zero) rarely
present practical problems, as most securities have strikes and other
interesting ‘features’ at levels strictly above zero. In particular, local
modifications of the power function around zero can easily eliminate
the absorption problem, while having little or no impact on pricing
and calibration. Moreover, dynamically, the CEV model is one of the
soundest local-volatility models, implying ‘self-similar’ behavior of
the underlying (see Hagan et al, 2002).
The use of so-called ‘mixture’ models has been suggested in the
literature for introducing volatility smiles. The merits of such models, and in particular their applicability to cancellable, knock-out or
other dynamics-linked securities are discussed elsewhere (see, for
example, Piterbarg, 2005a).
Risk May 2006
3/5/06 11:28:33
The model
An economy with two currencies, ‘domestic’ and ‘foreign’, is considered. Let P be the domestic risk-neutral measure. Let Pi(t, T), i = d,
f, be the prices, in their respective currencies, of the domestic and
foreign zero-coupon discount bonds. Also let ri(t), i = d, f, be the
short rates in the two currencies. Let S(t) be the spot forex rate, expressed in the units of domestic currency per one unit of the foreign
currency. For future purposes, the forward forex rate (the break-even
rate for a forward forex transaction) is denoted by F(t, T), with the
well-known relationship:
F (t , T ) =
Pf (t , T )
Pd (t , T )
S (t )
(1)
following from no-arbitrage arguments.
The following model is considered:
dPd (t , T ) / Pd (t , T ) = rd (t ) dt + σ d (t , T ) dWd (t )
dPf (t , T ) / Pf (t , T ) = r f (t ) dt − ρ fS σ f (t , T ) γ (t , S (t )) dt
+ σ f (t , T ) dW f (t )
(
(2)
)
dS (t ) / S (t ) = rd (t ) − r f (t ) dt + γ (t , S (t )) dWS (t )
where (Wd(t), Wf(t), WS(t)) is a Brownian motion under P with the
correlation matrix 〈dWi, dWj〉 = ρij, i, j = d, f, S (note the ‘quanto’ drift
adjustment for dPf(t, T) arising from changing the measure from the
foreign risk-neutral to the domestic risk-neutral). Gaussian dynamics
for the rates are specified:
s
T
⌠ e − ∫t χi (u ) du ds,
σ i (t , T ) = σ i (t ) 
⌡t
i = d, f
(3)
with σd(t), σf(t), χd(t), χf(t) deterministic functions. The forex skew
in the model is imposed via the local volatility function γ(t, x). The
‘standard’ Gaussian framework, as developed in, for example, Dempser & Hutton (1997), is recovered by using the function γ(t, x) that is
independent of x, γ(t, x) = γ(t).
For stability of calibration, it is essential to use a parametric form of
the local volatility function. The following parametrisation is used:
 x 
γ (t , x ) = ν (t ) 

