An Exact Bond Option Formula

Jamshidian, Farshid

doi:10.1111/j.1540-6261.1989.tb02413.x

Cited by 478 publications

(79 citation statements)

References 6 publications

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“…where Γ 1 and Γ 2 are the same as defined previously, andp k -s are given by (see, e.g., Jamshidian, 1989)…”

Section: Country-specific Factormentioning

confidence: 99%

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A Rating-Based Sovereign Credit Risk Model: Theory and Evidence

Yang

2014

SSRN Journal

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We develop a rating-based continuous-time model of sovereign credit risk with closed-form solutions for a wide range of credit derivatives. In our model, rating transition follows a continuoustime Markov chain, and countries with same credit rating share similar level of default risk. A parsimonious version of our model, with only 16 parameters, one common and one country-specific factor, can simultaneously capture the term structure of CDS spreads of 34 in-sample and 34 outof-sample countries well. On average, the common factor explains more than 60% of the variations of the CDS spreads of both the in-sample and out-of-sample countries, and 80% of the variations of the common factor is explained by the CBOE VIX index, the 5-year US Treasury rate, and the CDX NA IG Index. AbstractWe develop a rating-based continuous-time model of sovereign credit risk with closed-form solutions for a wide range of credit derivatives. In our model, rating transition follows a continuoustime Markov chain, and countries with the same credit rating share a similar level of default risk.A parsimonious version of our model, with only 16 parameters, as well as one common and one country-specific factor, can simultaneously capture the term structure of CDS spreads of 34 insample and 34 out-of-sample countries well. On average, the common factor explains more than 60% of the variations of the CDS spreads of both the in-sample and out-of-sample countries, and 80% of the variations of the common factor is explained by the CBOE VIX index, the 5-year US Treasury rate, and the CDX NA IG Index.

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“…where Γ 1 and Γ 2 are the same as defined previously, andp k -s are given by (see, e.g., Jamshidian, 1989)…”

Section: Country-specific Factormentioning

confidence: 99%

“…• The likelihood of y jt given y j(t−∆) , which is Gaussian (see, e.g., Jamshidian, 1989), across countries;…”

Section: Maximum Likelihood Estimationmentioning

confidence: 99%

A Rating-Based Sovereign Credit Risk Model: Theory and Evidence

Yang

2014

SSRN Journal

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“…Isto faz com que os modelos para o mercado norte-americano precisem ser adaptados para a realidade da BM&F, como fez Vieira e Pereira (2000). Nesse artigo, eles adaptaram, para o mercado brasileiro, os modelos dos trabalhos de Vasicek (1977) e Jamshidian (1989), que foram desenvolvidos para o mercado norte-americano. Este modelo tem a vantagem de possuir uma fórmula fechada para a avaliação de opções sobre IDI.…”

Section: Introductionunclassified

Apreçamento de opções de IDI usando o modelo CIR

Fajardo

Ornelas

2003

Estud. Econ.

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RESUMOA opção de IDI da BM&F possui características peculiares que torna o seu apreçamento diferente das opções de taxa de juros mais comuns, como as de títulos de renda fixa. Este artigo desenvolve uma fórmula para apreçamento dessas opções de IDI, utilizando a precificação livre de arbitragem. O modelo utilizado considera apenas um fator estocástico: a taxa de juros livre de risco de curto prazo. A equação diferencial usada para modelar o comportamento da taxa de juros é a do modelo CIR (COX INGERSOLL & ROSS, 1985), que possui reversão à média e não permite a existência de taxas de juros nominais negativas. Também é feita uma estimação dos parâmetros do modelo proposto baseando-se em dados históricos, para então comparar o preço teórico da opção baseado nestes parâmetros com os preços de mercado e com o preço teórico considerando a modelagem de Vasicek (1977). PALAVRAS-CHAVEtaxas de juros, estrutura a termo, opção de IDI, reversão à média ABSTRACTThe IDI option from the BM&F (Commodities and Futures Exchange) has unusual characteristics, that make its pricing different from common interest rate options. This paper develops a closed form formula for the pricing of these IDI options, using an arbitrage-free pricing approach. The model used considers only one stochastic factor: the short-term risk-free interest rate. The differential equation used to model the behavior of the interest rate comes from the CIR (COX INGERSOLL & ROSS, 1985) model, which has mean reversion property and does not allow negative nominal interest rates. It is also done a parameter estimation of the proposed model based on historic data, and then compares the theoretical price of the option based on these parameters with the market price and with the theoretical price considering the Vasicek (1977) model. KEY WORDSinterest rates, term structure, IDI option, mean reversion JEL ClassificationC51, G13

