Risk-neutral World Research Articles

Option Pricing Using a Skew Random Walk Binary Tree

We develop a binary tree pricing model with underlying asset price dynamics following Itô–McKean skew Brownian motion. Our work was motivated by the Corns–Satchell, continuous-time, option pricing model. However, the Corns–Satchell market model is incomplete, while our discrete-time market model is defined in the natural world, extended to the risk-neutral world under the no-arbitrage condition where derivatives are priced under uniquely determined risk-neutral probabilities, and is complete. The skewness introduced in the natural world is preserved in the risk-neutral world. Furthermore, we show that the model preserves skewness under the continuous-time limit. We provide empirical applications of our model to the valuation of European put and call options on exchange-traded funds tracking the S&P Global 1200 index.

Multi-Asset Option Pricing Using Normal Tempered Stable Processes with Stochastic Correlation

In this article, the authors develop a new multi-asset option pricing model where underlying dynamics are assumed to follow the normal tempered stable (NTS) process and its correlation structure evolves over time. The model is constructed by extending the constant correlation term in the previous NTS framework to the stochastic correlation making use of the Ornstein-Uhlenbeck process. We then derive its closed-form solution under the risk-neutral measure and apply it to an empirical study of a quanto option. The in-sample tests and calibration practices justify our model specification reflecting the empirical stylized facts on asset returns such as heavy tails, skewness, kurtosis, and stochastic correlation. Building on the empirical results, we can also identify the presence of stochastic correlation in the risk-neutral world.

Optimal management of a mega pumped hydro storage system under stochastic hourly electricity prices in the Iberian Peninsula

Concern for the environment led in some developed countries to planning future energy use that combines greater electrification with increasing use of Renewable Energy (RE) in the electricity generation mix. These scenarios can affect Security of Supply (SoS) because the intermittence of RE, its stochastic nature as an electricity generation sources and the demand's behaviour. It is necessary to increase storage capacity, among other measures. This paper analyses the effects of an optimal management strategy based on prices for Pumped Hydro Storage plants (PHES) using a daily mean reverting jump diffusion stochastic model of electricity prices in a risk neutral world including daily seasonality. Results show that a) income with this strategy under uncertainty may be insufficient compared to investment costs; b) the strategy does not usually provide proper guarantees as regards SoS at times of high demand for electricity; c) the technical characteristics of PHES such as the maximum upper & lower reservoir volume are highly significant. The effect of a minimum reservoir capacity at times of high demand is also analysed. PHES profitability can be improved under a Generating Company (GENCO) strategy coordinated with a wind farm and if the avoided CO2 emissions are taken into account.

Market complete option valuation using a Jarrow-Rudd pricing tree with skewness and kurtosis

Applying the Cherny-Shiryaev-Yor invariance principle, we introduce a generalized Jarrow-Rudd (GJR) option pricing model with uncertainty driven by a skew random walk. The GJR pricing tree exhibits skewness and kurtosis in both the natural and risk-neutral world. We construct implied surfaces for the parameters determining the GJR tree. Motivated by Merton’s pricing tree incorporating transaction costs, we extend the GJR pricing model to include a hedging cost. We demonstrate ways to fit the GJR pricing model to a market driver that influences the price dynamics of the underlying asset. We supplement our findings with numerical examples.

PERHITUNGAN HARGA OPSI EROPA MENGGUNAKAN MODEL BINOMIAL MULTIPERIODA (STUDI KASUS SAHAM TELKOM.JK)

Binomial Model is one of method to determine the option price. In this model, assumed that the stock price only move up or down in the each steps. In this journal will be reviewed the risk neutral world assumption and real world to determine option price and knowing how to calculate option payoff for european option from stock price use Binomial Multiperoid Model with different strike price.

