Mathematics in Modern Finance
Course Introduction: Financial Mathematics in Modern Finance
This is the second time the class is offered. Last year it was a smaller version, six units of credit, once a week, with practitioners from the industry (e.g., Morgan Stanley) sharing how math is applied in modern finance.
This year the course is expanded to 12 units of credit and runs twice a week (Tuesdays and Thursdays, 02:30–04:00 PM, in this classroom).
Instructors: expanded from two to four main instructors this year. New lecturers include Doctor Peter Kempstor and Doctor Chungbon Li; previously, Dr. Vasili Strela and Jake Shaw were the main instructors. The course aims to provide a foundation from linear algebra, probability, statistics, and some stochastic calculus to support the finance examples taught by industry practitioners.
Purpose of the course: to give you a sampling manual for how mathematics is applied in modern finance and to help you decide whether this is a field you would like to pursue as a career. The course also helps solidify math knowledge while introducing new finance content.
Prerequisites and math focus: prerequisite material will be discussed later; emphasis on mathematics that underpins financial modeling used in practice.
Open door policy and participation:
The course is open to students from various backgrounds and even from other universities (e.g., Harvard previously).
All instructors have MIT emails; contact via the class website if pace is too fast/slow or concepts are unclear.
Office hours and on-campus presence: instructors will be available; some may visit less often, others more regularly.
Course format and interaction:
The first lecture will include an introduction with a story and a lightweight quiz (not a real quiz; participation-based).
A focus on feedback: polling throughout the course to gauge pace and understanding.
The class website will host emails for feedback, course materials, and announcements.
Opening story: Vega vs Kappa anecdote
The instructor’s personal story centers on an early-career experience at Morgan Stanley on the options trading desk.
Question from the desk quant: “Where is the Vega report?”; the desk uses a sensitivity measure to volatility.
Vega is explained as a sensitivity of a book/portfolio to volatility changes; volatility is a measure of how much a price can change over time (standard deviation of price changes).
At Morgan Stanley, the same concept is referred to as Kappa in some contexts; Vega is a term used by many uneducated traders (from Solomon Brothers) who use “Vega” as a Greek letter, though at Morgan Stanley it is called Kappa. This illustrates a broader point: finance terms and terminology can evolve and differ across firms.
Takeaway: quantitative finance is relatively young (pricing models for options were developed starting with Black-Scholes in the 1970s) and has transformed trading floors over the last 20–30 years; practitioners increasingly have advanced math and computer science backgrounds.
Key lesson from the story: the field evolves rapidly; the core question is not merely right vs. wrong mathematically, but how concepts are defined, verified, and implemented in practice.
Background on financial markets: historical development of markets and products
Early markets involved exchanging goods (you have what I don’t, I have what you don’t).
Markets become centralized with stock exchanges, futures exchanges, etc.; products listed as securities on exchanges.
In recent years, electronic platforms and ECNs enable trading at higher volume and speed; OTC (over-the-counter) trading exists where counterparties trade directly with customized terms, often outside formal exchanges.
Markets are regionally segmented, with currencies traded across borders; products include local stocks, bonds, and currencies.
IPOs and the primary market: when a company transitions from private to public, it issues new shares through an IPO; trading then occurs in the secondary market after listing.
Debt instruments (bonds, notes, bills) are ubiquitous: governments issue sovereign debt (e.g., US Treasury), corporates issue debt for financing; debt can be securitized into asset-backed securities (ABS/CMBS/MBS).
Commodities include metals, energy, and agriculture; traded widely through futures (and some physical delivery).
Real estate markets play a major role in lending, mortgages, and asset-backed securities; the 2008 financial crisis highlighted the importance of real estate and mortgage-backed securities.
Derivatives have grown to include swaps, options, and structured products; these can be tailored for investors/borrowers and become increasingly complex to price and manage risk.
Market participants include a broad ecosystem of players; the relationship between lenders and borrowers is the core function of financial markets, enabling capital allocation and risk transfer.
Major market participants and their roles
Banks and dealers: key players in market making, providing liquidity and taking principal risk on trades; banks have institutional client businesses and asset management divisions; divisions include Fixed Income, Equity, and Investment Banking Division (IBD) covering corporate finance, IPOs, M&A, and advisory.
Asset managers: large force in markets managing money for others (mutual funds, pension funds, insurance companies, sovereign wealth funds, endowments, and more).
Hedge funds: strategies to profit from mispricings or inefficiencies; various approaches across sectors and instruments.
