By Nicolas Privault
Rate of interest modeling and the pricing of comparable derivatives stay topics of accelerating value in monetary arithmetic and hazard administration. This publication offers an available creation to those themes through a step by step presentation of techniques with a spotlight on specific calculations. each one bankruptcy is observed with routines and their entire options, making the e-book compatible for complicated undergraduate and graduate point scholars.
This moment version keeps the most beneficial properties of the 1st variation whereas incorporating an entire revision of the textual content in addition to extra routines with their suggestions, and a brand new introductory bankruptcy on credits chance. The stochastic rate of interest versions thought of diversity from normal brief cost to ahead price types, with a therapy of the pricing of comparable derivatives similar to caps and swaptions less than ahead measures. a few extra complex themes together with the BGM version and an method of its calibration also are lined.
Readership: complicated undergraduates and graduate scholars in finance and actuarial technology; practitioners enthusiastic about quantitative research of rate of interest types.
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Extra resources for An Elementary Introduction To Stochastic Interest Rate Modeling
1, construct a probability measure Q (α) becomes a standard Brownian under which the process Xt t∈[0,T ] motion. (2) Compute the expectation T IE exp (β − α) (α) (α) Xt dXt 0 + α2 2 T (α) Xt 2 dt 0 for all β < 1/T . (3) Compute the expectation IE exp α2 2 T (α) (Xt )2 dt 0 for all α < 1/T . 2. Consider the price process (St )t∈[0,T ] given by dSt = µdt + σdBt St and a riskless asset of value At = A0 ert , t ∈ [0, T ], with r > 0. Let (ζt , ηt )t∈[0,T ] a self-financing portfolio of value Vt = ηt At + ζt St , t ∈ [0, T ].
This leads to the identity 1 ∂2F ∂F ∂F (t, rt ) + σ 2 (t, rt ) 2 (t, rt ) + (t, rt ) = 0, −rt F (t, rt ) + µ ˜(t, rt ) ∂x 2 ∂x ∂t which can be rewritten as in the next proposition. 1. 5) ∂x 2 ∂x ∂t subject to the terminal condition F (T, x) = 1. 6) is due to the fact that P (T, T ) = $1. e. σ(t, rt ) ∂F dP (t, T ) ˆt = rt dt + (t, rt )dB P (t, T ) P (t, T ) ∂x ∂ log F ˆt . 5) by direct computation of the conditional expectation P (t, T ) = IEQ e− T t rs ds Ft . 7) We will assume that the short rate (rt )t∈R+ has the expression t rt = g(t) + h(t, s)dBs , 0 where g(t) and h(t, s) are deterministic functions, which is the case in particular in the [Vaˇsiˇcek (1977)] model.
4) Explicitly compute the strategy (ζt , ηt )t∈[0,T ] that hedges the contingent claim exp(ST ). March 27, 2012 14:11 World Scientific Book - 9in x 6in This page intentionally left blank main˙privault February 29, 2012 15:49 World Scientific Book - 9in x 6in Chapter 3 Short Term Interest Rate Models This chapter is a short introduction to some common short term interest rate models. g. [Brigo and Mercurio (2006)], [Carmona and Tehranchi (2006)], [James and Webber (2001)], [Kijima (2003)], [Rebonato (1996)], [Yolcu (2005)].