Structural Pricing of CoCos and Deposit Insurance with Regime Switching and Jumps

In this article, we develop a semi-analytical solution for a structural model that combines jump and regime switching risk. We use an Esscher transform that is applicable to regime switching double exponential jump diffusion to move from the historical world to the risk-neutral world. Further, we define and implement a matrix Wiener–Hopf factorization associated with the latter process, allowing us to price the various components of balance sheet. We illustrate the model with a study of a bank that issues contingent convertible bonds (CoCos). Thus, we obtain valuation formulas for the bank’s equity, debt, deposits, CoCos, and deposit insurance. We also show in an illustration the respective effects of the jump risk and of regime switching on the values of all of a bank’s balance sheet components.

Call Option Pricing Model and Recovery Theorem: A Specific Case of Pricing Warrants

Warrant is normally priced on the basis of Black and Scholes' model, which refers to calculations in a risk neutral world. Hence, it neither captures the market expectation nor being a good reference for the risk management process. This study examines a new way of pricing warrants under the real world probability by utilizing the recovered Vacisek short rate model. Applying Carr and Yu's recovery model, an extended version of Ross Recovery Theorem, we managed to recover the Vasisek process. Then, suppose that the economy is driven by this recovered Vacisek process, we point out a valuation model for the warrant of an underlying stock. We deduce that by applying the recovered Vacisek model we can derive the warrant price under the real world probability without the assumption of the market price of risk as in the risk neutral model.

Stochastic model genetic programming: Deriving pricing equations for rainfall weather derivatives

Abstract Rainfall derivatives are in their infancy since starting trading on the Chicago Mercantile Exchange (CME) in 2011. Being a relatively new class of financial instruments there is no generally recognised pricing framework used within the literature. In this paper, we propose a novel Genetic Programming (GP) algorithm for pricing contracts. Our novel algorithm, which is called Stochastic Model GP (SMGP), is able to generate and evolve stochastic equations of rainfall, which allows us to probabilistically transform rainfall predictions from the risky world to the risk-neutral world. In order to achieve this, SMGP's representation allows its individuals to comprise of two weighted parts, namely a seasonal component and an autoregressive component. To create the stochastic nature of an equation for each SMGP individual, we estimate the weights by using a probabilistic approach. We evaluate the models produced by SMGP in terms of rainfall predictive accuracy and in terms of pricing performance on 42 cities from Europe and the USA. We compare SMGP to 8 methods: its predecessor DGP, 5 well-known machine learning methods (M5 Rules, M5 Model trees, k-Nearest Neighbors, Support Vector Regression, Radial Basis Function), and two statistical methods, namely AutoRegressive Integrated Moving Average (ARIMA) and Monte Carlo Rainfall Prediction (MCRP). Results show that the proposed algorithm is able to statistically outperform all other algorithms.

Martingale Representation of Bond Price

In practice, rather than to presume a hypothetical condition that needs be validated theoretically, it is sufficient to accept a real condition that exists and can be directly validated for practical purposes. This paper presents a practical approach to formulating the theoretical price of a zero-coupon bond without the hassle of a change of probability measure - the martingale representation. The aim is threefold: theoretically, for modeling one need not presume a risk-neutral world which is born out of abstraction; conceptually, the notion of a fair price can be formulated by a verifiable condition - the martingale property; and computationally, the valuation of derivatives can be simplified.

Structural Pricing of CoCos and Deposit Insurance with Regime Switching and Jumps

In this article, we construct a structural model with jumps and regime switching to price banks' contingent convertible debt (CoCos) and deposit insurance. We use an Esscher transform that is applicable to regime switching double exponential jump diffusions to move from the historical world to the risk-neutral world. Further, we define and implement a matrix Wiener-Hopf factorization associated with the latter processes, allowing us to price the various components of a bank's balance sheet. Thus, we obtain valuation formulas for the bank's equity, debt, deposits, CoCos, and deposit insurance. We also show in an illustration the respective effects of the jump risk and of regime switching on the values of all of a bank's balance sheet components.

