Past Nomura Seminar

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.