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Long-Term Options in Foreign Exchange and Interest Rate Markets
Long dated derivatives require a flexible modelling framework.
The econometrics challenge is to embed historical and cross-sectional
estimations into derivative calibration. The engineering challenge is to structure a
model agnostic pricing engine whose performance depends only on the model size
but not on the process specification. The mathematical and numerical challenge
is to understand and use the smoothing mechanisms behind diffusion equations.
We illustrate through examples an efficient framework of this sort based
on direct kernel manipulations and operator algebraic methods.
We find that fully explicit discretization schemes provide a robust,
low-noise numerical valuation method for fundamental solutions of
diffusion equations and their derivatives.
Path dependent options are associated to an operator algebra
and can be classified into Abelian and non-Abelian:
block-diagonalizations and moment methods apply to the first
and block-factorization to the latter. Direct kernel manipulations
also allow one to correlate lattice models by means of dynamic
conditioning across even hundreds of factors without incurring
into the curse of dimensionality. Thanks to the internal smoothing mechanisms,
calculations are best executed in single precision floating point arithmetics
and staggering performance can be achieved by invoking BLAS Level-3 routines
on massively parallel chipsets such as GPUs and the Cell BE.
Examples to be discussed include the swaption volatility cube calibration,
CMSs and CMS spreads, snowballs, PRDCs, FX linked range accruals and volatility derivatives.
The list is long but the model agnostic math and engineering is in common.
[slides]
[abelian processes]
[credit correlation baskets]
[kernel convergence estimates]
[callable swaps, snowballs and videogames]
[stochastic integrals and abelian processes]
Effective approximation of FX/EQ options for the hybrid models: Heston and correlated Gaussian interest rates
We derive an effective approximation for FX/EQ options
for the Heston model coupled with correlated Gaussian interest rates.
The main technical result is an option evaluation for correlated
Heston/Lognormal processes. Unlike exactly solvable (affine) zero correlation case,
considered by J. Andreasen, non-trivial correlations
destroy affine structure / exact solvability.
Using powerful technique of the Markovian Projection
we come up with effective approximation and present its numerical confirmation.
[slides]
Using Compiler-Engineering Algorithms for Building Payoff Languages
During the last years the complexity of financial products has increased dramatically
(more complicated and/or hybrids payoffs) and the available development time
of pricing models for these products has decreased considerably.
Therefore, flexible payoff languages have gained importance in recent years
and one has to start thinking about parsing these languages.
But building a compiler is an old discipline in computer sciences and powerful algorithms are provided.
In this talk we describe how some of these compiler engineering algorithms
can be used for building general payoff languages for interest rate and hybrids products.
The talk will focus on algorithmic aspects and IT implementation/design issues will only be briefly sketched.
We will introduce basic concepts of compiler engineering (context-free grammars, Chomsky Hierachy),
describe some parsing algorithms (top-down algorithms, bottom up algorithms, LL/LR Parser)
and compiler-generators (YACC, SPIRIT), and finally use these concepts to explain how
powerful payoff languages can be built to treat features like
Foreign Exchange Derivatives: Market Conventions and Smile Dynamics
The conventions to quote the FX smile and the reasons why that created problems in the past will be uncovered.
For the liquid G10 FX options markets the smile behaviour will
be characterized and models which aim at consistency with these assumptions will be exhibited.
[slides]
The Continuous-Time Lattice Method --- Option Pricing through Matrix Diagonalization
The Continuous-Time Lattice Method for the pricing of derivatives
can be applied if the underlying random process is a combination
of a diffusion and a jump process. Instead of approximating the
underlying process directly, the Markov generator of the process
is approximated on a lattice while the time variable stays continuous.
Using matrix diagonalization, the probability kernel of the underlying
random process is obtained and used for pricing.
This method was made popular in finance by Claudio Albanese.
We demonstrate the simplicity of this method by applying
it to the case of European vanilla options.
The underying process is a combination of the CEV and Variance-Gamma Model.
The implied volatility smile of this model is obtained and shown to exhibit
an asymmetric smile and a flattening of the smile for longer times to maturity.
[slides]
Accelerating the Calibration of Stochastic Volatility Models
This paper compares the performance of three methods for
pricing vanilla options in models with known characteristic function:
(1) Direct integration, (2) Fast Fourier Transform (FFT), (3) Fractional FFT.
The most important application of this comparison is the choice
of the fastest method for the calibration of stochastic volatility models,
e.g. Heston, Bates, Barndorff-Nielsen-Shephard models or Levy models
with stochastic time. We show that using additional cache technique
makes the calibration with the direct integration method
at least seven times faster than the calibration with the fractional FFT method.
