Correlating Market Models

On the Information in the Interest Rate Term Structure and Option Prices

In this paper we empirically compare a wide range of different term structure models when it comes to the pricing and, in particular, hedging of caps and swaptions. We analyze the influence of the number of factors on the hedging and pricing results, and investigate which type of rate or derivative price data should be used to estimate the model parameters to obtain the best hedging and pricing results. We use on interest rates, and cap and swaption prices from 1995 to 1999. The empirical results on the hedging of caps and swaptions show that, if the number of hedge instruments is equal to the number of factors, the multi-factor models outperform one-factor models in hedging caps and swaptions. However, if one uses a large set of hedge instruments, one-factor models perform as well as multi-factor models. In terms of pricing, we find that models with two or three factors imply better out-of-sample predictions of cap and swaption prices than one-factor models. Also, estimation on the basis of current derivative prices leads to more accurate out-of-sample prediction of cap and swaption prices than estimation on the basis of interest rate data.

This chapter provides an overview of the most popular methods of inverse reinforcement learning (IRL) and imitation learning (IL). These methods solve the problem of optimal control in a data-driven way, similarly to reinforcement learning, however with the critical difference that now rewards are not observed. The problem is rather to learn the reward function from the observed behavior of an agent. As behavioral data without rewards is widely available, the problem of learning from such data is certainly very interesting. This chapter provides a moderate-level technical description of the most promising IRL methods, equips the reader with sufficient knowledge to understand and follow the current literature on IRL, and presents examples that use simple simulated environments to evaluate how these methods perform when the “ground-truth” rewards are known. We then present use cases for IRL in quantitative finance which include applications in trading strategy identification, sentiment-based trading, option pricing, inference of portfolio investors, and market modeling.

In mathematical finance diffusion models are widely used and a variety of different parametric models for the drift and diffusion coefficient coexist in the literature. Since derivative prices depend on the particular parametric model of the diffusion coefficient function of the underlying, a misspecification of this function leads to misspecified option prices. We develop two tests about a parametric form of the diffusion coefficient. The finite sample properties of the tests are investigated in a simulation study and the tests are applied to the 7 -day Eurodollar rate, the German stock market index DAX and five German stocks. For all observed processes, we find in the empirical analysis that our tests reject all tested parametric models. We conclude that affine diffusion processes might not be appropriate to model the evolution of financial time series and that a successful model for a financial market should incorporate the history of the observed processes of additional sources of randomness like stochastic volatility models.

Stochastic volatility of underlying assets has been shown to affect significantly the price of many financial derivatives. In particular, a fast mean-reverting factor of the stochastic volatility plays a major role in the pricing of options. This paper deals with the interest rate model dependence of the stochastic volatility impact on defaultable interest rate derivatives. We obtain an asymptotic formula of the price of defaultable bonds and bond options based on a quadratic term structure model and investigate the stochastic volatility and default risk effects and compare the results with those of the Vasicek model.

Options pricing is a critical problem and one of the fundamental building blocks in mathematical finance. Monte Carlo (MC) simulation is the most widely used solution for options pricing. In most cases options pricing must be performed in real time. Although today’s multi-core CPUs can provide a high computing power, the options pricing on today’s multi-core CPUs is far from responding in real time. In this paper, we use modern graphics processors (GPUs), which provide a much higher computing power than multi-core CPUs, to perform MC-based options pricing. The challenge is that GPUs only provide a small on-chip scratchpad while MC-based options pricing requires a large memory space and accesses the memory space irregularly. To make MC-based options pricing efficient on GPUs, we propose a recycling approach which compacts the data to shrink the memory space, and a crossing-path layout which reorganizes the data to make memory accesses GPU-friendly. We use real market data and benchmarks to evaluate our optimization approaches. The experimental result shows that our MC-based options pricing on the latest GPU is 43-fold faster than the latest multi-core CPU for single precision computation (with Intel C++ Compiler 11.1); compared with GCC 4.2.4, the speedup is as high as 145.

Lognormal vs Normal Volatilities and Sensitivities in Practice

Bermudan swaptions are options on interest rate swaps which can be exercised on one or more dates before the final maturity of the swap. Because the exercise boundary between the continuation area and stopping area is inherently complex and multi-dimensional for interest rate products, there is an inherent “tug of war” between the pursuit of calibration and pricing precision, tractability, and implementation efficiency. After reviewing the main ideas and implementation techniques underlying both single- and multi-factor models, we offer our own approach based on dimension reduction via Markovian projection. Specifically, on the theoretical side, we provide a reinterpretation and extension of the classic result due to Gyöngy which covers non-probabilistic, discounted, distributions relevant in option pricing. Thus, we show that for purposes of swaption pricing, a potentially complex and multidimensional process for the underlying swap rate can be collapsed to a one-dimensional one. The empirical contribution of the paper consists in demonstrating that even though we only match the marginal distributions of the two processes, Bermudan swaptions prices calculated using such an approach appear well-behaved and closely aligned to counterparts from more sophisticated models.

