# Plenary Talks

## Viability and hedging in continuous-path markets with infinite number of assets

We consider a continuous-time market market with potentially infinite number of liquid continuous-path assets. Such abstraction is needed, for example, in the study of bond markets (where there is a continuum of maturities), markets with traded options (where there is a continuum of maturities and/or strikes), or in the post-limit study of large financial markets. Under a spatial continuity requirement across assets in the volatility and drift coefficients, which is always satisfied in the case where the assets are at most countable, we provide exact versions of the fundamental theorem of asset pricing and hedging duality. Important in the development is a study of infinite-dimensional integration theory for continuous semimartingales which uses elements of Reproducing Kernel Hilbert Space theory.

## Simulating Risk Measures

Risk measures, such as value-at-risk and expected shortfall, are widely used in risk management, as exemplified in the Basel Accords proposed by Bank for International Settlements. We propose a simple general framework, allowing dependent samples, to compute these risk measures via simulation. The framework consists of two steps: In the C-step, we control the relative error in the simulation by computing the necessary sample size needed for simulation, using a newly derived asymptotic expansion of the relative errors for dependent samples; in the S-step, the risk measures are computed by using sorting algorithms. Numerical experiments indicate that the algorithm is easy to implement and fast, compared to existing methods, even at the 0.001 quantile level. We also give a comparison of the relative errors of value-at-risk and expected shortfall. This is a joint work with Wei Jiang.

## Fracking, Renewables & Mean Field Games

The dramatic decline in oil prices, from around $110 per barrel in June 2014 to around$30 in January 2016 highlights the importance of competition between different energy sources. Indeed, the price drop has been primarily attributed to OPEC's strategic decision not to curb its oil production in the face of increased supply of shale gas and oil in the US. We study how continuous time Cournot competitions, in which firms producing similar goods compete with one another by setting quantities, can be analyzed as continuum dynamic mean field games. We illustrate how the traditional oil producers may react in counter-intuitive ways in face of competition from alternative energy sources.

## Affine processes and non-linear differential equations

Affine processes have been used extensively to model financial phenomena since their marginal distributions are very tractable from an analytic point of view (up to the solution of a non-linear differential equation). It is well known by works of Dynkin-McKean-LeJan-Sznitman that one can turn this around and represent solutions of non-linear PDEs by affine processes. Recent advances in mathematical Finance in this direction have been made by Henry-Labordere and Touzi. We shall contribute some general theory and some new aspects to this field.

(Joint work with Georg Grafendorfer and Christa Cuchiero.)

# Contributed Talks

## Rare event simulation related to financial risks: efficient estimation and sensitivity analysis

We develop the reversible shaking transformation methods on path space of Gobet and Liu [GL15] to estimate the rare event statistics arising in different financial risk settings which are embedded within a unified framework of isonormal Gaussian process. Namely, we combine splitting methods with both Interacting Particle System (IPS) technique and ergodic transformations using Parallel-One-Path (POP) estimators. We also propose an adaptive version for the POP method and prove its convergence. We demonstrate the application of our methods in various examples which cover usual semi-martingale stochastic models (not necessarily Markovian) driven by Brownian motion and, also, models driven by fractional Brownian motion (non semi-martingale) to address various financial risks. Interestingly, owing to the Gaussian process framework, our methods are also able to efficiently handle the important problem of sensitivities of rare event statistics with respect to the model parameters. This is a joint work with Emmanuel Gobet, Stefano De Marco and Gang Liu.

## Optimal control under uncertainty and Bayesian parameters adjustments: Application to trading algorithms

We propose a general framework for the optimal control/design of trading algorithms in situations where market conditions or impact parameters are uncertain. Given a prior on the distribution of the unknown parameters, we explain how it should evolve according to the classical Bayesian rule after each sequence of trades. Taking these progressive prior-adjustments into account, we characterize the optimal policy through a quasi-variational parabolic equation, which can be solved numerically. From the mathematical point of view, we indeed treat a quite general impulse control problem with unknown parameters, and the derivation of the dynamic programming equation seems to be new in this context. The main difficulty lies in the nature of the set of controls which depends in a non trivial way on the initial data through the filtration itself. Typical examples of application are discussed.

