On p.42 of the book how has the formula for q under measure Q arrived at from its P equivalent. Thanks for the hint. On p.137 the equation for B(t) has the integral of W(s) but the equation for P(t,T) starts with product sigma(T-t)*W(t), even though the formula for f(t,T) an r(t) both have sigma*(W(t)). Thanks for the intuition. Kind regards, Szczepan
Hi I think I see what you mean. Let me know if I've not understood you correctly. The first term in the definition of \(P(t,T)\) includes \(\int_{t}^{T}\sigma W_t du\). But notice that there's no \(u\) in the integrand, and so we're essentially integrating a constant. Therefore that integral equals \(\sigma(T-t)W_t\). Notice that this is not the same as \(\int_{t}^{T}\sigma W_u du\) which depends on the path of the standard Brownian motion rather than its starting value \(W_t\).
Thanks Steve, integration is done over the maturity of the bond P(t,T) if that makes sense. How about the formula for the q on p.42 I'm struggling to derive the expression, manipulating the formulae shown before.
The risk-neutral up-step probability \(q\) is chosen so that the share price is expected to grow that the risk-free rate over each time-step. Algebraically this mean: \(q\times s_{up} + (1-q)\times s_{down} = s \times exp(r\delta)\) Rearranging will result in the page 42 definition for \(q\).
Thanks Steve. Should've been more precise in the original question: p.42 has two definitions of the q and itsv the second that gives me headaches. Precisely, it's q=(1/2)*(1-((delta*t)^(1/2))*[mju+(1/2)*sigma^2-r]/sigma) Thanks in advance
OK, I see what you mean. Start with the definition for \(q\), and replace \(s_{up}\) and \(s_{down}\) with their respective values found on page 41, which leads to: \[ q=\frac{exp(r\delta t)-exp(\mu\delta t - \sigma\sqrt{\delta t})}{exp(\mu\delta t + \sigma\sqrt{\delta t}) - exp(\mu\delta t - \sigma\sqrt{\delta t})} \] Now use Taylor's approximation in each of the four exponential functions: \(exp(x) \approx 1 + x +\tfrac{1}{2}x^2\). This then simplifies (remember that powers of \(\delta t\) higher than 1 go to zero) to the second expression for \(q\) on page 42.