Available at Frankfurt School:
http://www.frankfurt-school.de/dms/publications-cqf/CPQF_Arbeits6/CPQF_Arbeits6.pdf
[paper] [slides]
Options Pricing - From Theory to Practice
This session covers important aspects when valuing vanilla and
exotic equity related derivatives in a real world trading environment.
Requirements are addressed on the pricing models when structuring,
quoting and managing risk. Solutions are outlined on how to handle
specific equity related aspects such as discrete dividends
and local volatilities when using finite difference and Monte Carlo methods.
Agenda:
Effectiveness of CPPI Strategies under Discrete–Time Trading
The paper analyzes the effectiveness of the constant proportion portfolio
insurance (CPPI) method under trading restrictions. If the CPPI method
is applied in continuous time, the CPPI strategies provide a value above a
floor level unless the price dynamic of the risky asset permits jumps. The
risk of violating the floor protection is called gap risk. In practice, it is
caused by liquidity constraints and price jumps. Both can be modelled
in a setup where the price dynamic of the risky asset is described by a
continuous–time stochastic process but trading is restricted to discrete time.
We propose a discrete–time version of the continuous–time CPPI strategies
which satisfies three conditions. The resulting strategies are self–financing,
the asset exposure is non–negative and the value process converges. We
determine risk measures such as the shortfall probability and the expected
shortfall and discuss criteria which ensure that the gap risk does not increase
to a level which contradicts the original intention of portfolio insurance.
In addition, we introduce proportional transaction costs and analyze their
effects on the risk profile. This is joint work with
Sven Balder and Michael Brandl.
[paper] [slides]
Selected Applications of Optimization in Finance
In recent years optimization has gained considerable importance
in the area of financial mathematics.
During this talk we highlight several classes of
optimization problems appearing in applications and discuss
their numerical solution. In a first part of the talk
we show how modern optimization algorithms can be used
for the efficient calibration of financial market models.
In case the model does not allow to derive a semi-closed form solution,
we show that adjoint-based Monte Carlo methods can be employed
to successfully calibrate the model.
The second part of the talk discusses how optimization can
be applied to identify optimal hedging strategies,
with a particular focus on an uncertain
skew hedge for reverse barrier options.
[slides]
Optimization Software for Financial Mathematics
Information about available software to solve a large variety of
optimization problems is provided at plato.asu.edu/guide.html
while some of this software is evaluated at plato.asu.edu/bench.html.
Starting with these sources an overview will be given on codes
that are particularly useful for applications in mathematical
finance.
[slides]
Options Pricing - From Theory to Practice
This session covers important aspects when valuing vanilla and
exotic equity related derivatives in a real world trading environment.
Requirements are addressed on the pricing models when structuring,
quoting and managing risk. Solutions are outlined on how to handle
specific equity related aspects such as discrete dividends
and local volatilities when using finite difference and Monte Carlo methods.
Agenda:
High Performance Computing Techniques in Finance
We present a collection of techniques varying from the simple to the sophisticated.
We start by reviewing good coding practices. Then we illustrate that large benefits
are possible from reimplementation of core mathematical functions relevant to finance.
The next stage is to use hardware optimised libraries to
implement vectorised operations in time critical sections.
Finally we briefly review interpreted pay-off languages
and show how we may "compile without compiling".
[slides]
The British Option
We present a new put/call option where the buyer may exercise at any
time prior to maturity whereupon his payoff is the `best prediction'
of the European payoff under the hypothesis that the true drift of
the stock price equals a contract drift. Inherent in this is the
protection feature which is key to the British option. Should the
option holder believe the true drift of the stock price to be
unfavourable (based upon the observed price movements), he can
substitute the true drift with the contract drift and minimise his
losses. With the contract drift properly selected the British put
option becomes a more `buyer friendly' alternative to the American
put: when stock price movements are favourable, the buyer may
exercise rationally to very comparable gains; when price movements
are unfavourable he is afforded the unique protection described
above. Moreover, the British put option is always cheaper than the
American put. In the final part we present a brief review of optimal
prediction problems which preceded the development of the British
option.
This is a joint work with F. Samee (Manchester).
Option Pricing with No-Arbitrage Constraints
In the absence of arbitrage opportunities,
theory imposes the price of a call to be a
two times continuously differentiable,
decreasing and convex function of the strike price
with additional bounds for the first and second derivative.
Hence, any reliable call price function or state
price density - the compounded second derivative - should
satisfy these constraints. Unfortunately,
it is usually not unproblematic to extract
this function from market data, since further influences
like bid/ask spreads and illiquidity distort the shape of the curve.
Nevertheless, for several applications,
like the calibration of local volatility functions,
it is important to know the call price function.
