Past Nomura Seminar

We consider the market for a risky asset for which agents have interdependent private valuations. We study competitive rational expectations equilibria under the standard CARA-normal assumptions. Equilibrium is partially revealing even though there are no noise traders. Complementarities in information acquisition arise naturally in this setting. We characterize stable equilibria with endogenous information acquisition. Our framework encompasses the classical REE models in the CARA-normal tradition.

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We consider a contracting problem in which a principal hires an agent to manage a risky project. When the agent chooses volatility components of the output process and the principal observes the output continuously, the principal can compute the quadratic variation of the output, but not the individual components. This leads to moral hazard with respect to the risk choices of the agent. Using a very recent theory of singular changes of measures for Ito processes, we formulate the principal-agent problem in this context, and solve it in the case of CARA preferences. In that case, the optimal contract is linear in these factors: the contractible sources of risk, including the output, the quadratic variation of the output and the cross-variations between the output and the contractible risk sources. Thus, path-dependent contracts naturally arise when there is moral hazard with respect to risk management. This is a joint work with Nizar Touzi (CMAP, Ecole Polytechnique) and Jaksa Cvitanic (Caltech).

A sharp increase in the popularity of commodity investing in the past decade has triggered an unprecedented inflow of institutional funds into commodity futures markets. Such financialization of commodities coincided with significant booms and busts in commodity markets, raising concerns of policymakers. In this paper, we explore the effects of financialization in a model that features institutional investors alongside traditional futures markets participants. The institutional investors care about their performance relative to a commodity index. We find that if a commodity futures is included in the index, supply and demand shocks specific to that commodity spill over to all other commodity futures markets. In contrast, supply and demand shocks to a nonindex commodity affect just that commodity market alone. Moreover, prices and volatilities of all commodity futures go up, but more so for the index futures than for nonindex ones. Furthermore, financialization — the presence of institutional investors — leads to an increase in correlations amongst commodity futures as well as in equity-commodity correlations. Consistent with empirical evidence, the increases in the correlations between index commodities exceed those for nonindex ones. We model explicitly demand shocks which allows us to disentangle the effects of financialization from the effects of demand and supply (fundamentals). We perform a simple calibration and find that financialization accounts for 11% to 17% of commodity futures prices and the rest is attributable to fundamentals.

We consider evaluation methods for payoffs with an inherent

financial risk as encountered for instance for portfolios held

by pension funds and insurance companies. Pricing such payoffs

in a way consistent to market prices typically involves

combining actuarial techniques with methods from mathematical

finance. We propose to extend standard actuarial principles by

a new market-consistent evaluation procedure which we call `two

step market evaluation.' This procedure preserves the structure

of standard evaluation techniques and has many other appealing

properties. We give a complete axiomatic characterization for

two step market evaluations. We show further that in a dynamic

setting with continuous stock prices every evaluation which is

time-consistent and market-consistent is a two step market

evaluation. We also give characterization results and examples

in terms of $g$-expectations in a Brownian-Poisson setting.

We present a new model of financial markets that studies the evolution of wealth

among investment strategies. An investment strategy can be generated by maximizing utility

given some expectations or by behavioral rules. The only requirement is that any investment strategy

is adapted to the information filtration. The model has the mathematical structure of a random dynamical system.

We solve the model by characterizing evolutionary properties of investment strategies (survival, evolutionary stability, dominance).

It turns out that only a fundamental strategy investing according to expected relative dividends satisfies these evolutionary criteria.

We study lower- and dual upper-bounds for Bermudan options in a MonteCarlo/MC setting and provide four contributions. 1) We introduce a local least-squares MC method, based on maximizing the Bermudan price and which provides a lower-bound, which "also" minimizes (not the dual upper-bound itself, but) the gap between these two bounds; where both bounds are specified recursively. 2) We confirm that this method is near optimal, for both lower- and upper-bounds, by pricing Bermudan max-call options subject to an up-and-out barrier; state-of-the-art methods including Longstaff-Schwartz produce a large gap of 100--200 basis points/bps (Desai et al. (2012)), which we reduce to just 5--15 bps (using the same linear basis of functions). 3) For dual upper-bounds based on continuation values (more biased but less time intensive), it works best to reestimate the continuation value in the continuation region only. And 4) the difference between the Bermudan option Delta and the intrinsic value slope at the exercise boundary gives the sensitivity to suboptimal exercise (up to a 2nd-order Taylor approximation). The up-and-out feature flattens the Bermudan price, lowering the Bermudan Delta well below one when the call-payoff slope is equal to one, which implies that optimal exercise "really" matters.

