Chapter 6 · Stochastic Processes and Financial Dynamics
Part II · Dynamics in Continuous Time
Chapter at a glance
Modern Finance Laboratory · Module 6 · Week 6
Module 6 has three panels. The limit panel animates the scaled walk (6.1) with a slider for 𝑛, overlaying the running maximum and its reflection-principle prediction, and letting the reader swap the coin for skewed or heavy-tailed steps to watch Donsker invariance hold—and break when variance is infinite. The quadratic-variation panel loads a simulated or historical price path and computes realized variance across sampling frequencies from monthly to one-minute, plotting the estimate with its ±2 standard-error band against Theorem 6.4’s prediction and exposing the microstructure corruption at the finest scales. The jump panel superposes a Poi
Learning Outcome Statements
LOS 6.1 Construct Brownian motion as the scaling limit of random walks and state and use its defining properties.
LOS 6.2 Analyze the character of Brownian paths—continuity, non-differentiability, self-similarity—and compute first-passage probabilities with the reflection principle.
LOS 6.3 Prove that Brownian motion accumulates quadratic variation [𝑊, 𝑊] 𝑇 = 𝑇, and interpret realized variance as its estimator, with precision governed by sampling frequency.
LOS 6.4 Verify the three fundamental Brownian martingales and apply optional stopping in continuous time with appropriate care.
LOS 6.5 Model asset prices with geometric Brownian motion, derive its moments, and assess it honestly against the stylized facts of returns.
LOS 6.6 Model event risk with Poisson processes, construct compensated martingales, and quantify how jumps transform tail probabilities.
Laboratory · Module 6 (book §6.11)
Module 6: Paths, Quadratic Variation, and Jumps
Course Website · Week 6
Guided experiment (supports LOS 6.1–6.6). (i) Set 𝑛 = 4, 12, 52, 252 in the limit panel’s pricing overlay and reproduce the opening problem’s collar convergence to −2.76. (ii) In the QV panel, estimate annualized volatility from the same simulated year at monthly, daily, and five-minute sampling; record the three estimates and their error bands, and identify the frequency at which the bands stop shrinking on the historical series. (iii) Verify the reflection principle by Monte Carlo: fraction of driftless 17% paths touching +10% within the year versus the formula’s 0.556. (iv) In the jump panel, find two parameterizations—one pure GBM with inflated 𝜎, one jump-diffusion—that match annual volatility to within 0.1%, and report their respective waiting times for a −6% day; write one sentence on what a variance-matching model validation would have missed.
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Exercises
Exercises are grouped A–E throughout the book: A concept checks, B computations, C proofs and extensions, D modeling and application, E laboratory. Starred exercises (∗) are on the advanced track.
Part A — Concept checks
Exercise 6.1. √ In the scaling (6.1), explain in one sentence each: why dividing by 𝑛 instead of 𝑛 kills the limit; why dividing by 𝑛1/4 explodes it; and which hypothesis of the construction fails if the coin steps have infinite variance (relate to the Student-𝑡 tail-index discussion of Section 2.4).
Exercise 6.2. Classify as martingale, submartingale, supermartingale, or none, with 2 one line each: (a) 𝑊𝑡3 ; (b) |𝑊𝑡 |; (c) 𝑒 𝜎𝑊𝑡 ; (d) 𝑒 𝜎𝑊𝑡 − 𝜎 𝑡/2 ; (e) the running maximum 𝑀𝑡 ; (f) the compensated Poisson 𝑁𝑡 − 𝜆𝑡. Exercise 6.3. Meridian’s risk system converts daily volatility to annual by multiplying √ by 252. State the exact property of Brownian motion this rule instantiates, and give the two empirical violations from Chapter 2’s stylized facts under which the rule respectively understates and can misstate annual risk (tail thickness; volatility clustering with positive return autocorrelation). 150 6 Stochastic Processes and Financial Dynamics
Part B — Computations
Exercise 6.4. (The convergence table.) Using the CRR parameterization 𝑢 = 𝑒 𝜎 Δ𝑡 , 𝑑 = 1/𝑢 with 𝜎 = 17% and continuously compounded rate 3.98%: (a) price the 90/110 collar for 𝑛 = 4, 12, 52, 252 steps and reproduce the opening problem’s table; (b) describe the odd–even oscillation you observe and its cause (the strike’s position between lattice nodes); (c) the limit is −2.757: compute the dollar difference from the quarterly tree on $800M and say whether Chapter 5’s verdict survives.