 L (t ) 
β(t ) −1
(4)
where ν(t) is the relative volatility function, β(t) is a time-dependent
CEV parameter and L(t) is a time-dependent scaling constant (‘level’). Our choice of the parametric form is motivated by well-known
problems (see Andersen & Andreasen, 2002) with the dynamics,
and subsequent hedging performance, of models with ‘U-shaped’ local volatility functions. Among more ‘flat’ functions, most choices
are essentially equivalent, with the CEV specification being slightly
more technically convenient.
The volatility structures of zero-coupon discount bonds defined by
(3) can be recognised as arising from the one-factor Gaussian HeathJarrow-Morton model (sometimes called the Hull-White or extended
Vasicek model (see Hull & White, 1994)). Such a model admits a
Markovian representation in the short rate. Hence, the model (2)
admits a Markovian representation in three variables, the domestic
and foreign short rates and the forex spot. The value of any security
can then be calculated by solving a three-dimensional PDE (most
efficiently solved by utilising a level-splitting scheme, such as an
alternating-direction implicit scheme from Craig & Sneyd (1988)).
While valuation in such a model is (at least conceptually) simple,
calibration is much less so.
Calibration
■ Overview. The parameters defining the volatility structures of interest rates in both currencies, that is, the functions σd(t), σf(t), χd(t), χf(t),
are typically chosen to match European swaption values in the respective currencies. Methods for doing this are well known from (singlecurrency) interest rate modelling literature and are covered elsewhere
(see, for example, Brigo & Mercurio, 2001).1 The correlation parameters ρij, i, j = d, f, S, are typically chosen either by historical estimation
or from occasionally observed prices of ‘quanto’ interest rate contracts
(payouts made in domestic currency but linked to rates in foreign currency). Of particular importance to PRDC swaps is how to calibrate
the volatility function of the forex rate γ(t, x). Options on the forex
rate are traded across a wide range of maturities and strikes; this is the
information to which γ(t, x) should be calibrated. For PRDC swaps
and most other cross-currency derivatives, it is impossible to pick a
particular strike, or a particular maturity, of an forex option ‘relevant’
for the security in question. Nor can cancellable/knock-out PRDC
swaps be decomposed into simple forex options. Hence, the volatility
function γ(t, x) needs to be calibrated to prices of all available forex
options across maturities and strikes.
■ Forward forex rate. The value of a call option on the forex rate
can be represented as a European-style option on the forward forex
rate F (T, T) under the domestic T-forward measure PT. Moreover,
F(t, T) is a martingale under this measure. Given the model (2),
there exists a Brownian motion dWF(t) such that (under PT):
dF (t , T )
= Λ (t , F (t , T ) D (t , T )) dWF (t )
F (t , T )
(5)
where Λ(t, x) = (a(t) + b(t)γ(t, x) + γ2(t, x))1/2, a(t), b(t) are deterministic functions of model parameters (see Piterbarg, 2005b) and D(t, T)
@ Pd(t, T)/Pf(t, T).
We note that if γ(t, x) is a function of time t only, γ(t, x) = γ(t),
then F(T, T) is lognormally distributed, forex options can be valued
using the Black formula, and the function γ(.) can be implied from
the at-the-money forex option prices easily. In the general case of
γ(t, x) depending on x, the term distribution of F(., T) is not easy
to identify. The diffusion coefficient of dF/F depends not only on F,
but also on D(t, T), another stochastic variable. In what follows we
reduce the complexity of the diffusion coefficient step by step, ultimately bringing it into a recognisable form.
■ Markovian representation for forward forex rate dynamics. The
first step in simplifying the dynamics of the forward forex rate is ‘closing out’ the stochastic differential equation (SDE) for F, by which we
mean writing the SDE in a form that contains F as the only stochastic
variable. The simplest approach to this is to replace D(t, T) in (5) with
its ‘along the forward’ deterministic approximation:
( )
( )
D t , T ≈ D0 t , T ,
where
( )
D0 t , T =
( )
Pf ( 0, t , T )
Pd 0, t , T
(6)
and Pi(s, t, T), i = d, f, are forward prices of corresponding zero-coupon discount bonds. This approach, in addition to being rather ad
hoc, also produces approximations of low accuracy. Instead, an autonomous representation of the forward forex rate process that is exact
for European-style options can be derived. It follows from Gyöngy
(1986) (and can also be derived from the ideas of Dupire, 1994, as in
~
~
Piterbarg, 2005b) that, if we define Λ(t, x) by Λ2(t, x) = ET0(Λ2(t, F (t,
T)D(t, T))|F(t, T) = x), then European-style option prices in the model
(5) exactly match the ones in the following model:
For given mean-reversion functions χd (t), χ f (t), the fit of volatility functions σd (t), σf (t) may fail if the
market volatilities are steeply downward sloping. Such problems can always be addressed by increasing
mean reversions
1
risk.net
piterbarg.indd 67
67
3/5/06 11:28:36
CUTTING EDGE. CROSS-CURRENCY EXOTICS
A Market-implied Black volatilities of forex options for
strikes from equation (12) (%)
Expiry
Vol 1
Vol 2
Vol 3
Vol 4
Vol 5
Vol 6
Vol 7
6m
11.41
10.49
9.66
9.02
8.72
8.66
8.68
1y
12.23
10.98
9.82
8.95
8.59
8.59
8.65
3y
12.94
11.35
9.89
8.78
8.34
8.36
8.46
5y
13.44
11.84
10.38
9.27
8.76
8.71
8.83
7y
14.29
12.68
11.23
10.12
9.52
9.37
9.43
10y
16.43
14.79
13.34
12.18
11.43
11.07
10.99
15y
20.93
19.13
17.56
16.27
15.29
14.65
14.29
20y
22.96
21.19
19.68
18.44
17.50
16.84
16.46
25y
23.97
22.31
20.92
19.80
18.95
18.37
18.02
30y
25.09
23.48
22.17
21.13
20.35
19.81
19.48
Volatility
Skew
6m
9.02
–200
1y
8.94
–180
3y
8.78
–110
5y
9.25
–60
7y
10.09
–30
10y
12.11
0
15y
16.03
30
20y
18.05
50
25y
19.32
63
30y
20.56
72
T
( ) = Λˆ t , F t ,T dW t
( ( )) F ( )
F (t ,T )
1
σF = 
T
(7)
(8)
( ) ( ) ( ) ( ))
β( t )−1

