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1
Volatility Skews and Extensions of the Libor Market Model
1
Leif Andersen and Jesper Andreasen
General Re Financial Products,630 Fifth Avenue, Suite 450 New York, NY 10111First Version: August 12, 1997This Version: August 27, 1998
Abstract
This paper considers extensions of the
Libor
market model
(Brace
et al
(1997), Jamshidian(1997), Miltersen
et al
(1997)) to markets with volatility skews in observable option prices.We expand the family of forward rate processes to include diffusions with non-linear forwardrate dependence and discuss efficient techniques for calibration to quoted prices of caps andswaptions. Special emphasis is put on generalized CEV processes for which closed-formexpressions for cap and swaption prices are derived. We also discuss modifications of theCEV process which exhibit more appealing growth and boundary characteristics. The proposed models are investigated numerically through Crank-Nicholson finite differenceschemes and Monte Carlo simulations.
1. Introduction
In a significant new line of research, the recent papers by Brace
et al
(1997), Jamshidian (1997),and Miltersen
et al
(1997) introduce a novel approach to arbitrage-free term structure modeling.Rather than working with the continuously compounded instantaneous forward rates as in Heath
et al
(1992), or the continuously compounded spot interest rates as in Vasicek (1977) and Cox
et al
(1985), these papers take discretely compounded (Libor) forward rates as the model primitives.Unlike continuously compounded forward rates, log-normally diffused discrete forward rates turnout to be non-explosive and, significantly, allow for pricing of Libor caplets by the “marketconvention” Black (1976) formula. The log-normal models advocated by Brace
et al
(1997),Jamshidian (1997), and Miltersen
et al
(1997) are therefore often termed
Libor market models
.
1
The authors wish to thank Steven Shreve, Paul Glasserman, Wes Petersen, and Jakob Sidenius for insights anddiscussions.
2
While the Libor market models do not allow for usage of the Black (1976) formula in the pricing of swaptions, Brace
et al
(1997) derive good closed-form approximations for swaption prices under the log-normal market model assumptions. Availability of closed-form pricingformulas for both caps and swaptions enables efficient calibration of the model to market prices, akey feature of the model in terms of its usefulness in practical applications.The basic premise of the Libor market model -- log-normally distributed Libor rates -- is,however, increasingly being violated in many important cap and swaption markets. In particular,implied Black (1976) volatilities of caplet and swaption prices often tend to be decreasingfunctions of the strike and coupon, respectively, indicating a fat left tail of the empirical forwardrate distributions relative to log-normality. This so-called
volatility skew
is currently most pronounced in the Japanese Libor market, but also exists in US and German markets, amongothers. The presence of the volatility skew motivates the formulation of models where thediffusion coefficients of the discrete forward rates are non-linear functions of the rates themselves.In this paper we describe a general class of such models, which we will term
extended market models
. The models focused on here are characterized by a forward rate diffusion term that is
separable
, in the sense that it can be described as a product of a general time- and maturity-dependent function and a time-homogeneous non-linear function of the forward rate.The separable form of the diffusion coefficient is shown to be tractable and allows for quick calibration to caplets by numerical solution of one-dimensional forward or backward partialdifferential equations (PDEs). For this we suggest an efficient numerical routine based on adeterministic time-change and the Crank-Nicolson finite-difference scheme. Alternatively, for thecase where the forward-dependence of the diffusion term can be described by a power function,also termed the
CEV
(Constant Elasticity of Variance) model, we derive closed-form solutions for caplet prices. These results essentially extend the analysis of Schroder (1989) to the time-inhomogeneous case.As we will show, the CEV model is about as tractable as the log-normal market model butcan provide a much closer fit to observed caplet prices. To motivate our studies of the CEVmodel, below we show implied Black (1976) volatilities of CEV model caplet prices as functionsof strike plotted against bid and ask implied caplet volatilities from the Japanese Libor market(provided by the GRFP interest rate option desk, May 1998). We have included prices for 2- and9-year caplets; the CEV power (to be defined later) of the volatility coefficient is set to 0.6 for both maturities.