## Robust exponential hedging in discrete time

In this talk we focus on the robust exponential utility maximization problem with random endowment in discrete time. An investor invests dynamically in a market and maximizes his/her worst case expected exponential utility of the endowment plus terminal wealth with respect to a family of non-dominated probabilistic models. Under tightness of the reference measures and the existence of certain martingale measures we provide the existence of an optimal trading strategy, defined simultaneously under all models. Further, we characterize the dual problem and show duality for measurable endowments. The talk is based on joint work with Patrick Cheridito and Michael Kupper.

## Ito-Semi-Diffusions, a Tool to approximate Levy Processes

We consider a new class of continuous processes whose local evolution in time is modelled by an Ito diffusion. The increments of such a process that are associated with consecutive points of a given grid of times provide a sequence of independent random variables, each having a prescribed distribution. This class of processes, which we call Ito semi-diffusions, could provide an alternative to modelling asset returns by means of Levy processes that is associated with a simpler calculus as well as with market completeness. Given a Levy process, we construct a sequence of Ito semi-diffusions whose finite-dimensional distributions converge to the ones of the Levy process. Furthermore, we establish conditions under which a martingale measure exists for asset prices modelled by a class of Ito semi-diffusions. (Joint work with Mihail Zervos)

## Pricing European Options in a Multiplicative Impact Model with Transient Impact

In this talk, we will discuss modelling and optimization problems in a market model with a single risky asset and a large trader whose actions have impact on the asset’s price in a transitive way, i.e. the impact from a trade is decreasing in time. Her gains from trading can be uniquely defined for continuous strategies of bounded variation. We extend the model to more general (càdlàg) trading strategies by continuity considerations. In particular, the cost of a block trade is derived as a limit of continuous trades over shorter and shorter time interval. To obtain jumps in the limit of continuous processes, a suitable topology on the space of càdlàg paths is considered - the Skorokhod M1 topology. The gains process is shown to be continuous (on the input strategy/control) with respect to this topology. The model can be linked to a Marcus canonical equation for which a Wong-Zakai-type approximation results hold. Moreover, we show absence of arbitrage opportunities.
Having specified our model for a very general class of trading strategies/controls, we consider the optimization problem of pricing European options by superreplication that we formulate as a stochastic target problem. In special coordinates, a (geometric) dynamic programming principle holds that could be applied to show that the minimal superreplication price is the viscosity solution of a non-linear PDE with gradient constraints. When the PDE admits a sufficiently regular solution, a replicating strategy can be constructed.
This talk is based on joint works with Dirk Becherer and Peter Frentrup from Humboldt-Universität zu Berlin.

## Arbitrage and Hedging in model-independent markets with frictions

We provide a Fundamental Theorem of Asset Pricing and a Superhedging Theorem for a model independent discrete time financial market with proportional transaction costs. We consider a probability-free version of the No Robust Arbitrage condition introduced by Schachermayer in [6] and show that this is equivalent to the existence of Consistent Price Systems. Moreover, we prove that the superhedging price for a claim g coincides with the frictionless superhedging price of g for a suitable process in the bid-ask spread. We exploit this result to obtain that this price equals the supremum of the expectations over the set of Consistent Price Systems.

### References

[1]     Bayraktar E., Zhang Y., Fundamental Theorem of Asset Pricing under Transaction costs and Model uncertainty, to appear in Math. Oper. Res., 2015.

[2]    Bouchard B., Nutz M., Consistent Price Systems under Model Uncertainty, Fin. Stoch., 20(1), 83-98, 2015.

[3]    Burzoni M., Frittelli M., and Maggis M., Model-free Superhedging Duality, preprint, arXiv 1506.06608, 2015.

[4]    Cheridito P., Kupper M., Tangpi L., Duality for increasing convex functionals with countably many marginals constraints, preprint, 2015.

[5]    Dolinsky Y., Soner H. M., Robust hedging with proportional transaction costs, Fin. Stoch., 18 (2), 327-347, 2014.

[6]    Schachermayer W., The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time, Math. Fin., 14(1), 19-48, 2004.

