The St. Petersburg Paradox, an important topic in probability theory, has not been solved in the last 280 years. Since Nicolaus Bernoulli proposed the St. Petersburg Paradox in 1738, many people had tried to solve it and had proposed various explanations, but all were not satisfactory. In this paper we propose a new pricing theory with several rules, which incidentally resolves this paradox. The new pricing theory states that so-called fair (reasonable) pricing should be judged by the seller and the buyer independently. Reasonable pricing for the seller may not be appropriate for the buyer. The seller cares about costs, while the buyer is concerned about the realistic prospect of returns.The pricing theory we proposed can be applied to financial markets to solve the confusion that financial asset return with fat tails distribution will cause the option pricing formula to fail, thus making up the theoretical defects of quantitative financial pricing theory.

In this article, we tackle the problem of a market maker in charge of a book of options on a single liquid underlying asset. By using an approximation of the portfolio in terms of its vega, we show that the seemingly high-dimensional stochastic optimal control problem of an option market maker is in fact tractable. More precisely, when volatility is modeled using a classical stochastic volatility model -- e.g. the Heston model -- the problem faced by an option market maker is characterized by a low-dimensional functional equation that can be solved numerically using a Euler scheme along with interpolation techniques, even for large portfolios. In order to illustrate our findings, numerical examples are provided.

The unprecedented cessation of human activities during the COVID-19 pandemic has affected global energy use and CO2 emissions from fossil fuel use and cement production. Here we show that the decrease in global fossil CO2 emissions during the first quarter of 2020 was of 5.8% (542 Mt CO2 with a 20% 1-{\sigma} uncertainty). Unlike other emerging estimates, ours show the temporal dynamics of emissions based on actual emissions data from power generation (for 29 countries) and industry (for 73 countries), on near real time activity data for road transportation (for 132 countries), aviation and maritime transportation, and on heating degree days for commercial and residential sectors emissions (for 206 countries). These dynamic estimates cover all of the human induced CO2 emissions from fossil fuel combustion and cement production. The largest share of COVID-related decreases in emissions are due to decreases in industry (157.9 Mt CO2, -7.1% compared to 2019), followed by road transportation (145.7 Mt CO2, -8.3%), power generation (131.6 Mt CO2, -3.8%), residential (47.8 Mt CO2, -3.6%), fishing and maritime transport (35.5Mt CO2, -13.3%) and aviation (33.4 Mt CO2, -8.0%). Regionally, decreases in emissions from China were the largest and earliest (-10.3%), followed by Europe (EU-27 & UK) (-4.3%) and the U.S. (-4.2%). Relative decreases of regional CO2 emissions are consistent with regional nitrogen oxides concentrations observed by satellites and ground-based networks. Despite the unprecedented decreases in CO2 emissions and comparable decreases in economic activities, we monitored decreases in the carbon intensity (Emission per unit of GDP) in China (3.5%), the U.S. (4.5%) and Europe (5.4%) over the first quarter, suggesting that carbon-intensive activities have been disproportionally impacted.

In this paper I extend the work of Bernhardt and Donnelly (2019) dealing with modern explicit tontines, as a way of providing income under a specified bequest motive, from a defined contribution pension pot. A key feature of the present paper is that it relaxes the assumption of fixed proportions invested in tontine and bequest accounts. In making the bequest proportion an additional control function I obtain, hitherto unavailable, closed-form solutions for the fractional consumption rate, wealth, bequest amount, and bequest proportion under a constant relative risk averse utility. I show that the optimal bequest proportion is the product of the optimum fractional consumption rate and an exponentiated bequest parameter. Typical scenarios are explored using UK Office of National Statistics life tables, showing the behaviour of these characteristics under varying degrees of constant relative risk aversion.

