Risk-Neutral Pricing of a Treasury Rate Caplet in a Binomial Model

Summary

The document presents a one-period binomial pricing question for an at-the-money caplet on a future one-year Treasury rate. The caplet pays if the rate rises to the upper of two possible outcomes. The question compares treating the yield like an equity price in a stock-style binomial model with the supplied solution’s method for finding the risk-neutral probability.

The stated solution determines that probability by expressing each possible future Treasury rate as a discounted amount, then taking a probability-weighted average equal to the current rate. This produces a probability close to the one from the attempted calculation, but the two formulations differ because an interest rate is not itself a traded asset price with the same growth and discounting assumptions as an equity index. The document raises the key modeling distinction but does not include an answer explaining it in detail. Its example is limited to a single-period setup and does not establish a general interest-rate model.

Key ideas

• The example prices a caplet whose payoff depends on a future Treasury yield.
• The supplied solution finds the risk-neutral probability by averaging discounted future rates to match the current rate.
• A yield should not automatically be modeled as a traded equity-like asset in a binomial tree.
• The document poses the modeling issue but does not provide a complete explanation of the distinction.

Tags

# One Period Risk Neutral Probability for Caplet


# 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?

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