Free module · Quantitative toolkit
Stochastic Calculus
From Brownian increments and Itô integration to diffusion models, option prices and hedge sensitivities.
options · hedging
The building blocks
- A random variable has several possible values.
- Calculate its average and spread before introducing a random path.
- Review ordinary change and probability
- Build Brownian increments and stochastic sums
- Apply Itô’s rule and stochastic differential equations
Lessons in this module
- Start with one random step before continuous noise
- Before Itô: random walks, Brownian motion and information
- Itô integration and quadratic variation
- SDEs, Euler–Maruyama and an Ornstein–Uhlenbeck example
- Geometric Brownian motion: the model under Black–Scholes
- Itô's lemma: why gamma exists
- Black–Scholes and delta hedging in production
- Put–call parity and a full Greek P&L budget
Practice and apply
- Pricing a call by hand — S = 100, K = 105, T = 0.5 yr, r = 4%, σ = 25%
- Daily P&L of a hedged position from realised vs implied — Long 100 ATM calls, Γ = 0.03 per option (×100 multiplier), S = 100, implied vol 20%, today the stock moves 2%
- Scale a shock — For σ=3 and h=4, find the standard deviation σ√h.
Work through the practice exercises · Quant development tools