Interest rate trees: extensions and applications

This paper provides extensions to existing procedures for representing one-factor no-arbitrage models of the short rate in the form of a tree. It allows a wide range of drift functions for the short rate to be used in conjunction with a wide range of volatility assumptions. It shows that, if the market price of risk is a function only of the short rate and time, a single tree with two sets of probabilities on branches can be used to represent rate moves in both the real-world and risk-neutral world. Examples are given to illustrate how the extensions can provide modelling flexibility when interest rates are negative.

Option Portfolio Management in a Risk-Neutral World

The most commonly used strategy of the speculative investments in options is a statistical arbitrage between the objective underlying price distribution which the price is following and the risk-neutral distribution on the basis of which options were priced. This article investigates an alternative approach which does not demand these two distributions to be different. Instead, it uses a periodical roll-over of an investment horizon with including options of the next expirations in the portfolio. We consider a risk-neutral world where the real distribution coincides with the risk-neutral distribution as a model of the market. In such a market, the expected return from investments in any option portfolio corresponds with the risk-free rate. However, it is possible to construct and manage the portfolio dynamically in such a way that it provides higher return with a probability close to unity or lower return (possibly a large negative return) with a given very low probability. To optimize the portfolio a stochastic program with the approximative safety-first criterion for option portfolio was developed along with the corresponding multinomial scenario tree. The results of the Monte-Carlo simulation of the portfolio management are presented. The very low probability of loss during option portfolio management is provided by the strategy with periodical rolling horizon of the optimization. The developed portfolio management strategy can be used as a basis for constructing trading strategies for the real option markets.

Interest Rate Trees: Extensions and Applications

This paper provides extensions to existing procedures for representing one-factor no-arbitrage models of the short rate in the form of a tree. It allows a wide range of drift functions for the short rate to be used in conjunction with a wide range of volatility assumptions. It shows that, if the market price of risk is a function only of the short rate and time, a single tree with two sets of probabilities on branches can be used to represent rate moves in both the real-world and risk-neutral world. Examples are given to illustrate how the extensions can provide modeling flexibility when interest rates are negative.

О квадратичных моделях доходности в риск-нейтральной среде

О квадратичных моделях доходности в риск-нейтральной среде

On Quadratic Models of Yield in Risk-Neutral World

The paper presents a mathematically equivalent, but more compact description of the usually occurring quadratic model of yield. Equations for the functions of the term structure are obtained and general properties of their solutions are given. The main content of the paper is to consider the case when the probability properties of the model are subject to a risk-neutral setting, which allows to obtain the analytical solutions of the problem about the shape of the yield curves and forward curves in an explicit form. Equations of curves are presented in a compact form using hyperbolic functions. We have found families of yield curves and forward curves, which for a fixed short-term interest rate ensure the same limit long-term yield. The obtained results are illustrated by the numerical examples. The presented results illustrate the fact that quadratic models of the term structure of interest rate make it possible to obtain a richer spectrum of admissible yield curves and forward curves than affine models that allow only single curve to be obtained for a fixed starting interest rate, while quadratic models under conditions of fixed starting and long-term limiting rates and yields can give a fairly wide range of yield curves. Consequently, theoretically they have great opportunities for fitting theoretical curves to real market term structures. However, this will be realistically feasible if there is an opportunity to control the latent variables used in the quadratic model. So far, there is no description of these variables in the economic sense in the literature.

Forward-premium puzzle: is it time to abandon the usual regression?

ABSTRACTThe forward premium puzzle is usually evidenced by the rejection of the null hypothesis in the uncovered interest parity (UIP) regression. Because this parity need only hold in a risk-neutral world, a risk adjustment term is missing from the equation if speculation in foreign exchange markets is risky. We deal with this issue following the literature which assumes that discounted returns on foreign government bonds are log-normal, so we can linearize the Euler pricing equations (in level) and obtain a modified UIP system for which the risk adjustment term is obtained by applying to the pricing kernel-based relations a generalized autoregressive conditional heteroscedasticity-in-mean model. However, here we innovate by adopting a methodology which differs from all these related works. We construct and use a stochastic discount factor that does not depend on a specific model, by residing in the space of returns which we extract from the data by simply imposing the orthogonality restrictions represented by the Euler equations. So, we devise a purely statistical pricing kernel that performs well in in-sample level equations. Somewhat disappointingly, the risk premium inclusion in the conventional regression changes neither the significance nor the magnitude of the forecasting power of the forward premium for most currencies we study. The contrasting performance of the tests in level and in logs suggests that linearization may be to blame.

