Part XVIII — Applications to Quantitative Finance

Part XVII ends by observing that nothing in its last three pages "supplies a probability, a loss, a position size or a price," and hands the job here. This part does those four things and finds one failure underneath all of them: the assumption that makes a closed form available is, with remarkable consistency, the assumption the application violates — so the number arrives precise and about the wrong quantity. A growth-optimal fraction computed from two moments reads \(16.24\times\) on six laws whose placeable maximum falls from \(64.87\times\) to \(15.78\times\). A Monte Carlo delta returns \(0.000000\) with a standard error of \(0.000000\). A risk simulation reports an error bar \(27.3\) times too small. Two independent positions are charged \(101.00\) for diversifying. A tail-index estimator extrapolates a \(1\)-in-\(10{,}000\) loss to \(161.78\) against a truth of \(41.22\) on a law that has no tail index. And a regime detector finds \(2.82\) regimes a year in a process with one.

The dependencies run in file order and split into four blocks that share a spine. Kelly Criterion, Probability of Ruin and Drawdown Probabilities are the sizing block and read together: the first chooses a fraction and the other two price the path it produces, all three governed by the same two parameters and by the dimensionless horizon \(S^{2}T\) that decides what any record can settle. Hitting and First-Passage Times supplies the clock those three integrate out. Queue Models and Order Arrival Processes are a matched pair, each governed by a dimensionless feedback ratio that blows up hyperbolically at one — utilization \(\rho\) and branching ratio \(n\) — with a first moment that stays well behaved while the quantity anyone cares about diverges. Bayesian Signal Updating stands alone and prices evidence itself. Monte Carlo Option Pricing and Portfolio Risk Simulation are the simulation pair, both finding that the error a Monte Carlo run reports is the one it can see from inside itself rather than the one that matters. Value at Risk and Expected Shortfall must be read together, because their defects are complementary theorems: the first is elicitable and not coherent, the second coherent and not elicitable. Heavy-Tailed Returns and Extreme Value Theory are the tail pair, the first supplying the index the second's shape parameter inverts; Copulas needs the first of them and adds the dimension both lack; and Regime Detection drops the assumption every earlier page makes, that the parameters are constant.

Where this part stops is worth stating. It proves no ergodic theorem and builds no sampler, which is Part XVII; it constructs no Itô integral and measures no strong-order convergence, which is Stochastic Calculus; it builds no Kalman, unscented or particle filter, which is Particle and Kalman Filters; it derives no Black–Scholes formula, no Greek in closed form and no volatility smile, which is Options Pricing; it solves no execution schedule and estimates no impact function, which are Optimal Execution and Market Impact Models; it quotes no spread and controls no inventory, which is Market Making; and it reads no cached market data, so every number below is reproducible from the block above it on any machine.

What runs through all fifteen pages is that each failure is invisible to the diagnostic the method computes for itself, and each has a repair costing a single line. A convergence flag, a Monte Carlo standard error, a Kupiec breach count, a Hill plateau, a fitted correlation and an alarm rate are all correct answers to questions that stopped mattering; the free diagnostics that catch them — \(1/\lvert x_{\min}\rvert\) beside a Kelly fraction, \(1/\sqrt{2T}\) beside a simulated risk number, \(np\) beside a quantile, \((1+\hat\xi)/\sqrt k\) beside a shape parameter, a joint exceedance count beside a copula, an alarm rate under a simulated null — need no new data, no new model and no new assumption. Every one asks the same question: what would this number have been if there had been nothing there.

