Interest Rate Options Research Articles

Joint calibration of local volatility models with stochastic interest rates using semimartingale optimal transport

We develop and implement a non-parametric method for joint exact calibration of a local volatility model and a correlated stochastic short rate model using semimartingale optimal transport. The method relies on the duality results established in Joseph et al. (Calibration of local volatility models with stochastic interest rates using optimal transport, 2023). and jointly calibrates the whole equity-rate dynamics. It uses an iterative approach which starts with a parametric model and tries to stay close to it, until a perfect calibration is obtained. We demonstrate the performance of our approach on market data using European SPX options and European cap interest rate options. Finally, we compare the joint calibration approach with the sequential calibration, in which the short rate model is calibrated first and frozen.

Margin constraints and asset prices

Abstract I study the effects of regulatory policy changes on interest rate option prices: margin tightening from the introduction of mandatory interest rate swap clearing by the Dodd–Frank Act in 2010 and margin loosening from the counterbalance of voluntary swaption clearing and synthetic derivatives to the uncleared margin rule in 2016. Employing these variations as exogenous shocks for a quasi-experimental design, I show that swaption prices consistently respond to changes in margin requirements. The results are consistent with theories on the expected margin premium, where the constrained agent holds short positions in zero net supply.

Estimating time-varying factors’ variance in the string-term structure model with stochastic volatility

This paper presents a novel term structure of interest rate (TSIR) model with stochastic volatility and jumps (SVJ) that combines the market framework proposed by Brace et al. (1997) with the string-shock framework of Santa-Clara and Sornette (2001). In this model, the factors’ variance is estimated through the eigendecomposition of a variance–covariance matrix obtained with a measure of market volatility derived from out-of-the-money option prices and historical correlations of interest rates traded in the futures market. The stochastic evolution of the factors’ variance is governed by the 4/2 model developed by Grasselli (2017), including jumps. A novel method is employed to estimate the parameters of the SVJ model that minimizes the distance between the sample moments and the moments of a gamma distribution. The empirical application of the model in the Brazilian derivatives market demonstrates its effectiveness in accurately capturing the volatility smile of interest rate options.

A static replication approach for callable interest rate derivatives: mathematical foundations and efficient estimation of SIMM–MVA

The computation of credit risk measures such as exposure and Credit Value Adjustments (CVA) requires the simulation of future portfolio prices. Recent metrics, such as dynamic Initial Margin (IM) and Margin Value Adjustments (MVA) additionally require the simulation of future conditional sensitivities. For portfolios with non-linear instruments that do not admit closed-form valuation formulas, this poses a significant computational challenge. This problem is addressed by proposing a static replication algorithm for interest rate options with early-exercise features under an affine term-structure model. Under the appropriate conditions, we can find an equivalent portfolio of vanilla options that replicate these products. Specifically, we decompose the product into a portfolio of European swaptions. The weights and strikes of the portfolio are obtained by regressing the target option value with interpretable, feed-forward neural networks. Once an equivalent portfolio of European swaptions is determined, we can leverage on closed-form expressions to obtain the conditional prices and sensitivities, which serve as an input to exposure and SIMM-driven MVA quantification. For a consistent forward sensitivity estimation, this involves the differentiation of the portfolio-weights. The accuracy and convergence of the method is demonstrated through several representative numerical examples, benchmarked against the established least-square Monte Carlo method.

Hedging Interest Rate Options with Reinforcement Learning: an investigation of a heavy-tailed distribution

Purpose: The study intends to model an interest rate index option using a heavy-tailed distribution. The goal is to calculate the interest rate path-dependent option prices that are consistent with market data and to develop a reinforcement learning strategy to discretely hedge the position considering transaction costs. Methodology: This paper presents a mathematical framework to calculate the price of interest rate path-dependent options. The research adapted a Fourier cosine series formula to employ the characteristic function of the present value of the forward index, which is modeled as a variance-gamma process and uses deep Q-learning to hedge such options. Findings: There is market evidence that the implied volatility curve is not flat. The study demonstrated that the variance-gamma process generates an increasing volatility smile, which is consistent with market observations. Additionally, hedging results show that the path-dependent options generated from the variance-gamma process can be efficiently hedged with advanced Q-learning techniques. Research limitations/implications: The study comprised only the variance-gamma process. Other probability distributions, such as the Normal Inverse Gaussian model, should be investigated. Practical implications: This study reveals which type of probability distribution should be present in a pricing engine to be consistent with implied volatilities. The approach provided here can assist managers in evaluating and comprehending market pricing behavior as well as achieving discrete hedging with costs. Originality: The paper addressed the merging of a fast pricing method for the interest rate options with a heavy-tailed distribution and the discrete interest rate derivatives hedging with reinforcement learning.

