Research articles for the 2020-07-06

A Theory of Equivalent Expectation Measures for Expected Prices of Contingent Claims
Sanjay K. Nawalkha,Xiaoyang Zhuo

This paper introduces a theory of equivalent expectation measures, such as the R measure and the RT1 measure, generalizing the martingale pricing theory of Harrison and Kreps (1979) for deriving analytical solutions of expected prices - both the expected current price and the expected future price - of contingent claims. We also present new R-transforms which extend the Q-transforms of Bakshi and Madan (2000) and Duffie et al. (2000), for computing the expected prices of a variety of standard and exotic claims under a broad range of stochastic processes. Finally, as a generalization of Breeden and Litzenberger (1978), we propose a new concept of the expected future state price density which allows the estimation of the expected future prices of complex European contingent claims as well as the physical density of the underlying asset's future price, using the current prices and only the first return moment of standard European OTM call and put options.

Analytical scores for stress scenarios
Pierre Cohort,Jacopo Corbetta,Ismail Laachir

In this work, inspired by the Archer-Mouy-Selmi approach, we present two methodologies for scoring the stress test scenarios used by CCPs for sizing their Default Funds. These methodologies can be used by risk managers to compare different sets of scenarios and could be particularly useful when evaluating the relevance of adding new scenarios to a pre-existing set.

Coronavirus Perceptions And Economic Anxiety
Thiemo Fetzer,Lukas Hensel,Johannes Hermle,Christopher Roth

We provide one of the first systematic assessments of the development and determinants of economic anxiety at the onset of the coronavirus pandemic. Using a global dataset on internet searches and two representative surveys from the US, we document a substantial increase in economic anxiety during and after the arrival of the coronavirus. We also document a large dispersion in beliefs about the pandemic risk factors of the coronavirus, and demonstrate that these beliefs causally affect individuals' economic anxieties. Finally, we show that individuals' mental models of infectious disease spread understate non-linear growth and shape the extent of economic anxiety.

Deep Importance Sampling
Benjamin Virrion

We present a generic path-dependent importance sampling algorithm where the Girsanov induced change of probability on the path space is represented by a sequence of neural networks taking the past of the trajectory as an input. At each learning step, the neural networks' parameters are trained so as to reduce the variance of the Monte Carlo estimator induced by this change of measure. This allows for a generic path dependent change of measure which can be used to reduce the variance of any path-dependent financial payoff. We show in our numerical experiments that for payoffs consisting of either a call, an asymmetric combination of calls and puts, a symmetric combination of calls and puts, a multi coupon autocall or a single coupon autocall, we are able to reduce the variance of the Monte Carlo estimators by factors between 2 and 9. The numerical experiments also show that the method is very robust to changes in the parameter values, which means that in practice, the training can be done offline and only updated on a weekly basis.

Egalitarian and Just Digital Currency Networks
Gal Shahaf,Ehud Shapiro,Nimrod Talmon

Cryptocurrencies are a digital medium of exchange with decentralized control that renders the community operating the cryptocurrency its sovereign. Leading cryptocurrencies use proof-of-work or proof-of-stake to reach consensus, thus are inherently plutocratic. This plutocracy is reflected not only in control over execution, but also in the distribution of new wealth, giving rise to ``rich get richer'' phenomena. Here, we explore the possibility of an alternative digital currency that is egalitarian in control and just in the distribution of created wealth. Such currencies can form and grow in grassroots and sybil-resilient way. A single currency community can achieve distributive justice by egalitarian coin minting, whereby each member mints one coin at every time step. Egalitarian minting results, in the limit, in the dilution of any inherited assets and in each member having an equal share of the minted currency, adjusted by the relative productivity of the members. Our main theorem shows that a currency network, where agents can be members of more than one currency community, can achieve distributive justice globally across the network by joint egalitarian minting, whereby each agent mints one coin in only one community at each timestep. Specifically, we show that a sufficiently large intersection between two communities -- relative to the gap in their productivity -- will cause the exchange rates between their currencies to converge to 1:1, resulting in global distributive justice.

Impact of COVID-19 on Forecasting Stock Prices: An Integration of Stationary Wavelet Transform and Bidirectional Long Short-Term Memory
Daniel Štifanić,Jelena Musulin,Adrijana Miočević,Sandi Baressi Šegota,Roman Šubić,Zlatan Car

COVID-19 is an infectious disease that mostly affects the respiratory system. At the time of this research being performed, there were more than 1.4 million cases of COVID-19, and one of the biggest anxieties is not just our health, but our livelihoods, too. In this research, authors investigate the impact of COVID-19 on the global economy, more specifically, the impact of COVID-19 on financial movement of Crude Oil price and three U.S. stock indexes: DJI, S&P 500 and NASDAQ Composite. The proposed system for predicting commodity and stock prices integrates the Stationary Wavelet Transform (SWT) and Bidirectional Long Short-Term Memory (BDLSTM) networks. Firstly, SWT is used to decompose the data into approximation and detail coefficients. After decomposition, data of Crude Oil price and stock market indexes along with COVID-19 confirmed cases were used as input variables for future price movement forecasting. As a result, the proposed system BDLSTM+WT-ADA achieved satisfactory results in terms of five-day Crude Oil price forecast.