We propose a model where an algorithmic trader takes a view on the distribution of prices at a future date and then decides how to trade in the direction of her predictions using the optimal mix of market and limit orders. As time goes by, the trader learns from changes in prices and updates her predictions to tweak her strategy. Compared to a trader that cannot learn from market dynamics or form a view of the market, the algorithmic trader's profits are higher and more certain. Even though the trader executes a strategy based on a directional view, the sources of profits are both from making the spread as well as capital appreciation of inventories. Higher volatility of prices considerably impairs the trader's ability to learn from price innovations, but this adverse effect can be circumvented by learning from a collection of assets that co-move.

The risk of a financial position is usually summarized by a risk measure.

As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In

statistical decision theory, risk measures for which such verification and comparison is possible, are called elicitable. It is known that quantile based risk

measures such as value-at-risk are elicitable. However, the coherent risk measure expected shortfall is not elicitable. Hence, it is unclear how to perform

forecast verification or comparison. We address the question whether coherent and elicitable risk measures exist (other than minus the expected value).

We show that one positive answer are expectiles, and that they play a special role amongst all elicitable law-invariant coherent risk measures.

In this work, we want to construct the solution $(Y,Z,K)$ to the following BSDE

$$\begin{array}{l}

Y_t=\xi+\int_t^Tf(s,Y_s,Z_s)ds-\int_t^TZ_sdB_s+K_T-K_t, \quad 0\le t\le T, \\

{\mathbf E}[l(t, Y_t)]\ge 0, \quad 0\le t\le T,\\

\int_0^T{\mathbf E}[l(t, Y_t)]dK_t=0, \\

\end{array}

$$

where $x\mapsto l(t, x)$ is non-decreasing and the terminal condition $\xi$

is such that ${\mathbf E}[l(T,\xi)]\ge 0$.

This equation is different from the (classical) reflected BSDE. In particular, for a solution $(Y,Z,K)$,

we require that $K$ is deterministic. We will first study the case when $l$ is linear, and then general cases.

We also give some application to mathematical finance. This is a joint work with Philippe Briand and Romuald Elie.

We construct and study market models admitting optimal arbitrage. We say that a model admits optimal arbitrage if it is possible, in a zero-interest rate setting, starting with an initial wealth of 1 and using only positive portfolios, to superreplicate a constant c>1. The optimal arbitrage strategy is the strategy for which this constant has the highest possible value. Our definition of optimal arbitrage is similar to the one in Fenrholz and Karatzas (2010), where optimal relative arbitrage with respect to the market portfolio is studied. In this work we present a systematic method to construct market models where the optimal arbitrage strategy exists and is known explicitly. We then develop several new examples of market models with arbitrage, which are based on economic agents' views concerning the impossibility of certain events rather than ad hoc constructions. We also explore the concept of fragility of arbitrage introduced in Guasoni and Rasonyi (2012), and provide new examples of arbitrage models which are not fragile in this sense.

References:

Fernholz, D. and Karatzas, I. (2010). On optimal arbitrage. The Annals of Applied Probability, 20(4):1179–1204.

Guasoni, P. and Rasonyi, M. (2012). Fragility of arbitrage and bubbles in diffusion models. preprint.

There are many financial models used in practice (CIR/Heston, Vasicek,

Stein-Stein, quadratic normal) whose popularity is due, in part, to their

analytically tractable asset pricing. In this talk we will show that it is

possible to generalise these models in various ways while maintaining

tractability. Conversely, we will also characterise the family of models

which admit this type of tractability, in the spirit of the classification

of polynomial term structure models.

The finance literature documents a relation between labor income and

the cross-section of stock returns. One possible explanation for this

is the hedging decisions of investors with relative wealth concerns.