Exercise 6.5. Using the reflection principle for the driftless log-price 𝜎𝑊𝑡 with 𝜎 = 17%: (a) verify P(touch +10% within a year) = 0.556; (b) compute the probability of touching −15%—the trigger of Chapter 2’s opening problem—at some point during the year, and compare it with the endpoint probability computed there; (c) find the barrier level touched within the year with probability exactly 21 . Exercise Í 6.6. A year of daily log-returns on the equity book has sample mean 0.031% and 𝑟 𝑖2 = 0.02890. Compute realized variance and annualized realized volatility; attach the ±2 standard-error band using Var(RV) ≈ 2𝜎 4𝑇/𝑛; and state how many years of daily data would be needed to estimate the annual mean return to the same relative precision—the drift-versus-volatility asymmetry of Proposition 6.2 in estimation form.
Exercise 6.7. With the jump-diffusion of Example 6.10: (a) verify the once-per-3.2years figure; (b) compute the expected number of −6% days per decade under each model; (c) compute the total annual variance contributed by jumps, 𝜆(𝜇2𝐽 + 𝜎𝐽2 ), and confirm it is under 0.5% of the diffusion’s 𝜎 2 —the “invisible in variance, decisive in tails” arithmetic.
Part C — Proofs and extensions
Exercise 6.8. Prove that for 0 < 𝑠 < 𝑡: (a) Cov(𝑊𝑠 , 𝑊𝑡 ) = 𝑠 (write 𝑊𝑡 = 𝑊𝑠 + Δ); (b) E[𝑊𝑡 | 𝑊𝑠 ] equals 𝑊𝑠 while E[𝑊𝑠 | 𝑊𝑡 ] = 𝑠𝑡 𝑊𝑡 (use the joint Gaussian projection of Proposition 3.7); (c) the Brownian bridge interpretation of (b): conditioned on the endpoint, the path’s expectation interpolates linearly.
Exercise 6.9. Complete the two deferred steps of Proposition 6.9: (a) the induction giving the density of 𝑇𝑘 ; (b) independence and stationarity of increments from the e𝑡2 − 𝜆𝑡 is a martingale, the memorylessness of the exponential. Then verify that 𝑁 2 e 𝑁 e ]𝑡 . Poisson analogue of 𝑊𝑡 − 𝑡, and identify [ 𝑁,
Exercise 6.10. Using Theorem 6.4, prove that Brownian paths have infinite total variation on every interval [0, 𝑇]: assume finite total variation 𝑉, bound 𝑄 Π ≤ max𝑖 |Δ𝑖 | · 𝑉, ∫𝑇 and derive a contradiction from path continuity. Conclude that 0 𝑓 (𝑡) 𝑑𝑊𝑡 cannot be defined path-by-path as a classical Riemann–Stieltjes integral—the door through which Chapter 7 enters. 6.12 Exercises 151
Exercise 6.11. ∗ Prove the reflection principle exactly for the symmetric random walk (construct the reflection bijection and verify it is measure-preserving), and derive the discrete first-passage distribution P(max 𝑘 ≤𝑛 𝑋 𝑘 ≥ 𝑎). Compare numerically with (6.2) at 𝑛 = 252, 𝑎 = 16 to see the invariance principle’s accuracy at daily resolution.
Part D — Modeling and application
Exercise 6.12. Write the one-page volatility-measurement policy for Meridian’s risk system: which sampling frequency the official number should use and why (Theorem 6.4’s precision arithmetic); where microstructure noise caps the benefit; how the policy should treat volatility clustering (rolling windows versus full-sample); and the exact sentence that should accompany every reported volatility to prevent the trustee’s three-numbers confusion from recurring.
Exercise 6.13. Write the one-page model-risk note on the October day: the two candidate explanations (thin-tailed model versus jump mechanism) with the evidence each requires; the gap-risk implication for the collar’s dealer hedge and the question the CIO should put to the dealer about it; and your recommendation for which model the risk system should carry, with the single statistic you would monitor to revisit the choice.
Part E — Laboratory
Exercise 6.14. (Laboratory Module 6; supports LOS 6.1–6.6.) Perform the four-part guided experiment of Section 6.11. Submit: (a) the collar convergence overlay; (b) the three volatility estimates with error bands and the frequency at which historical bands stop shrinking; (c) the Monte Carlo reflection check; (d) the two variance-matched models with their −6% waiting times and your one-sentence validation moral.
Exercise 6.15. (Course Website, Week 6, Notebook 6.) Extend the notebook: simulate 2,000 GBM years and 2,000 jump-diffusion years at matched annual volatility; for each model report the frequency of daily moves beyond 3, 4, 5, 6 standard deviations with Monte Carlo standard errors, and produce the log-scale exceedance plot that a modelvalidation report should contain. Conclude with three sentences on which moments a validation suite must include beyond the second.
Full solutions are distributed to instructors in the Instructor’s Manual, Chapter 6; they are not posted here. Problem-set files are on the Course Website, Week 6.