3
Market and CEV caplet prices in Japanese Libor market, May 1998Figure 1
Though closed-form caplet prices and a good market fit makes the CEV model attractiveit also exhibits certain technical irregularities. These can be circumvented, however, by theintroduction of a 'regularized' version of the CEV process, here named the
LCEV
(
L
imited CEV)model. We show that the CEV closed-form caplet prices can be seen as a limiting case of those produced by the LCEV model. By numerical examples we illustrate that the CEV formulas can beused as very accurate approximations of caplet prices under the LCEV process.For swaptions, the market model is less tractable than is the case for caps and floors. Bymaking certain simplifying assumptions, however, we demonstrate that swaptions can be treatedin exactly the same way as caplets. In particular, we are able to construct highly accurate closed-form approximations for swaption prices in the CEV market model. Our analysis is based on theconcept of forward swap measures (see Jamshidian (1997)) which simplifies the development of closed-form approximations significantly compared to the approach taken in Brace
et al
(1997).In the final part of the paper, we consider schemes to implement the proposed framework in a Monte Carlo setting. Monte Carlo simulations are then used to examine some of our resultsthrough numerical examples. Particular emphasis is put on tests of the swaption approximationsand on quantifying discretization biases.The rest of this paper is organized as follows. In Section 2 we provide notation andintroduce the probability measures and stochastic processes necessary for later work. In Section3, we narrow the discussion to the class of separable forward rate processes. After provingcertain existence and uniqueness results, we describe a technique of deterministic time-change that
20%30%40%50%60%70%80%0.50%1.00%1.50%2.00%2.50%3.00%3.50%4.00%4.50%5.00%5.50%
Strike
I m p l i e d V o l a t i l i t y
2yr Offer2yr Mid2yr CEV2yr Bid9yr Offer9yr CEV9yr Mid9yr Bid
4
proves useful for this class of models. The section also introduces the CEV process and derives itstransition density. In Section 4 we consider the PDEs for pricing of caplets and derive closed-form formulas for caplet prices in the CEV model. We also introduce the LCEV model andconsider the convergence of the LCEV to the CEV model. Section 5 discusses the pricing of swaptions using closed-form approximations, and Section 6 is devoted to Monte-Carloimplementation of the extended market models and various numerical tests. Finally, Section 7contains our conclusions. For clarity, all significant proofs are deferred to an appendix.
2. Basic Setup
Consider an increasing maturity structure 0
011
= < < <
+
TTT
K
... and define a right-continuous
2
mapping function
nt
()
by
TtT
ntnt
()()
−
≤ <
1
.While we do not put any restrictions on the maturity structure other than it being increasing, in practice we would often use a nearly equidistant spacing between points (say 3 or 6 calendar months) to match conventions used in swap and futures markets. With
PtT
(,)
denoting the time
t
price of a zero-coupon bond maturing at time
T
, we define discrete forward rates on the maturitystructure as follows:
Ft PtT PtT
k k k k
()(,)(,)
≡ −
F H GI K J
+
11
1
δ
,
δ
kkk
TT
= −
+
1
,or
PtTPtTFt
kntjj jnt k
(,)(,)(())
()()
= +
−=−
∏
1
11
δ
.For this definition to be meaningful, we must require that
tT
k
≤
and
kK
≤
. For brevity, we willomit such obvious restrictions on time and indices in most of the equations that follow.The discrete forward rates constitute our model primitives and collectively determine thestate and evolution of interest rates. To state our assumptions about the stochastic processesdriving the forward rates, we first fix our probability measure to be the
T
k
+
1
forward measure
Q
k
+
1
, i.e. the equivalent probability measure induced by using the
T
k
+
1
-maturity zero-coupon
2
While some authors define
n
(
t
) to be left-continuous, we find our definition more convenient, particularly for discrete-time numerical work. In particular, our definition ensures that
n
(
t
) does not jump when we move forwardfrom a date that coincides with a point in the maturity structure.

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