 D0 ( t , T ) 


x
γˆ ( t , x ) = ν ( t )  x
− 1 
 1 + (β ( t ) − 1) r ( t ) 


 F ( 0, T )  

L ( t ) 
( )
(
1/ 2
and r(t) is some deterministic function (see Piterbarg, 2005b). It is
worth noting that had we used the simplistic approximation (6), the
corresponding formula would simply be:
( ) ( )

( ) 
 D t,T
γˆ t , x = ν t  x 0

L t
( ) ()
68
piterbarg.indd 68
β t −1
2 Λ̂
2
( ( )) ( ) dt
(t, F (0,T ))
(9)
 F (0, T )

1 − δF
,K +
c (T , K ) = Pd (0, T ) cBlack 
F (0, T ) , σ F δ F , T  (10)
δF
 δF

where:
ˆ t , x = a t + b t γˆ t , x + γˆ 2 t , x
Λ
() ()
b t η t + 2 γˆ t , F 0, T η t
and w(t), η(t) are simple functions of model parameters (see Piterbarg,
2005b, for details). In particular, F(·, T) follows a standard displaceddiffusion SDE with the skew parameter δF. The value of a call option
on the forex rate with maturity T and strike K is equal to:
This result is very intuitive – the Markovian dynamics is defined by
the diffusion coefficient that is the expected value of the original diffusion coefficient conditioned on the underlying. We emphasise that the
result is exact for all derivatives with European-style payouts.
To use it in practice, however, the conditional expectations have to
be calculated or, at least, approximated. The necessary calculations
are performed in Piterbarg (2005b), where it is shown that to a good
approximation, the following SDE can be used instead:
dF t , T
()
δ F = 1 + ∫0 w t
Note: as calibrated to market-implied Black volatilities of forex options from table A by matching the
value and the slope of the forex volatility smile for each expiry
dF (t , T ) 
= Λ (t , F (t , T )) dWF (t )
F (t , T )
 (t , F (0, T )) (δ F (t , T ) + (1 − δ ) F (0, T )) dW (t )
dF (t , T ) = Λ
F
F
F
where:
B Market volatilities and skews (σ*n, δ*n), n = 1, ... , 10, of
the displaced-diffusion model (%)
Expiry
Clearly, accounting for the stochastic nature of D(t, T) introduced an
adjustment to the slope of the local volatility function ^
γ (t, x) around
the forward x = F(0, T).
The next step is to be able to effectively price European-style
options with the model (8), a one-dimensional diffusion with timeand space-dependent volatility function.
■ Skew averaging. The approach of ‘parameter averaging’ (see Piterbarg, 2005c and 2005d) is well suited for the next step. With this
approach, time-dependent parameters are replaced with ‘effective’,
time-constant ones, thus allowing us to relate model and market parameters directly without actually performing any option calculations.
Applying the method, we find that to value options on the forex rate
with maturity T, the forward forex rate can be approximated by the
following SDE:
T
∫0
ˆ 2 t , F 0, T dt 
Λ

