Quantitative Markets Research Lab

Module 06 · Working Notes

Mathematical Notes

Six derivations that carry most of the weight in quantitative finance, written to be read rather than merely cited. Each note states what is being assumed, does the calculation, and says what the result is actually worth.

Note I

Itô's Lemma

Ordinary calculus fails for Brownian motion because \(W_t\) accumulates quadratic variation: over a partition of \([0,t]\), \(\sum (\Delta W)^2 \to t\), not zero. The squared increments of a Brownian path are not negligible — they behave, in the limit, like time itself. The heuristic multiplication table is

$$(dW)^2 = dt, \qquad dW\,dt = 0, \qquad (dt)^2 = 0.$$

Take an Itô process \(dX_t = \mu_t\,dt + \sigma_t\,dW_t\) and a smooth function \(f(t, x)\). A second-order Taylor expansion gives

$$df = \frac{\partial f}{\partial t}\,dt + \frac{\partial f}{\partial x}\,dX + \tfrac{1}{2}\frac{\partial^2 f}{\partial x^2}\,(dX)^2 + \cdots$$

In ordinary calculus the \((dX)^2\) term would vanish. Here \((dX)^2 = \sigma_t^2\,(dW)^2 + O(dt^{3/2}) = \sigma_t^2\,dt\), so a second-order term survives into the first-order differential:

$$df = \left(\frac{\partial f}{\partial t} + \mu_t\frac{\partial f}{\partial x} + \tfrac{1}{2}\sigma_t^2 \frac{\partial^2 f}{\partial x^2}\right) dt + \sigma_t \frac{\partial f}{\partial x}\,dW_t.$$

That extra \(\tfrac{1}{2}\sigma^2 f_{xx}\,dt\) — the Itô correction — is the single most consequential term in mathematical finance. It is why convexity has a price, why hedged option books bleed or earn theta, and why the drift of a log-price is not the drift of the price. Whenever a payoff is curved and the underlying is volatile, the correction term is where the money is.

Note II

Solving Geometric Brownian Motion

The SDE \(dS_t = \mu S_t\,dt + \sigma S_t\,dW_t\) is solved by applying Itô's lemma to \(f(S) = \ln S\), for which \(f' = 1/S\) and \(f'' = -1/S^2\):

$$d(\ln S_t) = \frac{1}{S_t}\,dS_t - \frac{1}{2}\frac{1}{S_t^2}(dS_t)^2 = \left(\mu - \tfrac{1}{2}\sigma^2\right)dt + \sigma\,dW_t.$$

The right-hand side no longer involves \(S_t\); it integrates directly:

$$S_t = S_0\,\exp\!\left[\left(\mu - \tfrac{1}{2}\sigma^2\right)t + \sigma W_t\right].$$

Log-prices are Gaussian; prices are lognormal; prices stay positive. Two readings of the \(-\tfrac{1}{2}\sigma^2\) term repay attention. First, \(\mathbb{E}[S_t] = S_0 e^{\mu t}\) — the correction exactly offsets the convexity of the exponential, by design. Second, the median path grows at \(\mu - \tfrac{1}{2}\sigma^2\), strictly less than the mean growth rate. A volatile asset's average outcome is dragged upward by a shrinking minority of enormous paths while the typical path does worse. This "volatility drag" is not a market imperfection; it is arithmetic, and it is the quantitative core of why compounding punishes variance.

Note III

Risk-Neutral Pricing, Sketched Honestly

Consider one period and two assets: a bond growing at \(r\), and a stock worth \(S\) that moves to \(uS\) or \(dS\) with \(d < e^{r\Delta t} < u\). To price a claim paying \(V_u\) or \(V_d\), build a portfolio of \(\Delta\) shares and \(B\) in bonds that replicates it in both states:

$$\Delta = \frac{V_u - V_d}{(u-d)S}, \qquad V = \Delta S + B = e^{-r\Delta t}\left[\,p^* V_u + (1-p^*) V_d\,\right], \qquad p^* = \frac{e^{r\Delta t} - d}{u - d}.$$

The real-world probability of the up-move never entered. The price is a discounted expectation under an artificial probability \(p^*\) — the unique one that makes the discounted stock a martingale. This is the whole content of risk-neutral pricing: no arbitrage plus replication implies prices are expectations under a measure \(\mathbb{Q}\) constructed for accounting convenience, not belief. In continuous time Girsanov's theorem plays the same role, shifting the drift of \(W_t\) so that \(dS_t = r S_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}}\), and

$$V_0 = e^{-rT}\,\mathbb{E}^{\mathbb{Q}}\!\left[\,\text{payoff}(S_T)\,\right].$$

The honesty clause: the argument requires that the payoff can actually be replicated — complete markets, continuous frictionless trading. With jumps, stochastic volatility, or transaction costs, replication is imperfect, \(\mathbb{Q}\) is no longer unique, and "the" arbitrage-free price becomes an interval. The market then selects a measure by supply and demand for risk, which is a polite way of saying the model hands the hard part back to economics.

