Sergey Nadtochiy

Local volatility dynamic models

This paper is concerned with the characterization of arbitrage-free dynamic stochastic models for the equity markets when Itô stochastic differential equations are used to model the dynamics of a set of basic instruments including, but not limited to, the underliers. We study these market models in the framework of the HJM philosophy originally articulated for Treasury bond markets. The main thrust of the paper is to characterize absence of arbitrage by a drift condition and a spot consistency condition for the coefficients of the local volatility dynamics.

OPTIMAL INVESTMENT FOR ALL TIME HORIZONS AND MARTIN BOUNDARY OF SPACE-TIME DIFFUSIONS

This paper is concerned with the axiomatic foundation and explicit construction of a general class of optimality criteria that can be used for investment problems with multiple time horizons, or when the time horizon is not known in advance. Both the investment criterion and the optimal strategy are characterized by the Hamilton-Jacobi-Bellman equation on a semi-infinite time interval. In the case when this equation can be linearized, the problem reduces to a time-reversed parabolic equation, which cannot be analyzed via the standard methods of partial differential equations. Under the additional uniform ellipticity condition, we make use of the available description of all minimal solutions to such equations, along with some basic facts from potential theory and convex analysis, to obtain an explicit integral representation of all positive solutions. These results allow us to construct a large family of the aforementioned optimality criteria, including some closed form examples in relevant financial models.

Static Hedging under Time-Homogeneous Diffusions

We consider the problem of semistatic hedging of a single barrier option in a model where the underlying is a time-homogeneous diffusion, possibly running on an independent stochastic clock. The main result of the paper is an analytic expression for the payoff of a European-type contingent claim, which has the same price as the barrier option up to hitting the barrier. We then consider some examples, such as the Black-Scholes, constant elasticity of variance, and zero-correlation SABR models. Finally, we investigate an approximation of the static hedge with options of at most two different strikes.

Tangent Lévy market models

In this paper, we introduce a new class of models for the time evolution of the prices of call options of all strikes and maturities. We capture the information contained in the option prices in the density of some time-inhomogeneous Lévy measure (an alternative to the implied volatility surface), and we set this static code-book in motion by means of stochastic dynamics of Itô’s type in a function space, creating what we call a tangent Lévy model. We then provide the consistency conditions, namely, we show that the call prices produced by a given dynamic code-book (dynamic Lévy density) coincide with the conditional expectations of the respective payoffs if and only if certain restrictions on the dynamics of the code-book are satisfied (including a drift condition à la HJM). We then provide an existence result, which allows us to construct a large class of tangent Lévy models, and describe a specific example for the sake of illustration.

A class of homothetic forward investment performance processes with non-zero volatility

We study forward investment performance processes with non-zero forward volatility. We focus on the class of homothetic preferences in a single stochastic factor model. The forward performance process is represented in a closed-form via a deterministic function of the wealth and the stochastic factor. This function is, in turn, given as a distortion transformation of the solution to a linear ill-posed problem. We analyze the solutions of this problem in detail. We, also, provide two examples for specific dynamics of the stochastic factor, specifically, log-mean reverting and Heston-type dynamics.

Local Variance Gamma and Explicit Calibration to Option Prices

In some options markets (e.g. commodities), options are listed with only a single maturity for each underlying. In others, (e.g. equities, currencies), options are listed with multiple maturities. In this paper, we provide an algorithm for calibrating a pure jump Markov martingale model to match the market prices of European options of multiple strikes and maturities. This algorithm only requires solutions of several one-dimensional root-search problems, as well as application of elementary functions. We show how to construct a time-homogeneous process which meets a single smile, and a piecewise time-homogeneous process which can meet multiple smiles.

An Approximation Scheme for Solution to the Optimal Investment Problem in Incomplete Markets

We provide an approximation scheme for the maximal expected utility and optimal investment policies for the portfolio choice problem in an incomplete market. Incompleteness stems from the presence of a stochastic factor which affects the dynamics of the correlated stock price. The scheme is built on the Trotter--Kato approximation and is based on an intuitively pleasing splitting of the Hamilton--Jacobi--Bellman (HJB) equation in two subequations. The first is the HJB equation of a portfolio choice problem with a stochastic factor but in a complete market, while the other is a linear equation corresponding to the evolution of the orthogonal (nontraded) part of the stochastic factor. We establish convergence of the scheme to the unique viscosity solution of the marginal HJB equation, and, in turn, derive a computationally tractable representation of the maximal expected utility and construct an

Robust Trading Of Implied Skew

\varepsilon

Asymptotic behavior of a random walk with interaction

-optimal portfolio in a feedback form.

Liquidity effects of trading frequency

In this paper, we present a method for constructing a (static) portfolio of co-maturing European options whose price sign is determined by the skewness level of the associated implied volatility. This property holds regardless of the validity of a specific model — i.e. the method is robust. The strategy is given explicitly and depends only on one’s beliefs about the future values of implied skewness, which is an observable market indicator. As such, our method allows the use of existing statistical tools to formulate the beliefs, providing a practical interpretation of the more abstract mathematical setting, in which the beliefs are understood as a family of probability measures. One of the applications of the results established herein is a method for trading one’s views on the future changes in implied skew, largely independently of other market factors. Another application of our results provides a concrete improvement of the model-independent super-replication and sub-replication strategies for barrier options proposed in [H. Brown, D. Hobson & L. C. G. Rogers (2001) Robust hedging of barrier options, Mathematical Finance 11 (3), 285–314.], which exploits the given beliefs on the implied skew. Our theoretical results are tested empirically, using the historical prices of S&P 500 options.