Sven Karbach

Teaching

Two MSc courses at the University of Amsterdam. One builds the theory; the other makes it compute.

Courses

2026–27

Portfolio Theory

A rigorous introduction to stochastic finance in finite discrete time, in four parts. The first models the market and proves the two theorems the subject rests on: arbitrage-freeness is equivalent to the existence of an equivalent martingale measure, and completeness to its uniqueness. The second asks what to do when replication is unavailable, and develops three criteria for choosing among the admissible answers — quadratic hedging, convex risk measures, and expected utility. The third turns those criteria into portfolio choice, by martingale methods and by dynamic programming. The fourth drops the frictionless assumption entirely: execution at a single quoted price gives way to spreads, market impact, uncertain fills and inventory risk, and the resulting control problems produce liquidation schedules and market-making quotes.

Two markets recur throughout. The Cox–Ross–Rubinstein model is the complete-market benchmark; a three-outcome trinomial model is the incomplete one, and it carries the hedging, utility and control experiments alike, so the criteria can be compared on identical data rather than described in the abstract.

The only prerequisite is a course like Measure Theoretic Probability. Neither functional analysis nor stochastic analysis is assumed; the discrete-time facts the arguments need are collected in an appendix. Thirteen of the figures in the lecture notes are produced by scripts that recompute a stated result by a route independent of its proof and assert the agreement — so running them tests the notes rather than illustrating them.

Lecture notes are the main source; Föllmer and Schied's Stochastic Finance is recommended alongside. Assessment is a written examination (70%) and take-home assignments (30%), the latter counting only when they help.

Part I — Modelling financial markets

01Financial markets in finite discrete time
  • The market as a filtered probability space; adapted and predictable processes
  • Trading strategies, the gains and cost processes, self-financing
  • Discounting, and the bank account as numéraire
  • Martingales and predictable martingale transforms
02Arbitrage and equivalent martingale measures
  • Arbitrage opportunities and the law of one price
  • Martingale measures characterised through terminal gains
  • The first fundamental theorem of asset pricing, proved by separating a simplex from the space of attainable gains
  • Exercise set 1 (PDF)
03Pricing and hedging European contingent claims
  • Attainable claims, replication, and risk-neutral pricing
  • The set of arbitrage-free prices: a claim is attainable exactly when that set is a single point
  • Exercise set 2 (PDF)
04Superhedging
  • The cheapest superhedge price, and superhedging duality
  • Why the cheapest superhedge sits on the boundary of the no-arbitrage interval and is not itself a price anyone would trade at
05Complete markets
  • The second fundamental theorem, and predictable representation
  • Branching bounds: at most (d+1)t atoms, so realistic return distributions force incompleteness
06The Cox–Ross–Rubinstein model
  • Arbitrage-free and complete exactly when D<r<U
  • The backward pricing recursion and the delta hedge, which replicates on every path rather than in expectation

Part II — Incomplete markets

01Variance-optimal hedging
  • Quadratic hedging as an orthogonal projection in L2
  • The bounded mean–variance trade-off, and closedness of the space of attainable gains
  • The Galtchouk–Kunita–Watanabe decomposition; the orthogonal residual is exactly what cannot be hedged
02Risk measures
  • Monotonicity, cash invariance, convexity, coherence
  • Acceptance sets, and risk as a capital requirement
  • Value-at-Risk, expected shortfall, the worst-case and entropic measures — and the diversification that raises reported VaR
  • Dual representation of convex risk measures (optional)
03Risk-optimal hedging and indifference prices
  • The seller's optimised capital requirement, and the risk-indifference price
  • For an attainable claim it collapses to the replication cost
04Expected utility and utility-indifference pricing
  • Certainty equivalents, Arrow–Pratt risk aversion, the CARA and CRRA families
  • Utility-based shortfall risk: exponential utility and entropic risk order positions identically

Part III — Portfolio optimisation

01Static portfolio optimisation
  • A maximiser exists precisely when the one-period market is arbitrage-free
  • Marginal utility at the optimum defines an equivalent martingale measure
02Dynamic utility optimisation and hedging
  • The marginal-utility martingale criterion, and its first-order converse
  • CRRA: constant-proportion strategies, Merton's ratio, and wealth-dependent indifference prices
  • CARA: the indifference price as a difference of relative entropies, computed by the minimal-entropy martingale measure
  • The risk-aversion limits — superhedging as risk aversion grows, the variance-optimal hedge as it vanishes (optional)
03Stochastic control and dynamic programming
  • The control problem, the Bellman recursion, and the optimality principle
  • The complete-market state-price method as the static counterpart (optional)
04Deep hedging as parameterised stochastic control
  • Replacing the admissible strategy class by a parameterised, predictable family
  • What it buys — frictions in the transition and reward — and what it costs: convexity and exact dual certificates

