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Cubature methods, a powerful alternative to Monte Carlo due to Kusuoka [Adv. Math. Econ. 6, 69–83, 2004] and Lyons–Victoir [Proc. R. Soc. Lond. Ser. A 460, 169–198, 2004], involve the solution to numerous auxiliary ordinary differential equations. With focus on the Ninomiya-Victoir algorithm [Appl. Math. Fin. 15, 107–121, 2008], which corresponds to a concrete level 5 cubature method, we study some parametric diffusion models motivated from financial applications, and exhibit structural conditions under which all involved ODEs can be solved explicitly and efficiently. We then enlarge the class of models for which this technique applies, by introducing a (model-dependent) variation of the Ninomiya-Victoir method. Our method remains easy to implement; numerical examples illustrate the savings in computation time.
We consider the problem of numerical approximation for forward-backward stochastic
differential equations with drivers of quadratic growth (qgFBSDE). To illustrate the significance
of qgFBSDE, we discuss a problem of cross hedging of an insurance related financial
derivative using correlated assets. For the convergence of numerical approximation schemes for
such systems of stochastic equations, path regularity of the solution processes is instrumental.
We present a method based on the truncation of the driver, and explicitly exhibit error estimates
as functions of the truncation height. We discuss a reduction method to FBSDE with globally
Lipschitz continuous drivers, by using the Cole-Hopf exponential transformation. We finally
illustrate our numerical approximation methods by giving simulations for prices and optimal
hedges of simple insurance derivatives.
A robust implementation of a Dupire type local volatility model is an important issue for every option trading floor. In the present note we provide new analytic insights into the asymptotic behavior of local volatility in the wings. We present a general approximation formula and specialize it to the Heston model, showing that local variance is linear in the wings. This further justifies the choice of certain local volatility parametrizations.
Primal-dual linear Monte Carlo algorithm for multiple stopping - An application to flexible caps
(2012)
In this paper we consider the valuation of Bermudan callable derivatives with
multiple exercise rights. We present in this context a new primal-dual linear
Monte Carlo algorithm that allows for ecient simulation of lower and upper price
bounds without using nested simulations (hence the terminology). The algorithm
is essentially an extension of a primal{dual Monte Carlo algorithm for standard
Bermudan options proposed in Schoenmakers et al. (2011), to the case of multiple
exercise rights. In particular, the algorithm constructs upwardly a system of dual
martingales to be plugged into the dual representation of Schoenmakers (2010).
At each level the respective martingale is constructed via a backward regression
procedure starting at the last exercise date. The thus constructed martingales are
nally used to compute an upper price bound. At the same time, the algorithm
also provides approximate continuation functions which may be used to construct
a price lower bound. The algorithm is applied to the pricing of
exible caps
in a Hull and White (1990) model setup. The simple model choice allows for
comparison of the computed price bounds with the exact price which is obtained
by means of a trinomial tree implementation. As a result, we obtain tight price
bounds for the considered application. Moreover, the algorithm is generically
designed for multi-dimensional problems and is tractable to implement.
We consider backward stochastic differential equations with drivers of quadratic growth (qgBSDE). We prove several statements concerning path regularity and stochastic smooth-
ness of the solution processes of the qgBSDE, in particular we prove an extension of Zhang's path regularity theorem to the quadratic growth setting. We give explicit convergence rates for the difference between the solution of a qgBSDE and its truncation, filling an important gap in numerics for qgBSDE. We give an alternative proof of second order Malliavin differentiability for BSDE with drivers that are Lipschitz continuous (and differentiable), and
then derive the same result for qgBSDE.
Optimal dual martingales, their analysis and application to new algorithms for Bermudan products
(2012)
In this paper we introduce and study the concept of optimal and surely
optimal dual martingales in the context of dual valuation of Bermudan
options, and outline the development of new algorithms in this context.
We provide a characterization theorem, a theorem which gives conditions
for a martingale to be surely optimal, and a stability theorem concerning martingales which are near to be surely optimal in a sense. Guided
by these results we develop a framework of backward algorithms for constructing such a martingale. In turn this martingale may then be utilized
for computing an upper bound of the Bermudan product. The methodology is pure dual in the sense that it doesn't require certain (input)
approximations to the Snell envelope.
In an Ito-Levy environment we outline a particular regression based
backward algorithm which allows for computing dual upper bounds with-
out nested Monte Carlo simulation. Moreover, as a by-product this algorithm also provides approximations to the continuation values of the
product, which in turn determine a stopping policy. Hence, we may obtain lower bounds at the same time.
In a first numerical study we demonstrate a backward dual regression algorithm in a Wiener environment that is easy to implement and
is regarding accuracy comparable with the method of Belomestny et. al.