( ( ))
1/ 2
(11)
where cBlack(F, K, σ, T) is the Black formula value for a call option with
forward F, strike K, volatility σ and time to maturity T.
This result fully resolves the problem of approximately pricing
options on the forex rate in the cross-currency model (2). For a given
expiry T and strike K, the pricing formula (10) is used with the ‘effective’ volatility of the forward forex rate σF defined by (11), and the
‘effective’ skew of the forward forex rate δF given by (9).
■ The algorithm. The first step of forex volatility calibration is to
fit the displaced-diffusion model (10) to forex options across strikes
separately for each maturity Tn, 0 = T0 < T1 < ... < TN. Then, from the
obtained set of market volatilities σ*n and market skew parameters δ*n,
n = 1, ... , N, the model parameters, the time-dependent functions
ν(t) and β(t) for t ∈ [0, TN] can be obtained by solving the equations
(11), (9) (assuming, for example, that ν(t) and β(t) are piecewise constant). The calculations can be organised into N sequential problems,
each one involving only a two-dimensional root search, with ν(Tn) and
β(Tn) found on step n and reused on consecutive steps. In practice, the
calibration is fast and robust. Fitting the model to, for example, forex
options for seven strikes and 10 maturities (as in the example in the
section below) takes about 0.1 seconds on a modern computer.
From (9), it follows that δF is a linear function of β(·); thus, the
equations for β(·) can always be solved. The fit of volatilities, however, may fail if the market volatilities do not increase fast enough,
just like for the ‘standard’ lognormal model. Intuitively, the market
volatility is a sum of volatilities of interest rate and forex components
of the forward forex rate. If the market volatility is too small compared with the interest rate volatility components (a situation possible especially for longer-dated maturities), then the forex spot volatility cannot be found. Such problems are usually solved by adjusting
one’s correlation assumptions.
Risk May 2006
3/5/06 11:28:41
Test results
To demonstrate the performance of the approximations we developed,
tests have been carried out for a standardised set of market data. The
interest rate curves in the domestic (yen) and foreign (dollar) economies are given by:
Pd (0, T ) = exp ( −0.02 × T ) ,
C Model volatilities and skews (νn, βn), n = 1, ... , N, as
calibrated to the market ones from table B (%)
Start period
Pf (0, T ) = exp ( −0.05 × T )
The volatility parameters for the interest rate evolution in both currencies are given by:
σ d (t ) ≡ 0.70%, χ d (t ) ≡ 0.0%, σ f (t ) ≡ 1.20%, χ f (t ) ≡ 5.0%
The correlation parameters are given by:
ρdf = 25.00%, ρdS = −15.00%, ρ fS = −15.00%
The initial spot forex rate (yen per dollar) is set at 105. For calibration,
we use the expiries T1, ... , T10 as in the first column of table A, and the
strikes Ki(Tn) of options as calculated by the formulas:
( )
(
)
K i Tn = F 0, Tn exp
(
0.1 × Tn1/ 2
× δi
)
where δ i = −1.5, −1.0, −0.5, 0.0, 0.5,1.0,1.5
Skew impact on PRDC swaps
A PRDC swap (see Sippel & Ohkoshi, 2002, and Jeffery, 2003, for extensive discussions) pays forex-linked coupons in exchange for floating-rate