Part IV — Algorithmic finance and optimal trading

01Market microstructure and execution accounting
  • Quotes and the limit order book; market orders, limit orders, and uncertain fills
  • Execution-price accounting: temporary impact and fees are explicit trading losses
  • Decision price against execution price, and why confusing them manufactures profit
02Optimal execution
  • Implementation shortfall, and its decomposition into impact and inventory risk
  • The discrete Almgren–Chriss schedule; TWAP as the risk-neutral corner of the frontier
  • The Bellman recursion with constraints, and what the separable objective quietly assumes
03Inventory-based market making
  • A discrete-time quoting model that reduces to inventory alone
  • Optimal quotes under exponential fill probabilities, and the inventory skew
  • Adverse selection: a maker can book positive spread on every fill and still lose money
04From a control model to a trading algorithm
  • Calibrating fill probabilities by a Poisson model on quoted exposure
  • Causality, and what a single period of look-ahead does to a backtest
  • A model-risk checklist: units, timing, accounting, constraints, support, benchmark

2026

Computational Finance

The numerical half of derivatives pricing. Every method in the course computes the same object — the discounted risk-neutral expectation of a payoff, its American analogue, or a derivative of it — by a different route. Monte-Carlo averages over simulated paths; finite differences solve the equivalent partial differential equation on a grid; Feynman–Kac is the bridge that makes those two the same question. Which route is right depends on dimension, on the payoff, and on whether you need the Greeks.

The course begins with data rather than a model. The stylized facts of returns — no linear autocorrelation, heavy tails, clustered volatility, the leverage effect — are what later motivate stochastic volatility and fix the sign of the correlation in the Heston model. They hold under the physical measure, and keeping that measure separate from the pricing measure is the discipline the whole course depends on.

Prerequisites are basic programming and foundational mathematics; everything is implemented in Python. Teaching is by lecture, computer lab, and sessions with practitioners from industry. Assessment is a written examination (40%, minimum grade 5) and three group lab assignments (20% each).

01Statistical foundations and stylized facts
  • Log-returns; absence of linear autocorrelation, heavy tails, gain/loss asymmetry
  • Volatility clustering, the leverage effect, aggregational Gaussianity
  • Why the physical and the risk-neutral measure must be kept apart from the first lecture
02Brownian motion and Itô calculus
  • Itô processes, Itô's lemma, and quadratic variation
  • Self-financing portfolios; the martingale representation theorem as the reason claims can be replicated at all
03Risk-neutral valuation and Girsanov's theorem
  • The market price of risk, and the change of measure it induces
  • The fundamental theorem of asset pricing; discounted prices as martingales
04Options, no-arbitrage bounds, and Black–Scholes
  • Put–call parity and model-free bounds; why early exercise of an American call is usually wasteful
  • The Black–Scholes formula, implied volatility, and the surface that the constant-volatility assumption cannot produce
05The Black–Scholes PDE, Feynman–Kac, and the Greeks
  • Delta hedging, and the PDE it forces
  • Feynman–Kac: the PDE and the expectation are two views of one object — which is what justifies both finite differences and Monte-Carlo
  • Delta, gamma, vega, theta, rho, and the PDE read as a balance relation
06Monte-Carlo pricing
  • The estimator, its O(N−1/2) error, and confidence intervals
  • Sampling error and discretisation error are separate approximations with separate controls
  • Why the dimension-independent rate makes Monte-Carlo the method for high-dimensional payoffs
07Discretisation of stochastic differential equations
  • Euler–Maruyama; strong and weak order of convergence; the Milstein correction
  • Exact simulation of the log-price under geometric Brownian motion, and why the same trick fails under stochastic volatility
  • Full-truncation Euler for a variance process that must not go negative
08Stochastic volatility: Heston and calibration
  • The model, the Feller condition, and the ODE for the expected variance
  • What each parameter does, and how the sign of the correlation produces the skew
  • Calibration by least squares, on prices or on implied volatilities, with the characteristic function in the inner loop
09Finite-difference methods
  • Explicit FTCS, and the stability condition that punishes a finer grid
  • Crank–Nicolson: unconditional stability and tridiagonal solves
10Variance reduction and Monte-Carlo Greeks
  • Antithetic and control variates
  • Bump-and-revalue, the pathwise derivative, and the likelihood-ratio method
  • Which one survives a discontinuous payoff, and which one does not
11American options
  • Optimal stopping; the obstacle problem, and the free boundary that emerges rather than being imposed
  • Longstaff–Schwartz by backward regression: why it gives a lower bound, and how a dual method brackets the answer
12Outlook: where the field is going
  • Rough volatility; deep BSDE solvers and neural calibration
  • Martingale optimal transport and model-free bounds; deep hedging and reinforcement learning
  • Neural SDEs and path signatures; XVA; quantum amplitude estimation; climate and transition risk

Supervision

I supervise PhD and MSc students in stochastic analysis, energy and commodity markets, and machine learning for financial time series. Current PhD projects are listed on the front page.

If a topic here interests you, write to me with a short note on your background and what you would like to work on.