This paper studies the hedging of derivatives whose pricing formulas are periodically recalibrated in the presence of model risk. We assume that the price and implied parameter processes are observed in the market but the true model of these processes is unknown. Given multiple candidates for the true model, we define a model set of candidates for the true model. We study the minimum hedging error and an optimal strategy under the worst situation and show the procedure for their calculation. Furthermore some numerical examples are provided to illustrate the impact of model risk on the optimal hedging.

This paper addresses errors in mean return estimates in continuous-time asset allocation models. A standard approach postulates that stochastic factors explain expected asset returns. The problem is then to estimate these factors from observed asset prices via filtering. Recent advances have also combined asset prices with expert opinions to improve the estimates. However, these methods have limitations: stocks prices favor momentum strategies, and expert opinions require careful debiasing. To resolve these issues, we propose a jump-diffusion risk-sensitive benchmarked asset management model in which investors estimate the factors from both traditional and alternative data. We show that this model admits a unique C^1,2 solution, and we derive the optimal investment policy in quasi-closed form. We find that investors construct their portfolios from a passive core and an active satellite. The passive core adds considerations for jump risk to a simple benchmark replication. The active satellite blends security selection, and factor tilts with event-driven strategies unique to jump-diffusion problems. Thus, our model explains the most popular investment strategies. Furthermore, the improved expert forecast model and the introduction of alternative data provide factor tilters with new tools to sharpen their asset allocation.

We propose a continuous‐time model in which investors use expert forecasts to construct a benchmark‐outperforming portfolio in two steps. The estimation step takes the form of a Kalman filter. The control step derives the optimal investment policy in closed form and establishes that the value function is the unique classical solution to the Hamilton‐Jacobi‐Bellman partial differential equation. We show that the optimal investment policy generates a continuum of investment strategies, from passive benchmark replication to fully active bets in the Kelly portfolio. However, our model warns against over‐betting on financial markets. Moreover, we find that the Kelly portfolio performs both security selection and factor tilt. A simulation study with market data confirms that factor choice is critical at every stage of the investment process. Finally, debiasing is equally crucial. Portfolios with debiased expert forecasts outperform portfolios with biased forecasts.

We investigate the links between various no-arbitrage conditions and the existence of pricing functionals in general markets, and prove the Fundamental Theorem of Asset Pricing therein. No-arbitrage conditions, either in this abstract setting or in the case of a market consisting of European Call options, give rise to duality properties of infinite-dimensional sub- and super-hedging problems. With a view towards applications, we show how duality is preserved when reducing these problems over finite-dimensional bases. We finally perform a rigorous perturbation analysis of those linear programming problems, and highlight numerically the influence of smile extrapolation on the bounds of exotic options.

Expert forecasts are an essential component of asset management and an important research topic. However, the effect of behavioral biases on expert forecasts is generally ignored. This paper examines the effect of biased expert forecasts on asset allocations. We find that biases have a significant impact on portfolios, explaining nearly 70% of excess risk-taking in our implementation. To address the effect of behavioral biases, we propose an integrated behavioral continuous-time portfolio selection model which we solve in closed form. The model applies general principles to identify and reduce the impact of five main behavioral biases. This paper concludes with a new personal fractional Kelly decomposition to account for the effect of opinions on the optimal asset allocation.

We propose a continuous-time portfolio selection model that explains the active-passive continuum. Our model illuminates the pivotal role of expert opinions and factors in the asset allocation process. In the model, investors aim to outperform a benchmark. As securities and benchmark's drift depends on unobservable factors, portfolio selection becomes a partial observation risk-sensitive control problem. We find that the optimal investment policy combines Kelly, benchmark-tracking, and intertemporal hedging portfolios. The Kelly portfolio sums two alpha-generating strategies. Our simulation reveals the importance of debiasing expert opinions and shows that factor choice is crucial at every stage of the investment process.

‘The classical theory of option pricing’ explains the theory of arbitrage pricing, which is closely related to the Dutch Book Arguments, but which brings in a new factor: prices in financial markets evolve over time and participants are able to trade at any time, instead of just taking bets and awaiting the result. In addition to the general theory, pricing models and methods have been developed for specific markets—foreign exchange, interest rates, and credit. The binomial and continuous-time mathematical models for stock prices are introduced along with the Black–Scholes formula, the volatility surface, the difference between European and American options, and the Fundamental Theorem of Asset Pricing.

‘Fund management’ discusses the objective to form portfolios of assets so as to maximize the investment return. A mathematical finance-oriented approach to optimal investment, in the context of the Black–Scholes price model, was proposed by Robert Merton in 1969. Fund management is a huge industry, and has become much more technical with the emergence of hedge funds deploying sophisticated strategies. There have been many attempts at constructing mathematical models for asset allocation that match real market behaviour more closely. The basic problem is that markets appear so erratic. Is there anything about them that is more invariant? The scenario tree model for long-term asset liability management is explained.

The risk management function of a financial company monitors a whole range of risks that the company faces: market risk, credit risk, liquidity risk, operational risk, reputational risk, and legal risk. Some of these are connected to regulatory requirements, while others are internal procedures designed to assist the management of the company’s assets and liabilities. ‘Risk management’ focuses on market risk, which is concerned with assessing how sensitive the value of the company’s trading book is to anticipated movements in the market prices of the assets it contains. Evaluations are carried out at various levels of aggregation from individual trading desk to the company as a whole.

