2011-14

Seminars in the Summer Term of 2013/14

16/5 Pavol Krasnovsky
30/5 Dejan Siraj (Mirror and Synchronous Couplings of Geometric Brownian Motions)
13/6 Jaka Gogala
20/6 Kamil Kladviko (Prague)

Seminars in Spring Term of 2013/2014

28th February: Harald Oberhauser (Oxford) SKOROKHOD EMBEDDINGS, NON-LINEAR PDES AND INTEGRAL EQUATIONS A classic problem in probability theory is the Skorokhod embedding problem. The question of how to construct solutions that are intuitive, computable and in some sense extremal has recently received more interest due to applications in model-independent pricing in finance. I will speak about recent work that is inspired by classic results due to Chacon, Root, Rost and many others. If time permits I will discuss also some connections with rough paths.
7th March: Yukihiro Tsuzuki (Tokyo) NO ARBITRAGE BOUNDS ON TWO ONE-TOUCH OPTIONS This talk investigates pricing bounds of two one-touch options with the same maturity but different barrier levels, where the pricing bound is a range within which a one-touch option can take a price when a price of another one-touch option is given. The upper bound or the lower bound is respectively a cost of super-replicating portfolio and sub-replicating portfolio, which consists of call options, put options, digital options and a one-touch option. We assume that the underlying process is a continuous martingale and do not postulate a model.
14th March: Beatrice Acciaio (LSE) ARBITRAGE OF THE FIRST KIND AND FILTRATION ENLARGEMENTS IN SEMIMARTINGALE FINANCIAL MODELS I will discuss the stability of the No Arbitrage of the First Kind (NA1) (or, equivalently, No Unbounded Profit with Bounded Risk) condition, in a general semimartingale financial model, both under initial and under progressive filtration enlargements. In both cases, I will provide a simple and general condition which is sufficient to ensure this stability for any fixed semimartingale model. Furthermore, I will give a characterization of the NA1 stability for all semimartingale models. (This talk is based on a joint work with C. Fontana and K. Kardaras.)

Seminars are at 2pm in C1.06

31st January 2014

Umut Cetin

Friday 15th November

15:00 Luciano Campi (LSE)

Utility indifference valuation for non-smooth payoffs with an application to power derivatives

Abstract: We consider the problem of exponential utility indifference valuation under the simplified framework where traded and nontraded assets are uncorrelated but where the claim to be priced possibly depends on both. Traded asset prices follow a multivariate Black and Scholes model, while nontraded asset prices evolve as generalized Ornstein-Uhlenbeck processes. We provide a BSDE characterization of the utility indifference price (UIP) for a large class of non-smooth, possibly unbounded, payoffs depending simultaneously on both classes of assets. Focusing then on European claims and using the Gaussian structure of the model allows us to employ some BSDE techniques (in particular, a Malliavin-type representation theorem due to Ma and Zhang (2002)) to prove the regularity of $Z$ and to characterize the UIP for possibly discontinuous European payoffs as a viscosity solution of a suitable PDE with continuous space derivatives. The optimal hedging strategy is also identified essentially as the delta hedging strategy corresponding to the UIP. Since there are no closed-form formulas in general, we also obtain asymptotic expansions for prices and hedging strategies when the risk aversion parameter is small. Finally, our results are applied to pricing and hedging power derivatives in various structural models for energy markets. This is based on a joint work with G. Benedetti.

Term 3 of 2012/2013

Thursday 25th April 2013

14:00 Philipp Strack (Bonn)

Optimal stopping with Private Information.

Term 2 of 2012/2013

Monday 25th February

14:00 Alex Mijatovic (Imperial)

Monday 4 February

14:00 Gechun Liang (Oxford)

Stochastic Control Representations for Penalized Backward Stochastic Differential Equations. Abstract: In this talk, We show that both reflected BSDE and its associated penalized BSDE admit both optimal stopping representation and optimal control representation. We also show that both multidimensional reflected BSDE and its associated multidimensional penalized BSDE admit optimal switching representation. The corresponding optimal stopping problems for penalized BSDE have the feature that one is only allowed to stop at Poisson arrival times.

