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Mathematical Finance's Blog

Mathematical Finance Header

Strict local martingales and optimal investment in a Black–Scholes model with a bubble

February 18, 2018 Comments (0)

Abstract
There are two major streams of literature on the modeling of financial bubbles: the strict local martingale framework and the Johansen–Ledoit–Sornette (JLS) financial bubble model. Based on a class of models that embeds the JLS model and can exhibit strict local martingale behavior, we clarify the connection between these previously disconnected approaches. While the original JLS model is never a strict local martingale, there are relaxations that can be strict local martingales and...

Backward SDEs for control with partial information

February 18, 2018 Comments (0)

Abstract
This paper considers a non‐Markov control problem arising in a financial market where asset returns depend on hidden factors. The problem is non‐Markov because nonlinear filtering is required to make inference on these factors, and hence the associated dynamic program effectively takes the filtering distribution as one of its state variables. This is of significant difficulty because the filtering distribution is a stochastic probability measure of infinite dimension, and therefore the...

Credit portfolio selection with decaying contagion intensities

February 12, 2018 Comments (0)

Abstract
We develop a fixed‐income portfolio framework capturing the exponential decay of contagious intensities between successive default events. We show that the value function of the control problem is the classical solution to a recursive system of second‐order uniformly parabolic Hamilton–Jacobi–Bellman partial differential equations. We analyze the interplay between risk premia, decay of default intensities, and their volatilities. Our comparative statics analysis finds that the investor...

Corporate security prices in structural credit risk models with incomplete information

February 12, 2018 Comments (0)

Abstract
The paper studies derivative asset analysis in structural credit risk models where the asset value of the firm is not fully observable. It is shown that in order to determine the price dynamics of traded securities, one needs to solve a stochastic filtering problem for the asset value. We transform this problem to a filtering problem for a stopped diffusion process and apply results from the filtering literature to this problem. In this way, we obtain an stochastic partial differential...

Robust Markowitz mean‐variance portfolio selection under ambiguous covariance matrix

February 11, 2018 Comments (0)

Abstract
This paper studies a robust continuous‐time Markowitz portfolio selection problem where the model uncertainty affects the covariance matrix of multiple risky assets. This problem is formulated into a min–max mean‐variance problem over a set of nondominated probability measures that is solved by a McKean–Vlasov dynamic programming approach, which allows us to characterize the solution in terms of a Bellman–Isaacs equation in the Wasserstein space of probability measures. We provide...

The limits of leverage

February 7, 2018 Comments (0)

Abstract
When trading incurs proportional costs, leverage can scale an asset's return only up to a maximum multiple, which is sensitive to its volatility and liquidity. In a model with one safe and one risky asset, with constant investment opportunities and proportional costs, we find strategies that maximize long‐term returns given average volatility. As leverage increases, rising rebalancing costs imply declining Sharpe ratios. Beyond a critical level, even returns decline. Holding the Sharpe...

The characteristic function of rough Heston models

February 7, 2018 Comments (0)

Abstract
It has been recently shown that rough volatility models, where the volatility is driven by a fractional Brownian motion with small Hurst parameter, provide very relevant dynamics in order to reproduce the behavior of both historical and implied volatilities. However, due to the non‐Markovian nature of the fractional Brownian motion, they raise new issues when it comes to derivatives pricing. Using an original link between nearly unstable Hawkes processes and fractional volatility...

Convex duality for Epstein–Zin stochastic differential utility

February 7, 2018 Comments (0)

Abstract
This paper introduces a dual problem to study a continuous‐time consumption and investment problem with incomplete markets and Epstein–Zin stochastic differential utilities. Duality between the primal and dual problems is established. Consequently, the optimal strategy of this consumption and investment problem is identified without assuming several technical conditions on market models, utility specifications, and agent's admissible strategies. Meanwhile, the minimizer of the dual...

Distribution‐constrained optimal stopping

February 7, 2018 Comments (0)

Abstract
We solve the problem of optimal stopping of a Brownian motion subject to the constraint that the stopping time's distribution is a given measure consisting of finitely many atoms. In particular, we show that this problem can be converted to a finite sequence of state‐constrained optimal control problems with additional states corresponding to the conditional probability of stopping at each possible terminal time. The proof of this correspondence relies on a new variation of the dynamic...

A unified approach to systemic risk measures via acceptance sets

February 7, 2018 Comments (0)

Abstract
We specify a general methodological framework for systemic risk measures via multidimensional acceptance sets and aggregation functions. Existing systemic risk measures can usually be interpreted as the minimal amount of cash needed to secure the system after aggregating individual risks. In contrast, our approach also includes systemic risk measures that can be interpreted as the minimal amount of cash that secures the aggregated system by allocating capital to the single institutions...