Unit 9 of 14

Interest rate models and measures

  • 8 lessons
  • Videos 65 to 72

Interest rate derivatives need a model for the whole term structure and a careful choice of pricing measure. This unit develops numeraires, rate models, rate options and collateral discounting.

A positive numeraire defines its own pricing measure, and bond numeraires give forward measures. Vasicek and Cox-Ingersoll-Ross models give affine bond prices, and Hull-White adds a time-dependent drift that fits the initial curve. The Heath-Jarrow-Morton drift restriction follows from absence of arbitrage, and forward rate market models gain drifts under a terminal measure. Caplets and swaptions are priced with Black and Bachelier formulas under forward and annuity measures, and SABR gives an asymptotic smile formula. With collateral, discount and projection curves have different roles.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Numeraires and short-rate models

  1. Numeraires and forward measures

    How a strictly positive traded numeraire defines its own pricing measure through a density process, why numeraire invariance keeps currency prices fixed, and how the bond numeraire gives the forward measure.

    Video 65

  2. Vasicek and CIR short rate models

    How the mean-reverting Vasicek and Cox-Ingersoll-Ross short-rate models price zero-coupon bonds with affine formulas, how they differ at the zero boundary, and why their parameters belong to the pricing measure.

    Video 66

  3. Hull White and initial curve fitting

    How the Hull-White model adds a time-dependent drift to a Gaussian short rate, how a deterministic shift fits an initial discount curve exactly, and how conditional bond prices follow from Gaussian integrals.

    Video 67

Forward-curve and market models

  1. Heath Jarrow Morton and the drift restriction

    How the Heath-Jarrow-Morton framework models the instantaneous forward curve, how bond price dynamics follow from it, and why the HJM drift restriction fixes forward drifts, with an exponential volatility kernel.

    Video 68

  2. Forward rate market models

    How a forward rate market model makes each simple forward rate a martingale under its own forward measure, which drift the terminal measure adds through forward-rate correlation, and why backward-looking overnight rates differ.

    Video 69

Rate options, smiles and collateral curves

  1. Caps floors and swaptions

    How caplets and floorlets are priced under the payment forward measure with Black's formula or the Bachelier formula, why caplet-floorlet parity needs no model, and how swaptions are priced under the annuity measure.

    Video 70

  2. SABR and smile asymptotics

    How the SABR model pairs a forward rate with a random volatility scale, what the elasticity parameter and the volatility of volatility do, and how the Hagan formula and its at-the-money limit approximate the smile.

    Video 71

  3. Collateral discounting and multiple curves

    Why full cash collateral makes the collateral rate the discount rate, how OIS discounting, projection curves and the tenor basis enter multi-curve swap valuation, and how a compounded overnight rate accrues.

    Video 72