## Nonlinear transaction costs, portfolio choice, and time-varying investment opportunities

We consider a market with one safe asset and one risky asset with general, not necessarily Markovian dynamics. In this setting, we solve a utility maximization problem, that simplifies to studying the tradeoff between expected returns, the variance of the corresponding positions, and nonlinear trading costs proportional to a power of the order flow. In the limit for small costs, the optimal strategy is given by the renormalized solution of an ordinary differential equation.

## An ergodic BSDE approach to maturity independent entropic risk measure and its large time behavior

Indifference pricing theory (see, for instance, Henderson (2002), Henderson and Hobson (2009), Hu et al. (2005), Mania and Schweizer (2005), Musiela and Zariphopoulou (2004)) not only resolves the pricing problem of a contingent claim in an incomplete market, but also incorporates the economic behavior of the agent via the utility function to the price. Most of the works in the literature on the risk measure and the indifference pricing assumed that the maturity of the risk position is known in prior. However, in reality, some of the risk exposure period is not noted in advance. Hence, a notion of risk measure which is maturity independent has to be developed. In this talk, under the framework of the forward utility, which was introduced and developed by Musiela and Zariphopoulou, following the idea in Zariphopoulou and Zitkovic (2010), and using an ergodic BSDE representation for an exponential forward utility in Liang and Zariphopoulou (2015), we define the maturity independent entropic risk measure and provide a BSDE representation for this measure. We also study the large time behavior of the T-normalized forward entropic risk measure via its BSDE representation. This is a joint work with Ying Hu, Gechun Liang and Thaleia Zariphopoulou.

## Skorokhod embedding and robust hedging with local time

In this work, we focus on robust hedging of options on local time when one or more marginals of the underlying price process are known. By using the stochastic control approach initiated by Galichon, Henry-Labordère and Touzi, we identify the optimal hedging strategies and the corresponding prices in the one-marginal case. Then we extend the analysis to the two-marginal case, where we provide candidates for the optimal superhedging strategies. To this end, we construct a new solution to the two-marginal SEP as a generalization of the Vallois embedding. Finally, a special multi-marginal case is studied, where the stopping times given by Vallois are well-ordered. In the full marginal setting, we construct a remarkable Markov martingale and compute its generator explicitly. Based on a joint work with Gaoyue Guo and Pierre Henry-Labordère.

## Credit risk with ambiguity on the default intensity

In this talk, we will introduce the concept of no-arbitrage in a credit risk market under ambiguity. We consider an intensity-based framework where we assume that the default intensity is strictly positive. This assumption is economically intuitive, as it is equivalent to an approach where at every time s credit risk is present and not negligible. However, we consider the realistic case where the intensity is not precisely known, but there is ambiguity on the intensity. By means of the Girsanov theorem, we start from the reference measure where the intensity is equal 1 and define the equivalent measures Ph where the intensity is h. Ambiguity is considered in the sense that h lies between an upper and lower bound. From this viewpoint, the credit risky case turns out to be similar to the case of drift uncertainty in the G-expectation framework.

## Portfolio optimization under fixed transaction costs

We consider an investor with constant absolute risk aversion trading in a market consisting of one safe and one risky asset with general Itô dynamics. We assume that she has to pay a fixed transaction cost ε for each trade regardless of its size. Using a non-Markovian dynamic programming approach we derive the leading order optimal trading strategy and state rigorous verification theorems. We give examples and present an application to utility indifference pricing. Our results verify the heuristics of [2, Section 5] in the absence of proportional costs, but for general Itô dynamics. Contrary to the related study of [1] in a different setup our derivation and verification rely on martingale methods and tools from stochastic calculus like [3] rather than homogenization and viscosity solutions.

This is a joint work with Jan Kallsen.

### References

[1]    Albert Altarovici, Johannes Muhle-Karbe, and Halil Mete Soner. Asymptotics for fixed transaction costs. Finance and Stochastics, 19(2):363–414, 2015.

[2]    Ralf Korn. Portfolio optimisation with strictly positive transaction costs and impulse control. Finance and Stochastics, 2(2):85–114, 1998.