Liquidity Provider Returns in Geometric Mean Markets
Alex Evans

Geometric mean market makers (G3Ms), such as Uniswap and Balancer, comprise a popular class of automated market makers (AMMs) defined by the following rule: the reserves of the AMM before and after each trade must have the same (weighted) geometric mean. This paper extends several results known for constant-weight G3Ms to the general case of G3Ms with time-varying and potentially stochastic weights. These results include the returns and no-arbitrage prices of liquidity pool (LP) shares that investors receive for supplying liquidity to G3Ms. Using these expressions, we show how to create G3Ms whose LP shares replicate the payoffs of financial derivatives. The resulting hedges are model-independent and exact for derivative contracts whose payoff functions satisfy an elasticity constraint. These strategies allow LP shares to replicate various trading strategies and financial contracts, including standard options. G3Ms are thus shown to be capable of recreating a variety of active trading strategies through passive positions in LP shares.

Markovian approximation of the rough Bergomi model for Monte Carlo option pricing
Qinwen Zhu,Grégoire Loeper,Wen Chen,Nicolas Langrené

The recently developed rough Bergomi (rBergomi) model is a rough fractional stochastic volatility (RFSV) model which can generate more realistic term structure of at-the-money volatility skews compared with other RFSV models. However, its non-Markovianity brings mathematical and computational challenges for model calibration and simulation. To overcome these difficulties, we show that the rBergomi model can be approximated by the Bergomi model, which has the Markovian property. Our main theoretical result is to establish and describe the affine structure of the rBergomi model. We demonstrate the efficiency and accuracy of our method by implementing a Markovian approximation algorithm based on a hybrid scheme.

Methods for forecasting the effect of exogenous risks on stock markets
Karina Arias-Calluari,Fernando Alonso-Marroquin,Morteza Nattagh-Najafi,Michael Harré

Markets are subjected to both endogenous and exogenous risks that have caused disruptions to financial and economic markets around the globe, leading eventually to fast stock market declines. In the past, markets have recovered after any economic disruption. On this basis, we focus on the outbreak of COVID-19 as a case study of an exogenous risk and analyze its impact on the Standard and Poor's 500 (S\&P500) index. We assumed that the S\&P500 index reaches a minimum before rising again in the not-too-distant future. Here we present two cases to forecast the S\&P500 index. The first case uses an estimation of expected deaths released on 02/04/2020 by the University of Washington. For the second case, it is assumed that the peak number of deaths will occur 2-months since the first confirmed case occurred in the USA. The decline and recovery in the index were estimated for the following three months after the initial point of the predicted trend. The forecast is a projection of a prediction with stochastic fluctuations described by $q$-gaussian diffusion process with three spatio-temporal regimes. Our forecast was made on the premise that any market response can be decomposed into an overall deterministic trend and a stochastic term. The prediction was based on the deterministic part and for this case study is approximated by the extrapolation of the S\&P500 data trend in the initial stages of the outbreak. The stochastic fluctuations have the same structure as the one derived from the past 24 years. A reasonable forecast was achieved with 85\% of accuracy.

Model order reduction for parametric high dimensional models in the analysis of financial risk
Andreas Binder,Onkar Jadhav,Volker Mehrmann

This paper presents a model order reduction (MOR) approach for high dimensional problems in the analysis of financial risk. To understand the financial risks and possible outcomes, we have to perform several thousand simulations of the underlying product. These simulations are expensive and create a need for efficient computational performance. Thus, to tackle this problem, we establish a MOR approach based on a proper orthogonal decomposition (POD) method. The study involves the computations of high dimensional parametric convection-diffusion reaction partial differential equations (PDEs). POD requires to solve the high dimensional model at some parameter values to generate a reduced-order basis. We propose an adaptive greedy sampling technique based on surrogate modeling for the selection of the sample parameter set that is analyzed, implemented, and tested on the industrial data. The results obtained for the numerical example of a floater with cap and floor under the Hull-White model indicate that the MOR approach works well for short-rate models.

Note on simulation pricing of $\pi$-options
Zbigniew Palmowski,Tomasz Serafin

In this work, we adapt a Monte Carlo algorithm introduced by Broadie and Glasserman (1997) to price a $\pi$-option. This method is based on the simulated price tree that comes from discretization and replication of possible trajectories of the underlying asset's price. As a result this algorithm produces lower and upper bounds that converge to the true price with the increasing depth of the tree. Under specific parametrization, this $\pi$-option is related to relative maximum drawdown and can be used in the real-market environment to protect a portfolio against volatile and unexpected price drops. We also provide some numerical analysis.