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“…A special case of the decomposition formula (5) can be found in Jamshidian (1989) who proves that in the Vasicek (1977) model, a European option on a portfolio of pure discount bonds (in particular, an option on a coupon-bearing bond) decomposes into a portfolio of European options on the individual discount bonds in the portfolio. This holds true because in the Vasicek model, the prices of all pure discount bonds at some future time T are decreasing functions of a single random variable, namely the spot rate at that time.…”

Section: Sums Of Comonotonic Random Variablesmentioning

confidence: 99%

Advanced Mathematical Methods for Finance

Nunno¹,

Øksendal²

2011

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Over the last decade, it has been shown that the concept of comonotonicity is a helpful tool for solving several research and practical problems in the domain of finance and insurance. In this paper, we give an extensive bibliographic overview -without claiming to be complete -of the developments of the theory of comonotonicity and its applications, with an emphasis on the achievements over the last five years. These applications range from pricing and hedging of derivatives over risk management to life insurance. ComonotonicityOver the last two decades, researchers in economics, financial mathematics and actuarial science have introduced results related to the concept of comonotonicity in their respective fields of interest. In this paper, we give an overview of the relevant literature in these research fields, with the main emphasis on the development of the theory and its applications in finance and insurance over the last five years. Although it is our intention to give an extensive bibliographic overview, due to the high number of papers on applications of comonotonicity, it is impossible to present here an exhaustive overview of the recent literature. Further, we restrict this paper to a description of how and where comonotonicity comes in and refer to the relevant papers for a detailed mathematical description. In order to make this paper self-contained, we also provide a short overview of the basic definitions and initial main results of comonotonicity theory, hereby referring to part of the older literature on this topic.The concept of comonotonicity is closely related to the following well-known result, which is usually attributed to both Hoeffding (1940) and Fréchet (1951): For any n-dimensional random vector X ≡ (X 1 , X 2 , . . . , X n ) with multivariate cumulative distribution function (cdf) F X and marginal univariate cdf's F X 1 , F X 2 , . . . , F Xn and for any x ≡ (x 1 , x 2 , . . . , x n ) ∈ R n it holds that F X (x) ≤ min (F X 1 (x 1 ) , F X 2 (x 2 ) , . . . , F Xn (x n )) .(1)In the sequel, the notation R n (F X 1 , F X 2 , . . . , F Xn ) will be used to denote the class of all random vectors Y ≡ (Y 1 , Y 2 , . . . , Y n ) with marginals F Y i equal to the respective marginals F X i . The set R n (F X 1 , F X 2 , . . . , F Xn ) is called the Fréchet class related to the random vector X.The upper bound in (1) is reachable in the Fréchet class R n (F X 1 , F X 2 , . . . , F Xn ) in the sense that it is the cdf of an n-dimensional random vector with marginals given by F X i , i = 1, 2, . . . , n. * Department of Mathematics, ECARES, Université Libre de Bruxelles, CP 210, 1050 Brussels, Belgium † Faculty of Business and Economics, Katholieke Universiteit, 3000 Leuven, Belgium. ‡ Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 S9, 9000 Gent, Belgium 1 In order to prove the reachability property, consider a random variable U , uniformly distributed on the unit interval (0, 1). Then one has that F −1 X 1 (U ), F −1 X 2 (U ), . . . , F −1 Xn (U ) ∈ ...

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An Exact Bond Option Formula

Abstract: This paper derives a closed‐form solution for European options on pure discount bonds, assuming a mean‐reverting Gaussian interest rate model as in Vasicek [8]. The formula is extended to European options on discount bond portfolios.

Cited by 478 publications

References 6 publications

A Rating-Based Sovereign Credit Risk Model: Theory and Evidence

A Rating-Based Sovereign Credit Risk Model: Theory and Evidence

Apreçamento de opções de IDI usando o modelo CIR

Advanced Mathematical Methods for Finance

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