Pricing of European Interest Rate Options Under Non-Affine Stochastic Volatility and Stochastic Interest Rates

Pricing of European Interest Rate Options Under Non-Affine Stochastic Volatility and Stochastic Interest Rates

Central Bank Policy Impacts on the Distribution of State Prices for Future Interest Rates, 2003–2022

In this article, we extend the 1978 Breeden–Litzenberger method of extracting state prices from option prices, showing how portfolios of butterfly spreads can be combined with right and left tail spreads to nonparametrically extract discrete state prices from option prices. We derive how those state prices should be biased estimates of true, objective probabilities. For interest rate options, we show that the biases can vary predictably over time (sometimes too high, sometimes too low), as the correlation of interest rates with consumption and wealth has changed signs over time. Consumption betas and proper risk premiums on bonds and of their state prices are at times predictably positive and at times predictably negative. We apply our technique to provide a brief 20-year history of central bank intervention impacts in the US, UK, and Eurozone from 2003 to 2022. Movements in state prices are quite large in the Financial Panic of 2008–2009, as well as in the European Sovereign Debt Crisis of 2010–2013, with Brexit and the Trump elections in 2016, and with the coronavirus pandemic in 2020–2021. Tapering in 2013 and 2022 and liftoffs in rates in 2015 and 2022 were shown to strongly shift state price distributions back toward the symmetry of 2003–2007. We show that central banks dramatically impacted entire state price distributions, not just levels of rates.

Black\u2019s model in a negative interest rate environment, with application to OTC derivatives

The most common application of Black’s formula is interest rate derivatives pricing. Black’s model, a variant of Black-Scholes option pricing model, was first introduced by Fischer Black in 1976. In recent market conditions, where global interest rates are at very low levels and in some markets are currently zero or negative, Black model—in its canonical form—fails to price interest rate options since positive interest rates are assumed in its formula. In this paper we propose a heuristic method that, without explicit assumptions about the forward rate generating process, extends the cumulative standard normal distribution domain to negative interest rates and allows Black’s model to work in the conventional way. Furthermore, we provide the derivations of the so called five Greek letters that enable finance professionals to evaluate the sensitivity of an option to various parameters. Along with the description of the methodology, we present an extensive simulation study and a comparison with the Normal model which is widely used in the negative environment option pricing problems.

Оценка влияния факторов на процентный риск коммерческого банка

In modern conditions of the rapid industrial development the banks have to forecast their risks and profitability precisely, to apply information technologies to assess their activities. To evaluate the bank's income, it is necessary to carry out an internal analysis of its assets and liabilities and determine the factors effecting the bank's profitability by managing interest rate risk. The hypothesis of the study is the analysis of the impact on the net interest income and interest rate risk of a commercial bank of factors such as the exchange rate and the key rate of the Bank of Russia (for example, Sberbank, PJSC). There has been studied the impact of the factors (exchange rate and key interest rate of Central Bank of Russia) on the bank's net interest income by using correlation and regression analysis and building a regression model. Many tools are found to be used by the experienced analysts. One of the main tools is GAP analysis of interest rate risk. There have been illustrated the graphs of changes in interest rates of savings and loan associations during the crisis in the United States in the 1950-1960, of realization of interest rate risk with an increase in interest rates, the distribution of assets and liabilities according to the maturity of the balance sheet structure, the impact of changes in the interest rate GAP on net interest income, etc. A matrix of correlations of all variables in the sample (rates of growing values) was constructed. Conclusions are drawn on the need to use hedging instruments (interest rate swaps, interest rate options), as well as of attracting the most reliable data on the state of interest rate risk in the commercial banks.

Brownian Path Generation and Polynomial Chaos

We introduce a new Brownian path generation method using a second degree polynomial chaos expansion, and use a conjugate gradient algorithm to find the orthogonal transformation that minimizes the mean truncation dimension of this polynomial. We combine the new path generation method with a control variate technique based on the same second degree polynomial chaos expansion, and apply the method to estimation of equity option prices under the Black--Scholes framework and interest rate option prices under the Libor Market Model. We use numerical results to compare the new method with some other path generation methods from the literature.