payments. Let us define it more formally.2 Suppose a tenor structure:
0 < T1  <  < TN ,
τ n = Tn +1 − Tn
is given. A forex-linked, or PRDC swap, coupon for the period
[Tn, Tn + 1], n = 1, ... , N – 1, pays the amount τnCn(S(Tn)) (on a unit
notional in domestic currency) at time Tn, where:
Volatility
Skew
0m
6m
9.03
–200
6m
1y
8.87
–172
1y
3y
8.42
–115
3y
5y
8.99
–65
5y
7y
10.18
–50
7y
10y
13.30
–24
10y
15y
18.18
10
15y
20y
16.73
38
20y
25y
13.51
38
25y
30y
13.51
38
D Differences, in implied Black volatilities, between
values of forex options (%)
(12)
For each expiry and strike, an implied Black volatility of the corresponding forex option is given in table A. This set of parameters is
regarded as ‘the market’.
For each expiry Tn, n = 1, ... , N, we fit a displaced-diffusion model
with a constant volatility σ*n and a constant skew parameter δ*n, n = 1,
... , 10 (see section above for notations). The skews are fitted to match
the slopes of the market forex volatility smiles for each expiry, and
the volatilities are fitted to the at-the-money volatilities. The resulting parameters are summarised in table B.
The cross-currency model is calibrated to these parameters, as outlined in the section above. The resulting parameters ((νn, βn), n = 1, ... ,
N) are summarised in table C.
The errors of the approximations are presented in table D as the
difference of implied Black volatilities, calculated by the approximations above and by using the PDE method, for maturities and strikes
as defined previously. The fit is excellent, with errors less than 0.1%
for at-the-money options and less than 0.2% for almost all others.
Let us next consider how well the model fits the market as given
by the implied Black volatilities in table A. We calculate the values of
forex options in the model (2) (with parameters given in table C) using
the PDE method, and compare them with the market volatilities. The
differences are reported in figure 1 for selected maturities. Clearly, the
calibration algorithm works as intended, with an excellent fit to the
at-the-money options and to the slopes of the volatility smiles for each
expiry. The fit to individual forex options is not very good for some
points, which is more of a limitation of the model specification used
than the calibration algorithm. Still, the smiles produced by the model
are much closer to the market than the ones generated by the lognormal model. In particular, the fit, compared with the lognormal model,
is much better for low-strike (in-the-money call) options.
End period
Expiry
Error 1
Error 2
Error 3
Error 4
Error 5
Error 6
Error 7
6m
–0.26
–0.16
–0.12
–0.10
–0.07
–0.02
0.05
1y
–0.06
–0.06
–0.07
–0.06
–0.03
0.03
0.14
3y
0.14
0.07
0.00
–0.02
0.01
0.11
0.28
5y
0.17
0.11
0.04
0.00
0.03
0.12
0.30
7y
0.21
0.16
0.07
0.02
0.03
0.12
0.29
10y
0.19
0.19
0.11
0.04
0.03
0.08
0.20
15y
–0.01
0.08
0.07
0.01
–0.04
–0.08
–0.09
20y
–0.39
–0.17
–0.08
–0.05
–0.06
–0.08
–0.11
25y
–0.62
–0.31
–0.15
–0.06
–0.03
–0.02
–0.02
30y
–0.82
–0.42
–0.18
–0.06
0.02
–0.05
–0.06
Note: calculated using the PDE method and the approximation method from ‘Skew averaging’
section


