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Expected short rate in the Vasicek model after ten years

Question

A short rate RR starts at R0=r>0R_0=r>0 and follows the Vasicek dynamics

dRt=(a−bRt) dt+σ dWtdR_t=(a-bR_t)\,dt+\sigma\,dW_t

with positive constants a,b,σa,b,\sigma and standard Brownian motion WW. Use parameters σ=0.1\sigma=0.1, b=0.1b=0.1, a=0.2a=0.2, r=1r=1 and horizon T=10T=10. First find E[RT]\mathbb{E}[R_T] in general, then evaluate it; the value has the form A+BeCA+Be^{C} with integers A,B,CA,B,C. Report A+B+CA+B+C.

Short answer

00. Applying Ito to ebtRte^{bt}R_t removes RR from the drift; taking expectations gives E[RT]=ab+(r−ab)e−bT\mathbb{E}[R_T]=\tfrac ab+\big(r-\tfrac ab\big)e^{-bT}. With bT=1bT=1 and a/b=2a/b=2 this is 2−e−12-e^{-1}, so A+B+C=2−1−1=0A+B+C=2-1-1=0.

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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