The principal portfolios of the standard Capital Asset Pricing Model (CAPM) are analyzed and found to have remarkable hedging and leveraging properties. Principal portfolios implement a recasting of any correlated asset set of N risky securities into an equivalent but uncorrelated set when short sales are allowed. While a determination of principal portfolios in general requires a detailed knowledge of the covariance matrix for the asset set, the rather simple structure of CAPM permits an accurate solution for any reasonably large asset set that reveals interesting universal properties. Thus for an asset set of size N, we find a market-aligned portfolio, corresponding to the market portfolio of CAPM, as well as N-1 market-orthogonal portfolios which are market neutral and strongly leveraged. These results provide new insight into the return-volatility structure of CAPM, and demonstrate the effect of unbridled leveraging on volatility.
This paper presents hedging strategies for European and exotic options in a Levy market. By applying Taylor's Theorem, dynamic hedging portfolios are con- structed under different market assumptions, such as the existence of power jump assets or moment swaps. In the case of European options or baskets of European options, static hedging is implemented. It is shown that perfect hedging can be achieved. Delta and gamma hedging strategies are extended to higher moment hedging by investing in other traded derivatives depending on the same underlying asset. This development is of practical importan…
We consider the problem of hedging a European interest rate contingent claim with a portfolio of zero-coupon bonds and show that an HJM type Markovian model driven by an infinite number of sources of randomness does not have some of the shortcomings found in the classical finite-factor models. Indeed, under natural conditions on the model, we find that there exists a unique hedging strategy, and that this strategy has the desirable property that at all times it consists of bonds with maturities that are less than or equal to the longest maturity of the bonds underlying the claim.
The optimal portfolio size for a venture capital (VC) fund is a topic often debated, but there is no consensus on the best strategy. This is because it is a function of many factors. It is not easy to find a general formula that can be applied to all situations, and it largely depends on the goal of the fund. In this report, we will go through the different factors step by step, studying how they affect fund returns and the optimal portfolio size, starting with some basic assumptions and then increasing the complexity of the model.
We investigate the portfolio selection problem against the systemic risk which is measured by CoVaR. We first demonstrate that the systemic risk of pure stock portfolios is essentially uncontrollable due to the contagion effect and the seesaw effect. Next, we prove that it is necessary and sufficient to introduce options to make the systemic risk controllable by the correlation hedging and the extreme loss hedging. In addition to systemic risk control, we show that using options can also enhance return-risk performance. Then, with a reasonable approximation of the conditional distribution of o…
We study a mean-field version of rank-based models of equity markets such as the Atlas model introduced by Fernholz in the framework of Stochastic Portfolio Theory. We obtain an asymptotic description of the market when the number of companies grows to infinity. Then, we discuss the long-term capital distribution. We recover the Pareto-like shape of capital distribution curves usually derived from empirical studies, and provide a new description of the phase transition phenomenon observed by Chatterjee and Pal. Finally, we address the performance of simple portfolio rules and highlight the inf…
Utility and risk are two often competing measurements on the investment success. We show that efficient trade-off between these two measurements for investment portfolios happens, in general, on a convex curve in the two dimensional space of utility and risk. This is a rather general pattern. The modern portfolio theory of Markowitz [H. Markowitz, Portfolio Selection, 1959] and its natural generalization, the capital market pricing model, [W. F. Sharpe, Mutual fund performance , 1966] are special cases of our general framework when the risk measure is taken to be the standard deviation and the…
This paper derives a portfolio decomposition formula when the agent maximizes utility of her wealth at some finite planning horizon. The financial market is complete and consists of multiple risky assets (stocks) plus a risk free asset. The stocks are modelled as exponential Brownian motions with drift and volatility being Ito processes. The optimal portfolio has two components: a myopic component and a hedging one. We show that the myopic component is robust with respect to stopping times. We employ the Clark-Haussmann formula to derive portfolio s hedging component.