[3]    Goran Peskir. A change-of-variable formula with local time on curves. Journal of Theoretical Probability, 18(3):499–535, 2005.

## A Variant of Strassen's Theorem with an Application to the Consistency of Option Prices

We want to check European call option quotes for consistency under the assumption of a positive bid-ask spread on the underlying. Under the assumption that the bid-ask spread is bounded by a constant we will state sufficient and necessary conditions on the option prices, which ensure that they are consistent with an arbitrage free model. Also we will derive model independent arbitrage strategies for single and multiple maturities in case these conditions fail. If the bid-ask spread is not bounded by a constant we will show that it suffices to check each maturity separately for consistency.
The condition we will derive relies on a extension of a well known result (Strassen's theorem), which asserts that a stochastic process is increasing in convex order if and only if there is a martingale with the same marginal distributions. Such processes, or families of measures, are nowadays known as peacocks. We extend Strassen's theorem in a novel direction, relaxing the requirement on the martingale. Instead of equal marginal laws, we just require the laws to be within closed balls, defined by some metric on the space of probability measures. In our main result -- which we will use to derive our consistency conditions -- the metric is the infinity Wasserstein distance, but we also study other metrics. Joint work with Stefan Gerhold.

## Model-Independent Bounds for Asian Options: a Dynamic Programming Approach

We consider the problem of finding model-independent bounds on the price of an Asian option, when the call prices at the maturity date of the option are known. Our method differ from most approaches to model-independent pricing in that we consider the problem as a dynamic programming problem, where the controlled process is the conditional distribution of the asset at the maturity date. By formulating the problem in this manner, we are able to determine the model-independent price through a PDE formulation. Notably, this approach does not require specific constraints on the payoff function (e.g. convexity), and would appear to be generalisable to many related problems. This is joint work with A.M.G. Cox.

## Good-Deal Bounds and Hedging under Drift- and Volatility Uncertainty

The talk is concerned with robust good-deal valuation and hedging in incomplete financial markets under drift- and volatility uncertainty. The approach does not only prevent arbitrage opportunities, but also excludes an economically meaningful notion of deals that are “too good”. The resulting good-deal valuation bounds are tighter than the classical no-arbitrage bounds; the latter being too costly to be useful in practice. Robust valuation corresponds to the worst-case over a family of subjective priors that may be mutually singular. Hedging strategies are derived as minimizers of suitable a-priori dynamic risk measures that are of no-good-deal type and allow for optimal risk sharing with the market. Robust hedging is obtained in relation to a supermartingale property of hedging errors under generalized scenarios, uniformly over all (uncertain) priors. To obtain constructive results, we rely on the theory of second-order backward stochastic differential equations (2BSDEs) and provide examples with explicit solutions that facilitate interpretation and are accessible to computations.

## Are American options European after all?

We call a given American option representable if there exists a European claim which dominates the American payoff at any time and such that the values of the two options coincide within the continuation set of the American claim. This concept has interesting implications from a probabilistic, analytic and financial point of view. Above that it provides a way towards numerically feasible American pricing problems, even in multivariate settings.
We aim at analyzing and linking together the mathematical notions of representable American claims, embedded American payoffs (in the sense of Jourdain and Martini, 2001) and cheapest dominating European options. This process reveals a new duality structure between European and American valuation problems which we deem as very fruitful for future research. Relying on methods from convex optimization on dual pairs, we make a first step towards verifying representability of certain American claims.
The talk is based on joint work with Sören Christensen (Gothenburg) and Jan Kallsen (Kiel).

## Optimal investment and consumption with downside risk constraint in jump-diffusion models

This paper extends the results of the article [C. Klüppelberg and S. M. Pergamenchtchikov. Optimal consumption and investment with bounded downside risk for power utility functions. In Optimality and Risk: Modern Trends in Mathematical Finance. The Kabanov Festschrift, pages 133-169, 2009] to a jump-diffusion setting. We show that under the assumption that only positive jumps in the asset prices are allowed, the explicit optimal strategy can be found in the subset of admissible strategies satisfying the same risk constraint as in the pure diffusion setting. When negative jumps probably happen, the regulator should be more conservative. In that case, we suggest to impose on the investor’s portfolio a stricter constraint which depends on the probability of having negative jumps in the assets during the whole considered horizon.

