# One Period Risk Neutral Probability for Caplet

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?

• @T123 the usual binomial short-rate model I've seen (as in Shreve Vol I) relies on having a full yield curve (spot rates), from which you can compute risk-neutral probabilities to preclude arbitrage. In the example problem, no 2-year spot rate at time 0 is provided, hence I'm confused at how they arrived at their conclusion. The lack of a 2-year spot rate also prompted my attempt to treat the rates as if they were an equity index. Jan 30 at 15:38
• Ok, i deleted the comment. I think you probably checked the books. already. Lets see which answers it attracts ;-)
– T123
Jan 30 at 15:54
• I got the solution key, and found that the problem was incomplete. It assumed we knew the 2-year spot rate, but never provided it in the question. It was an error in the homework. Feb 5 at 1:31