Topics

Topic	Focus
Kelly Criterion	Growth optimality as an almost-sure limit rather than a utility assumption, and the price of missing it — a fraction \(c\) of Kelly keeping exactly \(2c-c^{2}\) of the growth, measured at \(0.4375\), \(0.7500\), \(0.9375\) and \(0.0000\) with predicted and realized rates agreeing at \(68.91\%\) against \(68.81\%\), while median maximum drawdown runs \(-43.9\%\) to \(-95.5\%\); the formula as a second-order truncation blind to the support, so six laws with an identical \(8.49\%\) mean and \(7.23\%\) volatility all return \(16.24\times\) while the placeable maximum falls from \(64.87\times\) to \(15.78\times\) and the formula's own answer delivers \(37.46\%\), then \(-4.18\%\), then nothing; and dominance needing \(31.4\), \(125.6\) and \(785.2\) years to become decidable against half, three-quarters and nine-tenths of Kelly
Probability of Ruin	An exponential martingale settling first passage in two lines, giving \(\mathbf{P}(\text{ever reaching }xW_0)=x^{2/c-1}\) — a function of the Kelly multiple alone, with \(\mu\), \(\sigma\) and the horizon all cancelling, verified at \(0.5000\) against \(0.5015\) and \(0.1250\) against \(0.1278\); path continuity as the single load-bearing step, so a law with identical mean and volatility carrying one \(6\%\) jump a year takes the ninety-percent-loss probability from \(0.0988\) to \(0.9986\) and wipes the book out entirely on \(0.9811\) of paths against a diffusion probability of zero; and the exponent's only input being a drift, so a desk quoting \(0.1250\) on ten years of history is running a true value between \(0.0138\) and \(0.2957\)
Drawdown Probabilities	The stationary drawdown below a running peak as an exponential with mean \(\sigma/2S\) and expected recovery \(1/(2S^{2})\) years, so depth is bought with position size and duration with edge — doubling volatility at Sharpe \(0.65\) doubles depth from \(7.40\%\) to \(14.26\%\) and leaves recovery at \(1.18\) years; two horizon regimes for the maximum, \(\sqrt T\) without drift at a measured \(7.42\) against \(7.07\) and \(\log T\) with it, crossing at \(1/S^{2}\) years so a Sharpe-\(0.30\) book is driftless for its first eleven; and a realized maximum drawdown ranking a Sharpe-\(0.30\) book above a Sharpe-\(0.80\) one on \(0.1741\) of paired records while a \(30\%\) limit fires on \(0.0659\) of the good ones
Hitting and First-Passage Times	First passage under drift as inverse Gaussian, with a mean finite exactly when the drift points at the barrier; the Ornstein–Uhlenbeck half-life as the decay of a conditional expectation rather than a waiting time, so the published \(3.4\)-day fit goes with expected waits of \(4.5\) to \(8.8\) days across entry thresholds, matched by simulation, with a two-sigma median of \(5.6\) and a ninetieth percentile of \(13.6\); a two-barrier race whose zero-drift limit is \(L/(L+U)\), so win rates of \(0.5014\), \(0.3344\) and \(0.2501\) all give expected profits indistinguishable from zero; and a deadline at \(1.5\) half-lives removing \(28.61\%\) of the profit while the positions it closes are ahead by \(0.8174\) sds
Queue Models	The birth–death balance equations making the queue geometric and delay hyperbolic in idle capacity, verified at waits of \(0.998\) through \(52.794\), with the saturated row overshooting by \(7.7\%\) on four million arrivals because a queue near capacity is as hard to measure as it is slow; Little's law from a sample-path area argument with no distributional assumption at all; a passive order's fill probability \((\mu/(\mu+\nu))^{Q}\) halving every \(7.3\) units of queue while the wait grows linearly, so forty deep buys \(0.02\) of the probability for \(40.04\) times the wait; and Pollaczek–Khinchine making the wait read the second moment, so four laws with mean \(1.000\) and utilization \(0.80\) wait \(1.995\), \(3.998\), \(10.600\) and \(19.251\)
Order Arrival Processes	Self-excitation replacing a rate with a state, giving stationary intensity \(\lambda_0/(1-n)\), cluster size \(1/(1-n)\) and an endogenous share of exactly \(n\) — all three verified, with \(n=0.9\) leaving news unchanged at \(2\) while the observed rate is \(20\); the dispersion ratio's exact scale-dependence, running \(1.643\) at windows below the kernel's memory to \(43.667\) far above against a limit of \(44.44\), so its direction identifies the mechanism where its level cannot; and a Poisson fit recovering the rate to four significant figures while exceeding its own \(99.9\)th percentile \(131.7\) times too often, a misspecification the compensator's time change detects at \(0.0005\) against \(0.1602\)
Bayesian Signal Updating	Log posterior odds as a random walk whose drift is a Kullback–Leibler divergence, so conviction takes \(504\log(19)/S^{2}\) days — \(65.0\) years at Sharpe \(0.30\), verified at \(16{,}379\) against \(16{,}489\) — while Ville's inequality caps false conviction at \(e^{-L}=0.0526\) uniformly over every stopping rule, measured at \(0.0500\) to \(0.0527\); the per-observation influence as the likelihood's score, unbounded for a Gaussian and redescending for a \(t\), so a twenty-sigma day moves the posterior Sharpe by \(0.3147\) against \(0.0039\); and on \(t(3)\) returns the largest single day of twenty years carrying \(10.12\%\) of the evidence against \(0.79\%\), with robustness costing nothing when it is not needed
