Phase 13 - Lesson 13.5

Monte Carlo Option Pricing, Greeks, and Quasi-Monte Carlo

Discounted risk-neutral expectations, sensitivities by pathwise and likelihood-ratio methods, and low-discrepancy sequences.

⏱ 60 min● Advanced🔗 Prereqs: 13.2–13.4, risk-neutral pricing (Phase 9)

↖ Phase 13 hub

Builds on: 13.2 gave the estimator, 13.3 variance reduction, 13.4 path simulation; here we price and differentiate.
Leads to: Phase 14 develops the no-arbitrage theory that justifies the \(\mathbb{Q}\)-expectation used here.

Learning Objectives

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Express an option price as a discounted risk-neutral expectation and estimate it by Monte Carlo.
Compute Greeks by finite differences, the pathwise method, and the likelihood-ratio method, and state their trade-offs.
Explain quasi-Monte Carlo and the \(O((\log N)^d/N)\) discrepancy improvement over \(O(1/\sqrt N)\).
Price a path-dependent option (Asian) where no closed form exists.
Choose an appropriate estimator and variance-reduction strategy for a given payoff.

Key Vocabulary

Risk-neutral price: The no-arbitrage value \(V_0=e^{-rT}\E_{\mathbb{Q}}[\text{payoff}]\), an expectation under the pricing measure \(\mathbb{Q}\), not the real-world \(\Prob\).

Pathwise derivative: A Greek estimator differentiating the payoff along each path: \(\partial_\theta\E[Y]=\E[\partial_\theta Y]\) when the payoff is Lipschitz.

Likelihood-ratio method: A Greek estimator differentiating the density rather than the payoff: \(\partial_\theta\E[Y]=\E[Y\,\partial_\theta\log f_\theta]\).

Quasi-Monte Carlo (QMC): Replacing random points by a deterministic low-discrepancy sequence (Sobol, Halton) to fill space more evenly.

Discrepancy: A measure of how far a point set deviates from perfect uniformity; QMC error is bounded by discrepancy via Koksma–Hlawka.

Asian option: A path-dependent claim on the average price \(\bar S=\tfrac1m\sum S_{t_j}\); no simple closed form under GBM, a natural Monte Carlo target.

Intuition & Motivation

Intuition

Pricing by Monte Carlo is one sentence: simulate the underlying under \(\mathbb{Q}\), average the discounted payoff. The subtleties are three. First, measure: the drift must be the risk-free rate \(r\), never the real-world \(\mu\) - we are computing a price, not forecasting. Second, Greeks: hedging needs derivatives of the price, and bumping inputs (finite differences) is noisy, so cleverer pathwise and likelihood-ratio estimators are preferred. Third, efficiency: deterministic low-discrepancy points (quasi-Monte Carlo) fill the cube more evenly than random ones and can turn the \(1/\sqrt N\) rate into nearly \(1/N\) for smooth, low-effective-dimension payoffs.

The pricing recipe under \(\mathbb{Q}\)

For a European claim with payoff \(g(S_T)\) and constant rate \(r\), no-arbitrage (Phase 14) gives

\[V_0=e^{-rT}\,\E_{\mathbb{Q}}\!\big[g(S_T)\big],\qquad S_T=S_0\exp\!\Big(\big(r-\tfrac12\sigma^2\big)T+\sigma\sqrt{T}\,Z\Big),\ Z\sim\Normal(0,1).\] (13.14)

The Monte Carlo estimator draws \(Z_1,\dots,Z_N\), forms discounted payoffs, and averages - with the confidence interval of 13.2. For path-dependent claims (barriers, Asians) simulate the whole path (13.4) and apply \(g\) to it.

Key Idea

Under \(\mathbb{Q}\) the drift is \(r\), not the physical \(\mu\). The real-world measure \(\Prob\) governs how often outcomes actually occur; \(\mathbb{Q}\) is a pricing device that makes discounted traded assets martingales. Mixing \(\mu\) into a price is the canonical beginner error.