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

With the forthcoming introduction of SOFR benchmark rates in the US, market participants will need to adjust their interest rate option models to accommodate a variety of idiosyncrasies of the SOFR rate. The materiality of these changes for quoted options level is currently unknown, and will depend on market sentiment (as expressed in market risk premia, say), regulatory policies, and the rate fixing conventions ultimately available in the market. While we wait for liquidity in SOFR options to build, this paper pre-emptively considers two important characteristics of SOFR derivatives: the backward-looking settlement style of SOFR floating rate payments; and the “jagged” nature of SOFR evolution through time. The latter originates with liquidity conditions in the repo financing markets from which SOFR is constructed, where temporary demand-supply imbalances can result in the formation of short-term spikes of substantial magnitude. We construct a variety of mechanisms that allows us to build rich stochastic models for both “surprising” and anticipated (e.g., year-end) spikes, and demonstrate how to modify existing (smooth) term structure models to capture them. To accommodate high-efficiency pricing of vanilla derivatives in top-down models, we also develop several convenient numerical techniques that allow for effcient pricing of these structures. For instance, a novel scheme merges existing spike-free pricing formulas with a given spike characteristic function in a custom low-dimensional quadrature routine, enabling us to spike-enable standard valuation models (such as SABR) at minimal computational effort. Using SOFR-style caplets for illustration, we numerically demonstrate that the effect of spikes on implied caplet volatility levels and skews can be substantial, even at modest levels of risk premia in the spike model parameters. Besides being useful for the pricing of SOFR derivatives, our paper more broadly establishes a complete mathematical framework for rate spikes, applicable to pricing, scenario generation, and risk management in any rates market where spike phenomena exist.

Unspanned stochastic volatility from an empirical and practical perspective

I conduct a simulation study to address concerns raised in the empirical literature on unspanned stochastic volatility (USV, i.e., interest-rate-volatility risk that cannot be hedged with bonds or swaps). Regressions have been the popular method of identifying and measuring USV, and have led to a consensus that is in favour of USV models. Despite plausible challenges to this approach, my simulations show that regressions are able to correctly identify the presence and absence of USV. This relies on a number of methodological considerations which are inconsistent in the literature. Regression results from empirical data, from several modern interest-rate markets, resemble results from data simulated from USV models. I then assess the economic significance of USV. By comparing hedged and unhedged returns of market interest-rate options, I develop quantitative guidelines around how unspanned volatility risk compares to interest-rate risk.

Swap Rate à la Stock: Bermudan Swaptions Made Easy

We show how Markovian projection, together with some clever parameter freezing, can be used to reduce a full-fledged local volatility interest rate model – such as the Cheyette model – to a “minimal” form in which the swap rate evolved essentially like a dividend-paying stock. Using a number or numerical examples, we compare such a minimal “poor man's” model to a full-fledged Cheyette local volatility model and the market benchmark Hall–White one-factor model. Numerical tests demonstrate that the “poor man's” model is in fact sufficient to price Bermudian interest rate swaptions. The main practical implication of this finding is that – once local volatility, dividend, and short rate parameters are properly stripped from the volatility surface and interest rate curve – one can readily use the widely popular equity derivatives software for pricing exotic interest rate options such as Bermudans.

Impact of FOMC Cycle on Market Uncertainty: Evidence From Interest Rate Derivatives

This paper investigates how Federal Reserve (Fed) actions influence market uncertainty. We consider two kinds of Fed events: the day of the Federal Open Market Committee (FOMC) meeting -- which includes a policy statement, press conference and release of a Summary of Economic Projections -- and the day the minutes of the FOMC minutes are released -- which is typically set 3 weeks after the meeting. Unlike related papers focused on the issue, we measure market uncertainty by the implied volatility extracted from interest rate options, specifically swaptions. We use 1-month constant maturity volatility for swaptions over tenors ranging from one up to 30 years as they are reflective of how these volatilities are marked by dealers/market-makers and cover only one FOMC meeting/minutes release at a time. We use an event study approach along with extensive graphical analysis to evaluate the impact of Fed actions. The results show that 1-month constant maturity implied volatility increases marginally going into these events and falls much more significantly afterwards. Remarkably, the increase and reduction in uncertainty around a meeting is not reducing with increasing tenors, showing that the impact of information release has a similar impact over all horizons ranging from 1 to 30 years. If, on one side, this evidence witnesses the capacity of the central bank to control the long end of the curve, on the other it indicates the possibility that overreaction to news may generate the puzzling excess volatility which is observed in long term rates.

Loan Prime Rate Options

In this paper, we document vanilla interest-rate options (caps, floors and swaptions) newly introduced in China. The underlying rates are the RMB loan prime rates (LPRs), the foremost interest rates that matter to almost all businesses and households in China. They are digital with a tick size of five basis points, and the changes only occur at predetermined monthly announcement times. Although the current literature on interest-rate options is vast, these unique stylised features bring a new challenge for interest-rate option pricing. We propose a novel market model built upon the integer-valued Skellam distribution, named Skellam market model. It is simple and analytically tractable, which leads to pricing formulas in closed forms. We advocate that it is more meaningful to quote the LPR option prices in terms of the implied intensity rather than the conventional implied volatility. Our preliminary empirical work finds intensity frown implied from cap prices and intensity skew implied from swaption prices.