This article presents FVA and CVA of a bilateral derivative in a coherent manner, based on recent developments in fair value accounting and ISDA standards. We argue that a derivative liability, after primary risk factors being hedged, resembles in economics an issued variable funding note, and should be priced at the market rate of the issuer's debt. For the purpose of determining the fair value, the party on the liability side is economically neutral to make a deposit to the other party, which earns his current debt rate and effectively provides funding and hedging for the party holding the derivative asset. The newly derived partial differential equation for an option discounts the derivative's receivable part with counterparty's curve and payable part with own financing curve. The price difference from the counterparty risk free price, or total counterparty risk adjustment, is precisely defined by discounting the product of the risk free price and the credit spread at the local liability curve. Subsequently the adjustment can be broken into a default risk component -- CVA and a funding component -- FVA, consistent with a simple note's fair value treatment and in accordance with the usual understanding of a bond's credit spread consisting of a CDS spread and a basis. As for FVA, we define a cost -- credit funding adjustment (CFA) and a benefit -- debit funding adjustment (DFA), in parallel to CVA and DVA and attributed to counterparty's and own funding basis. This resolves a number of outstanding FVA debate issues, such as double counting, violation of the law of one price, misuse of cash flow discounting, and controversial hedging of own default risk. It also allows an integrated implementation strategy and reuse of existing CVA infrastructure.

While Indices, Index tracking funds and ETFs have grown in popularity during then last ten years, there are many structural problems inherent in Index calculation methodologies and the legal/economic structure of ETFs. These problems raise actionable issues of Suitability and fraud under US securities laws, because most Indices and ETFs are misleading, have substantial tracking errors and dont reflect what they are supposed to track. This article contributes to the existing literature by: a) introducing and characterizing the errors and Biases inherent in risk-adjusted index weighting methods and the associated adverse effects; b) showing how these biases/effects inherent in Index calculation methods reduce social welfare, and can form the basis for harmful arbitrage activities.

Derivative pricing is about cash flow discounting at the riskfree rate. This teaching has lost its meaning post the financial crisis, due to the addition of extra value adjustments (XVA), which also made derivatives pricing and valuation a very difficult task for investors. This article recovers a properly defined discount rate that corresponds to different collateral and margin schemes. A binomial tree model is developed, enabling end-users to price in counterparty default and funding risk. Coherent XVAs, if needed, naturally result from decomposing the discount rate, and can be computed on the same tree.

Recent studies concerning the point electricity price forecasting have shown evidence that the hourly German Intraday Continuous Market is weak-form efficient. Therefore, we take a novel, advanced approach to the problem. A probabilistic forecasting of the hourly intraday electricity prices is performed by simulating trajectories in every trading window to receive a realistic ensemble to allow for more efficient intraday trading and redispatch. A generalized additive model is fitted to the price differences with the assumption that they follow a mixture of the Dirac and the Student's t-distributions. Moreover, the mixing term is estimated using a high-dimensional logistic regression with lasso penalty. We model the expected value and volatility of the series using i.a. autoregressive and no-trade effects or load, wind and solar generation forecasts and accounting for the non-linearities in e.g. time to maturity. Both the in-sample characteristics and forecasting performance are analysed using a rolling window forecasting study. Multiple versions of the model are compared to several benchmark models. The study aims to forecast the price distribution in the German Intraday Continuous Market in the last 3 hours of trading, but the approach allows for application to other continuous markets. The results prove superiority of the mixture model over the benchmarks gaining the most from the modelling of the volatility.

We conduct a unique, Amazon MTurk-based global experiment to investigate the importance of an exponential-growth prediction bias (EGPB) in understanding why the COVID-19 outbreak has exploded. The scientific basis for our inquiry is the well-established fact that disease spread, especially in the initial stages, follows an exponential function meaning few positive cases can explode into a widespread pandemic if the disease is sufficiently transmittable. We define prediction bias as the systematic error arising from faulty prediction of the number of cases x-weeks hence when presented with y-weeks of prior, actual data on the same. Our design permits us to identify the root of this under-prediction as an EGPB arising from the general tendency to underestimate the speed at which exponential processes unfold. Our data reveals that the "degree of convexity" reflected in the predicted path of the disease is significantly and substantially lower than the actual path. The bias is significantly higher for respondents from countries at a later stage relative to those at an early stage of disease progression. We find that individuals who exhibit EGPB are also more likely to reveal markedly reduced compliance with the WHO-recommended safety measures, find general violations of safety protocols less alarming, and show greater faith in their government's actions. A simple behavioral nudge which shows prior data in terms of raw numbers, as opposed to a graph, causally reduces EGPB. Clear communication of risk via raw numbers could increase accuracy of risk perception, in turn facilitating compliance with suggested protective behaviors.