This implies a negative risk premium associated with stock returns

correlated with local undiversifiable wealth, since investors are

willing to pay more for stocks that help their hedging goals. We find

evidence that is consistent with these regularities. In addition, we

show that the effect varies across geographic areas depending on the

size and variability of undiversifiable wealth, proxied by labor income.

An investor trades a safe and several risky assets with linear price impact to maximize expected utility from terminal wealth.

In the limit for small impact costs, we explicitly determine the optimal policy and welfare, in a general Markovian setting allowing for stochastic market,

cost, and preference parameters. These results shed light on the general structure of the problem at hand, and also unveil close connections to

optimal execution problems and to other market frictions such as proportional and fixed transaction costs.

Worst-case portfolio optimization has been introduced in Korn and Wilmott

(2002) and is based on distinguishing between random stock price

fluctuations and market crashes which are subject to Knightian

uncertainty. Due to the absence of full probabilistic information, a

worst-case portfolio problem is considered that will be solved completely.

The corresponding optimal strategy is of a multi-part type and makes an

investor indifferent between the occurrence of the worst possible crash

and no crash at all.

We will consider various generalizations of this setting and - as a very

recent result - will in particular answer the question "Is it good to save

for bad times or should one consume more as long as one is still rich?"

We construct a model for asset price in a limit order book, which captures on one hand main stylized facts of microstructure effects, and on the other hand is tractable for dealing with optimal high frequency trading by stochastic control methods. For this purpose, we introduce a model for describing the fluctuations of a tick-by-tick single asset price, based on Markov renewal process.

We consider a point process associated to the timestamps of the price jumps, and marks associated to price increments. By modeling the marks with a suitable Markov chain, we can reproduce the strong mean-reversion of price returns known as microstructure noise. Moreover, by using Markov renewal process, we can model the presence of spikes in intensity of market activity, i.e. the volatility clustering. We also provide simple parametric and nonparametric statistical procedures for the estimation of our model. We obtain closed-form formulae for the mean signature plot, and show the diffusive behavior of our model at large scale limit. We illustrate our results by numerical simulations, and find that our model is consistent with empirical data on futures Euribor and Eurostoxx. In a second part, we use a dynamic programming approach to our semi Markov model applied to the problem of optimal high frequency trading with a suitable modeling of market order flow correlated with the stock price, and taking into account in particular the adverse selection risk. We show a reduced-form for the value function of the associated control problem, and provide a convergent and computational scheme for solving the problem. Numerical tests display the shape of optimal policies for the market making problem.

This talk is based on joint works with Pietro Fodra.

We extend Kyle's (1985) model of insider trading to the case where liquidity provided

by noise traders follows a general stochastic process. Even though the level of noise

trading volatility is observable, in equilibrium, measured price impact is stochastic.

If noise trading volatility is mean-reverting, then the equilibrium price follows a

multivariate stochastic volatility `bridge' process. More private information is revealed

when volatility is higher. This is because insiders choose to optimally wait to trade

more aggressively when noise trading volatility is higher. In equilibrium, market makers

anticipate this, and adjust prices accordingly. In time series, insiders trade more

aggressively, when measured price impact is lower. Therefore, aggregate execution costs

to uninformed traders can be higher when price impact is lower

We study optimal portfolio strategies in a market

where the drift is driven by an unobserved Markov chain. Information on

the state of this chain is obtained from stock prices and from expert

opinions in the form of signals at random discrete time points. We use

stochastic filtering to transform the original problem into an

optimization problem under full information where the state variable is

the filter for the Markov chain. This problem is studied with dynamic

programming techniques and with regularization arguments. Finally we

discuss a number of numerical experiments

We consider over-the-counter (OTC) transactions with bilateral default risk, and study the optimal design of the Credit Support Annex (CSA). In a setting where agents have access to a trading technology, default penalties and collateral costs arise endogenously as a result of foregone investment opportunities. We show how the optimal CSA trades off the costs of the collateralization procedure against the reduction in exposure to counterparty risk and expected default losses. The results are used to provide insights on the drivers of different collateral rules, including hedging motives, re-hypothecation of collateral, and close-out conventions. We show that standardized collateral rules can have a detrimental impact on risk sharing, which should be taken into account when assessing the merits of standardized vs. bespoke CSAs in non-centrally cleared OTC instruments. This is joint work with D. Bauer and L.R. Sotomayor (GSU).