Extending the G2++ Model: Fitting the Term Structure of Implied Volatility

This paper describes a fast and robust procedure for calibrating a two factor Gaussian model (G2++) with piecewise-constant volatility to a large set of options on interest rates swaps. As far as we know the algorithm implemented is novel as it overcomes some issues affecting highly parameterized models like model identification and the need to employ heuristic optimizations in order to get a good overall fitting. In this sense, our procedure extends the G2++ model ability to reproduce the term structure of volatility implied by interest rate options, similarly to how the Hull and White model extended the Vasicek model in order to perfectly fit the term structure of interest rates.

Valuation of interest rate ceiling and floor based on the uncertain fractional differential equation in Caputo sense

As an economic lever in financial market, interest rate option is not only the function of facilitating the bank to adjust the market fund supply and demand relation indirectly, but also provides the guarantee for investors to choose whether to exercise the right at the maturity date, thereby locking in the interest rate risk. This paper mainly studies the price of the interest rate ceiling as well as floor under the uncertain environment. Firstly, from the perspective of expert reliability, rather than relying on a large amount of historical financial data, to consider interest rate trends, and further assume that the dynamic change of the interest rate conforms to the uncertain process. Secondly, since uncertain fractional-order differential equations (UFDEs) have non-locality features to reflect memory and hereditary characteristics for the asset price changes, thus is more suitable to model the real financial market. We construct the mean-reverting interest rate model based on the UFDE in Caputo type. Then, the pricing formula of the interest rate ceiling and floor are provided separately. Finally, corresponding numerical examples and algorithms are given by using the predictor-corrector method, which support the validity of the proposed model.

ТОВАРНА БІРЖА ЯК ІННОВАЦІЙНО-ІНСТИТУЦІЙНИЙ ЕЛЕМЕНТ РОЗВИТКУ МІЖНАРОДНИХ ТОВАРНИХ РИНКІВ

The article examines the role of the commodity exchange as an innovative and institutional element in the development of international commodity markets. The current trends in the development of the international commodity stock market have been determined, compared to the volumes of world futures and options for 2018-2019, the number of outstanding contracts and their changes have been investigated. The transformation processes inherent in the international commodity exchange are considered and characterized. The structure of the international stock market by geographical regions is reflected. Trading volumes in different regions of 2019 are compared to 2018. Analysis of futures, options index of the capital index, interest rate trading, energy futures, options trading and precious metals trade have been done. The results of the world's largest stock exchanges by volume of trading in 2018-2019 are presented. One of the criteria examined is the volume of trading on the Intercontinental Exchange, which is due to the lower level of activity on all its subsidiaries in Europe, North America and the Asia-Pacific region. The current state of stock exchange trade in Ukraine is analysed, examples of obstacles for the effective functioning of stock exchanges of Ukraine are given. The development of commodity exchanges in Ukraine over recent years is considered, with an indication of the tendency of their development. Examples of commodities are indicated that will be used for the development of the commodity stock market of Ukraine. The statistical information on the number and structure of exchanges by specialization, their characteristics and role are investigated. The conclusions on the stock role market in an economy have been formed. The international experience of trading on the exchanges is generalized. The evolution of the stock market from the fair to the modern electronic stock market with its advantages and convenience is noted. The conclusions of the development of stock exchange trade in Ukraine are summarized: namely obstacles to its development and misunderstanding of advantages. Suggestions for successful development of stock exchange trading are given in accordance with world experience and national peculiarities. Key words: commodity exchange, futures, option, stock trading, international commodity exchanges, agricultural products, market infrastructure, stock exchange.

Implied volatility surface predictability: The case of commodity markets

Recent literature seek to forecast implied volatility derived from equity, index, foreign exchange, and interest rate options using latent factor and parametric frameworks. Motivated by increased public attention borne out of the financialization of futures markets in the early 2000s, we investigate if these extant models can uncover predictable patterns in the implied volatility surfaces of the most actively traded commodity options between 2006 and 2016. Adopting a rolling out-of-sample forecasting framework that addresses the common multiple comparisons problem, we establish that, for energy and precious metals options, explicitly modeling the term structure of implied volatility using the Nelson-Siegel factors produces the most accurate forecasts.

Interest rate option hedging portfolios without bank account

Purpose This paper aims to study the conditions for the hedging portfolio of any contingent claim on bonds to have no bank account part. Design/methodology/approach Hedging and Malliavin calculus techniques recently developed under a stochastic string framework are applied. Findings A necessary and sufficient condition for the hedging portfolio to have no bank account part is found. This condition is applied to a barrier option, and an example of a contingent claim whose hedging portfolio has a bank account part different from zero is provided. Originality/value To the best of the authors’ knowledge, this is the first time that this issue has been addressed in the literature.