Term 1 of 2012/2013

Thursday 15 November, Room C1.06

14:00 Igor V. Evstigneev (Manchester)

Strategic Behaviour and Evolution in Markets

The idea of this direction of work is to apply evolutionary dynamics (mutation and selection) to the analysis of the long-run performance of financial trading strategies. A stock market is understood as a heterogeneous population of frequently interacting investment strategies (portfolio rules) in competition for market capital. The aim of the work is to build a "Darwinian theory" of portfolio selection. To this end dynamic equilibrium models are developed combining ideas of evolutionary game theory and strategic games. The results obtained make it possible to identify an asymptotically unique strategy possessing the property of evolutionary stability, i.e. guaranteeing unconditional survival in the market selection process. This is a joint work with Rabah Amir, Thorsten Hens and Klaus R. Schenk-Hoppé.

Thursday 8 November

Room C1.06

14:00 Mathias Beiglböck (Vienna)

Optimal Transport, Robust Pricing, and Trajectorial Inequalities

Robust pricing of an exotic derivative with payoff $\Phi$ can be viewed as the task of estimating its expectation $E_Q \Phi$ with respect to a martingale measure $Q$ satisfying marginal constraints. It has proven fruitful to relate this to the theory of Monge-Kantorovich optimal transport. For instance, the duality theorem from optimal transport leads to new super-replication results. Optimality criteria from the theory of mass transport can be translated to the martingale setup and allow to characterize minimizing/maximizing models in the robust pricing problem. Moreover, the dual viewpoint provides new insights to the classical inequalities of Doob and Burkholder-Davis-Gundy.

Thursday 18 October, 2012

Room C1.06

14.00 Teemu Pennanen (King's College London)

Indifference pricing in illiquid markets

This paper studies portfolio optimization and indifference pricing in markets where illiquidity may affect the transfer of wealth over time and between investment classes. We extend well-known results on arbitrage bounds, attainable claims and duality to illiquid markets and general swap contracts where both claims and premiums may have multiple pay- out dates. In addition to classical frictionless markets and markets with transaction costs, our model covers nonlinear illiquidity effects that arise in limit order markets.

Monday 18 June, 2012

Room A1.01

14.00 : Kais Hamza (Monash University)

Mimicking selfsimilar processes

We construct a family of selfsimilar Markov martingales with given marginal distributions. This construction uses the selfsimilar and Markov properties of a reference process to produce a family of Markov processes that possess the same marginal distributions as the original process. The resulting processes are also selfsimilar with the same exponent as the original process. They can be chosen to be martingales under certain conditions. In this paper, we present two approaches to this construction, the transition-randomising approach and the time-change approach. We then compute the infinitesimal generators and obtain some path properties of the resulting processes. We also give some examples, including continuous Gaussian martingales, martingales of the squared Bessel process, stable L\'evy processes as well as an example of an artificial process having the marginals of $t^kV$ for some symmetric random variable $V$. We conclude by showing how to mimic a certain family of Brownian martingales and also extend the construction to non-Markovian continuous martingales.

Term 3 of 2011/2012

Wednesday 6 June, 2012 in room A1.01.

14.00 : Christoph Reisinger (University of Oxford)

Robust parameter estimation and valuation of financial derivatives under local and uncertain volatility

In this talk, we aim to combine two strands of research. First, we consider the calibration of a local volatility function in a Bayesian framework, and show how the posterior distribution can be used to estimate prices for exotic options. We work in great detail through numerical examples to clarify the construction of Bayesian estimators and their robustness to the model specification, number of calibration products, noisy data and misspecification of the prior (joint work with Alok Gupta). In the second part, we devise a numerical scheme to compute upper and lower price bounds under locally uncertain volatility, based on piecewise constant policy approximation of the corresponding control problem. We prove convergence of the fully discretised equations and give numerical examples for European and American options on a combination of two assets (joint work with Peter Forsyth).

Term 2 of 2011/2012

Tuesday 24 January, 2012 in room A1.01.

3.00 Alex Cox (University of Bath)

Optimal robust bounds for variance options

Robust, or model-independent properties of the variance swap are well-known, and date back to Dupire (1993) and Neuberger (1994), who showed that, given the price of co-terminal call options, the price of a variance swap was exactly specified under the assumption that the price process is continuous. In this talk we show that optimal upper and lower bounds on the prices of variance calls can be determined, and strategies can be obtained that that sub- and super-replicate in every model, and which hedge exactly in some model, strengthening results of Dupire (2005) and Carr & Lee (2010). In addition, we provide numerical examples, and observe some interesting consequences.