Libor and Swap Market Models for the Pricing of Interest Rate Derivatives: An Empirical Analysis

A long-standing issue in mathematical finance is the speed-up of option pricing, especially for multi-asset options. A recent study has proposed to use tensor train learning algorithms to speed up Fourier transform (FT)-based option pricing, utilizing the ability of tensor trains to compress high-dimensional tensors. In this study, we focus on another usage of the tensor train, which is to compress functions, including their parameter dependence. Here, we propose a pricing method, where, by a tensor train learning algorithm, we build tensor trains that approximate functions appearing in FT-based option pricing with their parameter dependence and efficiently calculate the option price for the varying input parameters. As a benchmark test, we run the proposed method to price a multi-asset option for the various values of volatilities or present asset prices. We show that, in the tested cases involving up to 11 assets, the proposed method outperforms Monte Carlo-based option pricing with 106 paths in terms of computational complexity while keeping better accuracy.

The thesis consists of three parts. In the first part, we establish a dual-curve market model. The new model accounts for the evolution of interest market after the 2007 credit crisis. The model can capture the spreads of swap rates between different tenors, and it can price caps, floors and swaptions in closed forms. In addition, we incorporate the stochastic volatility to the dual-curve market model, which allows us to capture volatility smiles or skews carried by interest rate caps and floors. In the second part, we create a market model for inflation rates using a Lévy process as driving process. An arbitrage-free condition is derived, which allows us to unite existing inflation-rate models developed under the approach of “foreign currency analogy”. Pricing formulae for inflation-linked swaps and options are developed, using moment generating function and fast Fourier transforms. In the third part, we develop two approximated option pricing formulas for Heston model. The first is based on generalized Hull-White pricing formula, while the second is obtained through weak Taylor expansions. The two methods can be utilized as well in time-dependent Heston model, where closed-form pricing is not available.

Compared with the developed countries, Chinese stock index option market developed late and accumulated less historical data. At the same time, it has the characteristics that many developed countries' stock option markets do not have. Black-Scholes Model is an important model in the field of option pricing. It is not enough for a financial model to be reasonable in theory, but it needs to be tested by the actual market. Therefore, based on the main transaction data of the CSI 300 stock index option contracts from 2010 to 2020, this paper analyzes the compliance of the CSI 300 stock index option market to the basic assumptions of the B-S option pricing model, the main parameters of the CSI 300 stock index options in the B-S option pricing model, and the fitting degree of the B-S model's prediction and actual prices. The result shows that there is a certain deviation between the pricing conclusions of the B-S model and the actual option prices. This paper analyzes the reasons for the deviation from many aspects, and puts forward the corresponding improvement strategies that the users of the model should adopt, so as to improve the applicability of the B-S option pricing model in Chinese stock index option market.

The SABR stochastic volatility model with β-volatility β ? (0,1) and an absorbing barrier in zero imposed to the forward prices/rates stochastic process is studied. The presence of (possibly) nonzero correlation between the stochastic differentials that appear on the right hand side of the model equations is considered. A series expansion of the transition probability density function of the model in powers of the correlation coefficient of these stochastic differentials is presented. Explicit formulae for the first three terms of this expansion are derived. These formulae are integrals of known integrands. The zero-th order term of the expansion is a new integral formula containing only elementary functions of the transition probability density function of the SABR model when the correlation coefficient is zero. The expansion is deduced from the final value problem for the backward Kolmogorov equation satisfied by the transition probability density function. Each term of the expansion is defined as the solution of a final value problem for a partial differential equation. The integral formulae that give the solutions of these final value problems are based on the Hankel and on the Kontorovich-Lebedev transforms. From the series expansion of the probability density function we deduce the corresponding expansions of the European call and put option prices. Moreover we deduce closed form formulae for the moments of the forward prices/rates variable. The moment formulae obtained do not involve integrals or series expansions and are expressed using only elementary functions. The option pricing formulae are used to study synthetic and real data. In particular we study a time series (of real data) of futures prices of the EUR/USD currency's exchange rate and of the corresponding option prices. The website: http://www.econ.univpm.it/recchioni/finance/w18 contains material including animations, an interactive application and an app that helps the understanding of the paper. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website:http://www.econ.univpm.it/recchioni/finance.

In this paper, we conduct a fast calibration in the jump-diffusion model to capture the Bitcoin price dynamics, as well as the behavior of some components affecting the price itself, such as the risk of pitfalls and its ambiguous effect on the evolution of Bitcoin’s price. In addition, in our study of the Bitcoin option pricing, we find that the inclusion of jumps in returns and volatilities are significant in the historical time series of Bitcoin prices. The benefits of incorporating these jumps flow over into option pricing, as well as adequately capture the volatility smile in option prices. To the best of our knowledge, this is the first work to analyze the phenomenon of price jump risk and to interpret Bitcoin option valuation as “exceptionally ambiguous”. Crucially, using hedging options for the Bitcoin market, we also prove some important properties: Bitcoin options follow a convex, but not strictly convex function. This property provides adequate risk assessment for convex risk measure.

In this paper we summarize the main idea and results of Yuen and Yang (2009, 2010a, 2010b) and provide some results on pricing of Parisian options under the Markov regime-switching model (MRSM). The MRSM allows the parameters of the market model depending on a Markovian process, and the model can reflect the information of the market environment which cannot be modeled solely by linear Gaussian process. However, when the parameters of the stock price model are not constant but governed by a Markovian process, the pricing of the options becomes complex. We present a fast and simple trinomial tree model to price options in MRSM. In recent years, the pricing of modern insurance products, such as Equity-Indexed annuity (EIA) and variable annuities (VAs), has become a popular topic. We show here that our trinomial tree model can been used to price EIA with strong path dependent exotic options in the regime switching model.