An option market maker incurs funding costs when carrying and hedging inventory. To hedge a net long delta inventory, for example, she pays a fee to borrow stock from the securities lending market. Because of haircuts, she posts additional cash margin to the lender which needs to be financed at her unsecured debt rate. This paper incorporates funding asymmetry (borrowed cash and invested cash earning different interest rates) and realistic stock financing cost into the classic option pricing theory. It is shown that an option position can be dynamically replicated and self financed in the presence of these funding costs. Noting that the funding amounts and costs are different for long and short positions, we extend Black-Scholes partial differential equations (PDE) per position side. The PDE's nonlinear funding cost terms create a free funding boundary and would result in the bid price for a long position on an option lower than the ask price for a short position. An iterative Crank-Nicholson finite difference method is developed to compute European and American vanilla option prices. Numerical results show that reasonable funding cost parameters can easily produce same magnitude of bid/ask spread of less liquid, longer term options as observed in the market place. Portfolio level pricing examples show the netting effect of hedges, which could moderate impact of funding costs.

Although not a formal pricing consideration, gap risk or hedging errors are the norm of derivatives businesses. Starting with the gap risk during a margin period of risk of a repurchase agreement (repo), this article extends the Black-Scholes-Merton option pricing framework by introducing a reserve capital approach to the hedging error's irreducible variability. An extended partial differential equation is derived with two new terms for expected gap loss and economic capital charge, leading to the gap risk economic value adjustment and capital valuation adjustment (KVA) respectively. Practical repo pricing formulae is obtained showing that the break-even repo rate decomposes into cost of fund and economic capital charge in KVA. At zero haircut, a one-year term repo on main equities could command a capital charge as large as 50 basis points for a 'BBB' rated borrower.

Haircutting non-cash collateral has become a key element of the post-crisis reform of the shadow banking system and OTC derivatives markets. This article develops a parametric haircut model by expanding haircut definitions beyond the traditional value-at-risk measure and employing a double-exponential jump-diffusion model for collateral market risk. Haircuts are solved to target credit risk measurements, including probability of default, expected loss or unexpected loss criteria. Comparing to data-driven approach typically run on proxy data series, the model enables sensitivity analysis and stress test, captures market liquidity risk, allows idiosyncratic risk adjustments, and incorporates relevant market information. Computational results for main equities, securitization, and corporate bonds show potential for uses in collateral agreements, e.g. CSAs, and for regulatory capital calculations.

This article introduces new models of disintermediation of the real estate broker by the buyer or the seller. The decision to retain a real estate broker is critical in the property purchase/sale process. The existing literature does not contain analysis of: 1) information asymmetry, 2) the conditions under which it will be optimal to disintermediate the broker, 3) social capital and reputation, 4) the impact of different types of real estate brokerage contracts. The article shows that dis-intermediation of the real estate broker by the seller or buyer may be optimal in certain conditions.

An uncollateralized swap hedged back-to-back by a CCP swap is used to introduce FVA. The open IR01 of FVA, however, is a sure sign of risk not being fully hedged, a theoretical no-arbitrage pricing concern, and a bait to lure market risk capital, a practical business concern. By dynamically trading the CCP swap, with the liability-side counterparty provides counterparty exposure hedge and swap funding, we find that the uncollateralized swap can be fully replicated, leaving out no IR01 leakage. The fair value of the swap is obtained by applying to swap's net cash flows a discount rate switching to counterparty's bond curve if the swap is a local asset or one's own curve if a liability, and the total valuation adjustment is the present value of cost of funding the risk-free price discounted at the same switching rate. FVA is redefined as a liquidity or funding basis component of total valuation adjustment, coherent with CVA, the default risk component. A Longstaff-Schwartz style least-square regression and simulation is introduced to compute the recursive fair value and adjustments. A separately developed finite difference scheme is used to test and find regression necessary to decouple the discount rate switch. Preliminary results show the impact of counterparty risk to swap hedge ratios, swap bid/ask spreads, and valuation adjustments, and considerable errors of calculating CVA by discounting cash flow or potential future exposure.

