Download e-book for iPad: An Elementary Introduction To Stochastic Interest Rate by Nicolas Privault

By Nicolas Privault

ISBN-10: 9814390852

ISBN-13: 9789814390859

Rate of interest modeling and the pricing of similar derivatives stay topics of accelerating value in monetary arithmetic and possibility administration. This ebook offers an available advent to those subject matters by means of a step by step presentation of recommendations with a spotlight on particular calculations. each one bankruptcy is followed with workouts and their whole strategies, making the booklet appropriate for complex undergraduate and graduate point scholars.

This moment variation keeps the most gains of the 1st version whereas incorporating a whole revision of the textual content in addition to extra routines with their strategies, and a brand new introductory bankruptcy on credits probability. The stochastic rate of interest versions thought of variety from commonplace brief fee to ahead cost types, with a therapy of the pricing of similar derivatives reminiscent of caps and swaptions lower than ahead measures. a few extra complex issues together with the BGM version and an method of its calibration also are coated.

Readership: complicated undergraduates and graduate scholars in finance and actuarial technological know-how; practitioners all for quantitative research of rate of interest versions.

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Extra info for An Elementary Introduction To Stochastic Interest Rate Modeling

Example text

The above change of variable also admits the following informal infinitedimensional formulation: 1 0

We have ∞ x2 dx (em+x − K)+ e− 2v2 √ 2πv 2 −∞ ∞ x2 dx = (em+x − K)e− 2v2 √ 2πv 2 −m+log K ∞ ∞ x2 x2 dx dx ex− 2v2 √ = em e− 2v2 √ −K 2 2πv 2πv 2 −m+log K −m+log K ∞ ∞ (v 2 −x)2 2 v2 dx dx e− 2v2 √ e−x /2 √ −K = em+ 2 2π 2πv 2 −m+log K (−m+log K)/v ∞ 2 2 x v dx = em+ 2 − KΦ((m − log K)/v) e− 2v2 √ 2 2πv 2 −v −m+log K IEQ [(em+X − K)+ ] = = em+ v2 2 Φ(v + (m − log K)/v) − KΦ((m − log K)/v). Moreover, still in the case of European options, the process (ξ)t∈[0,T ] can be computed via the next proposition.

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 ]. 1, construct a probability Q under which the process S˜t := St /At , t ∈ [0, T ] is an Ft -martingale. (2) Compute the arbitrage price C(t, St ) = e−r(T −t) IEQ [|ST |2 |Ft ], at time t ∈ [0, T ], of the contingent claim of payoff |ST |2 .

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An Elementary Introduction To Stochastic Interest Rate Modeling by Nicolas Privault


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