In recent years, the finance industry has mushroomed to become an important part of modern economies with many science and engineering graduates joining the industry as quantitative analysts, using mathematical and computational skills to solve complex problems of asset valuation and risk management. Mathematical Finance: A Very Short Introduction provides an overview of mathematical finance today. It introduces arbitrage theory, explaining why it works the way it does, and how it is key to pricing financial contracts, to credit trading, fund management, and the setting of interest rates. It also discusses developments to mathematical finance in the wake of the 2008 financial crash, and surveys the most pressing issues in mathematical finance today.

Credit risk is the risk that your counterparty might default on future obligations. There are a small number of credit rating agencies operating globally that assign a credit rating to each company under consideration. ‘Credit risk’ explains credit risk modelling and analysis, including credit default swaps, multi-asset credit risk, and collateralized debt obligations. Credit risk models are divided into two main categories: ‘structural form’ and ‘reduced form’. A pervasive problem in credit risk modelling is that while some parameters can be backed out by the calibration process, there are usually others about which the available data is insufficient for us to do anything more than take an educated guess.

We study a notion of local time for a continuous path, defined as a limit of suitable discrete quantities along a general sequence of partitions of the time interval. Our approach subsumes other existing definitions and agrees with the usual (stochastic) local times a.s. for paths of a continuous semimartingale. We establish pathwise version of the It\^o-Tanaka, change of variables and change of time formulae. We provide equivalent conditions for existence of pathwise local time. Finally, we study in detail how the limiting objects, the quadratic variation and the local time, depend on the choice of partitions. In particular, we show that an arbitrary given non-decreasing process can be achieved a.s. by the pathwise quadratic variation of a standard Brownian motion for a suitable sequence of (random) partitions; however, such degenerate behavior is excluded when the partitions are constructed from stopping times.

A new jump diffusion regime-switching model is introduced, which allows for linking jumps in asset prices with regime changes. We prove the existence and uniqueness of the solution to the risk-sensitive asset management criterion maximization problem in this setting. We provide an ODE for the optimal value function, which may be efficiently solved numerically. Relevant probability measure changes are discussed in the appendix. The recently introduced approach of Klebaner and Liptser (2013) is used to prove the martingale property of the relevant density processes.

Abstract This paper concerns sequential computation of risk measures for financial data and asks how, given a risk measurement procedure, we can tell whether the answers it produces are ‘correct’. We draw the distinction between ‘external’ and ‘internal’ risk measures and concentrate on the latter, where we observe data in real time, make predictions and observe outcomes. It is argued that evaluation of such procedures is best addressed from the point of view of probability forecasting or Dawid’s theory of ‘prequential statistics’ [12]. We introduce a concept of ‘calibration’ of a risk measure in a dynamic setting, following the precepts of Dawid’s weak and strong prequential principles, and examine its application to quantile forecasting (VaR – value at risk) and to mean estimation (applicable to CVaR – expected shortfall). The relationship between these ideas and ‘elicitability’ [24] is examined. We show in particular that VaR has special properties not shared by any other risk measure. Turning to CVaR we argue that its main deficiency is the unquantifiable tail dependence of estimators. In a final section we show that a simple data-driven feedback algorithm can produce VaR estimates on financial data that easily pass both the consistency test and a further newly-introduced statistical test for independence of a binary sequence.

In this article, the authors describe a simple procedure for combining statistical estimates with expert opinions to produce a view of future asset performance. The authors discuss the impact of behavioral bias on these views and propose general modeling principles to reduce this bias. They use standard linear filtering techniques to combine statistical estimates with expert opinions seamlessly and discuss applications to dynamic portfolio optimization. TOPICS:Performance measurement, portfolio construction

Utility indifference pricing is an effective method for investors to construct a strategy in an incomplete market. In fact, if an investor can trade a random endowment under the criteria shown by utility indifference pricing, they can devise financial contracts that are optimized according to their preferences. However, because it does not have the direct implication of equilibrium, the value of the random endowment given by indifference pricing is not necessarily the same as the market price. In this study, we attempt to derive the equilibrium of random endowment under the framework of indifference pricing. However, letting the utility function be of exponential type means that any trade involving random endowment will not appear in equilibrium. Thus, we show that non-zero trade in equilibrium appears by introducing uncertainty in a model, which is one of the sources of market incompleteness.

Stochastic optimisation has found a fertile ground for applications in finance. One of the greatest challenges remains to incorporate a set of scenarios that accurately model the behaviour of financial markets, and in particular their behaviour during crashes and crises, without sacrificing the tractability of the optimal investment policy. This paper shows how to incorporate return predictions and crash predictions as views into continuous time asset allocation models.

This paper considers the utility-based and indifference pricing in a market with transaction costs. The utility maximization problem, including contingent claims in the market with transaction costs, has been widely researched. In this paper, closely following the results of Bouchard (Financ Stoch 6:495–516, 2002), we consider the market equilibrium of contingent claims. This is done by specifying the utility function as exponential utility and, thus, determining equilibrium in the market with transaction costs. Unlike Davis and Yoshikawa (Math Finan Econ, 2015), we use the strong assumption to deduce the equilibrium at which trade does not occur (zero trade equilibrium). It implicitly shows that transaction costs may generate a non-zero trade equilibrium under a weaker assumption.