 
S (Tn )
Cn ( S ) = min  max  g f
− g d , bl  , bu  , n = 1,…, N − 1
s

 

Quantities gf and gd are called the foreign and the domestic coupons,
and bl and bu are the floor and the cap on the payout. In the classical
structure, bl = 0 and bu = +∞, that is, the payout is floored at zero and
there is no cap. The scaling factor s is often called the initial forex rate.
All the parameters can vary from coupon to coupon, that is, depend
on n, n = 1, ... , N – 1.
An initial fixed-rate coupon is often paid to the investor (the
receiver of PRDC swap coupons). It is not included in the definition
above as its valuation is straightforward.
With the domestic currency being the yen, and the foreign one being
the dollar, S(·) is expressed in yen per dollar. The investor receives a
positive coupon as long as S(·) is high, that is, as long as the dollar is
strong (or strong enough). Because of the significant interest rate differential between the two currencies, the forward forex rate curve F(0,
t), t ≥ 0, is strongly downward sloping, predicting a significant weakening of the dollar. Thus, the party receiving the structured coupon
in a PRDC swap is essentially betting that the dollar is not going to
weaken (or not as much) as predicted by the forward forex curve, a bet
that many Japanese investors are comfortable to make.
The standard PRDC swap can be seen as a collection of simple
forex options, and as such does not require a sophisticated model to
price. The varieties that are most popular, however, are more comA large variety of PRDC swaps is available. We only present the basic structure, as relevant for our
analysis
2
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CUTTING EDGE. CROSS-CURRENCY EXOTICS
1 Market and model values of forex options, in implied Black volatilities, versus strikes, for a number of expiries
12
16
14
10
12
10
6
%
%
8
8
6
4
4
2
Black volatility, market, expiry = 6m
Black volatility, model, expiry = 6m
0
80
90
100
110
0
120
80
100
120
140
20
%
%
15
10
15
10
5
5
Black volatility, market, expiry = 15y
Black volatility, model, expiry = 15y
20
40
60
80
100
120
140
plicated and do require a model. One type, a cancellable PRDC
swap, gives the issuer (the payer of the PRDC swap coupons) the
right to cancel the swap on any of the dates T1, ... , T N – 1 (or a subset
thereof). The other popular type is a knock-out PRDC swap, stipulating that the swap knocks out (disappears) if the spot forex rate on
any of the dates T1, ... , T N – 1 exceeds a pre-agreed level. Both features
are designed to limit the downside for the issuer, and also allow the
investor to monetise the options to cancel/knock-out in the form of
receiving high fixed coupons over the initial (no-call) period [0, T1].
Assuming bu = +∞, bl = 0 (the most commonly used settings), the
PRDC swap coupon can be represented as a call option on the forex
rate:
(
)
Cn ( S ) = h max S (Tn ) − k , 0 , k =
gf
sg d
, h=
gf
s
The option notional h determines the overall level of coupon payment,
and the strike k determines the likelihood of the coupon paying a nonzero amount. The relationship of the strike to the forward forex rate
to Tn determines the leverage of the PRDC swap. If the strike k is low,
then the coupon has a relatively high chance of paying a non-zero
amount. The option notional in this case is, typically, low. This is a
low-leverage situation. If the strike is high relative to the forward forex
rate, the probability that the coupon will pay a non-zero amount is
low. The notional h, in this case, is typically higher. This is a highleverage situation.
In the analysis below, PRDC swaps of different leverage are used to
demonstrate the smile impact. We consider cancellable and knock-out
versions of three PRDC swaps, a low-leverage, a medium-leverage and
a high-leverage one. The three swaps share many features, namely:
piterbarg.indd 70
60
25
20
70
40
30
25
0
Black volatility, market, expiry = 7y
Black volatility, model, expiry = 7y
2
0
Black volatility, market, expiry = 25y
Black volatility, model, expiry = 25y
0
20
40
60
80
100
120
■ They start in one year (T1 = 1) and have a total maturity of 30 years
(TN = 30).
■ All pay an annual PRDC swap coupon (τn = 1, n = 1, ... , N – 1) and
receive the domestic (yen) Libor rate.
■ The floor is zero and there is no cap, bl = 0, bu = +∞.
■ All have a time-dependent schedule of s = sn. In particular:
sn = F (0, Tn ) ,
n = 1,…, N − 1
This choice simplifies the structure of coupons, clarifying certain
conclusions.
■ Cancellable variants give a Bermuda-style option to cancel the swap on
each of the dates T1, ... , TN – 1 to the payer of the PRDC swap coupons.
■ Knock-out variants are up-and-out forex-linked barriers. In particular, the swap disappears on the first date among T1, ... , TN – 1 on which
the forex rate S(Tn) exceeds a given barrier. Note that the barriers are
different for the three cases.
The domestic and foreign coupons are chosen to provide different
amounts of leverage, while keeping the total value of the underlying swap roughly the same for all three cases. Table E provides the
remaining details of the securities.
The values of securities are in percentage points of the notional.
Valuation results for two models are presented. One is the standard
lognormal three-factor model calibrated to the at-the-money forex
options of expiries from table A. The other model is the skew-calibrated one as described above. The values (to the payer of PRDC
