Spike Modeling for Interest Rate Derivatives with an Application to SOFR Caplets

The space of probability distributions on a given sample space possesses natural geometric properties. For example, in the case of a smooth parametric family of probability distributions on the real line, the parameter space has a Riemannian structure induced by the embedding of the family into the Hilbert space of square-integrable functions, and is characterised by the Fisher-Rao metric. In the nonparametric case the relevant geometry is determined by the spherical distance function of Bhattacharyya. In the context of term structure modelling, we show that minus the derivative of the discount function with respect to the maturity date gives rise to a probability density. This follows as a consequence of the positivity of interest rates. Therefore, by mapping the density functions associated with a given family of term structures to Hilbert space, the resulting metrical geometry can be used to analyse the relationship of yield curves to one another. We show that the general arbitrage-free yield curve dynamics can be represented as a process taking values in the convex space of smooth density functions on the positive real line. It follows that the theory of interest rate dynamics can be represented by a class of processes in Hilbert space. We also derive the dynamics for the central moments associated with the distribution determined by the yield curve.

This paper aims to analyse the impact of behavioural biases on asset pricing by hypothesising that sentiment and momentum are relevant risk factors in Pakistan equity market. The paper also examines the influence of sentiment and momentum factors on market risk, size, and value premiums by estimating the interacted asset-pricing model. To carry out the empirical analysis, monthly stock returns of firms listed on Pakistan Stock Exchange are used for the period 2000–2013. The empirical results indicate that both investor sentiment and momentum factors have a significant impact on the required rate of returns. Specifically, it is found that the premium for both factors is positive and statistically significant. Further, the estimated results provide evidence that the inclusion of these two factors in the Fama-French three-factor model considerably increases the prediction power of the model. The results also reveal that the inclusion of sentiment factor in the Carhart four-factor model significantly increases the prediction power of the model. Yet, the estimation results indicate that the prediction power of the model further increases when the interaction terms are added to the model in order to examine the indirect effects of sentiment and momentum. The results of the interacted model provide evidence of a significant impact of investor sentiment and momentum factors on market risk, size, and value premiums. Although investor sentiment negatively affects all the three premiums, the effect of momentum is positive for both market risk and size premium, whereas, it is negative for value premium. The findings are helpful in explaining and understanding the effects of behavioural biases on stock returns in Pakistan. The findings of the indirect effects suggest that investor sentiment and momentum factors significantly increase the chance of mispricing in Pakistan equity market.

References and Further Reading

Required Market Risk Premium among countries in 2012

Measuring the Fundamental Value of a Stock Index through Dividend Future Prices

By means of a data set previously unavailable for academic research, actual trading histories of individual futures traders are examined. With this more detailed data, the author is able to (1) test the risk/return hypothesis directly; (2) include a much larger segment of the market than before; and (3) use actual instead of hypothetical t rading strategies. It is shown that the commercial (hedging) traders are most profitable, while noncommercial (speculative) traders earn n egative or zero profits. Because speculators are not receiving reward s for the risks they willingly absorb, the theory of normal backwarda tion and its extension can be rejected. Copyright 1987 by University of Chicago Press.

Financial mathematics plays a pivotal role in various aspects of modern economics and finance. This paper provides an introduction to the fundamental concepts, theories, and applications of financial mathematics. It begins by outlining the basic principles of financial mathematics, including the time value of money, interest rates, and compounding. Computational programs enhance these mathematical models, offering robust solutions and efficient computation for complex financial problems. This study explores the integration of computational programs with financial mathematics, their methodologies, and applications in the finance sector. The results underscore the significance of computational methods in improving the accuracy, speed, and scalability of financial models, ultimately contributing to better decision-making and risk management. We explore fundamental concepts, models, and techniques employed in financial mathematics, aiming to provide a comprehensive understanding of their applications and significance in real-world financial scenarios. This paper provides a comprehensive overview of the application of differential equations in financial mathematics, highlighting key models such as the Black-Scholes model, interest rate models, and optimal investment strategies.

