scholarly journals LÉVY–VASICEK MODELS AND THE LONG-BOND RETURN PROCESS

2018 ◽  
Vol 21 (03) ◽  
pp. 1850026
Author(s):  
DORJE C. BRODY ◽  
LANE P. HUGHSTON ◽  
DAVID M. MEIER
Keyword(s):  
Interest Rates ◽  
Kernel Method ◽  
Pricing Kernel ◽  
Vasicek Model ◽  
One Dimensional ◽  
Market Incompleteness ◽  
Exponential Moments ◽  
Bond Return ◽  
Classical Derivation ◽  
Return Process

The classical derivation of the well-known Vasicek model for interest rates is reformulated in terms of the associated pricing kernel. An advantage of the pricing kernel method is that it allows one to generalize the construction to the Lévy–Vasicek case, avoiding issues of market incompleteness. In the Lévy–Vasicek model the short rate is taken in the real-world measure to be a mean-reverting process with a general one-dimensional Lévy driver admitting exponential moments. Expressions are obtained for the Lévy–Vasicek bond prices and interest rates, along with a formula for the return on a unit investment in the long bond, defined by [Formula: see text], where [Formula: see text] is the price at time [Formula: see text] of a [Formula: see text]-maturity discount bond. We show that the pricing kernel of a Lévy–Vasicek model is uniformly integrable if and only if the long rate of interest is strictly positive.

scholarly journals PERHITUNGAN PREMI ASURANSI JOINT LIFE DENGAN MODEL VASICEK DAN CIR

2019 ◽  
Vol 8 (3) ◽  
pp. 246
Author(s):  
I MADE WAHYU WIGUNA ◽  
KETUT JAYANEGARA ◽  
I NYOMAN WIDANA
Keyword(s):  
Stochastic Process ◽  
Interest Rates ◽  
Life Insurance ◽  
Insurance Premium ◽  
Insurance Contract ◽  
Insurance Company ◽  
Vasicek Model ◽  
Cir Model ◽  
Premium Payment ◽  
Premium Price

Premium is a sum of money that must be paid by insurance participants to insurance company, based on  insurance contract. Premium payment are affected by interest rates. The interest rates change according to stochastic process. The purpose of this work is to calculate the price of joint life insurance premiums with Vasicek and CIR models. The price of a joint life insurance premium with Vasicek and CIR models, at the age of the insured 35 and 30 years has increased until the last year of the contract. The price of a joint life insurance premium with Vasicek model is more expensive than the premium price using CIR model.

Derivatives Pricing in the Positive Interest Rates

2004 ◽  
Vol 12 (2) ◽  
pp. 157-179
Author(s):  
Joon Hee Rhee
Keyword(s):  
Interest Rate ◽  
Interest Rates ◽  
Analytic Solutions ◽  
Infinitely Divisible ◽  
Pricing Kernel ◽  
Derivatives Pricing ◽  
Interest Rate Derivatives ◽  
Practical Applications ◽  
Kernel Approach

This paper examines the pricing of interest rates derivatives such as caps and swaptions in the pricing kernel framework. The underlying state variable is extended to the general infinitely divisible Levy process. For computational purposes, a simple pricing kernel as in Flesaker and Hughston (1996) and Jin and Glasserman (2001) is used. The main contribution or purpose of this paper is to find several proper positive martingales, which is key role of practical applications of the pricing kernel approach with interest rates guarantee to be positive. Particularly, this paper first finds and applies a quite general type of a positive martingale process to pricing interest rate derivatives such as swaptions and range notes in the incomplete market setting. Such interest rate derivatives are hard to find analytic solutions. Consequently, this paper shows that such a choice of the positive martingale in the kernel framework is a promising approach to price interest rate derivatives

scholarly journals Estimating the Domestic Short Rate in a Convergence Model of Interest Rates

2020 ◽  
Vol 75 (1) ◽  
pp. 33-48
Author(s):  
Zuzana Bučková ◽  
Zuzana Girová ◽  
Beáta Stehlíková
Keyword(s):  
Interest Rates ◽  
High Precision ◽  
Monetary Union ◽  
Optimization Problem ◽  
Vasicek Model ◽  
Dynamic Correlation ◽  
Zero Correlation ◽  
Original Case ◽  
Real Market ◽  
Convergence Model