Private equity: invests in private companies or takes companies private; seeks profitability improvements and exit opportunities.
Governments and policymakers: influence markets through monetary policy and policy announcements; interest rate decisions and expectations shape market environments.
Other players include corporate hedgers (corporations seeking to manage risk), insurers, and other institutional investors.
Core market concepts and purposes
Why markets exist: they connect lenders with borrowers, enabling capital allocation and risk sharing; investors seek better yields, while borrowers seek access to capital.
Market efficiency and risk-taking: many trades carry some degree of risk and potential return; some activities are effectively zero-sum, depending on the product and participants.
Hedging vs market making vs proprietary trading (three primary trading modes):
Hedging: reducing exposure to existing risk (e.g., currency, interest rate, or commodity risk); example includes locking in rates or currencies to protect future cash flows.
Market making: providing liquidity by quoting bid/ask prices and taking on principal risk to facilitate trades; profits mainly from the bid-ask spread while managing residual risk (
delta, gamma, theta, and vega).Proprietary trading: risk-taking to generate returns beyond the market benchmark (beta); focuses on alpha (excess return) and systematic strategies.
Key players’ incentives and constraints: credit risk, liquidity risk, regulatory constraints (e.g., post-crisis restrictions on proprietary trading), and capital requirements.
Foundational trading concepts: alpha, beta, and related ideas
Beta: captures the correlated movement of a portfolio with a benchmark (e.g., the market or an index).
Alpha: the excess return of a portfolio relative to its benchmark; a measure of performance not explained by market movement.
How these ideas arise: simple linear regression on time-series of returns; relation to portfolio construction and performance attribution.
Example: comparing a portfolio A with respect to the S&P 500 (B) to determine beta and alpha via regression of returns.
Practical examples and illustrations used in the lecture
Currency hedging example: borrowing in Japan to invest in Australia to exploit interest rate differentials; key risk is exchange rate movement affecting profits; exits can be problematic if many traders adopt the same strategy (crowding risk).
Cross-currency swap and currency forwards: hedging complexity goes beyond simple forwards; involves swaps and currency exposure management; real-world corporate risk management includes hedging revenue or costs in overseas currencies.
Corporate hedging in practice: Intel and other multinationals with overseas earnings must manage currency exposure; import/exporters face currency risk as well.
Market-making in opaque products: for options and other non-transparent instruments, market makers provide liquidity and manage their books using Greeks (delta, gamma, theta, vega) to balance risk.
Greeks in portfolio risk management:
Delta: sensitivity of price to the underlying asset, Δ = ∂P/∂S
Gamma: sensitivity of delta to the underlying, Γ = ∂²P/∂S²
Theta: sensitivity to time decay, Θ = ∂P/∂t
Vega: sensitivity to volatility, ν = ∂P/∂σ
Tail risk and capital considerations: Value at Risk (VaR), leverage (historically up to 40x before the 2008 crisis), and balance-sheet risk in derivatives portfolios.
Arbitrage and relative value strategies: exploiting price relationships across markets or instruments that should align (e.g., spot vs forward prices; cross-asset price relationships).
Systematic trading ideas: trend following, momentum, statistical arbitrage (stat arb), and fundamental analysis; a wide range of strategies, each with limitations as markets adapt.
Special situations and private equity: opportunities arising from corporate events or distressed assets; private equity and special situations firms look to unlock value.
The limits of the “holy grail” trading automation: automated trading is powerful but requires continuous updates and validation; no strategy is a perpetual money-making machine.
The mathematical toolkit in practice
Pricing models: crucial for pricing complex financial instruments; pricing drives risk management and trading decisions; solving differential equations is common in obtaining model prices.
Risk management: quantifying exposure, calculating risk metrics, and designing hedges; involves a mix of quantitative methods and judgment.
Trading strategies: quantitative researchers seek to develop and test strategies; caution against overreliance on single models or “one-size-fits-all” approaches; continuous refinement is essential.
Projects and applications from students (examples):
Noise in derivative pricing: estimating a derivative’s delta via Monte Carlo can produce noisy results; requires choosing an optimal shift size or numerical differentiation approach to reduce variance, balance computational cost, and achieve better accuracy.
Electronic trading and currency prediction: Kalman filters applied to non-uniformly spaced data for predicting exchange rates (e.g., ruble/USD) in a Moscow trading platform; integration into e-trading systems.
General workflow in the course: use of Monte Carlo methods, Kalman filters, and other numerical techniques; practical implementation discussions with Morgan Stanley and other industry projects.