Modeling Returns and Unconditional Variance in Risk Neutral World for Liquid and Illiquid Market

This article seeks to model daily asset returns using log-ARCH-Levy type model which is expected to reproduce most of the stylized features of financial time series data (such as volatility clustering, leptokurtic nature of log returns, joint covariance structure and aggregational Gaussianity) that are empirically found in different types of market. In addition, unconditional variance of daily log returns in risk neutral world of different conditional heteroscedastic models is derived. A key observation is that liquid markets and illiquid market may not have the same underlying dynamics. For instance empirical analysis based on S&P500 index log returns as a liquid market do not have autoregressive part in their first moments while in Nairobi Securities Exchange NSE20 index there is strong presence of autoregressive dynamics of order three, i.e. AR(3). Higher moments of both markets are serially correlated.

Does the Short Rate Revert to its Mean in the Risk-Neutral World?

We propose a methodology to calibrate the non-linear and non-affine short rate models under physical and risk-neutral probability measures, based on the U.S treasury yield curve data. For this purpose, we price the default-free bond using the Markov chain approximation method built on the well-established idea of local consistency. Integrating the standard recursive filtering algorithm and the Markov chain approximation bond price formula, we propose a Markov chain filter in which the transition equation is a general stochastic differential equation for the short rate dynamics, and the measurement equation can be based on observations of any interest rate derivatives. Our simulation experiments show that popular parameter estimators can generate spurious non-linear drift estimates. However, the proposed Markov chain filter does not. In contrast to the existing literature, through our empirical findings we argue that the short rate does not revert to its mean under the risk-neutral probability measure. Our findings corroborate the observations that yields on the short end term structure have remained near 0% for more than 30 months since the implementation of Quantitative Easing monetary policy.

Decisiones óptimas de portafolio cuando la tasa forward sigue el modelo Heath, Jarrow y Morton (HJM): un modelo de maximización de utilidad

In this paper, under the assumption that the interest rate is driven by the model of Heath-Jarrow-Morton (1990 and 1992), the optimal portfolio decisions of an agent who wishes to maximize his total expected utility of the type von Neumann-Morgenstern in infinite planning horizon are determined. To do this, the assumptions that settle on the dynamics of the short rate on a risk-neutral world are established and the first order conditions of the optimization problem of the consumer are derived from the equation of Hamilton-Jacobi-Bellman (HJB) of the continuous time stochastic dynamic programming

CVA, DVA, LVA, FVA, CSA and What Else?

It was upon a time, the Risk Neutral world. Under this world every payoff actualised was a martingale. The industry became more and more complex but still managed to provide prices for exotics, indeed via a Monte Carlo Method almost everything was possible under this measure. After Lehman Brothers defaults, the practitioner realised that the assumptions of this world were more important than the consequences of this magical world. The world was easy and the price fair but the market was not complete (far to be complete) ! It is not that the industry was not aware, it is just that it was easier and theoretically more beautiful to ignore some assumptions than make the uniqueness of the price disappears. Indeed how can we be sure of our pricing methodology if prices are not unique. Only some braves went to this road but the market is more powerful than any individual theory or conviction (it doesn't matter if you are right alone). In this article, which is part of an on going book, we established the base of the New Pricing Theory.The industry opts for adjustment of the traditional risk neutral price; CVA, DVA, FVA, LVA are part of the new pricing standard. The debate is not settled and many point of views are still open to discussion. We will first define the new adjustments, respecting all literature point of views then highlight our vision and the consequences in matter of pricing.