Self-financing Portfolio Research Articles

Dynamic Asset Pricing in a Unified Bachelier–Black–Scholes–Merton Model

We present a unified, market-complete model that integrates both Bachelier and Black–Scholes–Merton frameworks for asset pricing. The model allows for the study, within a unified framework, of asset pricing in a natural world that experiences the possibility of negative security prices or riskless rates. Unlike the classical Black–Scholes–Merton, we show that option pricing in the unified model differs depending on whether the replicating, self-financing portfolio uses riskless bonds or a single riskless bank account. We derive option price formulas and extend our analysis to the term structure of interest rates by deriving the pricing of zero-coupon bonds, forward contracts, and futures contracts. We identify a necessary condition for the unified model to support a perpetual derivative. Discrete binomial pricing under the unified model is also developed. In every scenario analyzed, we show that the unified model simplifies to the standard Black–Scholes–Merton pricing under specific limits and provides pricing in the Bachelier model limit. We note that the Bachelier limit within the unified model allows for positive riskless rates. The unified model prompts us to speculate on the possibility of a mixed multiplicative and additive deflator model for risk-neutral option pricing.

The Maximal and Minimal Distributions of Wealth Processes in Black–Scholes Markets

The Black–Scholes formula is an important formula for pricing a contingent claim in complete financial markets. This formula can be obtained under the assumption that the investor’s strategy is carried out according to a self-financing criterion; hence, there arise a set of self-financing portfolios corresponding to different contingent claims. The natural questions are: If an investor invests according to self-financing portfolios in the financial market, what are the maximal and minimal distributions of the investor’s wealth on some specific interval at the terminal time? Furthermore, if such distributions exist, how can the corresponding optimal portfolios be constructed? The present study applies the theory of backward stochastic differential equations in order to obtain an affirmative answer to the above questions. That is, the explicit formulations for the maximal and minimal distributions of wealth when adopting self-financing strategies would be derived, and the corresponding optimal (self-financing) portfolios would be constructed. Furthermore, this would verify the benefits of diversified portfolios in financial markets: that is, do not put all your eggs in the same basket.

No arbitrage and multiplicative special semimartingales

AbstractConsider a financial market with nonnegative semimartingales which does not need to have a numéraire. We are interested in the absence of arbitrage in the sense that no self-financing portfolio gives rise to arbitrage opportunities, where we are allowed to add a savings account to the market. We will prove that in this sense the market is free of arbitrage if and only if there exists an equivalent local martingale deflator which is a multiplicative special semimartingale. In this case, the additional savings account relates to the finite-variation part of the multiplicative decomposition of the deflator.

Predictable Losses of Liquidity Provision in Constant Function Markets and Concentrated Liquidity Markets

ABSTRACT We introduce a new comprehensive and model-free measure for the unhedgeable and predictable loss (PL) incurred by liquidity providers (LPs) in constant function markets (CFMs) and in concentrated liquidity markets. PL compares the value of the LP's holdings in the CFM liquidity pool (assuming no fee revenue) with that of a self-financing portfolio that (i) continuously replicates the dynamic holdings of the LP in the pool to offset the market risk of the LP's position, and (ii) invests in a risk-free account. We provide closed-form formulae for PL in CFMs with and without concentrated liquidity, and show that the losses stem from two sources: convexity cost, which depends on liquidity taking activity and the convexity of the pool's trading function; and opportunity cost, which is due to locking the LP's assets in the pool. For liquidity providers, PL is the appropriate measure to assess the cost of liquidity provision in CFMs, so that fees and compensation to LPs provide the right incentives for a well-functioning market. When prices form outside of the pool, we show that PL is reduced when liquidity taking is costly, i.e., when the convexity of the pool's trading function is high. On the other hand, when prices form in the pool, PL is reduced when liquidity taking is cheap, i.e., when the convexity of the trading function is low. Finally, we use Uniswap v3 and Binance transaction data to compute PL and fees collected by LPs and show that, at present, liquidity provision in CFMs is a loss-leading activity.