AbstractIn this paper we study the convergence model of interest rates by Corzo and Schwartz. It models the situation when a country is going to enter a monetary union, for example the eurozone. We are interested in estimating the underlying short rate, which is a theoretical variable, not observed on the market. We use the procedure already employed for the Vasicek model to the eurozone data and for the case of a zero correlation we show that a similar procedure can be used also for the estimation of the domestic parameters and the short rate values. The assumption of the zero correlation allows us to simplify the optimization problem, but using simulations we show that our algorithm is robust to the specification of the correlation. It estimates the short rate with a high precision also in the original case of a nonzero correlation, as well as in the case of a dynamic correlation, when the correlation is modelled as a function of time. Finally, we use the algorithm to real market data and estimate the short rate before adoption of the euro currency in Slovakia, Estonia, Latvia and Lithuania.

scholarly journals EXTENSION OF VASICEK MODEL TO THE MODELLING OF INTEREST RATE

2020 ◽  
Vol 4 (2) ◽  
pp. 151-155
Author(s):  
Adaobi Udoye ◽  
Lukman Akinola ◽  
Eka Ogbaji
Keyword(s):  
Stochastic Process ◽  
Interest Rate ◽  
Interest Rates ◽  
Regression Method ◽  
Least Square ◽  
Interesting Aspect ◽  
Vasicek Model ◽  
Least Square Regression ◽  
Ito's Lemma ◽  
Random Times

Interest rate modelling is an interesting aspect of stochastic processes. It has been observed that interest rates fluctuates at random times, hence the need for its modelling as a stochastic process. In this paper, we apply the existing Vasicek model, Itô’s lemma and least-square regression method in the modelling and providing dynamics for a given interest rate.

scholarly journals Estimating the Short Rate from the Term Structures in the Vasicek Model

2014 ◽  
Vol 61 (1) ◽  
pp. 87-103
Author(s):  
Jana Halgašová ◽  
Beáta Stehlíková ◽  
Zuzana Bučková
Keyword(s):  
Interest Rate ◽  
Interest Rates ◽  
Calibration Method ◽  
Current Level ◽  
Model Parameters ◽  
Bond Prices ◽  
Vasicek Model ◽  
Term Structures ◽  
Short Rate Models

Abstract In short rate models, bond prices and term structures of interest rates are determined by the parameters of the model and the current level of the instantaneous interest rate (so called short rate). The instantaneous interest rate can be approximated by the market overnight, which, however, can be influenced by speculations on the market. The aim of this paper is to propose a calibration method, where we consider the short rate to be a variable unobservable on the market and estimate it together with the model parameters for the case of the Vasicek model

Korean Treasury Bond Futures Pricing Model

2004 ◽  
Vol 12 (1) ◽  
pp. 1-22
Author(s):  
Youngsoo Choi ◽  
Se Jin O ◽  
Jae Yeong Seo
Keyword(s):  
Interest Rate ◽  
Interest Rates ◽  
Market Price ◽  
Alternative Methods ◽  
Vasicek Model ◽  
Market Participants ◽  
Treasury Bond ◽  
Cost Of Carry ◽  
The Interest Rate ◽  
The Cost

This paper proposes two alternative methods which are used for pricing the theoretical value of the KTB futures on the non-traded underlying asset; first method is to use the CKLS model, under which the volatility of interest rate changes is highly sensitive to the level of the interest rate, and then employ binomial trees to compute the theoretical value of futures, second one is to use the multifactor Vasicek model considering correlations between yields-to-maturity and then employ the Monte Carlo simulation to compute it. In the empirical study on KTB303 and KTB306, an CKLS methodology is superior to the conventional KORFX method based on the cost-of-carry model in terms of the size of difference between market price and theoretical price. However, the phenomena, the price discrepancy using the KOFEX methodology is very small for all test perlod, implies that the KOFEX one is being used for the most market participants. The reasons that an multifactor Vasicek methodlogy is performed poorly in comparison to another methods are 1) the Vasicek model might be not a good model for explaining the level of interest rates, or 2) the important point considered by the most market participants may be on the volatility or interest rate, not on the correlations between yields-to-maturity.