We analyse the impact of market makers' risk aversion on the equilibrium in a speculative market consisting of a risk neutral informed trader and noise traders. The unwillingness of market makers to bear risk causes the informed trader to absorb large shocks in their inventories. The informed trader's optimal strategy is to drive the market price to its fundamental value while disguising her trades as the ones of an uninformed strategic trader. This results in a mean reverting demand, price reversal, and systematic changes in the market depth. We also find that an increase in risk aversion leads to lower market depth, less efficient prices, stronger price reversal and slower convergence to fundamental value. The endogenous value of private information, however, is non-monotonic in risk aversion. We will mainly concentrate on the case when the private signal of the informed is static. If time permits, the implications of a dynamic signal will be discussed as well.

Based on a joint work with Albina Danilova.

The performance of the shares of a closed end bond fund is based on the returns of an underlying portfolio of bonds. The returns on closed end bond funds are typically higher than those of comparable open ended bond funds and this result is attributed to the use of leverage by closed end bond funds. This talk develops a simple model to assess the impact of leverage on the expected return and riskiness of a closed end bond fund. We illustrate the model with some examples

Advanced models such as Lévy models require advanced numerical methods for developing efficient pricing algorithms. Here we focus on PIDE based methods. There is a large arsenal of numerical methods for solving parabolic equations that arise in this context. Especially Galerkin and Galerkin inspired methods have an impressive potential. In order to apply these methods, what is required is a formulation of the equation in the weak sense.

We therefore classify Lévy processes according to the solution spaces of the associated parabolic PIDEs. We define the Sobolev index of a Lévy process by a certain growth condition on the symbol. It follows that for Lévy processes with a certain Sobolev index b the corresponding evolution problem has a unique weak solution in the Sobolev-Slobodeckii space with index b/2. We show that this classification applies to a wide range of processes. Examples are the Brownian motion with or without drift, generalised hyperbolic (GH), CGMY and (semi) stable Lévy processes.

A comparison of the Sobolev index with the Blumenthal-Getoor index sheds light on the structural implication of the classification. More precisely, we discuss the Sobolev index as an indicator of the smoothness of the distribution and of the variation of the paths of the process.

An application to financial models requires in particular to admit pure jump processes as well as unbounded domains of the equation. In order to deal at the same time with the typical payoffs which can arise, the weak formulation of the equation has to be based on exponentially weighted Sobolev-Slobodeckii spaces. We provide a number of examples of models that are covered by this general framework. Examples of options for which such an analysis is required are calls, puts, digital and power options as well as basket options.

The talk is based on joint work with Ernst Eberlein.

The martingale optimal transportation problem is motivated by

model-independent bounds for the pricing and hedging exotic options in

financial mathematics.

In the simplest one-period model, the dual formulation of the robust

superhedging cost differs from the standard optimal transport problem by

the presence of a martingale constraint on the set of coupling measures.

The one-dimensional Brenier theorem has a natural extension. However, in

the present martingale version, the optimal coupling measure is

concentrated on a pair of graphs which can be obtained in explicit form.

These explicit extremal probability measures are also characterized as

the unique left and right monotone martingale transference plans, and

induce an optimal solution of the kantorovitch dual, which coincides

with our original robust hedging problem.

By iterating the above construction over n steps, we define a Markov

process whose distribution is optimal for the n-periods martingale

transport problem corresponding to a convenient class of cost functions.

Similarly, the optimal solution of the corresponding robust hedging

problem is deduced in explicit form. Finally, by sending the time step

to zero, this leads to a continuous-time version of the one-dimensional

Brenier theorem in the present martingale context, thus providing a new

remarkable example of Peacock, i.e. Processus Croissant pour l'Ordre

Convexe. Here again, the corresponding robust hedging strategy is

obtained in explicit form.