Point-to-point stochastic control of a self-financing portfolio

This paper aims at computing optimal control policies to drive a self-financing portfolio of financial assets from a given initial financial state to a final state in a given time horizon such that for the first case, the functional portfolio financial risk is minimized and, for the second case, the functional portfolio profit is maximized. The optimal control policies are the optimal investment allocation processes, the optimal state process is the optimal investor’s wealth process, also called the system response to the input control and is obtained by solving the combined system of differential equations formed by the state and costate system of differential equations derived and extracted from Pontryagin’s Minimum Principle. Computational simulations are provided to show the effectiveness and the reliability of the approach.

Implied Volatility Spread and Stock Mispricing

Stocks can be mispriced for at least two reasons: value-relevant information is not timely incorporated or investor sentiment can induce mispricing. Using the mispricing measure proposed by Stambaugh, Yu, and Yuan (2015), we show that informed trading in the options market, proxied by the implied volatility spread, can substantially alleviate stock mispricing. Higher implied volatility spread reliably predicts subsequently lower stock mispricing after controlling for an array of economic variables including firm size, illiquidity, idiosyncratic volatility, institutional ownership, and investor’s divergence of opinions. In addition, this effect is more pronounced when the options trading volume is higher, consistent with the notion that higher options trading volume provides better camouflage for informed trading in the spirit of Kyle (1985). Our findings highlight the importance of information incorporation to reduce asset mispricing. We further show that a self-financing monthly portfolio that goes long on most underpriced stocks and short on most overpriced stocks when the implied volatility spread is the lowest yields statistically and economically significant abnormal returns.

Bayesian Portfolio Selection: Application to Tactical Asset Allocation

This article discusses the portfolio selection problem from a Bayesian perspective. In doing so, I first provide an overview of the portfolio problem and motivate the decision-making process from an expected utility point of view. Then, I demonstrate the analytical solution to the problem and stress the intuition behind the Bayesian application. In particular, in the case of risky assets, the optimal portfolio corresponds to three funds. The first is the minimum variance portfolio, whereas the others denote two self-financing portfolios corresponding to the mean returns. The combination between the first and the second funds is consistent with the conventional mean-variance portfolio. Furthermore, with the inclusion of the third fund, the portfolio incorporates the priors/beliefs of the decision-making into the portfolio selection. Based on the analytical insights, I conduct a small empirical experiment using two ETFs. The experiment emulates a tactical asset allocation problem. All the empirical analysis is conducted using R.

Past Stock Returns and the MAX Effect

This paper investigates how past stock returns affect investors’ lottery demand. We show that the maximum daily return (MAX) effect, first documented by Bali et al. (Maxing Out: Stocks as Lotteries and the Cross-Section of Expected Returns) is predominantly concentrated among stocks that have worst returns in the recent past. In contrast, for stocks that have recorded best past returns, the MAX effect becomes much weaker or even disappears. Our findings hold in the presence of a variety of control variables including the conventional return predictors, stock mispricing, stock price proximity to its 52-week high, and capital gains overhang. We interpret it as evidence that investors’ lottery demand intensifies in the presence of investment losses, consistent with the notion of time varying and state dependent nature of investors’ lottery preference. We further show that a self-financing portfolio that goes long in stocks with the worst past returns and low MAX and short in stocks with worst past returns and high MAX yields statistically and economically significant abnormal returns after risk adjustment by standard asset pricing models.

The role of net stable funding ratio on the bank lending channel: evidence from European Union

This paper is motivated by the ongoing debate about the Basel III impact on the efficient functioning of the banking sector. We empirically examine the effect that the implementation of the net stable funding ratio has on real economy. Using data from the EU banking sector, we conduct a retrospective analysis by simulating and investigating historically the NSFR index and its role in the implementation of a common monetary policy. We intervene on the traditional bank lending channel of Bernanke and Blinder (Am Econ Rev 82:901–921, 1992) by incorporating the interaction term between liquidity and interest rates. The analysis is conducted both at an aggregated loan supply level and by loan category while it incorporates, additionally to the interaction term, conventional asset pricing approaches with the adoption of self-financing trading strategies detecting nonlinearities in the relationship between liquidity provisions and bank lending channel. According to our findings, there is evidence of a heterogeneous response of financial intermediaries’ loan supply (due to changes of interest rates) across different NSFR levels. Banks with higher NSFR respond positively to an interest rate increase, by restructuring their loans’ portfolios to achieve higher risk-adjusted returns, conditional on the presence of an efficient asset allocation. On the contrary, low NSFR banks reduce loan supply as a response to higher interest rates.

