I am studying some financial modeling put together by the Society of Actuaries in the USA. In it, the following practice problem was given:
Find the Risk Neutral price of an at-the-money interest rate caplet with expiry in one year with a notional amount of 10,000 on the underlying U.S. 1 year treasury rate a year from now. Assume a single period binomial model, where the 1-year treasury yield at expiry may be either 3.5% or 2.5% exclusively with some probability greater than 0. The current Treasury rate is 3%. Hence, the derivative may pay 50 (if the future rate rises to 3.5%) or 0 (in the other case).
In my attempted solution, I tried to have ${u=3.5/3}$, ${d=2.5/3}$, ${r=.03}$. So we have ${\tilde{p}=\frac{1.03-\frac{2.5}{3}}{\frac{3.5}{3}-\frac{2.5}{3}}=0.59}$. Here, I've treated the yield as a "stock index" of sorts, and the risk-free rate as the current 1-year treasury rate.
In the provided solution, they solve for ${\tilde{p}}$ via: ${0.03 = \tilde{p}\frac{.035}{1.035} + (1-\tilde{p})\frac{.025}{1.025}\rightarrow \tilde{p}\approx 0.59513}$.
Why is my formulation (treating the rates like an equity index) incorrect? How did they arrive at their equation for the risk neutral probability?