Note IV

Feynman-Kac and the Black-Scholes PDE

The Feynman-Kac theorem is the bridge between expectations and differential equations. If

$$V(t,x) = \mathbb{E}\!\left[\left. e^{-r(T-t)}\,\varphi(X_T)\,\right|\, X_t = x\right], \qquad dX_s = a(X_s)\,ds + b(X_s)\,dW_s,$$

then \(V\) solves

$$\frac{\partial V}{\partial t} + a(x)\frac{\partial V}{\partial x} + \tfrac{1}{2}b(x)^2\frac{\partial^2 V}{\partial x^2} - rV = 0, \qquad V(T,x) = \varphi(x).$$

The proof idea is one line of Itô: apply the lemma to \(e^{-rt}V(t, X_t)\); since a conditional expectation of a fixed terminal payoff is a martingale, its \(dt\) term must vanish, and that vanishing is the PDE. Now specialize to the risk-neutral stock \(a = rS\), \(b = \sigma S\):

$$\frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \tfrac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - rV = 0.$$

This is the Black-Scholes equation, and Feynman-Kac explains why it and the risk-neutral expectation of Note III are the same object viewed from opposite sides: the expectation is the PDE's stochastic representation; the PDE is the expectation's infinitesimal description. Solving it with the call payoff boundary condition yields the closed form in the pricing laboratory. The same bridge carries the physics intuition: the equation is a heat equation in disguise (substitute \(x = \ln S\) and rescale time), so option value diffuses — kinks in payoffs get smoothed exactly the way heat smooths a temperature spike.

Note V

Monte Carlo Error Scales as \(O(1/\sqrt{N})\)

Estimate \(\theta = \mathbb{E}[f(X)]\) by the sample mean \(\hat\theta_N = \frac{1}{N}\sum_{i=1}^N f(X_i)\) over independent draws. The estimator is unbiased, and its standard error follows from nothing deeper than the variance of a mean:

$$\operatorname{se}(\hat\theta_N) = \frac{\sigma_f}{\sqrt{N}}, \qquad \sigma_f^2 = \operatorname{Var}[f(X)],$$

with the central limit theorem supplying asymptotic normality and hence confidence intervals. Three consequences structure all practical Monte Carlo work:

1. The tyranny of the square root. One more decimal digit of accuracy costs a factor of 100 in samples. Monte Carlo is the method of choice not because it converges fast but because its rate does not deteriorate with dimension — a 100-dimensional basket option converges at the same \(N^{-1/2}\) as a one-dimensional one, while grid methods collapse under \(O(h^{-d})\) nodes.

2. The constant is attackable. Since the rate is fixed, all craft goes into shrinking \(\sigma_f\): antithetic variates, control variates (price the arithmetic Asian against the geometric one, which is known in closed form), importance sampling for tail events, and quasi-random sequences that trade independence for low discrepancy and near-\(N^{-1}\) behavior.

3. Tail estimates inherit the worst constants. Estimating \(\mathbb{P}[L > \ell] = p\) by naive simulation has relative standard error \(\approx \sqrt{(1-p)/(pN)}\) — for a \(p = 0.1\%\) event, ten percent relative accuracy needs on the order of \(10^7\) paths. Rare-event estimation without importance sampling is mostly the production of confident noise, which links this note directly to the next one.

Note VI

Why Fat Tails Break Normal VaR

Parametric-normal VaR takes a mean and standard deviation and reads the quantile off the Gaussian: \( \mathrm{VaR}_\alpha = -( \mu + z_{1-\alpha}\,\sigma )\). The procedure is exact when returns are normal and quietly catastrophic when they are not, because the Gaussian density dies like \(e^{-x^2/2}\) while empirical return distributions die like a power law, \(\mathbb{P}[|r| > x] \sim x^{-\alpha}\) with tail index \(\alpha\) around 3 to 4 for daily equity returns. Every moment of the comparison fails in the same direction:

The counting argument. Daily equity moves of five standard deviations should occur, under normality, about once per 14,000 years. The realized record produces them every few years; October 19, 1987 was, on a Gaussian yardstick, roughly a 20σ event — a probability so small it has no physical interpretation. The model is not slightly wrong in the tail; it is wrong by factors of \(10^{3}\) to \(10^{50}\), depending how far out you look.

The moment argument. Excess kurtosis of daily index returns is far above the Gaussian's zero. Since sample variance is dominated by the very observations the Gaussian deems nearly impossible, \(\hat\sigma\) is inflated by past crises while the normal quantile formula simultaneously understates how much worse than \(z\hat\sigma\) the next crisis will be. The errors do not cancel; at high confidence levels the understatement wins.

The structural argument. Volatility clustering means returns are a mixture of distributions — calm-regime and stress-regime — and mixtures of normals with different variances are themselves fat-tailed. So even if each day were conditionally Gaussian, unconditional normal VaR would still be miscalibrated. Worse, the stress regime arrives with correlations lurching toward one, so the portfolio-level tail is fatter than any asset-level analysis suggests.

The disciplined responses, in ascending order of humility: historical-simulation VaR (let the empirical quantile speak, as on the risk dashboard); Expected Shortfall (average the tail rather than poking it at one point); extreme value theory (fit the tail's own asymptotic family, GPD/GEV, instead of the body's); and the standing admission that any tail estimate at the 99.9% level is an extrapolation wearing the costume of a measurement.

Sources

Standing references for these notes

  • Øksendal, Stochastic Differential Equations — Notes I, II, IV.
  • Shreve, Stochastic Calculus for Finance II — Notes III, IV.
  • Glasserman, Monte Carlo Methods in Financial Engineering — Note V.
  • McNeil, Frey & Embrechts, Quantitative Risk Management — Note VI.