Fixed Income Deep Dive

Module 1 in the curriculum. This module builds on an already solid foundation: bond pricing, yield curves, spot/forward rates, bootstrapping, and hands-on numerical optimization of no-arbitrage yield curves from the Gottex Brokers years (2010—2014), plus a partial fixed-income course at HEC Lausanne. The goal here is to fill in the theoretical scaffolding that sits underneath that practical experience.

Why this module matters

Every fixed-income instrument is, at its core, a bundle of cash flows discounted at rates that encode the market’s beliefs about the future. Understanding why those discount rates behave the way they do --- and what constraints no-arbitrage imposes on them --- is the bridge between curve-fitting engineering and rigorous pricing theory.

Learning roadmap

Core ideas to internalize

Duration and convexity as Taylor expansions

The price-yield relationship for a bond is nonlinear. Duration and convexity are simply the first- and second-order terms in a Taylor expansion of price $P$ around a yield $y$:

$\frac{dP}{P}\approx -D_{\text{mod}}dy+\frac{1}{2}C(dy{)}^2$

where $D_{\text{mod}}$ is modified duration and $C$ is convexity. Key rate durations generalize this to shifts at individual maturities rather than parallel moves.

Term structure models

Short-rate models like Vasicek and CIR specify the dynamics of the instantaneous rate $r(t)$:

$dr=\kappa (\theta -r)dt+\sigma dW\text{(Vasicek)}$

$dr=\kappa (\theta -r)dt+\sigma \sqrt{r}dW\text{(CIR)}$

The HJM framework takes a different approach --- modeling the entire forward curve $f(t,T)$ directly and deriving the no-arbitrage drift condition. This connects naturally to the numerical curve optimization done at Gottex.

No-arbitrage and FTAP

The Fundamental Theorem of Asset Pricing provides the theoretical anchor: the absence of arbitrage is equivalent to the existence of a risk-neutral measure $\mathbb{Q}$ under which discounted bond prices are martingales. This is the “why” behind the no-arbitrage constraints imposed during yield curve bootstrapping.

Prerequisites

• Bond pricing fundamentals (already solid)
• Probability Theory --- measure-theoretic foundations
• Stochastic Processes --- Ito calculus for the continuous-time models

Key references

• Tuckman & Serrat --- Fixed Income Securities (the practical standard)
• Fabozzi --- The Handbook of Fixed Income Securities
• MIT OCW 15.401 --- Finance Theory I, sessions 4—7 (bond pricing, term structure)