A theoretical generalization of the Markowitz model incorporating skewness and kurtosis

This paper proposes a generalization of Markowitz model that incorporates skewness and kurtosis into the classical mean–variance allocation framework. The principal appeal of the present approach is that it provides the closed-form solution of the optimization problem. The four moments optimal portfolio is then decomposed into the sum of three portfolios: the mean–variance optimal portfolio plus two self-financing portfolios, respectively, accounting for skewness and kurtosis. Theoretical properties of the optimal solution are discussed together with the economic interpretation. Finally, an empirical exercise on real financial data shows the contribution of the two portfolios accounting for skewness and kurtosis when financial returns depart from Normal distribution.

Hedging Longevity Risk in Defined Contribution Pension Schemes

Pension schemes all over the world are under increasing pressure to efficiently hedge the longevity risk posed by ageing populations. In this work, we study an optimal investment problem for a defined contribution pension scheme which decides to hedge the longevity risk using a mortality-linked security, typically a longevity bond. The pension scheme invests in the risky assets available in the market, including the longevity bond, by using the contributions from a representative scheme member to ensure a minimum guarantee such that the member is able to purchase a lifetime annuity upon retirement. We transform this constrained optimal investment problem into an unconstrained problem by replicating a self-financing portfolio of future contributions from the member and the minimum guarantee provided by the scheme. We solve the resulting optimisation problem using the dynamic programming principle and through a series of numerical studies reveal that the longevity risk has an important impact on the performance of investment strategies. Our results provide mathematical evidence supporting the use of mortality-linked securities for efficient hedging of the longevity risk.

On a multi-asset version of the Kusuoka limit theorem of option superreplication under transaction costs

We consider, using the geometric description, a sequence of models of multi-asset financial markets with proportional transaction costs vanishing in the limit. We assume that the price processes are He-type multinomial approximations of a process whose components are correlated geometric Brownian motions. For a given vector-valued contingent claim, defined as a continuous function of the price trajectories, we consider for each model the hedging set, that is, the set of all vector-valued initial endowments permitting to superreplicate the contingent claim by the final position of a self-financing portfolio. We calculate the limit of the hedging sets in the closed topology, obtaining in this way a set-valued version of the Kusuoka limit theorem.

OPTIMAL MEAN–VARIANCE PORTFOLIO SELECTION WITH NO-SHORT-SELLING CONSTRAINT

In this paper, the objective is to study the continuous mean–variance portfolio selection with a no-short-selling constraint and obtain a time-consistent solution. We assume that there is a self-financing portfolio with wealth process [Formula: see text], in which [Formula: see text] represents the fraction of wealth invested in the risk asset under the short selling prohibition. We investigate the mean–variance optimal constrained problem defined by obtaining the supremum over all admissible controls of the difference between the expectation of the value process at some designated terminal time [Formula: see text] and a positive constant times the variance of [Formula: see text]. To envisage the quadratic nonlinearity introduced by the variance, the method of Lagrangian multipliers reduces the nonlinear problem into a set of linear problems which can be solved by applying the Hamilton–Jacobi–Bellman equation and change of variables formula with local time on curves. Solving the HJB system provides the time-inconsistent solution and from there, we derive the time-consistent optimal control.

Relative pricing of French Treasury inflation-linked and nominal bonds: an empirical approach using arbitrage strategies

This paper investigates whether arbitrage opportunities exist between inflation-linked bonds and nominal bonds on the French Treasury market. Following arbitrage theory, we apply the risk hedging concept: we set up self-financing portfolios hedged against risks through durations of different orders. Perfectly hedged portfolios are those with a zero initial and a zero final value. The results show arbitrage gains when the first three duration orders are implemented, but they are not significantly different from zero when a fourth-order duration is added. Furthermore, a regression of arbitrage gains on the illiquidity measure of nominal and index Treasury bonds provides evidence that the illiquidity of inflation-linked bonds significantly explains arbitrage gains, whereas the illiquidity measure of nominal bonds does not.