(2009).
Optimal dual martingales, their analysis and application to new algorithms for Bermudan products
(2012)
In this paper we introduce and study the concept of optimal and surely
optimal dual martingales in the context of dual valuation of Bermudan
options, and outline the development of new algorithms in this context.
We provide a characterization theorem, a theorem which gives conditions
for a martingale to be surely optimal, and a stability theorem concerning martingales which are near to be surely optimal in a sense. Guided
by these results we develop a framework of backward algorithms for constructing such a martingale. In turn this martingale may then be utilized
for computing an upper bound of the Bermudan product. The methodology is pure dual in the sense that it doesn’t require certain (input)
approximations to the Snell envelope.
In an Ito-Levy environment we outline a particular regression based
backward algorithm which allows for computing dual upper bounds without nested Monte Carlo simulation. Moreover, as a by-product this algorithm also provides approximations to the continuation values of the
product, which in turn determine a stopping policy. Hence, we may obtain lower bounds at the same time.
In a first numerical study we demonstrate a backward dual regression algorithm in a Wiener environment that is easy to implement and
is regarding accuracy comparable with the method of Belomestny et. al.
(2009).
In this article we propose a novel approach to reduce the computational complexity
of the dual method for pricing American options. We consider a sequence of
martingales that converges to a given target martingale and decompose the original
dual representation into a sum of representations that correspond to dierent levels
of approximation to the target martingale. By next replacing in each representation
true conditional expectations with their Monte Carlo estimates, we arrive at what
one may call a multilevel dual Monte Carlo algorithm. The analysis of this algorithm
reveals that the computational complexity of getting the corresponding target upper
bound, due to the target martingale, can be signicantly reduced. In particular, it
turns out that using our new approach, we may construct a multilevel version of the
well-known nested Monte Carlo algorithm of Andersen and Broadie (2004) that is,
regarding complexity, virtually equivalent to a non-nested algorithm. The performance
of this multilevel algorithm is illustrated by a numerical example.
We study a nonlinear operator defined via minimal supersolutions of backward stochastic differential equations with generators that are monotone in y, convex in z, jointly lower semicontinuous, and bounded below by an affine function of the control variable. We show existence, uniqueness, monotone convergence, Fatou’s Lemma and lower semicontinuity of this functional. We provide a comparison principle for the underlying minimal supersolutions of BSDEs, which we illustrate by maximizing expected exponential utility.
We study minimal supersolutions of backward stochastic differential equations. We show the existence and uniqueness of the minimal supersolution, if the generator is jointly lower semicontinuous, bounded from below by an affine function of the control variable, and satisfies a specific normalization property. Semimartingale convergence is used to establish the main result.
Density expansions for hypoelliptic diffusions (X1^,...,X^d) are revisited. In particular, we are interested in density expansions of the projection (X^1_T,...,X^l_T) at time $T>0$, with $l \le d$. Global conditions are found which replace the well-known ”not-in-cutlocus” condition known from heat-kernel asymptotics; cf. G. Ben Arous (88). Our small noise expansion allows for a ”second order” exponential factor. Applications include tail and implied volatility asymptotics in some correlated stochastic volatility models; in particular, we solve a problem left open by A. Gulisashvili and E.M. Stein (2009).
We consider a dynamical system described by the differential equation $\dot{Y}_t = -U^'(Y_t)$
with a unique stable point at the origin. We perturb the system by L\'evy noise of
intensity $\varepsilon$, to obtain the stochastic differential equation $dX^\varepsilon_t = -U^'(X^\varepsilon_{t-})dt + \varepsilon dL_t}.
The process $L$ is a symmetric L\'evy process whose jump measure $\nu$ has exponentially
light tails, $\nu([u, \infty))\sim exp(-u^\alpha), \alpha > 0, u \to\infty$. We study the first exit problem for
the trajectories of the solutions of the stochastic differential equation from the interval
$[-1, 1]$. In the small noise limit $\varepsilon\to 0$ we determine the law and the mean value of the
first exit time, to discover an intriguing phase transition at the critical index $\alpha = 1$.
With an emphasis on generators with quadratic growth in the control variable we consider
measure solutions of BSDE, a solution concept corresponding to the notion of risk neutral
measure in mathematical finance. In terms of measure solutions, solving a BSDE reduces
to martingale representation with respect to an underlying filtration. Measure solutions
related to measures equivalent to the historical one provide classical solutions. We derive
the existence of measure solutions in scenarios in which the generating functions are just
continuous, of at most linear growth in the control variable (corresponding to generators of
at most quadratic growth in the usual sense), and with a random bound in the time parameter
whose stochastic integral is a BMO martingale. Our main tools include a stability property
of sequences of measure solutions, for which a limiting solution is obtained by means of the
weak convergence of measures.