Numerical Scheme for Game Options in Local Volatility models
Benjamin Gottesman Berdah

In this paper we introduce a numerical method for optimal stopping in the framework of one dimensional diffusion. We use the Skorokhod embedding in order to construct recombining tree approximations for diffusions with general coefficients. This technique allows us to determine convergence rates and construct nearly optimal stopping times which are optimal at the same rate. Finally, we demonstrate the efficiency of our scheme with several examples of game options.

Optimal Consumption with Reference to Past Spending Maximum
Shuoqing Deng,Xun Li,Huyen Pham,Xiang Yu

This paper studies an infinite horizon optimal consumption problem under exponential utility, together with non-negativity constraint on consumption rate and a reference point to the past consumption peak. The performance is measured by the distance between the consumption rate and a fraction $0\leq\lambda\leq 1$ of the historical consumption maximum. To overcome its path-dependent nature, the consumption running maximum process is chosen as an auxiliary state process that renders the value function two dimensional depending on the wealth variable $x$ and the reference variable $h$. The associated Hamilton-Jacobi-Bellman (HJB) equation is expressed in the piecewise manner across different regions to take into account constraints. By employing the dual transform and smooth-fit principle, the classical solution of the HJB equation is obtained in an analytical form, which in turn provides the feedback optimal investment and consumption. For $0<\lambda<1$, we are able to find four boundary curves $x_1(h)$, $\breve{x}(h)$, $x_2(h)$ and $x_3(h)$ for the wealth level $x$ that are nonlinear functions of $h$ such that the feedback optimal consumption satisfies: (i) $c^*(x,h)=0$ when $x\leq x_1(h)$; (ii) $0<c^*(x,h)<\lambda h$ when $x_1(h)<x<\breve{x}(h)$; (iii) $\lambda h\leq c^*(x,h)<h$ when $\breve{x}(h)\leq x<x_2(h)$; (iv) $c^*(x,h)=h$ but the running maximum process remains flat when $x_2(h)\leq x<x_3(h)$; (v) $c^*(x,h)=h$ and the running maximum process increases when $x=x_3(h)$. Similar conclusions can be made in a simpler fashion for two extreme cases $\lambda=0$ and $\lambda=1$. Numerical examples are also presented to illustrate some theoretical results and financial insights.

Optimal portfolios for different anticipating integrals under insider information
Carlos Escudero,Sandra Ranilla-Cortina

We consider the non-adapted version of a simple problem of portfolio optimization in a financial market that results from the presence of insider information. We analyze it via anticipating stochastic calculus and compare the results obtained by means of the Russo-Vallois forward, the Ayed-Kuo, and the Hitsuda-Skorokhod integrals. We compute the optimal portfolio for each of these cases. Our results give a partial indication that, while the forward integral yields a portfolio that is financially meaningful, the Ayed-Kuo and the Hitsuda-Skorokhod integrals do not provide an appropriate investment strategy for this problem.

Pricing and hedging American-style options with deep learning
Sebastian Becker,Patrick Cheridito,Arnulf Jentzen

In this paper we introduce a deep learning method for pricing and hedging American-style options. It first computes a candidate optimal stopping policy. From there it derives a lower bound for the price. Then it calculates an upper bound, a point estimate and confidence intervals. Finally, it constructs an approximate dynamic hedging strategy. We test the approach on different specifications of a Bermudan max-call option. In all cases it produces highly accurate prices and dynamic hedging strategies with small replication errors.

Repo Haircuts and Economic Capital: A Theory of Repo Pricing
Wujiang Lou

A repurchase agreement lets investors borrow cash to buy securities. Financier only lends to securities' market value after a haircut and charges interest. Repo pricing is characterized with its puzzling dual pricing measures: repo haircut and repo spread. This article develops a repo haircut model by designing haircuts to achieve high credit criteria, and identifies economic capital for repo's default risk as the main driver of repo pricing. A simple repo spread formula is obtained that relates spread to haircuts negative linearly. An investor wishing to minimize all-in funding cost can settle at an optimal combination of haircut and repo rate. The model empirically reproduces repo haircut hikes concerning asset backed securities during the financial crisis. It explains tri-party and bilateral repo haircut differences, quantifies shortening tenor's risk reduction effect, and sets a limit on excess liquidity intermediating dealers can extract between money market funds and hedge funds.

Robust fundamental theorems of asset pricing in discrete time
Huy N. Chau

This paper is devoted to the study of robust fundamental theorems of asset pricing in discrete time and finite horizon settings. The new concept "robust pricing system" is introduced to rule out the existence of model independent arbitrage opportunities. Superhedging duality and strategy are obtained.