By Nicolas Privault
Rate of interest modeling and the pricing of comparable derivatives stay matters of accelerating significance in monetary arithmetic and probability administration. This ebook presents an available creation to those subject matters by way of a step by step presentation of recommendations with a spotlight on specific calculations. every one bankruptcy is followed with workouts and their entire ideas, making the ebook compatible for complicated undergraduate and graduate point scholars.
This moment version keeps the most gains of the 1st variation whereas incorporating a whole revision of the textual content in addition to extra workouts with their suggestions, and a brand new introductory bankruptcy on credits danger. The stochastic rate of interest types thought of variety from ordinary brief cost to ahead expense versions, with a therapy of the pricing of comparable derivatives similar to caps and swaptions lower than ahead measures. a few extra complicated subject matters 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 interested in quantitative research of rate of interest types.
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Extra resources for An Elementary Introduction To Stochastic Interest Rate Modeling
T At = A0 exp rs ds , t ∈ R+ . e. in integral form: t t σu Su dBu , µu Su du + St = S0 + 0 0 t ∈ R+ , February 29, 2012 18 15:49 World Scientific Book - 9in x 6in main˙privault An Elementary Introduction to Stochastic Interest Rate Modeling with solution t St = S0 exp t σu dBu + 0 0 1 µu − σu2 du , 2 t ∈ R+ , cf. 2. Let ηt and ζt be the numbers of units invested at time t, respectively in the assets (St )t∈R+ and (At )t∈R+ . The value of the portfolio Vt at time t is given by Vt = ζt At + ηt St , t ∈ R+ .
4). 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.
Cox-Ingerson-Ross model. 4) which models the variations of the short rate process rt , where α, β, σ and r0 are positive parameters. 4) in integral form. (2) Let u(t) = IE[rt | Fs ], 0 ≤ s ≤ t. 4), that u(t) satisfies the differential equation u (t) = α − βu(t), 0 ≤ s ≤ t. (3) By an application of Itˆ o’s formula to rt2 , show that 3/2 drt2 = rt (2α + σ 2 − 2βrt )dt + 2σrt dBt . 5), find a differential equation satisfied by v(t) = E[rt2 |Fs ], 0 ≤ s ≤ t, and compute E[rt2 |Fs ], 0 ≤ s ≤ t. You may assume that a = 0 to simplify the computation.