## The opportunity process for utility maximization and applications

We address the utility maximization problem with stock S driven by a càdlàg semimartingale. We generalized the notion of the opportunity process Y introduced for power utilities [1] to utility functions U defined on the positive real line as a reduced form of the dual optimizer. Under suitable conditions on U, we show that the boundedness of log Y guarantees that the dual optimizer satisfies the probabilistic Muckenhoupt’s (Ar) condition for some r > 1. We also identify cases for which the (Ar) condition is equivalent to the boundedness of the log Y. Our sufficient condition is shown to be weaker than the recent condition obtained in [2].

This is based on a joint work with Peter Imkeller.

### References

[1]    M. Nutz, The opportunity process for optimal consumption and investment with power utility, Mathematics and Financial Economics, Vol. 3, No. 3-4, 139-159, 2010.

[2]    D. Kramkov and K. Weston, Muckenhoupt’s (Ap) condition and the existence of the optimal martingale measure, arXiv preprint arXiv:1507.05865, 2015.

## Exhaustible Resources with Production Adjustment Costs

We develop a general equilibrium model of exhaustible resources with production adjustment costs and show that Hotelling’s rule does not hold in the presence of adjustment costs. Demand uncertainty combined with adjustment costs can naturally explain many economic phenomena observed in the real markets, such as backwardation and contango. Exploration can generate a U-shaped price profile, while adjustment costs will significantly prolong the period of price staying at the bottom. Compared with the model by Carlson, Khokher and Titman (2007), our model is not only more analytically tractable but also gives rise to some economic phenomena in a more significant fashion.

## Efficient hedging under ambiguity

The superhedging price of a claim is the minimal price necessary to form a hedging portfolio that completely eliminates the risk. It is well known that this price is typically too high for practical purposes. In this talk, we discuss a general way of combining risk-tolerance and model uncertainty. Our results allows for transaction costs and trading constraints, and are based on representation formulas for increasing convex functionals. The talk is based on joint works with P. Cheridito and M. Kupper.

## Can probability weighting help prospect theory explain the disposition effect?

We consider a behavioural asset liquidation problem under prospect theory. Some recent results on optimal stopping with probability weighting are strengthened which in turn enable us to derive the optimal strategy with clear financial interpretations. Numerical studies are performed to demonstrate how inclusion of probability weighting can lead to more realistic prediction of sale strategies. Utilising tools from stochastic analysis, we also explain how to extract measures of the price disposition effect (the tendency of an investor to hold a losing stock too long but to sell a winning stock too early) within our theoretical model. This is a joint work with Vicky Henderson and David Hobson.

## Inference from high-frequency data: A subsampling approach

In this paper, we show how to estimate the asymptotic (conditional) covariance matrix, which appears in many central limit theorems in high-frequency estimation of asset return volatility. In a semimartingale model, we provide an estimator of this matrix by subsampling; an approach that computes rescaled copies of the original statistic based on local stretches of high-frequency data, and then it studies the sampling variation of these. We show that our estimator is consistent both in frictionless markets and models with additive microstructure noise. We derive a rate of convergence for it and are also able to determine an optimal rate for its tuning parameters (e.g., the number of subsamples). As a variance-covariance matrix estimator, it has the attractive feature that it is positive semi-definite by construction. Moreover, the subsampler tends to adapt to the problem at hand and be robust against misspecification of the noise process. We highlight the finite sample properties of the subsampler in a Monte Carlo study, while some initial empirical work demonstrates its use to draw feasible inference about volatility in financial markets.

