Stochastic Processes

Stochastic Processes (Modules 0.3 + 0.4)

This is a two-part module. Part I builds the objects (random walks, Markov chains, Brownian motion). Part II develops the calculus needed to work with them (Ito’s lemma, SDEs, change of measure). Together they provide the mathematical engine behind derivatives pricing, term structure models, and credit risk.

Dependency Chain

probability-theory
       |
       v
stochastic-processes (Part I: Foundations)
       |
       v
stochastic-processes (Part II: Ito Calculus)
       |
       +---> derivatives-pricing (Black-Scholes, Greeks, vol models)
       +---> fixed-income (Vasicek, CIR, HJM)
       +---> credit-risk (reduced-form models, intensity processes)
       +---> portfolio-theory (continuous-time optimization, Merton)

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Part I (0.3): Foundations

Prerequisite: Probability

Random Walks (Discrete)

Symmetric and asymmetric random walks on $\mathbb{Z}$. Connection to the binomial asset pricing model: each step is an up/down move in the stock price tree. Properties: recurrence (1D), expected hitting times, reflection principle.

Markov Chains

Transition matrices $P$, stationary distributions $\pi P=\pi$, ergodicity, absorbing states, expected hitting times.

Finance application: credit rating migration matrices ARE Markov chains. The probability that a BBB-rated issuer is in default within 5 years is just the $(BBB,D)$ entry of $P^5$.

Brownian Motion (Wiener Process)

Definition and key properties:

• $W_0=0$
• Continuous sample paths (a.s.)
• Independent increments: $W_t-W_s\perp W_s-W_u$ for $u<s<t$
• Normal increments: $W_t-W_s\sim N(0,t-s)$
• Nowhere differentiable (a.s.)
• Quadratic variation: $⟨W{⟩}_t=t$

Construction: Brownian motion arises as the scaled limit of random walks (Donsker’s invariance principle / functional CLT). This is more than a curiosity — it justifies using continuous models for discrete trading.

Geometric Brownian Motion

The standard stock price model:

$S_t=S_0\exp \left(\left(\mu -\frac{{\sigma }^2}{2}\right)t+\sigma W_t\right)$

Properties: log-normal distribution, always positive, multiplicative structure. The $-\frac{{\sigma }^2}{2}$ correction (Ito’s correction) is what makes $E[S_t]=S_0{e}^{\mu t}$ rather than $S_0{e}^{(\mu +{\sigma }^2/2)t}$.

Connection to experience: at Gottex, the Vasicek and CIR models for interest rates are SDEs driven by Brownian motion — they are just GBM’s cousins with mean reversion.

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Part II (0.4): Ito Calculus

Prerequisite: Part I above.

Why New Calculus?

Classical calculus breaks down for Brownian motion:

• $W_t$ is nowhere differentiable, so $dW/dt$ does not exist
• Quadratic variation $⟨W{⟩}_t=t\ne 0$, so second-order terms in Taylor expansions do not vanish

This means the chain rule is wrong. Ito’s lemma is the corrected chain rule.

Ito Integral

Definition of ${\int }_0^T{H}_{s}d{W}_s$ as the $L^2$ limit of sums. Key properties: zero mean, Ito isometry $E\left[{\left({\int }_0^T{H}_{s}d{W}_s\right)}^2\right]=E\left[{\int }_0^T{H}_s^{2}ds\right]$, and the result is a martingale (under integrability conditions).

Ito’s Lemma

For $f\in C^2$ and an Ito process $X_t$:

$df(X_t)=f^{\prime }(X_t)d{X}_t+\frac{1}{2}{f}^{\prime \prime }(X_t)(d{X}_t{)}^2$

The extra $\frac{1}{2}{f}^{\prime \prime }(X_t){\sigma }^{2}dt$ term is what makes Black-Scholes work. Without it, you cannot derive the PDE for option prices.

Stochastic Differential Equations

Existence and uniqueness (Lipschitz + linear growth conditions). Euler-Maruyama numerical scheme for simulation:

$X_{t+\Delta t}=X_t+\mu (X_t)\Delta t+\sigma (X_t)\sqrt{\Delta t}Z,Z\sim N(0,1)$

Girsanov’s Theorem

Change of measure from $\mathbb{P}$ (real-world) to $\mathbb{Q}$ (risk-neutral):

• Under $\mathbb{P}$: $dS/S=\mu dt+\sigma d{W}^{\mathbb{P}}$
• Under $\mathbb{Q}$: $dS/S=rdt+\sigma d{W}^{\mathbb{Q}}$

The drift changes from $\mu$ to $r$. The volatility $\sigma$ does not change. This is the formal justification for risk-neutral pricing: we do not need to know $\mu$ (expected return) to price options — only $\sigma$ matters.

Martingale Representation Theorem

Every $\mathbb{Q}$-martingale adapted to the Brownian filtration can be written as a stochastic integral with respect to $W^{\mathbb{Q}}$. This is why you can hedge: any contingent claim can be replicated by a dynamic trading strategy.

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What This Module Unlocks

• Black-Scholes derivation: apply Ito’s lemma to $f(S_t,t)$, set up the hedging portfolio, get the PDE. See derivatives-pricing.
• Risk-neutral pricing: Girsanov + martingale representation = the fundamental theorems of asset pricing.
• Term structure models: Vasicek, CIR, Hull-White are all SDEs. See fixed-income.
• Credit risk models: intensity-based default models use Poisson processes driven by stochastic intensities. See credit-risk.
• Monte Carlo simulation: Euler-Maruyama is the workhorse for pricing exotic derivatives numerically.