We provide results on the existence and uniqueness of equilibrium in dynamically incomplete financial markets in discrete time. Our framework allows for heterogeneous agents, unspanned random endowments and convex trading constraints. In the special case where all agents have preferences of the same type and all random endowments are replicable by trading in the financial market we show that a one-fund theorem holds and give an explicit expression for the equilibrium pricing kernel. If the underlying noise is generated by finitely many Bernoulli random walks, the equilibrium dynamics can be described by a system of coupled backward stochastic difference equations, which in the continuous-time limit becomes a multi-dimensional backward stochastic differential equation. If the market is complete in equilibrium, the system of equations decouples, but if not, one needs to keep track of the prices and continuation values of all agents to solve it. As an example we simulate option prices in the presence of stochastic volatility, demand pressure and short-selling constraints.
The LIBOR market model is very popular for pricing inter-
est rate derivatives, but is known to have several pitfalls. In addition, if
the model is driven by a jump process, then the complexity of the drift
term is growing exponentially fast (as a function of the tenor length). In
this work, we consider a Levy-driven LIBOR model and aim at developing accurate and efficient log-Levy approximations for the dynamics of
the rates. The approximations are based on truncation of the drift term
and Picard approximation of suitable processes. Numerical experiments
for FRAs, caps and swaptions show that the approximations perform
very well. In addition, we also consider the log-Levy approximation of
annuities, which offers good approximations for high volatility regimes.
In this paper, we study the dual representation for generalized multiple stopping problems,
hence the pricing problem of general multiple exercise options. We derive a dual representation which allows for cashflows which are subject to volume constraints modeled by
integer valued adapted processes and refraction periods modeled by stopping times. As
such, this extends the works by Schoenmakers (2010), Bender (2011a), Bender (2011b),
Aleksandrov and Hambly (2010), and Meinshausen and Hambly (2004) on multiple exercise
options, which either take into consideration a refraction period or volume constraints, but
not both simultaneously. We also allow more flexible cashflow structures than the additive
structure in the above references. For example some exponential utility problems are covered
by our setting. We supplement the theoretical results with an explicit Monte Carlo algorithm
for constructing confidence intervals for the price of multiple exercise options and exemplify
it by a numerical study on the pricing of a swing option in an electricity market.
We consider a class of generalized capital asset pricing models in continuous time with a finite number of agents and tradable securities. The securities may not be sufficient to span all sources of uncertainty. If the agents have exponential utility functions and the individual endowments are spanned by the securities, an equilibrium exists and the agents’ optimal trading strategies are constant. Affine processes, and the theory of information-based asset pricing are used to model the endogenous asset price dynamics and the terminal payoff. The derived semi-explicit pricing formulae are applied to numerically analyze the impact of the agents’ risk aversion on the implied volatility of simultaneously-traded European-style options.
In the paradigm of VON N EUMANN AND M ORGENSTERN, a representation of affine pref-
erences in terms of an expected utility can be obtained under the assumption of weak continu-
ity. Since the weak topology is coarse, this requirement is a priori far from being negligible.
In this work, we replace the assumption of weak continuity by monotonicity. More precisely,
on the space of lotteries on an interval of the real line, it is shown that any affine preference
order which is monotone with respect to the first stochastic order admits a representation in
terms of an expected utility for some nondecreasing utility function. As a consequence, any
affine preference order on the subset of lotteries with compact support, which is monotone
with respect to the second stochastic order, can be represented in terms of an expected util-
ity for some nondecreasing concave utility function. We also provide such representations
for affine preference orders on the subset of those lotteries which fulfill some integrability
conditions. The subtleties of the weak topology are illustrated by some examples.
We develop a model for the dynamic evolution of default-free and defaultable interest rates in a LIBOR framework. Utilizing the class of affine processes, this model produces positive LIBOR rates and spreads, while the dynamics are analytically tractable under defaultable forward measures. This leads to explicit formulas for CDS spreads, while semi-analytical formulas are derived for other credit derivatives. Finally, we give an application to counterparty risk.
We solve Skorokhod's embedding problem for Brownian motion with linear drift $(W_t+ \kappa t)_{t\geq 0}$ by means of techniques of stochastic control theory. The search for a stopping time
$T$ such that the law of $W_T + \kappa T$ coincides with a prescribed law $\mu$ possessing the first
moment is based on solutions of backward stochastic differential equations of quadratic
type. This new approach generalizes an approach by Bass [Bas] of the classical version of
Skorokhod's embedding problem using martingale representation techniques.