## Hedging with stochastic price impact

We consider the problem of hedging a European contingent claim in a model with stochastic volatility and stochastic temporary price impact as proposed by Almgren and Chriss (2001). Following the approach of Rogers and Singh (2010) and Naujokat and Westray (2011), the hedging problem can be regarded as a cost optimal tracking problem of a frictionless hedging strategy. We also consider a constraint version of the problem which allows to, e.g., account for physical delivery or liquidation at maturity. It turns out that, rather than towards the current target position, the optimal policy trades towards an optimal tracking signal consisting of a convex combination of a weighted average of expected future target positions as well as the expected value of the predetermined terminal position which is computed under a new measure. A key role in the description of the optimal frictional hedge is played by the value process of a related optimal liquidation problem which was characterized in Kruse and Popier (2015) as a solution to a Backward Stochastic Riccati equation with possible singular terminal condition. Our framework allows for target hedging strategies with jumps (as, e.g., for Barrier options) which are of particular interest in illiquid markets, but not covered by previous studies in the literature where the frictionless reference hedge is confined to continuous diffusion-type processes. In a model with constant price impact and volatility, explicit solutions are available.
This is based on joint work with Peter Bank and H. Mete Soner.

## Mean-Field stochastic control problem

We consider the stochastic optimal control problem of McKean-Vlasov stochastic differential equation. Such problem arises typically when considering asymptotic cooperative equilibrium for a large population of controlled particles (players, financial agents, ...) in mutual interaction of mean-field type. By using feedback controls, we reformulate the problem into a deterministic control problem with only the marginal distribution as controlled state variable, and prove that dynamic programming principle holds in its general form. Then, by relying on the notion of differentiability with respect to probability measures recently introduced by P.L. Lions, and a special Itô formula for flows of probability measures, we derive the (dynamic programming) Bellman equation for mean-field stochastic control problem. This Bellman equation in the Wassertein space of probability measures reduces to the classical finite dimensional partial differential equation in the case of no mean-field interaction. We prove a verification theorem in our McKean-Vlasov framework, and give explicit solutions to the Bellman equation for the linear quadratic mean-field control problem, with applications to the mean-variance portfolio selection and a systemic risk model. Finally, we consider a notion of lifted viscosity solutions for the Bellman equation, and show the viscosity property and uniqueness of the value function to the McKean-Vlasov control problem.

Joint work with Huyên Pham, University Paris Diderot.

## Expert Opinions and Logarithmic Utility Maximization for Multivariate Stock Returns with Gaussian Drift

Investors in financial markets make trading decisions based on the available information about current market developments. We investigate optimal trading strategies in a financial market with multidimensional stock returns where the drift is an unobservable multivariate Ornstein-Uhlenbeck process. Information about the drift is obtained by observing stock returns and expert opinions. The latter provide unbiased estimates on the current state of the drift at discrete points in time.
The optimal trading strategy of investors maximizing expected logarithmic utility of terminal wealth depends on the conditional expectation of the drift given the available information. For this conditional expectation, the filter, we state filtering equations. Furthermore, we investigate properties of the conditional covariance matrices of the filter. Firstly, we consider the asymptotic behaviour of the covariance matrices for an increasing number of expert opinions on a finite time horizon. Secondly, we state conditions for the convergence of the covariance matrices on an infinite time horizon with regularly arriving expert opinions. In the context of filtering we make use of the Kalman filter. The conditional covariance matrices follow ordinary differential equations of Riccati type. We rely on basic theory about matrix Riccati equations for the analysis.
Finally, we derive the optimal trading strategy of an investor. The optimal expected logarithmic utility of terminal wealth, the value function, is a functional of the conditional covariance matrices. Hence, our analysis of the covariance matrices allows us to deduce properties of the value function. For instance, we can show that the value function of an investor who observes expert opinions converges to that of a fully informed investor if we let the number of expert opinions go to infinity on a finite time horizon.

## Feynman-Kac Representation for Periodic Problems

We establish a Feynman-Kac type representation for the solutions to a class of periodic parabolic terminal-boundary value problems, whose terminal and boundary conditions depend on the unknown function itself. In particular, the solution to the periodic problem is represented as the expectation of functionals of a diffusion process with periodic interventions at first passage times, which can be interpreted as the expected present value of a perpetual stochastic cash flow with periodic continuous and discrete payments. As an application in finance, we discuss the pricing of the dual-purpose fund, a recently popular structured mutual fund in China. Our representation result provides a rigorous mathematical characterization of the fund's value in terms of a periodic PDE, as well as an efficient numerical evaluation procedure. Our results suggest that the capital shares of the dual-purpose fund in China are overpriced by the market, which agrees with the findings in the literature.