scholarly journals DETERMINATION OF THE LÉVY EXPONENT IN ASSET PRICING MODELS

2019 ◽  
Vol 22 (01) ◽  
pp. 1950008
Author(s):  
GEORGE BOUZIANIS ◽  
LANE P. HUGHSTON
Keyword(s):  
Lévy Process ◽  
Complete Information ◽  
Levy Process ◽  
Pricing Kernel ◽  
Market Prices ◽  
Risky Assets ◽  
Asset Pricing Models ◽  
Benchmark Portfolio ◽  
Exponential Moments

We consider the problem of determining the Lévy exponent in a Lévy model for asset prices given the price data of derivatives. The model, formulated under the real-world measure [Formula: see text], consists of a pricing kernel [Formula: see text] together with one or more non-dividend-paying risky assets driven by the same Lévy process. If [Formula: see text] denotes the price process of such an asset, then [Formula: see text] is a [Formula: see text]-martingale. The Lévy process [Formula: see text] is assumed to have exponential moments, implying the existence of a Lévy exponent [Formula: see text] for [Formula: see text] in an interval [Formula: see text] containing the origin as a proper subset. We show that if the prices of power-payoff derivatives, for which the payoff is [Formula: see text] for some time [Formula: see text], are given at time [Formula: see text] for a range of values of [Formula: see text], where [Formula: see text] is the so-called benchmark portfolio defined by [Formula: see text], then the Lévy exponent is determined up to an irrelevant linear term. In such a setting, derivative prices embody complete information about price jumps: in particular, the spectrum of the price jumps can be worked out from current market prices of derivatives. More generally, if [Formula: see text] for a general non-dividend-paying risky asset driven by a Lévy process, and if we know that the pricing kernel is driven by the same Lévy process, up to a factor of proportionality, then from the current prices of power-payoff derivatives we can infer the structure of the Lévy exponent up to a transformation [Formula: see text], where [Formula: see text] and [Formula: see text] are constants.

Control of price acceptability under the univariate Vasicek model

2016 ◽  
Vol 03 (03) ◽  
pp. 1650014
Author(s):  
S. Dang-Nguyen ◽  
Y. Rakotondratsimba
Keyword(s):  
Interest Rates ◽  
Point Of View ◽  
Hitting Times ◽  
Spot Rate ◽  
Rate Process ◽  
Vasicek Model ◽  
Numerical Examples ◽  
Financial Contract ◽  
Future Date ◽  
Price Constraints

The valuation of the probability of a financial contract to be lower or higher than a given price under the univariate Vasicek model is discussed in this paper. This price restriction can be justified by consistency reasons, since some prices may not be coherent on a financial point of view, e.g. they imply negative yields, or thought as unreachable by the asset manager. At first, assuming that the pricing functions is monotone, the price constraints are formulated in terms of a threshold on the value of the spot rate process. Since this process is Gaussian, these limits are reformulated in terms of a barrier of the Gaussian increments. Next, once the thresholds are identified, the probability to satisfy the price restriction after the generation of the spot rate at one future date can be computed. Then, assuming that the bounds on the spot rate are constant during a Monte-Carlo simulation, the probability of generating a path of this process that does not satisfy the constraint is valued using some results related to the hitting times. Lastly, the proposed approach is applied to various interest rates sensitive contracts and is illustrated by some numerical examples.

For which functions are 𝑓(𝑋_{𝑡})-𝔼𝕗(𝕏_{𝕥}) and 𝕘(𝕏_{𝕥})/𝔼𝕘(𝕏_{𝕥}) martingales?

2021 ◽  
Vol 105 (0) ◽  
pp. 79-91
Author(s):  
F. Kühn ◽  
R. Schilling
Keyword(s):  
Lebesgue Measure ◽  
Lévy Process ◽  
Levy Process ◽  
Content Type ◽  
One Dimensional ◽  
Exponential Moments ◽  
Bounded Functions

Let X = ( X t ) t ≥ 0 X=(X_t)_{t\geq 0} be a one-dimensional Lévy process such that each X t X_t has a C b 1 C^1_b -density w. r. t. Lebesgue measure and certain polynomial or exponential moments. We characterize all polynomially bounded functions f : R → R f\colon \mathbb {R}\to \mathbb {R} , and exponentially bounded functions g : R → ( 0 , ∞ ) g\colon \mathbb {R}\to (0,\infty ) , such that f ( X t ) − E f ( X t ) f(X_t)-\mathbb {E} f(X_t) , resp. g ( X t ) / E g ( X t ) g(X_t)/\mathbb {E} g(X_t) , are martingales.

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