3.3.6 Advanced Extensions

Basic Prerequisites: 03 · Pricing Formulas and 05 · Failure Modes.

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1. Intuition & Practical Objective

Every real-market defect of BSM (the smile, jumps, fat tails, early exercise) motivates an extension. This page is the launchpad: it shows the two immediate extensions that stay inside the BSM family - American early exercise (still lognormal, still one asset) and Merton jump-diffusion (keeps a closed form as a Poisson mixture of BS) - and then hands off to the dedicated topic-folder pages for stochastic/local volatility and numerical methods.

Why these first? American options are the same PDE with a free boundary, so they are the closest extension. Jump-diffusion is the minimal incomplete-market extension and still prices in closed form - the perfect bridge to why one-factor models fail. Everything farther (Heston, SABR, FDM, MC) is linked from here.

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2. Mathematical Ground Truth & Derivations

2.1 American options & the early-exercise boundary (Haug Ch 3; Björk §7.8)

An American option adds the right to exercise before $T$, giving the free-boundary (variational) problem

$$V\ge \max (\text{intrinsic},0),\frac{\partial V}{\partial t}+rS\frac{\partial V}{\partial S}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2}-rV\le 0,$$

with equality where it is optimal to continue. The optimal exercise boundary $S^{\ast }(t)$ solves a smooth-pasting condition. Key facts:

• American call, no dividends: never optimal to exercise early, so $C^{\text{Am}}=c^{\text{BSM}}$ (Haug §1.2).
• American put / dividend-paying call: early exercise is optimal; no closed form - solve numerically (binomial tree, finite-difference, or analytic approximations like Barone–Adesi–Whaley or Bjerksund–Stensland, all in Haug Ch 3).

2.2 Merton (1976) jump-diffusion - the minimal incomplete-market extension

Under $\mathbb{Q}$,

$$d{S}_t/S_t=(r-\lambda \kappa )dt+\sigma d{W}_t+(J-1)d{N}_t,$$

where $N$ is a Poisson process with intensity $\lambda$, $J$ the multiplicative jump size ($\ln J\sim N({\mu }_J,{\sigma }_J^2)$), and $\kappa =\mathbb{E}[J-1]=e^{{\mu }_J+\frac{1}{2}{\sigma }_J^2}-1$ the compensator that keeps $e^{-rt}{S}_t$ a martingale. Because a Poisson-mixture of lognormals is lognormal, the call prices in closed form as a Poisson-weighted average of BSM calls at inflated vol:

$$C_{\text{Merton}}=\sum _{n=0}^{\infty }\frac{e^{-{\lambda }^{\prime }T}({\lambda }^{\prime }T{)}^n}{n!}{c}_{\text{BSM}}\left(S,X,T,r,{\sigma }_n\right),{\sigma }_n=\sqrt{{\sigma }^2+\frac{n{\sigma }_J^2}{T}},{\lambda }^{\prime }=\lambda (1+\kappa ).$$

(Glasserman Ch 3.5, eq. 3.79–3.81, gives the pathwise simulation form.)

2.3 Beyond: toward local & stochastic volatility (bridges)

• Local volatility (Dupire): replace constant $\sigma$ with a deterministic function $\sigma (S,t)$ calibrated to the entire smile; the model stays complete (one driver). → Implied Volatility Surfaces.
• Stochastic volatility (Heston/SABR): add a second driver to variance, breaking completeness; characteristic-function/Fourier pricing. → Heston & SABR.
• Numerical methods: finite-difference schemes (Duffy) for the PDE, Monte Carlo (Glasserman) for path-dependent exotics.

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3. Computational Implementation - two extensions, both stdlib

A. American put via CRR tree (Haug §4.2; Hull Ch 13) - the early-exercise premium is the gap vs the European value.

(Haug §4.2 verifies the CRR American put $4.692$ against the same inputs.)

B. Merton jump-diffusion vs pure BSM - a Poisson mixture of lognormals priced by Monte Carlo. Constant-vol BSM (no jumps) under-prices when jumps are present.

The jump model prices the call materially higher ($13.35$ vs $10.45$): the fat left tail makes the payoff right-skewed, and constant-vol BSM cannot represent that tail risk - exactly the structural failure of 05 · Failure Modes.

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4. Failure Modes & First-Principles Breakdowns

1. American pricing is a free-boundary problem, not a closed form. Applying the European formula to an American put understates value by the early-exercise premium ($0.24$ here). Use the tree/analytic approximations (Haug Ch 3).
2. Jump-diffusion markets are incomplete. One stock cannot hedge the jump source of randomness, so prices are not unique - the Merton price is one (risk-neutral) choice, not a no-arbitrage necessity. This is the theoretical cliff-edge beyond BSM.
3. Numerical methods carry their own errors. Trees have discretization (oscillating) error; MC has $O(n^{-1/2})$ error and needs variance reduction (Glasserman Ch 4); FDM has stability/consistency constraints (Duffy Ch 3). “More steps” is not automatically better - see the failure pages of the linked topics.
4. Stochastic-vol parameters are hard to identify. Heston/SABR free parameters trade off against each other and are notoriously hard to calibrate stably (see Heston & SABR).

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5. References

• Haug, The Complete Guide to Option Pricing Formulas
• Hull, Options, Futures, and Other Derivatives
• Glasserman, Monte Carlo Methods in Financial Engineering
• Duffy, Finite Difference Methods in Financial Engineering (FDM for the BSM PDE).
• Björk, Arbitrage Theory in Continuous Time

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6. Connected Graph Bridges

• Back: 05 · Failure Modes · Index Hub
• Forward topic-folder pages: Heston & SABR · Implied Volatility Surfaces · Interest Rate Models
• Base: Stochastic Calculus & Itô · No-Arbitrage & Binomial Trees