4.00 Johannes Ruf (University of Oxford)

Föllmer's measure, Novikov's condition and options on exploding exchange rates

In the first part of this talk, I will present a proof of Novikov's condition by means of the Föllmer measure. In the second part, I will
discuss an application of the Föllmer measure to Foreign Exchange options. Strict local martingale models have been suggested to model the underlying exchange rate. In such models, put-call parity does not hold if one assumes minimal superreplicating costs as contingent claim prices. I will illustrate how put-call parity can be restored by changing the definition of a contingent claim price. More precisely, I will discuss a change of numeraire technique when the underlying is only
a local martingale. Then, the new (Föllmer) measure is not necessarily equivalent to the old measure. If one now defines the price of a contingent claim as the minimal superreplicating costs under both measures, then put-call parity holds. I will discuss properties of this new pricing operator. This talk is based on joint work with Peter Carr and Travis Fisher.

Tuesday 13 March in room A1.01

3.00 Mike Tehranchi (University of Cambridge)

Put-call symmetry and self-duality

We discuss generalisations of the notions of put-call symmetry and
self-duality. These notions have found applications in the pricing and
hedging of certain path-dependent contingent claims. Our results include a
classification of the possible forms of self-duality in one-dimension: in
addition to the arithmetic and geometric duality already appearing the
literature, there exists exactly one other type among continuous models. We
also give a description of the possible forms of put-call symmetry for
common models: in dimension greater than two, interesting new symmetries
appear.

4.00 Pavel Gapeev (LSE)

Pricing of perpetual American options in diffusion-type models with running maxima and drawdowns

We derive closed form solutions to the problem of rational valuation of
several perpetual American options in a diffusion-type financial market
model with path-dependent coefficients. The asset price dynamics are
described by a geometric diffusion-type process with local drift and
diffusion coefficients depending on the current states of the running
maximum and the maximum drawdown process. The problem is embedded into
the associated optimal stopping problem for a necessarily
multi-dimensional continuous Markov process having the asset price, the
running maximum, and the maximum drawdown process as its state space
components. The method of proof is based on the reduction of the
resulting optimal stopping problem to an equivalent free-boundary
problem with smooth-fit/normal-boundary conditions for the value
functions at the exercise boundary and normal-reflection conditions at
the maximum and maximum drawdown, and subsequent martingale verification
using a local time-space formula.

We obtain explicit expressions for the rational prices of perpetual
American standard put and call and maximum drawdown put options with
fixed and floating strikes. We show that the structure of the value
functions and exercise regions of the perpetual American put and call
options becomes essentially more complicated when the coefficients of
the diffusion-type model depend on the running maximum and maximum
drawdown process. The perpetual American maximum drawdown put options
represent protections for the holders of particularly risky assets that
can fall deeply after achieving their historical maxima. Such contracts
are realised if the running minimum of the asset price falls below a
certain fixed value (fixed strike) or below the current value of a
certain number of assets (floating strike). We derive first-order
ordinary differential equations for the optimal exercise boundaries of
the perpetual American options mentioned above, in the cases in which
the coefficients of the model essentially depend on the running maximum
and the maximum drawdown processes. Based on joint work with Neofytos
Rodosthenous (LSE).

PAST SEMINARS:

Term 1 of 2011/2012

Tuesday 25 October, 2011 in room A1.01.

3.00 Vladimir Vovk (Royal Holloway, University of London)

Finance without probability: some recent results

The standard approach to finance starts from postulating a statistical model for the prices
of securities (such as the Black-Scholes model). Since such models are often difficult to
justify, it is interesting to explore what can be done without making any stochastic assumptions.
To my knowledge, the first results of this kind were obtained (in the case of discrete time)
by Thomas Cover in 1991 and David Hobson in 1998, and their work has been developed in various
directions. In this talk I will discuss a different kind of results: probability-type properties
of price paths emerging without a statistical model. I will only consider the simplest case of
one security, and instead of stochastic assumptions will make some analytic assumptions. If
the price path is known to be cadlag without huge jumps, its quadratic variation exists unless
a predefined trading strategy earns infinite capital without risking more than one monetary unit.
This makes it possible to apply the known results of Ito calculus without probability (Follmer 1981,
Norvaisa) in the context of idealized financial markets. If, moreover, the price path is known to
be continuous, it becomes Brownian motion when physical time is replaced by quadratic variation;
this is a probability-free version of the Dubins-Schwarz theorem.

