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Financial Mathematics
Explore time and risk models; arbitrage and discrete pricing; continuous models and limits. Concepts and connections are available independently of video coverage.
Prerequisites and connections
Brownian motion
Brownian motion has continuous paths and independent Gaussian increments with variance equal to elapsed time.
Video explanations to be addedExpectation
Expectation is a probability-weighted average when defined; not every random variable has a finite expectation.
3 reviewed resources
Time and risk models
Time value of money
Discounting compares amounts at different dates under a specified rate model.
Video explanations to be addedCompound interest
Interest accrues on both principal and previously accumulated interest.
Video explanations to be addedPortfolio risk
Covariances link individual asset variability to the variability of weighted portfolios.
Video explanations to be addedMean–variance optimization
Expected return and covariance describe a constrained portfolio optimization model.
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Arbitrage and discrete pricing
No-arbitrage
A market model excludes zero-cost trades with nonnegative and sometimes positive payoff.
Video explanations to be addedReplicating portfolios
A trading strategy replicates a claim when its terminal payoff matches the claim under the model.
Video explanations to be addedBinomial pricing models
Up-and-down price moves support backward valuation under no-arbitrage assumptions.
Video explanations to be addedRisk-neutral measures
Discounted traded-price processes are martingales under a suitable pricing measure.
Video explanations to be added
Continuous models and limits
Geometric Brownian motion
A multiplicative stochastic differential model keeps positive initial asset prices positive.
Video explanations to be addedBlack–Scholes model
Option pricing follows a diffusion market with restrictive volatility, trading and friction assumptions.
Video explanations to be addedValue at Risk
VaR is a loss quantile at a stated horizon and confidence level, not a maximum possible loss.
Video explanations to be addedModel risk
Incorrect assumptions, calibration or use can invalidate model-based conclusions.
Video explanations to be added