# Poster Session

## Heterogeneous Risk Preferences and General Equilibrium in Financial Markets

This paper considers a continuous time economy populated by an arbitrarily large number of agents whose CRRA preferences differ in their risk aversion parameter. I derive closed form expressions for the interest rate, the market price of risk, and the dynamics of consumption weights. I find that agents dynamically self select into one of three groups depending on their preferences: leveraged investors, diversified investors, and saving divestors. The thresholds for selection are driven by a wedge between the market price of risk and the risk free rate. I simulate the economy and compare the features to real world observations. The model replicates a falling dividend to price ratio over time, low risk free rate, and rising inequality. However, changes in the initial condition have non-trivial effects on the short run outcomes of the model, which points towards the need for a more coherent theory of the distribution of risk preference.

## The Optimal Investment and consumption for financial market with financial assets of type "spread"

The aim of this project is to consider the optimization problem for investment and consumption of the financial markets generated by the differences in risky financial assets. As usual in the portfolio optimization problems, it is considered the financial assets of geometric Brownian motion type. In this project we use the model of financial markets "spread" generated by Ornstein - Uhlenbeck process. This model was proposed by Boguslavsky and Boguslavskaya (2004) for the optimal investment problem without consumption. The aim of this project is to apply the approach of Berdjane and Pergamenschikow (2013) to study the Hamilton - Jacobi - Bellman equation and to build optimal strategies. The Hamiton - Jacobi - Bellman equation for this problem was obtained. Also, the existence and uniqueness theorem for the classical solutions for this problem was shown.

## Pricing Option on Commodity Futures under String Shocks

Commodity futures curve exhibits a variety of shapes ranging from contango to backwardation. As the price of commodity linked contingent claim depends on the shape of future curve, it is imperative for the term structure of instantaneous future convenience yield to have a richer class of dynamics and shapes. This article offers a new approach to value options on commodity futures using the framework of “string shock”. This shock can be thought of as a string whose shape varies stochastically with time. It is used to perturb the term structure of instantaneous future convenience yield. Such models are capable of producing any correlation pattern among future convenience yields of various maturities in a parsimonious way. Unlike predecessors, our model allows for specification of separate volatility and correlation functions which make this model easier to parameterize. The dynamics of the commodity future price process is obtained and a formula for pricing a European call option on commodity futures is derived by assuming the log-normality of spot price, the normality of instantaneous future convenience yields, and deterministic interest rates. The call option formula has a closed simple-to-use form. We operationalize our model by using Ornstein-Uhlenbeck sheet as string shock to the future convenience yield. Implied model parameters are obtained by calibrating the model to the 50 delta ATM implied volatility of Light Sweet Crude Oil (WTI) options spanning across 22 maturities. Under the assumption of deterministic interest rates, we find that our model achieves a better fit for the out of the sample 50 delta call prices than the Miltersen-Schwartz model.

## Properties and Extensions of the Heath-Platen Algorithm

Heston model is one of the most popular stochastic volatility models, in which closed-form solutions exist only for few option types. Barrier options are path dependent options which only provide a final payment depending on whether the path of the underlying asset hits some certain barriers during the lifetime of the option. This barrier level is reflected as an indicator function in the payoff of the relevant barrier option. Due to this additional feature, it becomes difficult to obtain a closed-form solution of the barrier options even in the well known Black Scholes setting. To the best of our knowledge, there is no closed-form solution for barrier options in the Heston setting. Therefore, Monte Carlo methods are really favorable in pricing barrier options in the Heston setting. In case of barrier options, due to the knock-out feature of the barrier, crude Monte Carlo methods face the problem of high statistical and discretization (bias) errors. The Heath-Platen (HP) algorithm is a variance reduction technique that is particularly suited for barrier options in the Heston setting. In the present study, we consider the properties and extensions of the HP algorithm to value the barrier options by Monte Carlo methods in the Heston setting. The HP algorithm performs superior regarding both high variance reduction and best accuracy for the standard European call option. However, it is necessary to examine the convergence rate of the weak error for the HP algorithm in order to determine the optimal computational cost of the method. In this study, we aim to present the results about the weak convergence of the HP algorithm. In addition, we show that the HP algorithm can be applied to a wider range of valuation problems, particularly for some other option types and some other models as well.

