Chapter 8 · Derivatives, PDEs, and the Feynman–Kac Bridge
Part II · Dynamics in Continuous Time
Chapter at a glance
Modern Finance Laboratory · Module 8 · Week 8
Module 8 has four panels. The PDE panel solves (8.1) by finite differences before the reader’s eyes, marching the payoff backward through time with the solution surface rendered live, and a toggle overlays the Monte Carlo estimate of (8.3) on the same axes— Feynman–Kac as a split-screen demonstration. The Greeks panel displays 𝑉, Δ, Γ, vega, Θ for any user portfolio of vanillas across (𝑆, 𝑡), with a jump-day cursor that reads off 1 2 2 Γ(Δ𝑆) at any point of Figure 8.3(a)’s map, and a live check of the identity (8.5). The smile panel loads a strike ladder of market quotes, inverts each to implied volatility, and reprices the user’s book at fla
Learning Outcome Statements
LOS 8.1 Derive the Black–Scholes partial differential equation twice: by constructing the replicating portfolio, and by the martingale (drift-kill) argument.
LOS 8.2 State and prove the Feynman–Kac theorem, and use it to move in both directions between conditional expectations and parabolic PDEs, including for bond pricing.
LOS 8.3 Derive the Black–Scholes formula from the risk-neutral lognormal expectation, and deploy it for pricing, strike selection, and audit.
LOS 8.4 Compute and interpret the Greeks, prove the theta–gamma identity from the pricing PDE, and quantify a hedged book’s jump-day P&L by sign and region of gamma.
LOS 8.5 Define implied volatility, explain why it is well defined, and read the volatility skew as a quantitative verdict on the model’s assumptions.
LOS 8.6 Price barrier options in closed form via the reflection principle and change of measure, and evaluate knockout features in hedging programs.
Laboratory · Module 8 (book §8.11)
Module 8: The Pricing Machine
Course Website · Week 8
Guided experiment (supports LOS 8.1–8.6). (i) Price the collar in the PDE panel; verify −2.7574 against the closed form and watch the Monte Carlo band tighten around both. (ii) Reproduce Example 8.4: read the dealer’s gamma at 𝑆 = 92 and 𝑆 = 100, and the jump-day P&L at each. (iii) In the smile panel, reprice the collar with the put at 18.4% and the call at 16.5%; report the move in fair value and in the zero-cost strike. (iv) Price the dealer’s knockout put (𝐵 = 80) and compute the discount to the vanilla 90-put; slide monitoring from continuous to daily and record the change.
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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 8.1. The drift 𝜇 appears in the index’s dynamics and nowhere in (8.1) or (8.4). Trace exactly where it disappeared in each of Section 8.1’s two derivations, and reconcile with Chapter 1’s two-state version of the same disappearance. 8.12 Exercises 193
Exercise 8.2. A junior analyst computes an option price by simulating index paths with drift 7% and averaging discounted payoffs. Name the error, the theorem it violates, the direction of the bias for a call, and the one-word fix.
Exercise 8.3. “Theta pays for gamma.” State the identity (8.5), explain each term’s sign for a long straddle, and say precisely under what realized-variance condition a delta-hedged long-gamma book breaks even—connecting to Theorem 6.4.
Part B — Computations
Exercise 8.4. (The permanent record.) With the chapter’s parameters, verify by direct computation: put(90) = 1.7228; call(110) = 4.4803; the collar at −2.7574; the zerocost strike 𝐾 ∗ = 122.0; and the signed 90/118.3 collar’s residual value −0.63. Confirm put–call parity numerically at both strikes.
Exercise 8.5. Compute the full Greek panel of the signed collar (delta, gamma, vega, theta at 𝑆0 = 100), verify the identity (8.5) to machine precision, and translate vega into dollars per volatility point on the $800M book.
Exercise 8.6. An at-the-money call (𝐾 = 100) trades at 9.60. Compute its implied volatility by bisection; then compute the vega at that volatility and use it for the onestep Newton estimate from a starting guess of 17%, comparing the two answers.
