Pricing Rate Options in Short-Rate Models: Trees and Closed Forms

Short-rate models price all fixed-income derivatives by backward induction on a tree: at each node, the derivative value is the risk-neutral expected value of next-period payoffs, discounted at the node's short rate. This universal pricing machine handles the instruments that Black's formula cannot: callable bonds (the issuer has the right to redeem at par before maturity), Bermudan swaptions (exercise at any reset date), and bond options with American exercise. For European-exercise instruments in the Hull-White model, analytical closed forms exist and serve as benchmark checks for the tree. The option-adjusted spread (OAS) — a constant spread added to all discount rates that makes a callable bond's model price equal its market price — is the primary risk metric for fixed-income portfolio managers. This lesson develops the full pricing and OAS workflow.

1. 1. European Bond Options on Trees
2. 1. Closed-Form Bond Option Prices in Hull-White
3. 1. Callable Bonds and OAS
4. 1. Swaption Pricing in Hull-White: Closed Form
5. 1. Bermudan Swaptions on Trees
6. 1. Where This Goes Next