3.7.1 Exotic & Path-Dependent Options from Zero

Basic Prerequisites: Stochastic Calculus & Itô’s Lemma.

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1. Intuition & Practical Objective

A vanilla European option is priced from one number: the terminal stock price $S_T$. Everything - the whole distribution - collapses into a single draw at expiry. An exotic option is anything whose payoff needs more than $S_T$. The moment it references the path’s running maximum or minimum (lookback), its average (Asian), whether it touched a level (barrier), or a second asset (exchange/quanto), the pricing problem changes character, because the payoff is now a functional of the whole path, not a function of its endpoint.

The practical objective of this page: build the intuition for why this change matters, with no formula beyond one idea - the risk-neutral expectation extends from "$S_T$" to “any functional of the path.” Three “aha”s:

1. Path-dependence = the payoff ignores nothing. Two different paths can end at the same $S_T$ yet pay different amounts (one touched the barrier, one didn’t; one set a lower minimum, one didn’t). So the terminal distribution alone is useless - you need the joint law of the path statistics and $S_T$.
2. The pricing measure is unchanged. You still compute $V(0)=e^{-rT}{\mathbb{E}}^{\mathbb{Q}}[\text{payoff}]$ (Glasserman 1.39; Shreve Ch 7). The only difference is that the payoff is now a function of more of the trajectory, so the expectation is over the full path - which is precisely what Monte Carlo simulates.
3. The underlying’s drift under $\mathbb{Q}$ is the cost-of-carry $b$. For path-dependent products the distribution of the running max/min/average depends on the drift of $\log S$, which under $\mathbb{Q}$ is $b-\frac{1}{2}{\sigma }^2$ (not $r$ in general). Simulating with the wrong drift misprices every path statistic.

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2. Mathematical Ground Truth & Derivations

The shared object: a path-functional. Under the risk-neutral measure the stock is

$$dS=bSdt+\sigma Sd{W}^{\mathbb{Q}},$$

and every exotic here is $V(0)=e^{-rT}{\mathbb{E}}^{\mathbb{Q}}[f(\text{path})]$ for a different functional $f$:

Exotic	Path functional $f$
Barrier (down-and-out call)	$(S_T-X{)}^{+}\cdot 1\{{\min }_{0\le t\le T}{S}_t>H\}$
Cash-or-nothing digital	$K\cdot 1\{S_T>X\}$
Lookback (floating call)	$S_T-{\min }_{0\le t\le T}{S}_t$
Asian (average-rate put)	$(X-\frac{1}{T}{\int }_0^T{S}_{t}dt{)}^{+}$
Quanto	payoff in domestic currency of a foreign equity

The reflection principle - the workhorse for barrier & lookback closed forms. For driftless Brownian motion, the probability that the maximum exceeds a level $m$ by time $T$ equals twice the probability that the terminal value exceeds $m$: $\mathbb{P}({\bar{W}}_T>m)=2\mathbb{P}(W_T>m)$. This “reflects” every path that touched $m$ into a path ending above $m$ (Shreve II Thm 7.2.1 gives the exact joint density of $({\bar{W}}_T,W_T)$). It is the seed of every barrier/lookback closed form: price the vanilla, subtract (or add) the reflected “image” terms.

Shreve’s unifying structure (Ch 7). Knock-out barriers solve the BSM PDE on a shrunk domain $0\le x\le B$ with $v(t,B)=0$; lookbacks add a running-max state $Y(t)$ and a smooth-pasting condition $v_y(t,y,y)=0$; Asians add an integral state $Y(t)={\int }_0^{t}Sdu$ and the degenerate PDE $v_t+rx{v}_x+x{v}_y+\frac{1}{2}{\sigma }^2{x}^2{v}_{xx}=rv$. In every case the state vector grows - and that is exactly the dimensionality that forces Monte Carlo.

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3. Computational Implementation - path-dependence made visible

The single most convincing way to feel the divide: simulate one batch of paths, record both the terminal values and the running minima, and price a lookback (path-dependent) against a vanilla (terminal-only) that share the exact same terminal draws. The lookback price cannot be recovered from $S_T$ alone. Stdlib only.

The vanilla MC lands on the BSM closed form (terminal-only: the closed form is the answer), while the lookback put on the same paths is more than twice as valuable - it pays off on every path whose running minimum (not just endpoint) dipped below $X$. Same terminal draws, radically different prices: that is path-dependence.

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4. Failure Modes & First-Principles Breakdowns

1. The “terminal distribution is enough” trap. For an exotic you cannot compress the path to $S_T$; you need the joint law of $(\text{path statistic},S_T)$. This is the single biggest conceptual error.
2. Wrong drift under $\mathbb{Q}$. Simulate with the cost-of-carry $b$ (e.g. $r-q$ for an index) in $d\log S=(b-\frac{1}{2}{\sigma }^2)dt+\sigma dW$, and discount at $r$. Using $r$ as the drift when $b\ne r$ (dividend-paying underlying) biases every running-min/max statistic.
3. Monitoring specification is part of the contract. Continuous vs discrete monitoring changes the price (see 06 · Advanced Extensions); for a knock-in a barrier checked daily is cheaper than one checked continuously; for a knock-out the daily check is more expensive, because the grid misses touches the continuous monitor would catch. Ignoring this is a first-principles mis-specification.

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5. References

• Shreve, Stochastic Calculus for Finance II
• Glasserman, Monte Carlo Methods in Financial Engineering
• Haug, The Complete Guide to Option Pricing Formulas

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6. Connected Graph Bridges

• Base: Stochastic Calculus & Itô · Black–Scholes–Merton & Feynman–Kac
• Continue: 02 · Barriers & Digitals · Index Hub
• Theory: Multivariable Calculus