In an equity market with stable capital distribution, a capitalization-weighted index of small stocks tends to outperform a capitalization-weighted index of large stocks.} This is a somewhat careful statement of the so-called "size effect", which has been documented empirically and for which several explanations have been advanced over the years. We review the analysis of this phenomenon by Fernholz (2001) who showed that, in the presence of (a suitably defined) stability for the capital structure, this phenomenon can be attributed entirely to portfolio rebalancing effects, and will occur regardless of whether or not small stocks are riskier than their larger brethren. Collision local times play a critical role in this analysis, as they capture the turnover at the various ranks on the capitalization ladder.

We shall provide a rather complete study of this phenomenon in the context of a simple model with stable capital distribution, the so-called ``Atlas model" studied in Banner et al.(2005).

This is a Joint work with Adrian Banner, Robert Fernholz, Vasileios Papathanakos and Phillip Whitman.

In this paper we prove a weak necessary and sufficient maximum principle for Markov regime switching stochastic optimal control problems. Instead of insisting on the maximum condition of the Hamiltonian, we show that 0 belongs to the sum of Clarke's generalized gradient of the Hamiltonian and Clarke's normal cone of the control constraint set at the optimal control. Under a joint concavity condition on the Hamiltonian and a convexity condition on the terminal objective function, the necessary condition becomes sufficient. We give four examples to demonstrate the weak stochastic maximum principle.

We discuss the superhedging problem under model uncertainty based on existence

and duality results for minimal supersolutions of backward stochastic differential equations.

The talk is based on joint works with Samuel Drapeau, Gregor Heyne and Reinhard Schmidt.

Suppose we are given a double continuum (in time and strike) of discounted

option prices, or equivalently a set of measures which is increasing in

convex order. Given sufficient regularity, Dupire showed how to construct

a time-inhomogeneous martingale diffusion which is consistent with those

prices. But are there other martingales with the same 1-marginals? (In the

case of Gaussian marginals this is the fake Brownian motion problem.)

In this talk we show that the answer to the question above is yes.

Amongst the class of martingales with a given set of marginals we

construct the process with smallest possible expected total variation.

The Ross Recovery Theorem gives sufficient conditions under which the

market’s beliefs

can be recovered from risk-neutral probabilities. His approach places

mild restrictions on the form of the preferences of

the representative investor. We present an alternative approach which

has no restrictions beyond preferring more to less,

Instead, we restrict the form and risk-neutral dynamics of John Long’s

numeraire portfolio. We also replace Ross’ finite state Markov chain

with a diffusion with bounded state space. Finally, we present some

preliminary results for diffusions on unbounded state space.

In particular, our version of Ross recovery allows market beliefs to be

recovered from risk neutral probabilities in the classical Cox

Ingersoll Ross model for the short interest rate.

The original transport problem is to optimally move a pile of soil to an excavation.

Mathematically, given two measures of equal mass, we look for an optimal bijection that takes

one measure to the other one and also minimizes a given cost functional. Kantorovich relaxed

this problem by considering a measure whose marginals agree with given two measures instead of

a bijection. This generalization linearizes the problem. Hence, allows for an easy existence

result and enables one to identify its convex dual.

In robust hedging problems, we are also given two measures. Namely, the initial and the final

distributions of a stock process. We then construct an optimal connection. In general, however,

the cost functional depends on the whole path of this connection and not simply on the final value.

Hence, one needs to consider processes instead of simply the maps S. The probability distribution

of this process has prescribed marginals at final and initial times. Thus, it is in direct analogy

with the Kantorovich measure. But, financial considerations restrict the process to be a martingale

Interestingly, the dual also has a financial interpretation as a robust hedging (super-replication)

problem.

In this talk, we prove an analogue of Kantorovich duality: the minimal super-replication cost in

the robust setting is given as the supremum of the expectations of the contingent claim over all

martingale measures with a given marginal at the maturity.

This is joint work with Yan Dolinsky of Hebrew University.

A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities:

(i) The optimal terminal wealth X*(T) := Xφ* (T) of the classical problem to

maximise the expected U-utility of the terminal wealth Xφ(T) generated by admissible

portfolios φ(t); 0 ≤ t ≤ T in a market with the risky asset price process modeled as a semimartingale;

(ii) The optimal scenario dQ*/dP of the dual problem to minimise the expected

V -value of dQ/dP over a family of equivalent local martingale measures Q. Here V is

the convex dual function of the concave function U.