Research on Self-financing Effective Portfolio Based on Fuzzy Genetic Algorithm

Research on Self-financing Effective Portfolio Based on Fuzzy Genetic Algorithm

Hedging Longevity Risk in Defined Contribution Pension Schemes

Pension schemes all over the world are under increasing pressure to efficiently hedge the longevity risk posed by ageing populations. In this work, we study an optimal investment problem for a defined contribution pension scheme which decides to hedge the longevity risk using a mortality-linked security, typically a longevity bond. The pension scheme invests in the risky assets available in the market, including the longevity bond, by using the contributions from a representative scheme member to ensure a minimum guarantee such that the member is able to purchase a lifetime annuity upon retirement. We transform this constrained optimal investment problem into an unconstrained problem by replicating a self-financing portfolio of future contributions from the member and the minimum guarantee provided by the scheme. We solve the resulting optimisation problem using the dynamic programming principle and through a series of numerical studies reveal that the longevity risk has an important impact on the performance of investment strategies. Our results provide mathematical evidence supporting the use of mortality-linked securities for efficient hedging of the longevity risk.

Portfolio Selection Based on Bayesian Theory

The traditional portfolio selection model seriously overestimates its theoretic optimal return. Aiming at this problem, two portfolio selection models are proposed to modify the parameters and enhance portfolio performance based on Bayesian theory. Firstly, a Bayesian-GARCH(1,1) model is built. Secondly, Markov Chain is applied to curve the parameters’ state transfer, and a Bayesian Markov regime-Switching-GARCH(1,1) model is constructed. Both the two models can handle the overestimation problem and can obtain self-financing portfolios. In the numerical experiments, both the models are examined with data from China stock market, and their performances are compared and analyzed. The results show that BMS-GARCH(1,1) model is superior to the Bayesian-GARCH(1,1) model.

Cross-Predictability of Industry Return in Trade Network: Using LASSO

This study tests the predictability power of returns across economically related industries, using monthly data over 1990–2010 and a machine learning model, the Least Absolute Shrinkage and Selection Operator (LASSO). This article presents evidence that LASSO identifies significant predictors of the returns of an individual industry among its interdependent industries. The results indicate cross-predictability of industry return by showing a significant relation between an industry’s returns and the lagged returns of its LASSO-selected trade partners. The study finds that industries that are more central in the input-output network have a greater effect in predicting related industry return. It also computes out of sample (2011–2016) the mean return forecasts of the portfolio of industries based on the predictability effect of the lagged returns of trade partners. The self-financing trading strategy of buying (selling) a high (low) portfolio based on related industry lagged returns yields about 10.44 percent annual excess return (0.83 percent monthly), with an annual Sharpe ratio of about 0.86. The study shows that after controlling for the Fama-French (1993) three factors, the monthly returns from the self-financing trading strategies generate a significant alpha of 0.7 percent monthly over the period 2011–2016. TOPICS:Big data/machine learning, factor-based models, portfolio construction, performance measurement

The French Treasury Inflation Linked Bond Puzzle

Mispricing between Treasury inflation linked bond and nominal bonds is an issue under attention. This paper examines whether mispricing exists on the French bond market by constructing arbitrage strategies between these two types of assets. We consider self-financing portfolios hedged against the different types of interest rate risks, represented by different order durations. We find that arbitrage profits remain even after controlling for level, slope and hump risks. An econometric analysis shows that changes in the difference between forward nominal and real rates explain the detected profits. The implementation of arbitrage strategies are supported by an innovative zero-coupon pricing methodology based on piecewise constant forward rates.

Stop-loss and leverage in optimal statistical arbitrage with an application to energy market

In this paper, we develop a statistical arbitrage trading strategy with two key elements in hi-frequency trading: stop-loss and leverage. We consider, as in Bertram (2009), a mean-reverting process for the security price with proportional transaction costs; we show how to introduce stop-loss and leverage in an optimal trading strategy.We focus on repeated strategies using a self-financing portfolio. For every given stop-loss level we derive analytically the optimal investment strategy consisting of optimal leverage and market entry/exit levels.First we show that the optimal strategy á la Bertram depends on the probabilities to reach entry/exit levels, on expected First-Passage-Times and on expected First-Exit-Times from an interval. Then, when the underlying log-price follows an Ornstein-Uhlenbeck process, we deduce analytical expressions for expected First-Exit-Times and we derive the long-run return of the strategy as an elementary function of the stop-loss.Following industry practice of pairs trading we consider an example of pair in the energy futures' market, reporting in detail the analysis for a spread on Heating-Oil and Gas-Oil futures in one year sample of half-an-hour market prices.