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DerivativesPortfolio & RiskFixed IncomeCorporate FinanceStochastic ModelsGuide
Guide
Derivatives pricing
01Forwards, calls and puts02Combining positions03Arbitrage and the forward price04Put–call parity05The one-step binomial model06Risk-neutral probabilities07Multi-step trees and backward induction08American options and early exercise09Delta and hedging10Geometric Brownian motion11The Black–Scholes formula12The Greeks: gamma, vega, theta and rho13Implied volatility and the volatility smile14Stochastic volatility and the Heston model15Static replication: Breeden–Litzenberger and Carr–Madan16Exotic options: barriers, Asians and digitals
Portfolio & risk
01Returns and volatility02Diversification and correlation03The efficient frontier and the capital market line04Risk aversion and the optimal complete portfolio05CAPM, beta and the security market line06Performance measures: Sharpe, Treynor, Jensen and information ratio07Factor models and systematic risk08Value at Risk and expected shortfall
Fixed income
01Bond prices and yields02Duration and convexity03Immunization and duration matching04Spot rates, forward rates and the yield curve05Bond futures and the cheapest-to-deliver06Interest rate swaps07Callable bonds and negative convexity08Credit risk: default probabilities and credit spreads
Corporate finance
01Interest, compounding and discounting02Annuities, perpetuities and loan amortization03Net present value and internal rate of return04The weighted average cost of capital05Capital structure and the Modigliani–Miller theorems06Discounted cash flow valuation07Relative valuation with multiples08Real options: the value of flexibility
Stochastic models
01Random walks and Brownian motion02Probability spaces, σ-algebras and random variables03Filtrations, adapted processes and stopping times04Conditional expectation and martingales05Properties of Brownian motion06The reflection principle and first passage times07The Brownian bridge08The Itô integral09Itô's lemma10Kolmogorov equations and the Feynman–Kac formula11Change of measure and Girsanov's theorem12Monte Carlo option pricing13Simulating SDEs: the Euler–Maruyama scheme14Mean reversion: Ornstein–Uhlenbeck and Vasicek15Jumps and the Merton jump-diffusion model
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Guide

Lessons meant to be read in order. Each one only uses what came before it.

Fixed income

Bond pricing, yield curves, duration and convexity.

Start
0 of 8 read
  1. 01Bond prices and yieldsA bond as a stream of cash flows, yield to maturity, par, premium and discount.
  2. 02Duration and convexityMacaulay and modified duration, first- and second-order approximations of interest-rate risk.
  3. 03Immunization and duration matchingProtecting a liability against rate moves by matching present value and duration, and why convexity and rebalancing matter.
  4. 04Spot rates, forward rates and the yield curveZero-coupon rates, bootstrapping from par yields, implied forward rates and the shape of the curve.
  5. 05Bond futures and the cheapest-to-deliverDeliverable baskets, conversion factors, invoice price, net delivery cost and why the cheapest-to-deliver bond switches with the yield.
  6. 06Interest rate swapsFixed and floating legs, the swap as two bonds, the par swap rate from discount factors, valuing an existing swap and DV01.
  7. 07Callable bonds and negative convexityThe issuer's call option, price compression near the call price, yield to call and negative convexity.
  8. 08Credit risk: default probabilities and credit spreadsHazard rates, survival probabilities, recovery, pricing defaultable bonds and the CDS spread.

Check yourself

Questions from the lessons of this series. Open a lesson's group whenever you want to test yourself.

Immunization and duration matching2 questionsRe-read the lesson →
Check yourself

A liability has duration 7. You immunize it with a short bond of duration 2 and a long bond of duration 10. What share of the money goes into the short bond? Answer in percent.

Check yourself

A duration-matched barbell has more convexity than the liability (CA>CLC_A > C_LCA​>CL​). Yields shift in parallel by a small amount. What happens to the surplus?

Bond futures and the cheapest-to-deliver2 questionsRe-read the lesson →
Check yourself

At delivery the futures price is F=95F = 95F=95. A deliverable bond with conversion factor CF=1.10CF = 1.10CF=1.10 trades at a clean price of 106.00. What is its net delivery cost per 100 of face?

Check yourself

Yields fall from 6% to 3%. Which of the three bonds becomes the cheapest to deliver?

Interest rate swaps2 questionsRe-read the lesson →
Check yourself

Annual payments, discount factors P(0,1)=0.96P(0,1) = 0.96P(0,1)=0.96 and P(0,2)=0.92P(0,2) = 0.92P(0,2)=0.92. What is the two-year par swap rate? Answer in percent.

Check yourself

You entered a payer swap (you pay fixed, receive floating) at the par rate. The next day the whole curve falls. What is your swap worth now?

Callable bonds and negative convexity2 questionsRe-read the lesson →
Check yourself

The market starts to expect much more volatile interest rates; today's yields and everything else stay the same. What happens to the price of the callable bond?

Check yourself

At a yield of 5% the callable bond has an effective convexity of about −68-68−68. Yields move 1 percentage point up or down. Compared with a bond of the same duration and zero convexity, the callable bond…

Credit risk: default probabilities and credit spreads2 questionsRe-read the lesson →
Check yourself

A bond trades at a credit spread of 150 basis points. Assuming a recovery rate of 40%, what hazard rate does the credit triangle imply? Answer in percent per year.

Check yourself

Hazard rates implied by market spreads are usually well above the historical default rates of similar issuers. Why?

About this tool

Narrative guides that connect the interactive tools into a story: financial instruments and their payoffs, the no-arbitrage principle, risk-neutral pricing, delta hedging and the road to the Black–Scholes formula. Each guide links to the labs where you can try the ideas yourself.

Frequently asked questions

In what order should I read the guides?

Start with instruments and no-arbitrage, then risk-neutral pricing and delta hedging; together they lead to the Black–Scholes formula.

Who are the guides for?

Students of financial and insurance mathematics and anyone who wants intuition behind derivatives pricing, not just the formulas.