swap coupons) of underlying PRDC swaps, cancellable PRDC swaps
and knock-out PRDC swaps are reported.
The value of cancellable and knock-out swaps increases with leverage. This is of course not surprising, as higher volatility in the value of
Risk May 2006
3/5/06 11:28:47
E Parameters of PRDCs and their values, in percentage
points of the notional, using the standard lognormal
model and the skew-calibrated model
Leverage
Low
Medium
High
4.50%
6.25%
9.00%
2 Value of a cancellable PRDC as a function of the forex
rate at T = 5 (in percentage points of the notional)
50
Trade details
Domestic coupon
2.25%
4.36%
8.10%
Barrier
110.00
120.00
130.00
Underlying
–8.66
–9.24
–9.35
Cancellable
13.61
17.13
23.16
Knock-out
4.16
8.08
14.12
Underlying
–10.67
–11.66
–10.86
Cancellable
11.90
14.62
20.37
Knock-out
1.52
2.89
6.32
Underlying
–2.01
–2.43
–1.52
Cancellable
–1.71
–2.51
–2.79
Knock-out
–2.64
–5.19
–7.80
PV, lognormal model
40
Cancellable value
Foreign coupon
30
20
10
Forex rate
0
PV, skew model
Diff, skew - lognormal
the underlying swap (resulting from higher leverage) increases the value
of the option to cancel it (and the knock-out option in this case).
PRDC swaps consist of (short) forex call options with low strikes
(‘low’ means ‘weak dollar’ for the analysis in this section). Introduction of the skew increases the implied volatility of low-strike options,
thus pushing the value of PRDC swaps down (for the issuer). The
same holds true for the cancellable swaps as well, with the (negative) impact uniformly increasing with the increased leverage. The
effect is quite substantial, comparable with typical profits booked
by the issuer. Hence, not accounting for the forex skew can easily
show a profit on a trade that was actually a loss. The conclusion is
not surprising: accounting for the forex skew, for example in the way
developed in this article, is absolutely critical for proper pricing and
risk-managing the PRDC swap book.
To understand the skew impact on cancellable PRDCs, let us look
at its value at an intermediate date as a function of the spot forex
0
25
50
75
100
125
150
–10
rate on that date. A sample payout is presented in figure 2. Clearly,
it is optimal to cancel the swap if the forex rate is high enough. The
payout is concave for S < forward, reflecting the negative convexity of short forex option positions, the PRDC swap coupons due to
the issuer. For S > forward, the payout is convex, reflecting the cancel option at higher strike. In particular, accounting for the skew
affects the value of a cancellable swap in two ways. First, the higher
volatility for lower strikes means that the ‘left’, concave side of the
payout is valued lower. The lower volatility for high strikes means
that the ‘right’, convex side of the payout is also valued lower. Hence,
the skew impact on cancellable PRDCs is compounded. The profile
of the cancellable PRDC is similar to a call spread, a payout that
is very sensitive to the skew of the volatility smile. With the model
developed in this article, the skew of the smile can be nicely controlled, thus allowing us to capture the main risk factors affecting these
types of security very well. ■
Vladimir Piterbarg is head of fixed-income quantitative research at Barclays
Capital in London. He would like to thank Jesper Andreasen and Leif Andersen
for their thoughtful comments, and anonymous referees who made suggestions that greatly improved the article. E-mail: vladimir.piterbarg@barcap.com
References
Andersen L and J Andreasen, 2000
Volatility skews and extensions of the
Libor market model
Applied Mathematical Finance 7, March,
pages 1–32
Andersen L and J Andreasen, 2002
Volatile volatilities
Risk December 2002, pages 163–168
Balland P, 2005
Stochastic volatility for hybrids
World Business Strategies workshop
on hybrids
Brigo D and F Mercurio, 2001
Interest-rate models – theory and
practice
Springer Verlag
Craig J and A Sneyd, 1988
An alternating-direction implicit
scheme for parabolic equations with
mixed derivatives
Computers and Mathematics with
Applications 16(4), pages 341–350
Dempster M and J Hutton, 1997
Numerical valuation of cross-currency
swaps and swaptions
In Mathematics of Derivative Securities,
edited by M Dempster and S Pliska,
pages 473–503, Cambridge University
Press
Dupire B, 1994
Pricing with a smile
Risk January 1994, pages 18–20
Gyöngy I, 1986
Mimicking the one-dimensional
marginal distributions of processes
having an Ito differential
Probability Theory and Related Fields
71, pages 501–516
Hagan P, D Kumar, A Lesniewski and
D Woodward, 2002
Managing smile risk
Wilmott Magazine, September, pages
84–108
Hull J and A White, 1994
Numerical procedures for implementing
term structure models I: Single-factor
models
Journal of Derivatives 2, pages 7–16
Jeffery C, 2003
The problem with power-reverse duals
Risk October 2003, pages 20–22
Piterbarg V, 2005a
Mixture of models: a simple recipe for a
... hangover?
Wilmott Magazine, January, pages
72–77
Piterbarg V, 2005b
A multi-currency model with FX
volatility skew
SSRN working paper
Piterbarg V, 2005c
Stochastic volatility model with timedependent skew
Applied Mathematical Finance 12(2),
June, pages 147–185
Piterbarg V, 2005d
Time to smile
Risk May 2005, pages 71–75
Schlogl E, 2002
A multicurrency extension of the
lognormal interest rate models
Finance and Stochastics 6(2), pages
173–196
Sippel J and S Ohkoshi, 2002
All power to PRDC notes
Risk October, pages S31–S33
risk.net
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