US risk premia under emerging markets constraints

Discount rates are essential to applied finance, especially in setting prices for regulated utilities and valuing the liabilities of insurance companies and defined benefit pension plans. This paper reviews the basic building blocks for estimating discount rates. It also examines market risk premiums, as well as what constitutes a benchmark fair or required rate of return, in the aftermath of the financial crisis and the U.S. Federal Reserve’s bond-buying program. Some of the results are disconcerting. In Canada, utilities and pension regulators responded to the crash in different ways. Utilities regulators haven’t passed on the full impact of low interest rates, so that consumers face higher prices than they should whereas pension regulators have done the opposite, and forced some contributors to pay more. In both cases this is opposite to the desired effect of monetary policy which is to stimulate aggregate demand. A comprehensive survey of global finance professionals carried out last year provides some clues as to where adjustments are needed. In the U.S., the average equity market required return was estimated at 8.0 per cent; Canada’s is 7.40 per cent, due to the lower market risk premium and the lower risk-free rate. This paper adds a wealth of historic and survey data to conclude that the ideal base long-term interest rate used in risk premium models should be 4.0 per cent, producing an overall expected market return of 9-10.0 per cent. The same data indicate that allowed returns to utilities are currently too high, while the use of current bond yields in solvency valuations of pension plans and life insurers is unhelpful unless there is a realistic expectation that the plans will soon be terminated.

ABSTRACTWhile many studies document that the market risk premium is predictable and that betas are not constant, the dividend discount model ignores time‐varying risk premiums and betas. We develop a model to consistently value cashflows with changing risk‐free rates, predictable risk premiums, and conditional betas in the context of a conditional CAPM. Practical valuation is accomplished with an analytic term structure of discount rates, with different discount rates applied to expected cashflows at different horizons. Using constant discount rates can produce large misvaluations, which, in portfolio data, are mostly driven at short horizons by market risk premiums and at long horizons by time variation in risk‐free rates and factor loadings.

Many studies on hedge fund returns are formulated in a static framework, resorting to a multifactor model using mimicking portfolios or benchmarks accounting for the style of different hedge fund strategies. While classical linear regressions represent a parsimonious approach to the modeling of hedge fund strategies, time-invariant factor loadings might be misleading (Berkelaar, Kobor, and Kouwenberg, The VaR Modeling Handbook, 2009). Actually, hedge fund strategies are essentially dynamic processes. Since conditional models appear promising to explain portfolio managers dynamic strategies, we study in this paper how conditional alphas and betas react to financial market variables such as market risk premium, interest rate, market volatility, or macroeconomic variables related to the business cycle (Racicot and Theoret, Journal of Wealth Management, 2007). An interesting assumption tested in the literature (Swinkels and van der Sluis, Working Paper, Tilburg University, 2001; Bollen and Whaley, Working Paper, Vanderbilt University, 2006; Racicot and Theoret, Journal of Wealth Management, 2008) is the conjecture that the conditional alpha and beta follow a random walk. Actually, those parameters might also react to conditioning financial market or macroeconomic news. Using the Kalman filter setting, our study explores the reaction of HFR indices conditional alpha and beta to financial market variables, especially interest rates, market risk premia, and squared market risk premia, a proxy for volatility. Our results show that the alphas of most of the indices computed over strategies are not very sensitive to financial market variables. A pure recursive process seems, therefore, more relevant for the alpha. The data generating process of the alpha is thus more similar to a random walk process than to a conditional model, Atl Econ J (2010) 38:377–378 DOI 10.1007/s11293-010-9230-6