Homework, reading, and course resources
Optional but recommended: build a personal glossary of financial concepts from the course website; read course materials and external references to deepen understanding.
The course website hosts syllabi, literature, and lecture slides; announcements and sign-ups occur through the site and a sign-up sheet distributed in class.
The instructor emphasizes staying engaged and asking questions, with email as a primary channel for feedback and clarification.
Practical takeaways and mindsets
Finance is a field that has evolved rapidly; mathematical rigor is essential, but real-world applications involve judgment, interpretation, and ongoing validation.
The interplay between mathematics, economics, and computation defines modern finance; understanding the limits of models and the importance of risk management is critical.
Students come from diverse backgrounds (math, statistics, finance, engineering, etc.); the course is designed to bridge theory and practice.
Quick reminders about logistics and contacts
The course website and sign-up sheets are the primary channels for announcements and updates.
Office hours and accessibility: instructors are available for discussions; use email to reach out with questions or feedback.
Note on the overarching theme
The lecture combines storytelling, intuition-building, and quantitative rigor to introduce how mathematics underpins pricing, hedging, risk management, and trading strategies in modern finance.
Final takeaways from the introductory session
Expect a broad survey of the financial markets, from basic terminology to advanced models and practical applications.
Be prepared to engage with both theory (pricing models, risk metrics, derivatives) and practice (case studies, projects, and industry-relevant tools).
The course aims not only to teach techniques but also to help you decide if quantitative finance is a field you want to pursue.
Notable numerical references and formulas mentioned in passing
Delta: oxed{\n ext{Delta} = rac{\, ext{d}P}{ ext{d}S}
a}
Gamma: oxed{ ext{Gamma} = rac{ ext{d}^2 P}{ ext{d}S^2}}oxed{ ext{Theta} = rac{ ext{d}P}{ ext{d}t}}oxed{
u = rac{ ext{d}P}{ ext{d}oldsymbol{
u}} ext{ or } rac{ ext{d}P}{ ext{d}oldsymbol{\sigma}}}Example EV calculations for a choice problem:
Scenario 1 (first choice): A = 0.2 × 500 + 0.8 × (−500) = −$300, B = −$280
Scenario 2 (second choice): A = 0.8 × 500 + 0.2 × (−500) = $240, B = 280
Interpretation: different risk/return profiles lead to different preferences based on risk tolerance and potential payoffs.
Leverage reference: before 2008, leverage as high as 40× was common; such leverage amplified losses when markets moved against positions.
Asset pricing and risk concepts: VaR (Value at Risk) as a risk measure; balance sheet considerations when derivatives are present; capital requirements for banks and trading desks.
Closing note
The instructor signals that more detailed lectures will cover deeper mathematical topics and practical applications, including projects from the previous year and ongoing work in e-trading, pricing, and risk management.
Instructors and Class Structure
Instructors: Jake Shaw; Vasili Strela; Peter Kempstor; Chungbon Li (Chun Boon Li). The team expanded to cover new math lectures (linear algebra, probability, statistics, stochastic calculus) to ground students for the finance applications.
Class format: two 2-hour sessions per week; emphasis on both math foundations and real-world industry applications.
Course objectives: help students decide if finance is a field to pursue; deepen math knowledge; bridge academia with industry practices.
Foundational Background and Prerequisites
Core mathematical topics highlighted for this course:
Linear algebra
Probability
Statistics
Basic stochastic calculus
The course includes mathematical rigor while tying it to financial concepts, models, and risk management.
Market Structure and Participants Overview
Markets: centralized exchanges, ECNs, and OTC trading; products include equities, debt, commodities, currencies, and derivatives.
Primary vs secondary markets:
Primary market: IPOs and initial listing.
Secondary market: trading of listed securities after IPO.
Key market players and roles:
Banks and dealers (market makers): provide liquidity, quote prices, take principal risk.
Brokers: match buyers and sellers, earn commissions, do not take principal risk.
Asset managers (mutual funds, pension funds, insurers, sovereign wealth funds, endowments): manage public or private capital; seek returns to meet liabilities or fund objectives.
Hedge funds: active capital allocators seeking alpha via various strategies.
Private equity: invest in companies, sometimes take them private, with the aim of value creation and eventual exit.
Governments and policymakers: influence markets through macro policy, rate decisions, and regulation.
Corporate treasuries and other institutions: hedging exposure and managing liability cash flows.
Instruments and Markets: A Quick Taxonomy
Equities (stocks): ownership interest in a company; traded on exchanges and via index products.