Monte Carlo Option Pricing	Weak order as the rate a price consumes, so trading discretization bias against sampling error under one budget puts the optimum at \(C^{1/3}\) steps and degrades the achievable error to \(C^{-1/3}\); the optimal split at four million path-steps sitting at two steps, with both ends costing \(8.13\) and \(11.21\) times the best; the pathwise estimator unbiased exactly when the payoff is Lipschitz and the likelihood-ratio estimator needing only integrability, agreeing on a call at \(0.579134\), \(0.579205\) and \(0.579182\) against \(0.579260\); and on a digital the pathwise estimator returning \(0.000000\) with a standard error of \(0.000000\) against true deltas of \(0.019552\) to \(0.199431\)
Portfolio Risk Simulation	The law of total variance splitting a simulated risk number into a sampling term falling like \(1/N\) and a parameter term that does not contain \(N\), verified by quadrature to the fourth decimal; the parameter term under fixed weights being a relative \(1/\sqrt{2(T-1)}\) independent of dimension and of the confidence level, measured at \(0.0624\) to \(0.0139\) and accounting for \(0.9935\) of the variance at a hundred thousand paths; the crossing at \(832\), \(1{,}298\), \(4{,}651\) and \(23{,}253\) paths for the four standard levels; and past it, an actual error settling at \(0.078090\%\) while the reported Monte Carlo error falls to \(0.002868\%\), a ratio of \(27.3\)
Value at Risk	The sample quantile's variance \(p(1-p)/(nf(q)^{2})\) making the effective sample \(np\) — two observations at a year — with asymptotic standard deviations of \(0.5125\), \(0.2563\) and \(0.1146\) matching \(0.4624\), \(0.2539\) and \(0.1153\); a fitted normal converging to \(2.3244\) against a truth of \(2.6495\) with a flat bias and a shrinking standard error, which is the published \(2.50\times\) breach rate; elicitability under the pinball loss against a breach-counting test whose size is \(0.096\) rather than \(0.05\) on a year and whose power against a ten-percent error is \(0.181\); and subadditivity holding on every elliptical law and failing by \(101.00\) on two independent positions whose individual VaRs are gains
Expected Shortfall	Coherence from the representation as a maximum of expectations over reweighted probabilities, a supremum of linear functionals being sublinear; the coherent definition against the tail conditional expectation usually coded in its place, agreeing to every printed digit on continuous losses and failing subadditivity at \(99.24\) against \(-0.39\) on an atomic one where the coherent version gives \(99.38\) against \(159.50\); the integral representation costing \(1.52\), \(1.81\), \(2.82\) and \(3.92\) times the VaR estimator's standard deviation as tails thicken; and non-convex level sets — two laws with an ES of exactly \(2.3378\) mixing to \(2.3646\) — ruling out any scoring rule while the matched-VaR control holds \(1.9600\) to the last digit
Heavy-Tailed Returns	Regular variation as the definition that makes a tail scale-free, with the moment boundary and max-sum equivalence as corollaries, the largest of forty losses taking \(0.9331\) of an extreme total at \(\alpha=1.5\) against \(0.0960\) at \(\alpha=6.0\) and \(0.1068\) for an exponential; the Hill estimator as a maximum-likelihood rate with standard error \(\alpha/\sqrt k\), settling onto \(2.502\), \(3.993\) and \(6.014\) and flattening to within \(0.021\); and the same estimator returning \(3.005\), \(2.764\), \(2.401\), \(2.119\) and \(1.795\) on a lognormal that has no tail index, extrapolating a \(1\)-in-\(10{,}000\) loss to \(161.78\) against \(41.22\), while even a correctly specified \(t(3)\) drifts and overstates by \(2.6\times\)
Extreme Value Theory	Max-stability forcing exactly three limit types, unified by one shape whose sign separates a power law, an exponential-like tail and a hard ceiling — fitted at \(0.3349\), \(0.2386\), \(0.0005\) and \(-1.0014\) against \(1/3\), \(1/4\), \(0\) and \(-1\); the Gaussian reaching its limit an order of magnitude slower than any other row and in the complacent direction, at \(-0.0688\) where the truth is zero; threshold choice as a bias–variance dial whose optimum climbs from the \(90.0\)th to the \(99.0\)th percentile as the sample grows hundredfold; and on Gaussian data a fitted shape of \(-0.8192\) with a standard deviation of \(0.9032\), so the sign — which is the entire model — is undetermined
Copulas	Sklar's theorem factoring every joint law uniquely into margins and a copula invariant under monotone rescaling, so rank correlations are copula functionals and Pearson's is not; four families at Kendall's \(\tau=0.40\) with Pearson correlations spanning \(0.5767\) to \(0.5876\) while lower tail dependence runs \(0.1175\), \(0.3400\), \(0.5575\) and \(0.0350\) and the Archimedean pair is one-sided in opposite directions; the coefficient estimated from one corner, so two books differing by \(0.081\) are ranked backwards on \(0.1314\) of four-year samples; and a Gaussian copula matching Kendall's \(\tau\), Pearson and the marginal Kolmogorov–Smirnov statistic exactly while understating joint first-percentile breaches by \(1.89\times\)
Regime Detection	CUSUM as a maximum of sequential likelihood ratios over changepoint locations, with average run length growing exponentially in the threshold — \(88\), \(721\), \(5{,}263\), \(40{,}954\) days — while detection delay grows linearly at \(3.1\), \(4.3\), \(5.5\), \(6.7\), and Lorden's theorem making the trade a property of the problem; the delay therefore irreducible, so the gap between a smoothed backtest and a causal one is a lower bound, with a causal detector capturing \(0.9026\) of an oracle's gain on real regimes; and the identical detector on a single GARCH process with no regimes raising \(2.82\) alarms a year, sitting out \(7.46\%\) of days, and moving the Sharpe by \(-0.0178\) while helping on \(0.4050\) of paths