Exercise 8.7. Price the down-and-out call (𝐾 = 100, 𝐵 = 90) from (8.6) and verify the chapter’s 7.75; recompute for 𝐵 = 80 and 𝐵 = 95, and explain the direction of the change in one sentence each. Then compute the down-and-in call for 𝐵 = 90 by in–out parity (𝐶DI = 𝐶 − 𝐶DO ).
Part C — Proofs and extensions
Exercise 8.8. Derive the put analogue of (8.4) directly from the risk-neutral expectation (not via parity), and differentiate the call formula to prove Δ = Φ(𝑑1 ) exactly—showing why the naive term-by-term differentiation happens to give the right answer (the two correction terms cancel; exhibit the cancellation).
Exercise 8.9. Prove the Feynman–Kac converse in detail for the Black–Scholes case: define 𝑉 (𝑡, 𝑆) by the conditional expectation, assume smoothness, and show the drift computation forces (8.1). Then verify the theorem on the bond example: check that 𝑃 = 𝑒 𝐴−𝐵𝑥 with 𝐵 = (1 − 𝑒 −𝜅 𝜏 )/𝜅 satisfies the Vasicek PDE term by term, deriving the ODE that 𝐴 must solve.
Exercise 8.10. ∗ (Numeraire change.) Show that the first term of the call price can be written 𝑆0 Q𝑆 (𝑆𝑇 > 𝐾), where 𝑑Q𝑆 /𝑑Q = 𝑆𝑇 /(𝑆0 𝑒𝑟𝑇 ) is the measure using the stock as numeraire; verify via Girsanov that under Q𝑆 the log-index drifts at 𝑟 + 21 𝜎 2 , and rederive Φ(𝑑1 ) as a probability. 194 8 Derivatives, PDEs, and the Feynman–Kac Bridge
Exercise 8.11. ∗ (Barrier, completely.) Prove the joint reflection identity for driftless Brownian motion—P(min𝑡 ≤𝑇 𝑋𝑡 > 𝑏, 𝑋𝑇 ∈ 𝑑𝑦) = 𝜑𝑇 (𝑦 − 𝑥0 ) − 𝜑𝑇 (𝑦 − (2𝑏 − 𝑥 0 )) 𝑑𝑦 for 𝑦 > 𝑏, with 𝜑𝑇 the Normal(0, 𝜎 2𝑇) density—by the discrete path-flipping bijection and passage to the limit; then carry out the Girsanov drift restoration to complete the proof of Proposition 8.5.
Part D — Modeling and application
Exercise 8.12. Write the file-closing memorandum for the CFO: the one-page analytic record (Exercise 8.4’s numbers with formulas cited); the model-resolution finding of Example 8.3 and its $5.1M consequence, with the two remediation options; and the standing governance rule you propose for when tree-resolution audits must be escalated to closed-form or high-resolution numerical pricing before contract terms are set.
Exercise 8.13. Write the recommendation memo on the dealer’s knockout collar: price the down-and-out 90-put with barrier 80 (adapt (8.6)); state the premium saved versus the vanilla structure; describe the exact scenario set in which the saved premium is repaid (path touches 80, protection dies, index continues down), using the reflection principle to estimate that scenario’s risk-neutral probability; and give your recommendation with the skew consideration of Section 8.5 explicitly weighed.
Part E — Laboratory
Exercise 8.14. (Laboratory Module 8; supports LOS 8.1–8.6.) Perform the four-part guided experiment of Section 8.11. Submit: (a) the PDE-versus-Monte-Carlo split screen at the collar’s parameters; (b) the gamma map readings and jump P&L at 𝑆 = 92 and 100; (c) the skew repricing of the collar with the moved zero-cost strike; (d) the knockout put’s price at continuous and daily monitoring.
Exercise 8.15. (Course Website, Week 8, Notebook 8.) Extend the notebook: implement the delta hedge of a short 90-put under (a) the Black–Scholes model and (b) the jump-diffusion of Example 6.10, both hedged at daily frequency with the Black–Scholes delta. Report the two hedging-error distributions, the left-tail quantiles, and the premium markup (in volatility points) that would compensate the jump book’s shortfall at the 1% quantile—your first jump-risk premium, hand-computed.
Full solutions are distributed to instructors in the Instructor’s Manual, Chapter 8; they are not posted here. Problem-set files are on the Course Website, Week 8.