38 References
39 References
This page collects the main references for Mathematical Finance: Pricing, Hedging, Risk, and Computation. The bibliography emphasizes sources that support the mathematical development of the book and the Google Colab computer labs.
The references are organized by topic for students who want to read beyond the notes. The complete bibliography appears at the end of the page.
39.1 How to use the references
Students should not try to read all references at once. A practical approach is:
- Use the book chapters and labs as the primary path.
- Use the references below when a topic deserves a deeper theoretical or computational treatment.
- For independent projects, start from one foundational reference and one computational reference.
For example, a project on Monte Carlo option pricing may combine the theory of risk-neutral valuation with simulation methods from Glasserman (2004) and the option-pricing background in Black and Scholes (1973), Merton (1973), and Cox et al. (1979).
39.2 Foundational option pricing and no-arbitrage
The classical option-pricing framework begins with no-arbitrage, replication, and risk-neutral valuation. The Black–Scholes–Merton model is developed in Black and Scholes (1973) and Merton (1973). The binomial model of Cox et al. (1979) provides a discrete-time entry point that connects naturally to martingale pricing and numerical approximation.
The modern martingale formulation of no-arbitrage pricing is shaped by Harrison and Kreps (1979) and Harrison and Pliska (1981). More advanced treatments can be found in Karatzas and Shreve (1998), Duffie (2001), Musiela and Rutkowski (2005), and Delbaen and Schachermayer (2006).
39.3 Stochastic calculus and continuous-time finance
For Brownian motion, stochastic integrals, Ito’s formula, stochastic differential equations, and continuous-time asset-pricing models, useful references include Shreve (2004b), Björk (2009), Øksendal (2003), and Revuz and Yor (1999).
A good learning path is to pair the mathematical exposition in Shreve (2004a) and Shreve (2004b) with the computational experiments in the labs on Brownian motion, Ito integrals, change of measure, and Black–Scholes hedging.
39.4 Monte Carlo methods and numerical pricing
Monte Carlo pricing is introduced historically in Boyle (1977) and surveyed in Boyle et al. (1997). A systematic computational treatment is given in Glasserman (2004). Least-squares Monte Carlo for American-style derivatives is developed in Longstaff and Schwartz (2001).
For finite-difference methods and PDE-based pricing, useful references include Tavella and Randall (2000), Brandimarte (2006), Higham (2004), Wilmott et al. (1995), and Kwok (2008).
39.5 Volatility, smiles, stochastic volatility, and jumps
The volatility surface is central in modern derivatives markets. Practical and theoretical perspectives are given in Gatheral (2006) and Natenberg (1994). The Heston stochastic-volatility model was introduced in Heston (1993). Jump-diffusion and jump-stochastic-volatility models are represented by Merton (1976), Bates (1996), and Cont and Tankov (2004).
These references connect directly to the labs on implied volatility, volatility smiles, exotic options, and the Heston model.
39.6 Fixed income and term-structure models
The short-rate and term-structure part of the book uses the Vasicek model Vasicek (1977), the Cox–Ingersoll–Ross model Cox et al. (1985), and the Heath–Jarrow–Morton framework Heath et al. (1992). Related interest-rate models include Ho and Lee (1986) and Hull and White (1990).
For broader fixed-income modeling and market practice, see Brigo and Mercurio (2006) and Fabozzi (2012). Principal-component ideas for yield-curve movements are motivated by Litterman and Scheinkman (1991).
39.7 Portfolio theory and asset pricing
The portfolio section begins with mean–variance analysis from Markowitz (1952) and Markowitz (1959). The CAPM is associated with Sharpe (1964) and Lintner (1965). Multi-factor equity models are represented by Fama and French (1993) and Carhart (1997).
For a broader graduate-level perspective on asset pricing and empirical finance, see Campbell et al. (1997) and Cochrane (2005). A computational and optimization-oriented treatment of investments can be found in Luenberger (1998).
39.8 Risk management and financial time series
The risk-management labs use Value at Risk, Expected Shortfall, backtesting, volatility clustering, heavy tails, and stress testing. Useful references include Jorion (2007), McNeil et al. (2015), Embrechts et al. (1997), Alexander (2008), and Tsay (2010).
These sources are especially useful for final projects involving portfolio losses, model validation, or risk forecasting.
39.9 Market microstructure and execution
Although execution is not the main focus of the first version of the book, optimal execution provides an important bridge between mathematical finance and practical trading systems. The Almgren–Chriss model is developed in Almgren and Chriss (2000) and Almgren and Chriss (2001).
39.10 General textbooks and practitioner references
For a broad derivatives overview, Hull (2022) is a standard reference. For students who want a mathematically accessible computational introduction, Higham (2004) is especially useful. For practitioner-style option intuition, Natenberg (1994) provides a complementary perspective.
39.11 Suggested reading by lab
| Lab | Main topic | Suggested references |
|---|---|---|
| 1 | Returns and one-period no-arbitrage | Cox et al. (1979); Shreve (2004a) |
| 2 | Binomial trees and risk-neutral pricing | Cox et al. (1979); Shreve (2004a) |
| 3 | Brownian motion, GBM, Monte Carlo | Boyle (1977); Glasserman (2004) |
| 4 | Black–Scholes, Greeks, Delta hedging | Black and Scholes (1973); Merton (1973); Hull (2022) |
| 5 | Implied volatility and smile | Gatheral (2006); Natenberg (1994) |
| 6 | Monte Carlo variance reduction | Glasserman (2004); Boyle et al. (1997) |
| 7 | Exotic options | Kwok (2008); Hull (2022) |
| 8 | Finite-difference PDE methods | Tavella and Randall (2000); Brandimarte (2006); Higham (2004) |
| 9 | Ito integrals and Ito formula | Shreve (2004b); Øksendal (2003) |
| 10 | Martingales and change of measure | Harrison and Pliska (1981); Karatzas and Shreve (1998); Björk (2009) |
| 11 | Bonds, duration, convexity | Fabozzi (2012); Brigo and Mercurio (2006) |
| 12 | Vasicek and CIR models | Vasicek (1977); Cox et al. (1985) |
| 13 | HJM and yield-curve simulation | Heath et al. (1992); Brigo and Mercurio (2006) |
| 14 | Portfolio optimization | Markowitz (1952); Markowitz (1959); Luenberger (1998) |
| 15 | CAPM and factor models | Sharpe (1964); Lintner (1965); Fama and French (1993); Carhart (1997) |
| 16 | VaR, ES, and backtesting | Jorion (2007); McNeil et al. (2015); Alexander (2008) |
| 17 | Heston stochastic volatility | Heston (1993); Gatheral (2006) |
| 18 | Integrated pricing, hedging, calibration, and risk | Black and Scholes (1973); Glasserman (2004); McNeil et al. (2015) |