In SP5, Chapter 13, page 13, the question asks to price a caplet. I understand everything except one part of the solution. The volatility of the interest rate is given as "10% pa". The solution (indirectly) defines k = 2 but then says sigma_k = 0.1 (sigma_k being the interest rate volatility). Why is the volatility given as a pa figure? I initially thought that this meant that for k = 2 then sigma_k = sigma_2 = 2 * 0.1 = 0.2. Is the fact that the volatility is pa irrelevant?
I dont think I fully understand your query here. The volatility in the formula is an annual volatility. The assumption is that the rate R(k) is assumed to be lognormally distributed with volatility sigma (pa is in-said, but assumed) So that is what goes into the formula. In this particular case, the caplet for the third year (t=2 to 3) expires at t=2. But if you had wanted a volatility over a two year period it would have been 0.1 (2^0.2) rather than 0.1 * 2. But as mentioned, its the annual figure that goes in, even if the caplet option expires after 20 years.