In this talk we consider markets modeled by Itô-Lėvy processes, and we present

in a first part a new proof of the above result in this setting, based on the maximum

principle in stochastic control theory. An advantage with our approach is that it also

gives an explicit relation between the optimal portfolio φ* and the optimal scenario

Q*, in terms of backward stochastic differential equations. In a second part we present

robust (model uncertainty) versions of the optimization problems in (i) and (ii), and

we prove a relation between them. We illustrate the results with explicit examples.

The presentation is based on recent joint work with Bernt ¬Oksendal, University of

Oslo, Norway.

The banking industry lost a trillion dollars during the global financial crisis. Some of these losses, if not most of them, were attributable to complex derivatives or securities being incorrectly priced and hedged. We introduce a new methodology which provides a better way of trying to hedge and mark-to-market complex derivatives and other illiquid securities which recognise the fundamental incompleteness of markets and the presence of model uncertainty. Our methodology combines elements of the No Good Deals methodology of Cochrane and Saa-Requejo with the Robustness methodology of Hansen and Sargent. We give some numerical examples for a range of both simple and complex problems encompassing not only financial derivatives but also “real options”occurring in commodity-related businesses.

We discuss sparse portfolio optimization in continuous time.

Optimization objective is to maximize an expected utility as in the

classical Merton problem but with regularizing sparsity constraints.

Such constraints aim for asset allocations that contain only few assets or

that deviate only in few coordinates from a reference benchmark allocation.

With a focus on growth optimization, we show empirical results for various

portfolio selection strategies with and without sparsity constraints,

investigating different portfolios of stock indicies, several performance

measures and adaptive methods to select the regularization parameter.

Sparse optimal portfolios are less sensitive to estimation

errors and performance is superior to portfolios without sparsity

constraints in reality, where estimation risk and model uncertainty must

not be ignored.

We study portfolio optimization and contingent claim valuation in markets where illiquidity may affect the transfer of wealth over time and between investment classes. In addition to classical frictionless markets and markets with transaction costs, our model covers nonlinear illiquidity effects that arise in limit order markets. We extend basic results on arbitrage bounds, attainable claims and optimal portfolios to illiquid markets and general swap contracts where both claims and premiums may have multiple payout dates. We establish the existence of optimal trading strategies and the lower semicontinuity of the optimal value of portfolio optimization under conditions that extend the no-arbitrage condition in the classical linear market model.

A multi-dimensional extension of the structural default model with firms' values driven by diffusion processes with Marshall-Olkin-inspired

correlation structure is presented. Semi-analytical methods for solving

the forward calibration problem and backward pricing problem in three

dimensions are developed. The model is used to analyze bilateral counter- party risk for credit default swaps and evaluate the corresponding credit and debt value adjustments.

The models of Brownian motion, Poisson processes, Levy processes and martingales are frequently used as basic formulations of prices in financial market. But probability and/or distribution uncertainties cause serious problems of robustness. Nonlinear expectations (G-Expectations) and the corresponding martingales are useful tools to solve them.

In Kuznetsov et al. (2011) a new Monte Carlo simulation technique was introduced for a large family of L\'evy processes that is based on the Wiener-Hopf decomposition. We pursue this idea further by combining their technique with the recently introduced multilevel Monte Carlo methodology. We also provide here a theoretical analysis of the new Monte Carlo simulation technique in Kuznetsov et al. (2011) and of its multilevel variant. We find that the rate of convergence is uniformly with respect to the ``jump activity'' (e.g. characterised by the Blumenthal-Getoor index).

We derive an exact implied volatility expansion for any model whose European call price can be expanded analytically around a Black-Scholes call price. Two examples of our framework are provided (i) exponential Levy models and (ii) CEV-like models with local stochastic volatility and local stochastic jump-intensity.

We analyze the fluctuation of the loss from default around its large portfolio limit in a class of reduced-form models of correlated default timing. We prove a weak convergence result for the fluctuation process and use it for developing a conditionally Gaussian approximation to the loss distribution. Numerical results illustrate the accuracy of the approximation.

This is joint work with Kostas Spiliopoulos (Boston University) and Justin Sirignano (Stanford).

