A numerical method to price European derivatives based on the one factor LIBOR Market Model of interest rates

We consider an interest rate model with log-normally distributed rates in the terminal measure in discrete time. Such models are used in financial practice as parametric versions of the Markov functional model, or as approximations to the log-normal Libor market model. We show that the model has two distinct regimes, at low and high volatilities, with different qualitative behavior. The two regimes are separated by a sharp transition, which is similar to a phase transition in condensed matter physics. We study the behavior of the model in the large volatility phase, and discuss the implications of the phase transition for the pricing of interest rates derivatives. In the large volatility phase, certain expectation values and convexity adjustments have an explosive behavior. For sufficiently low volatilities the caplet smile is log-normal to a very good approximation, while in the large volatility phase the model develops a non-trivial caplet skew. The phenomenon discussed here imposes thus an upper limit on the volatilities for which the model behaves as intended.

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Extension of SABR Libor Market Model to handle negative interest rates

Quantitative Finance and Economics ◽

10.3934/qfe.2020007 ◽

2020 ◽

Vol 4 (1) ◽

pp. 148-171

Author(s):

Jie Xiong ◽

◽

Geng Deng ◽

Xindong Wang

Keyword(s):

Interest Rates ◽

Market Model ◽

Libor Market Model ◽

Negative Interest Rates

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MATHEMATICAL ANALYSIS AND NUMERICAL METHODS FOR A PARTIAL DIFFERENTIAL EQUATIONS MODEL GOVERNING A RATCHET CAP PRICING IN THE LIBOR MARKET MODEL

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202511005465 ◽

2011 ◽

Vol 21 (07) ◽

pp. 1479-1498 ◽

Cited By ~ 5

Author(s):

A. PASCUCCI ◽

M. SUÁREZ-TABOADA ◽

C. VÁZQUEZ

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Interest Rates ◽

Time Discretization ◽

Fundamental Solutions ◽

Market Model ◽

Computational Time ◽

Cauchy Problems ◽

Partial Differential ◽

Libor Market Model

In this paper, we present a mathematical model for pricing a particular financial product: the ratchet cap. This derivative product depends on certain interest rates (whose dynamics we assume that follow the LIBOR market model). The pricing model is rigorously posed in terms of a sequence of nested Cauchy problems associated to uniformly parabolic partial differential equations. First, for each problem the existence and uniqueness of solution is obtained. Next, this analysis allows to propose a new and more efficient numerical method based on the approximation by computable fundamental solutions of constant coefficient operators. The advantage in terms of computational time of this new modeling and analytically based approach is illustrated by comparison with the classically used Monte Carlo simulation and a characteristics Crank–Nicolson time discretization combined with finite elements strategy.

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LIBOR market model with multiplicative basis

International Journal of Financial Engineering ◽

10.1142/s2424786318500147 ◽

2018 ◽

Vol 05 (02) ◽

pp. 1850014 ◽

Cited By ~ 1

Author(s):

Yangfan Zhong

Keyword(s):

Interest Rate ◽

Interest Rates ◽

Slight Modification ◽

Cash Flows ◽

Market Model ◽

Interest Rate Models ◽

Forward Rates ◽

Libor Market Model ◽

Multiplicative Basis ◽

Negative Interest Rates

The study on the multiple-curve interest rate models becomes increasingly active since the 2007 credit crunch, for which one curve, typically the OIS curve, is used for discounting purpose, while the LIBOR curves (associated with various market tenors) are used for projecting the future cash flows. In this work, we extend the standard LIBOR market model to accommodate such multiple-curve setting by means of a multiplicative basis. The multiplicative basis is modeled as an exponential function of multi-factor square-root processes. Under the multiplicative basis setup, the OIS forward rates are correlated with the implied (additive) LIBOR-OIS spreads. We then derive closed-form pricing formulas for caplet, swaption, and interest rate futures in the multiplicative basis framework. In particular, we show that the valuation of caplet and swaption can be easily computed by a proper integral of real-valued functions, which facilitates the calibration of our model. Finally, we discuss a slight modification of our model to allow for negative interest rates.

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The Demand For Money in Pakistan: Reply (Notes and Comments)

The Pakistan Development Review ◽

10.30541/v14i3pp.370-375 ◽

1975 ◽

Vol 14 (3) ◽

pp. 370-375

Author(s):

M. A. Akhtar

Keyword(s):

Interest Rates ◽

Money Demand ◽

Strong Support ◽

Physical Capital ◽

Careful Examination ◽

Demand For Money ◽

Weak Support ◽

Complementarity Hypothesis ◽

The Difference ◽

The One

I am grateful to Abe, Fry, Min, Vongvipanond, and Yu (hereafter re¬ferred to as AFMVY) [1] for obliging me to reconsider my article [2] on the demand for money in Pakistan. Upon careful examination, I find that the AFMVY results are, in parts, misleading and that, on the whole, they add very little to those provided in my study. Nevertheless, the present exercise as well as the one by AFMVY is useful in that it furnishes us with an opportunity to view some of the fundamental problems involved in an empi¬rical analysis of the demand for money function in Pakistan. Based on their elaborate critique, AFMVY reformulate the two hypo¬theses—the substitution hypothesis and the complementarity hypothesis— underlying my study and provide us with some alternative estimates of the demand for money in Pakistan. Briefly their results, like those in my study, indicate that income and interest rates are important in deter¬mining the demand for money. However, unlike my results, they also suggest that the price variable is a highly significant determinant of the money demand function. Furthermore, while I found only a weak support for the complementarity between money demand and physical capital, the results obtained by AFMVY appear to yield a strong support for that rela¬tionship.1 The difference in results is only a natural consequence of alter¬native specifications of the theory and, therefore, I propose to devote most of this reply to the criticisms raised by AFMVY and the resulting reformulation of the two mypotheses.

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