This article prices OTC derivatives with either an exogenously determined initial margin profile or endogenously approximated initial margin. In the former case, margin valuation adjustment (MVA) is defined as the liability-side discounted expected margin profile, while in the latter, an extended partial differential equation is derived and solved for an all-in fair value, decomposable into coherent CVA, FVA and MVA. For uncollateralized customer trades, MVA can be transferred to the customer via an extension of the liability-side pricing theory. For BCBS-IOSCO covered OTC derivatives, a market maker has to charge financial counterparties a bid-ask spread to transfer its funding cost. An IM multiplier is applied to calibrate to external IM models to allow portfolio incremental pricing. In particular, a link to ISDA SIMM for equity, commodity and fx risks is established through the PDE with its vega and curvature IM components captured fully. Numerical examples are given for swaps and equity portfolios and offer a plausible attribution of recent CME-LCH basis spread widening to elevated MVA accompanying dealers' hedging of customer flows.

We study a sequential-learning model featuring a network of naive agents with Gaussian information structures. Agents apply a heuristic rule to aggregate predecessors' actions. They weigh these actions according the strengths of their social connections to different predecessors. We show this rule arises endogenously when agents wrongly believe others act solely on private information and thus neglect redundancies among observations. We provide a simple linear formula expressing agents' actions in terms of network paths and use this formula to characterize the set of networks where naive agents eventually learn correctly. This characterization implies that, on all networks where later agents observe more than one neighbor, there exist disproportionately influential early agents who can cause herding on incorrect actions. Going beyond existing social-learning results, we compute the probability of such mislearning exactly. This allows us to compare likelihoods of incorrect herding, and hence expected welfare losses, across network structures. The probability of mislearning increases when link densities are higher and when networks are more integrated. In partially segregated networks, divergent early signals can lead to persistent disagreement between groups.

Utilizing a generative regime switching framework, we perform Monte-Carlo simulations of asset returns for Value at Risk threshold estimation. Using equity markets and long term bonds as test assets in the global, US, Euro area and UK setting over an up to 1,250 weeks sample horizon ending in August 2018, we investigate neural networks along three design steps relating (i) to the initialization of the neural network, (ii) its incentive function according to which it has been trained and (iii) the amount of data we feed. First, we compare neural networks with random seeding with networks that are initialized via estimations from the best-established model (i.e. the Hidden Markov). We find latter to outperform in terms of the frequency of VaR breaches (i.e. the realized return falling short of the estimated VaR threshold). Second, we balance the incentive structure of the loss function of our networks by adding a second objective to the training instructions so that the neural networks optimize for accuracy while also aiming to stay in empirically realistic regime distributions (i.e. bull vs. bear market frequencies). In particular this design feature enables the balanced incentive recurrent neural network (RNN) to outperform the single incentive RNN as well as any other neural network or established approach by statistically and economically significant levels. Third, we half our training data set of 2,000 days. We find our networks when fed with substantially less data (i.e. 1,000 days) to perform significantly worse which highlights a crucial weakness of neural networks in their dependence on very large data sets ...

A phase plot of the oil economy is built using the literature data of world oil production, price, and EROEI (Energy Returned on Energy Invested). An analogy between the oil economy and the Benard convection is proposed; some methods of interpretation and forecast of the system behavior are also shown based on "phase portrait" using as main variables the price, production and EROEI values. A scenery is proposed on this basis.

This article introduces decision models for commercial real estate leasing. The concepts and models developed in the article can also be applied to equipment leasing and other types of leasing.

Over sixty percent of employees at a large South African financial services company select the minimum rate of 7.5 percent for their monthly retirement contributions, far below the recommended rate of 15 percent. I use a field experiment to investigate whether providing employees with a retirement calculator, which shows projections of retirement income, leads to increases in contributions. The average treatment effect is positive but very small and not statistically different from zero.