Neural Networks for Portfolio-Level Risk Management: Portfolio Compression, Static Hedging, Counterparty Credit Risk Exposures and Impact on Capital Requirement
Despite the fact that the Euler allocation principle has been adopted by many financial institutions for their internal capital allocation process, a comprehensive description of Euler allocation seems still to be missing. We try to fill this gap by presenting the theoretical background as well as practical aspects. In particular, we discuss how Euler risk contributions can be estimated for some important risk measures. We furthermore investigate the analysis of CDO tranche expected losses by means of Euler's theorem and suggest an approach to measure the impact of risk factors on non-linear p…
In this paper, we construct a solution to the optimal contract problem for delegated portfolio management of the fist-best (risk-sharing) type. The novelty of our result is (i) in the robustness of the optimal contract with respect to perturbations of the wealth process (interpreted as capital injections), and (ii) in the more general form of principals objective function, which is allowed to depend directly on the agents strategy, as opposed to being a function of the generated wealth only. In particular, the latter feature allows us to incorporate endogenous trading constraints in the contra…
This paper studies the finite horizon portfolio management by optimally tracking a ratcheting capital benchmark process. It is assumed that the fund manager can dynamically inject capital into the portfolio account such that the total capital dominates a non-decreasing benchmark floor process at each intermediate time. The tracking problem is formulated to minimize the cost of accumulated capital injection. We first transform the original problem with floor constraints into an unconstrained control problem, however, under a running maximum cost. By identifying a controlled state process with r…
The mean-variance hedging (MVH) problem is studied in a partially observable market where the drift processes can only be inferred through the observation of asset or index processes. Although most of the literatures treat the MVH problem by the duality method, here we study a system consisting of three BSDEs derived by Mania and Tevzadze (2003) and Mania et.al.(2008) and try to provide more explicit expressions directly implementable by practitioners. Under the Bayesian and Kalman-Bucy frameworks, we find that a relevant BSDE yields a semi-closed solution via a simple set of ODEs which allow …
We maximize the expected utility from terminal wealth for an HARA investor when the market price of risk is an unobservable random variable. We compute the optimal portfolio explicitly and explore the effects of learning by comparing it with the corresponding myopic policy. In particular, we show that, for a market price of risk constant in sign, the ratio between the portfolio under partial observation and its myopic counterpart increases with respect to risk tolerance. As a consequence, the absolute value of the partial observation case is larger (smaller) than the myopic one if the investor…
Given a set of assets and an investment capital, the classical portfolio selection problem consists in determining the amount of capital to be invested in each asset in order to build the most profitable portfolio. The portfolio optimization problem is naturally modeled as a mean-risk bi-criteria optimization problem where the mean rate of return of the portfolio must be maximized whereas a given risk measure must be minimized. Several mathematical programming models and techniques have been presented in the literature in order to efficiently solve the portfolio problem. A relatively recent pr…
We introduce a new set of consistent measures of risks, in terms of the semi-invariants of pdf's, such that the centered moments and the cumulants of the portfolio distribution of returns that put more emphasis on the tail the distributions. We derive generalized efficient frontiers, based on these novel measures of risks and present the generalized CAPM, both in the cases of homogeneous and heterogeneous markets. Then, using a family of modified Weibull distributions, encompassing both sub-exponentials and super-exponentials, to parameterize the marginal distributions of asset returns and the…
We consider the problem of minimizing capital at risk in the Black-Scholes setting. The portfolio problem is studied given the possibility that a correlation constraint between the portfolio and a financial index is imposed. The optimal portfolio is obtained in closed form. The effects of the correlation constraint are explored; it turns out that this portfolio constraint leads to a more diversified portfolio.
The Black-Scholes model, defined under the assumption of a perfect financial market, theoretically creates a flawless hedging strategy allowing the trader to evade risks in a portfolio of options. However, the concept of a "perfect financial market," which requires zero transaction and continuous trading, is challenging to meet in the real world. Despite such widely known limitations, academics have failed to develop alternative models successful enough to be long-established. In this paper, we explore the landscape of Deep Neural Networks(DNN) based hedging systems by testing the hedging capa…
We propose and study a simple model of dynamical redistribution of capital in a diversified portfolio. We consider a hypothetical situation of a portfolio composed of N uncorrelated stocks. Each stock price follows a multiplicative random walk with identical drift and dispersion. The rules of our model naturally give rise to power law tails in the distribution of capital fractions invested in different stocks. The exponent of this scale free distribution is calculated in both discrete and continuous time formalism. It is demonstrated that the dynamical redistribution strategy results in a larg…
In this paper we develop a novel methodology for estimation of risk capital allocation. The methodology is rooted in the theory of risk measures. We work within a general, but tractable class of law-invariant coherent risk measures, with a particular focus on expected shortfall. We introduce the concept of fair capital allocations and provide explicit formulae for fair capital allocations in case when the constituents of the risky portfolio are jointly normally distributed. The main focus of the paper is on the problem of approximating fair portfolio allocations in the case of not fully known …
Cryptocurrency markets have attracted many interest for global investors because of their novelty, wide online availability, increasing capitalization and potential profits. In the econophysics tradition we show that many of the most available cryptocurrencies have return statistics that do not follow Gaussian distributions but heavy--tailed distributions instead. Entropy measures are also applied showing that portfolio diversification is a reasonable practice for decreasing return uncertainty.
Banks must manage their trading books, not just value them. Pricing includes valuation adjustments collectively known as XVA (at least credit, funding, capital and tax), so management must also include XVA. In trading book management we focus on pricing, hedging, and allocation of prices or hedging costs to desks on an individual trade basis. We show how to combine three technical elements to radically simplify XVA management, both in terms of the calculations, and the implementation of the calculations. The three technical elements are: trade-level regression; analytic computation of sensitiv…
We design an optimal strategy for investment in a portfolio of assets subject to a multiplicative Brownian motion. The strategy provides the maximal typical long-term growth rate of investor's capital. We determine the optimal fraction of capital that an investor should keep in risky assets as well as weights of different assets in an optimal portfolio. In this approach both average return and volatility of an asset are relevant indicators determining its optimal weight. Our results are particularly relevant for very risky assets when traditional continuous-time Gaussian portfolio theories are…
