# An Elementary Introduction To Stochastic Interest Rate by Nicolas Privault By Nicolas Privault

Rate of interest modeling and the pricing of comparable derivatives stay matters of accelerating value in monetary arithmetic and possibility administration. This ebook presents an obtainable advent to those subject matters via a step by step presentation of thoughts with a spotlight on particular calculations. every one bankruptcy is observed with workouts and their whole options, making the booklet appropriate for complicated undergraduate and graduate point scholars.

This moment version keeps the most positive factors of the 1st version whereas incorporating an entire revision of the textual content in addition to extra workouts with their options, and a brand new introductory bankruptcy on credits possibility. The stochastic rate of interest types thought of variety from ordinary brief fee to ahead expense types, with a therapy of the pricing of similar derivatives corresponding 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 coated.

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Extra info for An Elementary Introduction To Stochastic Interest Rate Modeling

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18) of F = φ(ST ). x Let now (St,s )s∈[t,∞) be the price process solution of the stochastic differential equation x dSt,s ˆs , = rs ds + σs dB x St,s s ∈ [t, ∞), x = x ∈ (0, ∞). 5) in Appendix A. 21) we recover the fact that C(t, x) solves the Black-Scholes PDE:  ∂C 1 ∂2C ∂C    (t, x) + x2 σ 2 (t) 2 (t, x) + xr(t) (t, x) = r(t)C(t, x), ∂t 2 ∂x ∂x    C(T, x) = φ(x). e. 16) and the following lemma. 3. Let X be a centered Gaussian random variable with variance v 2 . We have IEQ [(em+X − K)+ ] = em+ Proof.

3). 9) where A and C are functions to be determined under the conditions A(0) = 0 and C(0) = 0. 5) yields the system of Riccati and linear differential equations  σ2 2   C (s)  −A (s) = −aC(s) − 2    −C (s) = bC(s) + 1, which can be solved to recover σ 2 − 2ab σ 2 − ab −bs σ 2 −2bs 4ab − 3σ 2 + s + e − e A(s) = 4b3 2b2 b3 4b3 and 1 C(s) = − (1 − e−bs ). b As a verification we easily check that C(s) and A(s) given above do satisfy bC(s) + 1 = −e−bs = −C (s), and aC(s) + a σ2 σ 2 C 2 (s) = − (1 − e−bs ) + 2 (1 − e−bs )2 2 b 2b σ 2 − 2ab σ 2 − ab −bs σ2 = − e + 2 e−2bs 2 2 2b b 2b = A (s).

1. Exponential Vasicek model. Consider a short rate interest rate proces (rt )t∈R+ in the exponential Vasicek model: drt = rt (η − a log rt )dt + σrt dBt , where η, a, σ are positive parameters. 3) as a function of the initial condition y0 , where θ, a, σ are positive parameters. Hint. Let Zt = Yt − θ/a, t ∈ R+ . (2) Let Xt = eYt , t ∈ R+ . Determine the stochastic differential equation satisfied by (Xt )t∈R+ . 2) in terms of the initial condition r0 . (4) Compute the conditional mean IE[rt |Fu ] of rt , 0 ≤ u ≤ t, where (Fu )u∈R+ denotes the filtration generated by the Brownian motion (Bt )t∈R+ .