In this talk a new method for the estimation of the call
price function is proposed. The approach is nonparametric,
which means that no explicitly given form is assumed.
The call price function is estimated in a two stage procedure.
First, an estimation for the function from the market data
is performed such that monotonicity and convexity constraints are satisfied.
In a second step this estimate is modified with subject
to the boundary conditions for the derivatives.
It can be shown that under certain conditions
the estimation converges asymptotically almost surely to the true
call price function and the finite sample behavior
of this procedure is demonstrated by an application to real data.
[slides]
The Law of the Minimal Price
The paper introduces a general market setting under which the Law of One Price
does no longer hold. Instead the Law of the Minimal Price will be derived,
which for a range of contingent claims provides lower prices than suggested
under the currently prevailing approach.
This new law only requires the existence of the numeraire portfolio,
which turns out to be the portfolio that maximizes expected logarithmic utility.
In several ways the numeraire portfolio cannot be outperformed by any nonnegative portfolio.
The new Law of the Minimal Price leads directly to the real world pricing formula,
which uses the numeraire portfolio as numeraire and the real world probability measure
as pricing measure when computing conditional expectations.
The pricing and hedging of extreme maturity bonds illustrates
that the price of a zero coupon bond, when obtained under the Law of the Minimal Price,
can be far less expensive than when calculated under the risk neutral approach.
[slides]
Auto-Static for the People: Risk-Minimizing Hedges of Barrier Options
We present a straightforward method for computing risk-minimizing
static hedge strategies under general asset dynamics.
Experimental investigations for barrier options show
that in a stochastic volatility model with jumps the resulting hedges
have superior performance to previous suggestions in the literature.
We also illustrate that the risk-minimizing static hedges work
in an infinite intensity Levy-driven model,
and that the performance of the hedges are robust with respect to model risk.
This is joint work with Johannes Siven from Lund University.
Live link:
http://www.math.ku.dk/~rolf/Siven/AutoStatic.pdf
[paper] [slides]
Dynamic Hedging of Variable Annuities – TwinStar: The AXA Way
Looking at trends in the European life insurance market,
the ability of insurers to generate at-tractive risk/reward
profiles for their clients will obviously continue to
predominantly deter-mine success in the retirement/savings market.
This stresses the importance of guarantees and other innovative,
flexible product features. At the same time however,
the importance of as-set/liability management (ALM) techniques
has increased substantially due to lower risk capi-tal levels
and the changing supervisory regime.
Under these circumstances, profitable growth with innovative products
will highly depend on the question which parts of the value
creation chain can be dealt with internally.
This may lead to a (re)shift of focus from macro to micro ALM
like when hedging embedded guaran-tees dynamically with derivatives.
Using AXA's TwinStar – the first variable annuity product
in the European market – as an ex-ample,
basic techniques for this shall be outlined.
[slides]
High Performance Computing Techniques in Finance
We present a collection of techniques varying from the simple to the sophisticated.
We start by reviewing good coding practices. Then we illustrate that large benefits
are possible from reimplementation of core mathematical functions relevant to finance.
The next stage is to use hardware optimised libraries to
implement vectorised operations in time critical sections.
Finally we briefly review interpreted pay-off languages
and show how we may "compile without compiling".
[slides]
Generalized Swap Market Model and the Valuation of Interest Rate Derivatives
In this paper we will establish a generalized Swap Market Model (GSMM)
by unifying the stochastic processes of swap rates with constant tenors under
a single swap measure. GSMM is a natural extension of Libor Market Model (LMM)
for swap rates, and LMM can be considered as a special case of GSMM
since Libor is a special swap rate with the constant tenor of one period.
GSMM can be applied for pricing and hedging any interest rate derivatives,
and is suited especially for CMS and swap rate products.
There are a number of advantages of GSMM: (1). GSMM models swap rates directly,
and therefore achieves the best match between products and model. (2).
GSMM can be calibrated to the term structure of swaption volatilities
easily and quickly. (3) There is no translation of risk sensitivities
with respect to swap rates within GSMM. In contrast, risk sensitives
such as Vega for swap rates can not be derived directly,
and must be translated in an inefficient, inaccurate and
non-transparent manner in the most existing interest rate models.
(4) All smile modelings for LMM can be taken over for GSMM since
GSMM and LMM share an almost identical mathematical structure.
(5) GSMM avoids the inconsistency of the market conventions
in cap and swaptions markets. Accompanied by these favourite features,
GSMM should be a promising interest rate model for pricing and
hedging most traded swap rate structures in financial market.
Available at SSRN: http://ssrn.com/abstract=1028710
[slides] [paper]
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