Debt (bonds, notes, bills): issuer (government, corporates) borrows money and pays interest; can be securitized into asset-backed securities.
Commodities: metals, energy, agriculture; traded via futures; some physical delivery possible.
Real estate and asset-backed securities (ABS, CMBS, MBS): lending linked to real assets and cash flows; 2008 crisis highlighted systemic risk from real estate-linked securitization.
Derivatives: swaps, options, and structured products; can be tailored to investor/borrower needs; pricing and risk management become increasingly complex.
Currencies and FX: exchange rate markets; hedging via forwards, swaps, and options; cross-currency exposures are common for multinational corporations.
Hedging, Market Making, and Proprietary Trading
Hedging: reducing exposure to existing risk (e.g., currency risk from overseas revenues, interest rate risk from floating-rate debt).
Market making: providing liquidity in opaque markets or for complex products; profit from bid-ask spread while managing risk exposure (delta, gamma, theta, vega).
Proprietary trading: taking on risk to generate returns beyond the benchmark; emphasis on alpha; post-2008 regulation tightened some proprietary activities.
Risk Concepts and Quantitative Tools
Core risk metrics and concepts:
Value at Risk (VaR): risk measure indicating potential loss over a time horizon at a given confidence level.
Leverage: use of borrowed funds to amplify exposure; extreme leverage can amplify losses (historically up to 40× before the crisis).
Tall risk measures and stress testing: evaluating tail risks and extreme events.
Greeks and risk management on options books:
Delta: sensitivity to the underlying asset; oxed{ ext{Delta} = rac{ ext{d}P}{ ext{d}S} }oxed{ ext{Gamma} = rac{ ext{d}^2P}{ ext{d}S^2} }oxed{ ext{Theta} = rac{ ext{d}P}{ ext{d}t} }oxed{
u = rac{ ext{d}P}{ ext{d}oldsymbol{
u}} ext{ or } rac{ ext{d}P}{ ext{d}oldsymbol{\sigma}} }$$
Balance sheet and capital: risk management requires consideration of assets, liabilities, and capital adequacy when derivatives and leverage are involved.
Pricing, Risk Management, and Trading Strategies
Pricing models: essential for pricing complex financial instruments; model calibration to market data; risk parameters derived from pricing models.
Risk management: quantify exposure, assess risk, design hedges, and monitor model risk and assumptions.
Trading strategies (overview):
Arbitrage: exploit deterministic relationships or mispricings across markets/instruments (e.g., spot vs forward; cross-asset opportunities).
Relative value / value strategies: long-term perspectives on underlying value rather than short-term mispricings.
Systematic trading: trend-following, momentum, statistical arbitrage; relies on data and computational models.
Fundamental analysis: macroeconomic factors, company fundamentals, policy changes; requires broader qualitative assessment.
Special situations/private equity: opportunistic investments around corporate events or distressed assets.
The role of math in practice:
Pricing models provide model prices and risk metrics; risk management quantifies exposure to these instruments.
Trading strategies rely on mathematical and computational tools; however, continuous adaptation is required as markets evolve.
Real-World Projects and Practical Applications
Student projects from last year included:
Estimating a noisy derivative (delta) via Monte Carlo; exploring how to reduce noise and determine an optimal shift size for numerical differentiation.
Electronic trading and currency predictions using Kalman filters on non-uniform data; application to forecast exchange rates for a Moscow office.
The course website serves as a hub for syllabi, literature, and lecture slides; instructors encourage students to sign up for announcements and participate in ongoing discussions.
Homework, Reading, and Course Website
Suggested optional homework: compile your own financial glossary from the course website; read glossary entries and other course materials to build a working vocabulary.
The course website provides syllabi, literature list, and lecture materials (including Jake’s slides). Sign up on the sheet for email updates and announcements; you can also add yourself on the website for notices.
Practical Takeaways and Learnings from the Intro Session
Finance is a relatively young field that has transformed dramatically over the past few decades due to quantitative methods and computing power.
Mathematics plays a central role in pricing, risk management, and strategy development, but practical implementation requires judgment, risk controls, and an understanding of market dynamics.
The course aims to give you a comprehensive view of how math applies to modern finance, with emphasis on both theory and real-world practice, and to help you decide if this is the right path for you.
Quick Reminders for Class Logistics
The course is open to attendees from various backgrounds and universities; pace and content can be adjusted based on feedback.
Contact through the class website and email; office hours are available; sign-up sheets help organize communications.
Expect a mix of storytelling, case studies, and rigorous mathematical content, with opportunities to work on real-world projects and experiments.