## Stability of utility maximization problem with transaction costs

We first observe the static stability of the utility maximization problem in the market with proportional transaction costs when the stock price process is càdlàg. The primal and dual value functions as well as the optimizers are continuous of initial capitals, utility functions and physical probability measures. Then, we consider the convergence of optimal dual processes and shadow price processes by assuming the continuity of the stock price process. Under the strict positivity of liquidation value process of an optimal trading strategy, all optimal dual processes which induce the dual optimizer are local martingales. Disturbing the initial investments, the investor's preferences, the sequence of optimal dual processes converges in the sense of [1, Theorem 2.7] to an optimal dual process in the original market. The limit process defines a shadow price in the original market.

### References

[1]    C. Czichowsky and W. Schachermayer. Strong supermatingales and limits of nonnegative martingales. Annals of Probability, Vol. 44 , No. 1, pp. 171-205, 2016.

## Error analysis in Fourier methods for option pricing

We provide a bound for the error committed when using a Fourier method to price European options when the underlying follows an exponential Lévy dynamic. The price of the option is described by a partial integro-differential equation (PIDE). Applying a Fourier transformation to the PIDE yields an ordinary differential equation that can be solved analytically in terms of the characteristic exponent of the Lévy process. Then, a numerical inverse Fourier transform allows us to obtain the option price. We present a novel bound for the error and use this bound to set the parameters for the numerical method. We analyse the properties of the bound for a dissipative and pure-jump example. The bound presented is independent of the asymptotic behaviour of option prices at extreme asset prices. The error bound can be decomposed into a product of terms resulting from the dynamics and the option payoff, respectively. The analysis is supplemented by numerical examples that demonstrate results comparable to and superior to the existing literature.

## Time-consistent control in discrete time for Mean-Variance problem

We study the time-consistent equilibrium control for mean-variance (MV) problem in discrete time. We have characterised the discrete time condition for open-loop control and compared it with the closed-loop control. The result turns out to be that the open-loop control is always more risky than the closed-loop at any time. Since the equilibrium control is derived backwards from the terminal time, we also study the behaviour as time goes backwards infinite. The recent study on equilibrium control for both open-loop and closed-loop both indicate that, as time goes back infinite, the control would decrease to 0, and increase rapidly as the time gets closed to the maturity. To deal with this situation, we suggest several different methods to overcome the obstacle: Firstly, we recognise that the variance term in MV problem mistakenly penalise the perturbation as the investor is far from the maturity. This leads to re-scale the original MV functional by adjusting the risk-aversion. In the meantime, we show that the stability of equilibrium control could under a different wealth model.

## An Extension to the Azema martingale and drawdown options

In this paper, we prove an extension of the Azema martingale. To our knowledge, this is the first martingale which links the time elapsed since the last maximum of a Brownian motion with the maximum process itself. Furthermore, the martingale also has a representation in terms of the local time at 0, and the time elapsed since the last 0. We derive explicit formulas for the joint densities of the rst time the drawdown period hits a pre-specied duration, and the maximum up to this time. We apply the results to price a new type of drawdown option.

## Analysis of the split tree for pricing American and European options

We are investigating a discrete-time model for valuing options. Cox, Ross and Rubinstein have shown that binomial trees for pricing options converge to the Black-Scholes value in European case, but the drawback is that the convergence is oscillatory. Tian added a new parameter into the model, the tilting parameter, which helped to get smooth convergence. Chang and Palmer have shown that the speed of convergence is O, when the time step tends to zero.
We are working on the model suggested by Joshi which has two different drifts. This model, also known as a Split tree, consists of two trees. For the first steps we tilt the tree to have a node on the strike value and after that we continue without drift. Firstly, we explain why is it good to continue without a drift. Secondly, we are interested in why to split tree after steps or if another split time is better.
Finally, we check the convergence for European vanilla options and prove that this tree admits smooth convergence at least O, which can be improved by extrapolation. This makes Split tree faster than others. For American case we again show that convergence is smooth.