Abstract: Sharp asymptotic lower bounds of the expected quadratic

variation of discretization error in stochastic integration are given.

The theory relies on inequalities for the kurtosis and skewness of a

general random variable which are themselves seemingly new.

Asymptotically efficient schemes which attain the lower bounds are

constructed explicitly. The result is directly applicable to practical

hedging problem in mathematical finance; it gives an asymptotically

optimal way to choose rebalancing dates and portofolios with respect

to transaction costs. The asymptotically efficient strategies in fact

reflect the structure of transaction costs. In particular a specific

biased rebalancing scheme is shown to be superior to unbiased schemes

if transaction costs follow a convex model. The problem is discussed

also in terms of the exponential utility maximization.

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.

We construct explicitly a bridge process whose distribution, in its own filtration, is the same as the difference of two independent Poisson processes with the same intensity and its time 1 value satisfies a specific constraint. This construction allows us to show the existence of Glosten-Milgrom equilibrium and its associated optimal trading strategy for the insider. In the equilibrium the insider employs a mixed strategy to randomly submit two types of orders: one type trades in the same direction as noise trades while the other cancels some of the noise trades by submitting opposite orders when noise trades arrive. The construction also allows us to prove that Glosten-Milgrom equilibria converge weakly to Kyle-Back equilibrium, without the additional assumptions imposed in \textit{K. Back and S. Baruch, Econometrica, 72 (2004), pp. 433-465}, when the common intensity of the Poisson processes tends to infinity. This is a joint work with Umut Cetin.

We consider a class of stochastic control problems with fuel constraint that are closely connected to the problem of finding adaptive mean-variance-optimal portfolio liquidation strategies in the Almgren-Chriss framework. We give a closed-form solution to these control problems in terms of the log-Laplace transforms of certain J-functionals of Dawson-Watanabe superprocesses. This solution can be related heuristically to the superprocess solution of certain quasilinear parabolic PDEs with singular terminal condition as given by Dynkin (1992). It requires us to study in some detail the blow-up behavior of the log-Laplace functionals when approaching the singularity.

We consider a broker who has to place a large order which consumes a sizable part of average daily trading volume. By contrast to the previous literature, we allow the liquidity parameters of market depth and resilience to vary deterministically over the course of the trading period. The resulting singular optimal control problem is shown to be tractable by methods from convex analysis and, under

minimal assumptions, we construct an explicit solution to the scheduling problem in terms of some concave envelope of the resilience adjusted market depth.

In an incomplete continuous-time securities market with uncertainty generated by Brownian motions, we derive closed-form solutions for the equilibrium interest rate and market price of risk processes. The economy has a finite number of heterogeneous exponential utility investors, who receive partially unspanned income and can trade continuously on a finite time-interval in a money market account and a single risky security. Besides establishing the existence of an equilibrium, our main result shows that if the investors' unspanned income has stochastic countercyclical volatility, the resulting equilibrium can display both lower interest rates and higher risk premia compared to the Pareto efficient equilibrium in an otherwise identical complete market. This is joint work with Peter Ove Christensen.

Given a diffusion in R^n, we prove a small-noise expansion for its density. Our proof relies on the Laplace method on Wiener space and stochastic Taylor expansions in the spirit of Benarous-Bismut. Our result applies (i) to small-time asymptotics and (ii) to the tails of the distribution and (iii) to small volatility of volatility.

We shall study applications of this result to stochastic volatility models, recovering the Berestycki- Busca-Florent formula (using (i)), the Gulisashvili-Stein expansion (from (ii)) and Lewis' expansions (using (iii)).

This is a joint work with J.D. Deuschel (TU Berlin), P. Friz (TU Berlin) and S. Violante (Imperial College London).

In this talk, approximations to utility indifference prices for a contingent claim in the large position size limit are provided. Results are valid for general utility functions and semi-martingale models. It is

shown that as the position size approaches infinity, all utility functions with the same rate of decay for large negative wealths yield the same price. Practically, this means an investor should price like an exponential investor. In a sizeable class of diffusion models, the large position limit is seen to arise naturally in conjunction with the limit of a complete model and hence approximations are most appropriate in this setting.