On the Term Structure of Interest Rates in the Presence of Reflecting and Absorbing Boundaries

Goldstein, Robert S.; Keirstead, William P.

doi:10.2139/ssrn.19840

Cited by 36 publications

(37 citation statements)

References 22 publications

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“…5 The second equality in (26) follows from the deÞnitions of the scale and speed densities (21) and (22 Proposition 2 generalizes Proposition 1 to the case of arbitrary continuous functions b(x), a(x) > 0, and R(x) ≥ 0 (in the example of Section 2 we have b(x) = (r − q)x, a(x) = σx and R(x) = r). It unbundles any European-style, double-barrier contingent claim with the L 2 payoff into a portfolio of eigensecurities with eigenpayoffs ϕ n .…”

Section: Spectral Methods For Options On Scalar Diffusionsmentioning

confidence: 82%

“…Hansen, Scheinkman, and Touzi (1998) develop spectral methods for econometric applications (estimation of scalar diffusions). Goldstein and Keirstead (1997) apply the eigenfunction expansion approach to the pricing of bonds under the short-rate processes with reßecting and absorbing boundaries. In an interesting recent paper, Lewis (1998) applies the eigenfunction expansion approach to solve two problems in continuous-time Þnance: pricing options on stocks that pay dividends at a constant dollar rate and pricing bonds under a short-rate process with non-linear drift.…”

Section: Introductionmentioning

confidence: 99%

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Pricing Options on Scalar Diffusions: An Eigenfunction Expansion Approach

Давыдов¹,

Linetsky

2003

Operations Research

145

122

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This paper develops an eigenfunction expansion approach to pricing options on scalar diffusion processes. All derivative securities are unbundled into portfolios of primitive securities termed eigensecurities. Eigensecurities are eigenvectors of the pricing operator (present value operator). Pricing is then immediate by the linearity property of the pricing operator and the eigenvector property of eigensecurities. To illustrate the computational power of the method, we develop two applications: pricing vanilla, single-and double-barrier options under the constant elasticity of variance (CEV) process and interest rate knock-out options in the Cox-IngersollRoss (CIR) term-structure model.

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Section: Spectral Methods For Options On Scalar Diffusionsmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Pricing Options on Scalar Diffusions: An Eigenfunction Expansion Approach

Давыдов¹,

Linetsky

2003

Operations Research

145

122

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“…20 Note the difference with, e.g., an alternative model r t = |σW t + χ(t)|. The latter is harder to tackle for general χ(t).…”

Section: "Modulus" Ho and Lee Modelmentioning

confidence: 99%

“…Our starting point is (5) together with the terminal condition v(z, T, T ) = Y (z). When σ(r t , t) and ν(r t , t) have no explicit time dependence, reflecting barriers for r t can be introduced as in, e.g., (Goldstein and Keirstead, 1997) and (Gorovoi and Linetsky, 2004). 36 Even if σ(r t , t) has no explicit time dependence, for general, explicitly time-dependent ν(r t , t) things are trickier.…”

Section: B Reflecting Barriers For the Short-rate Processmentioning

confidence: 99%

“…In all the models studied in (Goldstein and Keirstead, 1997) and (Gorovoi and 3 One can circumvent this via apparently unrealistic time dependence for the drift in r t . 4 Mathematically, the "spectrum" is the set of eigenvalues for the static Schrödinger equation to which the pricing PDE reduces, e.g., when the underlying parameters are time-independent.…”

Section: Introductionmentioning

confidence: 99%

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Coping With Negative Short Rates

Kakushadze¹

2016

Wilmott

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We discuss a simple extension of the Ho and Lee model with generic timedependent drift in which: 1) we compute bond prices analytically; 2) the yield curve is sensible and the asymptotic yield is positive; and 3) our analytical solution provides a clean and simple way of separating volatility from the drift in the short-rate process. Our extension amounts to introducing one or two reflecting barriers for the underlying Brownian motion (as opposed to the short-rate), which allows to have more realistic time-dependent drift (as opposed to constant drift). In our model the spectrum -or, roughly, the set of short-rate values contributing to bond and other claim prices -is discrete and positive. We discuss how to calibrate our model using empirical yield data by fitting three parameters and then read off the time-dependent drift.

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A Two-Factor Model for Low Interest Rate Regimes

Miller

Platen

2004

Asia-Pacific Finan Markets

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This paper derives a two-factor model for the term structure of interest rates that segments the yield curve in a natural way. The first factor involves modelling a non-negative short rate process that primarily determines the early part of the yield curve and is obtained as a truncated Gaussian short rate. The second factor mainly influences the later part of the yield curve via the market index. The market index proxies the growth optimal portfolio (GOP) and is modelled as a squared Bessel process of dimension four. Although this setup can be applied to any interest rate environment, this study focuses on the difficult but important case where the short rate stays close to zero for a prolonged period of time. For the proposed model, an equivalent risk neutral martingale measure is neither possible nor required. Hence we use the benchmark approach where the GOP is chosen as numeraire. Fair derivative prices are then calculated via conditional expectations under the real world probability measure. Using this methodology we derive pricing functions for zero coupon bonds and options on zero coupon bonds. The proposed model naturally generates yield curve shapes commonly observed in the market. More importantly, the model replicates the key features of the interest rate cap market for economies with low interest rate regimes. In particular, the implied volatility term structure displays a consistent downward slope from extremely high levels of volatility together with a distinct negative skew. Copyright Springer Science + Business Media, Inc. 2004fair pricing, growth optimal portfolio, interest rate caps, interest rate term structure, total market price for risk,

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On the Term Structure of Interest Rates in the Presence of Reflecting and Absorbing Boundaries

Cited by 36 publications

References 22 publications

Pricing Options on Scalar Diffusions: An Eigenfunction Expansion Approach

Pricing Options on Scalar Diffusions: An Eigenfunction Expansion Approach

Coping With Negative Short Rates

A Two-Factor Model for Low Interest Rate Regimes

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