This paper explores the multifaceted phenomenon of stock market volatility, a critical aspect of financial markets with profound implications for investors, businesses, and policymakers. The study begins by introducing the concept of stock market volatility, defining it as the degree of price fluctuations and highlighting its significance in risk assessment, investment decision-making, and market stability. The paper then delves into the historical evolution of volatility studies, tracing its roots from early observations to the development of sophisticated models like GARCH and stochastic volatility models. It examines key theoretical frameworks, including the Capital Asset Pricing Model (CAPM), Arbitrage Pricing Theory (APT), Behavioral Finance, and GARCH models, that provide insights into the drivers of volatility. Furthermore, the paper explores the various determinants of volatility, encompassing macroeconomic factors (economic growth, interest rates, inflation, exchange rates), market-specific factors (company-specific news, industry trends, market sentiment, liquidity), and global factors (geopolitical events, global economic conditions).The study investigates the implications of volatility for different stakeholders, including investors (risk assessment, portfolio diversification, emotional decision-making), businesses (financial planning, cost of capital, investment decisions), policymakers (monetary policy, regulatory frameworks), and market participants (trading activity, market liquidity, market sentiment).A case study on stock market volatility during the COVID-19 pandemic is presented, highlighting the impact of economic uncertainty, market sentiment, and the oil price shock on market fluctuations. The paper concludes by emphasizing the importance of understanding and managing volatility, and proposes future research directions, including the application of machine learning, the integration of behavioral finance insights, and the analysis of regional and sectoral volatility. This research contributes to a deeper understanding of stock market volatility, enabling investors, policymakers, and researchers to make more informed decisions and navigate the complexities of financial markets.

Recent events have highlighted the potential importance of nonlinear efiects of flscal variables (notably debt and deflcits) on interest rates: While in times when government solvency is not a concern the standard crowding-out efiects are of moderate magnitude, in times when default risk becomes an issue the interest rate efiects can become very large. This paper provides new evidence on the magnitude of these efiects. For the case when default risk is not a concern, it uses an arbitrage-free term structure model to estimate the dynamic efiects of flscal policy shocks on interest rates along the entire maturity spectrum. For the case when default risk becomes a concern (thereby violating a central assumption of the term structure model), I present evidence based on EMU government bond spread regressions on time-varying efiects of national flscal policies on spreads as well as the time-varying sensitivity of yield spreads to international risk aversion as a function of the state of flscal policy. JEL classiflcation: E6, H6.

After the replacement of Libor with alternative reference rates across the world, market participants must now adjust to a variety of changes to both contract terms and market rate dynamics. Focusing mostly on the new SOFR rate in the US, this paper considers the interaction of two important characteristics of SOFR derivatives: the backward-looking settlement style of SOFR floating rate payments; and the historically ‘jagged’ nature of SOFR time series. We introduce mechanisms that allow us to modify existing term structure models to incorporate rich dynamics for both ‘surprising’ and anticipated spikes and hikes in overnight rates. Numerical tests on SOFR-style caplets show that such model enhancements can produce significant effects on implied caplet volatility levels and skews. Besides establishing a practitioner-friendly framework for derivatives pricing, our paper is broadly applicable to scenario generation and risk management in any market with discontinuous rates dynamics.

Investors having an understanding of investment statistics are important. Especially quantitative tools related to investment risk measurement. Value-at-Risk Adjusted is one of the investment risk measurement tools, which assumes that returns are not normally distributed.This paper intends to measure investment risk based onValue-at-Risk Adjustedor called Modified Value-at-Risk under the Capital Asset Pricing Model. It is assumed that the return of the market index has a non-constant average and there is a long memory effect. The average of the return of the market index is estimated using ARFIMA models.It is also assumed that the stock risk premium correlates with market risk premiums, and stock risk premiums some time before. The correlation will be analyzed using the ARMAX-GARCH model approach. The Modified Value-at-Risk was then formulated based on the Capital asset Pricing Model with the ARMAX-GARCH model approach.To measure the performance of Modified Value-at-Risk that has been formulated is done with back testing. Back testing is carried out based on the Lopez II method. As a case study, analyzed some data on 10 stocks traded on the capital market in Indonesia.The results of the analysis show that the market index return risk premium significantly follows the ARFIMA model, and the 10 share risk premium significantly follows the ARMAX-GARCH model. Based on the results of back testing calculations indicate that the Value-at-Risk Adjustedor Modified Value-at-Risk is very suitable to be used to measure investment risk in the 10 stocks analyzed.