Although the CML (Capital Market Line), the Intertemporal-CAPM, the CAPM/SML (Security Market Line) and the Intertemporal Arbitrage Pricing Theory (IAPT) are widely used in portfolio management, valuation and capital markets financing; these theories are inaccurate and can adversely affect risk management and portfolio management processes. This article introduces several empirically testable financial theories that provide insights, and can be calibrated to real data and used to solve problems, and contributes to the literature by: i) explaining the conditions under which ICAPM/CAPM, IAPT and CML may be accurate, and why such conditions are not feasible; and explaining why the existence of incomplete markets and dynamic un-aggregated markets render CML, IAPT and ICAPM inaccurate; ii) explaining why the Consumption-Savings-InvestmentProduction framework is insufficient for asset pricing and analysis of changes in risk and asset values; and introducing a unified approach to asset pricing that simultaneously considers six factors, and the conditions under which this approach will work; iii) explaining why leisure, taxes and housing are equally as important as consumption and investment in asset pricing; iv) introducing the Marginal Rate of Intertemporal Joint Substitution (MRIJS) among Consumption, Taxes, Investment, Leisure, Intangibles and Housing - this model incorporates Regret Theory and captures features of reality that dont fit well into standard asset pricing models, and this framework can support specific or very general finance theories and or very complicated models; v) showing why the Elasticity of Intertemporal Substitution (EIS) is inaccurate and is insufficient for asset pricing and analysis of investor preferences.

This article develops a haircut model by treating repos as debt investments and seeks haircuts to control counterparty contingent exposure to asset price gap risk. It corroborates well with empirically stylized facts, explains tri-party and bilateral repo haircut differences, recasts haircut increases during the financial crisis, and sets a limit on access liquidity dealers can extract while acting as funding intermediaries between money market funds and hedge funds. Once a haircut is set, repo's residual risk becomes a pricing challenge, as is neither hedgeable nor diversifiable. We propose a capital pricing approach of computing repo economic capital and charging the borrower a cost of capital. Capital charge is shown to be countercyclical and a key element of repo pricing and used in explaining the repo pricing puzzle and maturity compression phenomenon.

Securitization has become prevalent in many countries, and has substantial impact on government monetary policy and fiscal policy which have not yet been adequately analyzed in the existing literature. This article develops optimal conditions for efficient securitization, identifies constraints on securitization, and analyzes the interactions of capital-reserve requirements and securitization. This article introduces new decision models and theories of asset-securitization.

Identifying risk spillovers in financial markets is of great importance for assessing systemic risk and portfolio management. Granger causality in tail (or in risk) tests whether past extreme events of a time series help predicting future extreme events of another time series. The topology and connectedness of networks built with Granger causality in tail can be used to measure systemic risk and to identify risk transmitters. Here we introduce a novel test of Granger causality in tail which adopts the likelihood ratio statistic and is based on the multivariate generalization of a discrete autoregressive process for binary time series describing the sequence of extreme events of the underlying price dynamics. The proposed test has very good size and power in finite samples, especially for large sample size, allows inferring the correct time scale at which the causal interaction takes place, and it is flexible enough for multivariate extension when more than two time series are considered in order to decrease false detections as spurious effect of neglected variables. An extensive simulation study shows the performances of the proposed method with a large variety of data generating processes and it introduces also the comparison with the test of Granger causality in tail by [Hong et al., 2009]. We report both advantages and drawbacks of the different approaches, pointing out some crucial aspects related to the false detections of Granger causality for tail events. An empirical application to high frequency data of a portfolio of US stocks highlights the merits of our novel approach.

A negative basis trade enters a long bond position and buys protection on the issuer of the bond through credit default swap (CDS), aiming at arbitrage profit due to the bond-CDS basis. To classic reduced form model theorists, the existence of the basis is an abnormality or merely liquidity noise. Such a view, however, fails to explain large basis trading losses incurred during the financial crisis. Employing a bond continuously hedged by CDS under a dynamic spread model with bond repo financing, we find that there is unhedged and unhedgeable residual jump to default risk that can't be diversified because of credit correlation. An economic capital approach has to apply and a charge on the use of capital follows. Together with the hedge funding cost, it allows us to better understand the basis's economics and to predict its fair level.

Based on data from the European banking stress tests of 2014, 2016 and the transparency exercise of 2018 we demonstrate for the first time that the latent geometry of financial networks can be well-represented by geometry of negative curvature, i.e., by hyperbolic geometry. This allows us to connect the network structure to the popularity-vs-similarity model of Papdopoulos et al., which is based on the Poincar\'e disc model of hyperbolic geometry. We show that the latent dimensions of `popularity' and `similarity' in this model are strongly associated to systemic importance and to geographic subdivisions of the banking system. In a longitudinal analysis over the time span from 2014 to 2018 we find that the systemic importance of individual banks has remained rather stable, while the peripheral community structure exhibits more (but still moderate) variability.