Tuesday 1 November in room A1.01

3.00 Martin Schweizer (ETH Zurich)

Measure changes and preservation of local integrability for martingale densities

An equivalent sigma-martingale measure (EsMM) for a givenstochastic process S is a
probability measure R equivalent to the original measure P such that S is an R-sigma-martingale.
Existence of an EsMM is equivalent to a classical absence-of-arbitrage property of S, and is
invariant if we replace the reference measure P with an equivalent measure Q. Now suppose
that there exists an EsMM for S such that the density dR/dP has some integrability under P. Does
this property also remain invariant if we replace P by some equivalent Q? We prove that
(surprisingly!) the answer is Yes if one imposes instead of a global only a local integrability
requirement. This is joint work with Tahir Choulli (University of Alberta, Edmonton).

4.00 Miklos Rasonyi (University of Edinburgh)

Behavioural investors in multiperiod market models

We consider the problem of optimal investment for an investor whose behaviour
is described by cumulative prospect theory. Most of previous research focussed
on either one-period or complete models. It turns out that the multiperiod
incomplete case exhibits a number of new phenomena. We provide easily verifiable
conditions for the well-posedness of this problem and show the existence of
optimal strategies in a discrete-time multiperiod setting. Then we prove
similar results for certain continuous-time models and point out the difficulties
of further generalizations.

Tuesday 15 November, 2011 in room A1.01.

3.00 Peter Tankov (Paris VII)

Asymptotically optimal discretization of hedging strategies with jumps

In this work, we consider the hedging error due to discrete trading in models based on pure
jump semimartingales. Extending an approach developed by Fukasawa (2009) for continuous
processes, we propose a framework enabling to (asymptotically) optimize the discretization
times. More precisely, a discretization rule is said to be optimal if for a given cost function,
no strategy has (asymptotically, for large costs) a lower mean square discretization error for
a smaller cost. We focus on discretization strategies based on the exit times of the hedging
strategy process out of random intervals and characterize explicitly the asymptotic behavior
of the associated errors and costs. This allows us to determine the optimal intervals. In
particular, we show that in the case where the cost functional is simply the expected number
N of discretization dates, the error associated to our optimal strategy with the cost equal to
N, converges to zero at a faster rate than the error obtained by readjusting at N equally spaced dates.

Academic year 2010/11

Wednesday 8 June, 2011 in room MS.04.

1.00 Freddy Delbaen (ETH Zurich)

Uniqueness of solutions of BSDE with quadratic driver and unbounded terminal value

This is joint work with Hu and Richou, Univ de Rennes.

Tuesday 24 May, 2011 in room A1.01.

2.00 Martin Keller-Ressel (TU Berlin)

Convex order properties of discrete realized variance and applications to variance options

I consider a square-integrable semimartingale with conditionally independent increments
and symmetric jump measure, and show that its discrete realized variance dominates its quadratic
variation in increasing convex order. The result has immediate applications to the pricing of
options on realized variance. For a class of models including time-changed L ́evy models and Sato
processes with symmetric jumps the results show that options on variance are typically underpriced,
if continuously sampled quadratic variation is substituted for the discretely sampled realized variance.
I will also discuss a counterexample that shows that the conditionally independent increments of
the process are crucial for the result and present some arguments for the conjecture that the imposed
conditions on the process in fact necessary and sufficient.

paper link: http://arxiv.org/PS_cache/arxiv/pdf/1103/1103.2310v1.pdf

Wednesday 18 May, 2011 in room A1.01

2.00 Constantinos Kardaras (Boston University)

On random times

In this talk, a study of random times on filtered probability spaces is undertaken.
One of the main messages is that, as long as distributional properties of adapted
processes up to the random time are involved, there is no loss of generality in assuming
that the random time is actually a randomized stopping time. This perspective sheds an
intuitive light on results in the theory of progressive enlargement of filtrations, as
is the semimartingale decomposition result of Jeulin and Yor. Financial applications of
the previous theory include the role of the numeraire portfolio in stochastic finance as
an indicator of overall market performance, as well as the problem of expected utility
maximization from terminal wealth with a random time-horizon. Further applications in
distributional properties of one-dimensional transient diffusions up to certain random
times will be discussed.

Tuesday 3 May, 2011 in room A1.01.

2.00 Peter Bank (TU Berlin)

A large investor trading at market indifference prices

We discuss a financial model where asset prices depend on the demand
generated by a large investor via a utility indifference principle. The
continuous-time dynamics of this model turn out to be best described by a
nonlinear stochastic differential equation in the sense of Kunita (1991).
We discuss existence and uniqueness of solutions to this SDE in terms of
Malliavin differentiablitiy of claims and in terms of Sobolev embeddings
for stochastic integrals and smootheness of utility functions. (This is
joint work with Dimitry Kramkov).

3.00 Reception (Statistics common room)

Everyone welcome!