Greeks: three estimators

A Greek is a derivative of the price with respect to a parameter \(\theta\) (e.g. \(S_0,\sigma,r\)). Three approaches:

Finite difference (bump-and-revalue): \(\hat\Delta\approx\tfrac{V(S_0+h)-V(S_0-h)}{2h}\). Simple and general, but variance blows up like \(1/h^2\) unless the same random numbers are reused for both bumps (common random numbers).
Pathwise method: differentiate the payoff along each path, \(\Delta=\E_{\mathbb{Q}}[e^{-rT}g'(S_T)\,\partial S_T/\partial S_0]\). Low variance, but requires a (a.e.) differentiable payoff - fails for a digital’s jump.
Likelihood-ratio method: differentiate the density, \(\Delta=\E_{\mathbb{Q}}[e^{-rT}g(S_T)\,\partial_{S_0}\log f(S_T)]\). Handles discontinuous payoffs, but has higher variance and needs the density in closed form.

For the Black–Scholes delta, the pathwise estimator uses \(\partial S_T/\partial S_0=S_T/S_0\), giving \(\Delta=\E_{\mathbb{Q}}[e^{-rT}\ind\{S_T\gt K\}\,S_T/S_0]\), which matches the closed-form \(N(d_1)=0.5987\) for our benchmark.

Worked Example - Delta of a call by the pathwise method

Call payoff \(g(S_T)=(S_T-K)^+\) has \(g'(S_T)=\ind\{S_T\gt K\}\) (a.e.).

Chain rule: \(\partial_{S_0}(S_T-K)^+=\ind\{S_T\gt K\}\,\partial_{S_0}S_T=\ind\{S_T\gt K\}\,S_T/S_0\) since \(S_T\propto S_0\).

Pathwise delta estimator: \(\hat\Delta=e^{-rT}\tfrac1N\sum_i \ind\{S_T^{(i)}\gt K\}\,S_T^{(i)}/S_0\).

For \(S_0=K=100,r=0.03,\sigma=0.2,T=1\) this converges to \(N(d_1)=0.5987\), the exact BS delta.

Quasi-Monte Carlo

Random points clump; a deterministic low-discrepancy sequence (Sobol, Halton) is engineered to fill \([0,1)^d\) evenly. The Koksma–Hlawka inequality bounds the QMC integration error by the product of the integrand’s variation and the point set’s discrepancy:

\[\Big|\frac1N\sum_{i=1}^N f(u_i)-\int_{[0,1)^d} f\Big|\ \le\ V(f)\,D_N^*,\qquad D_N^*=O\!\Big(\frac{(\log N)^d}{N}\Big).\] (13.15)

So QMC error can approach \(O(N^{-1})\) - nearly a full power of \(N\) better than Monte Carlo’s \(O(N^{-1/2})\) - for smooth integrands of low effective dimension. Caveats: the bound degrades in nominal dimension \(d\), and plain QMC gives no error estimate, so practitioners use randomized QMC (scrambled Sobol) to recover an unbiased estimator with a variance you can measure.

Interactive: Monte Carlo the call and read off convergence

Common Mistakes to Avoid

Using the physical drift \(\mu\) instead of \(r\) in the price - the answer is then not a no-arbitrage value at all.
Naive bump-and-revalue Greeks with independent random numbers for each bump; always use common random numbers or the variance is enormous.
Applying the pathwise method to a discontinuous payoff (digital, barrier); its derivative has a delta-function you cannot sample - use the likelihood-ratio method there.
Expecting QMC to beat Monte Carlo in every problem; its advantage erodes for high effective dimension or non-smooth integrands, and plain QMC offers no error bar.

Quant Practitioner Tips

Slogan: price = \(e^{-rT}\E_{\mathbb{Q}}[\text{payoff}]\). Simulate under \(\mathbb{Q}\), discount at \(r\), report a confidence interval.
Pair a vanilla control variate (with known BS price) with an exotic target - often a 10×+ variance cut for Asians and baskets.
For smooth, low-dimensional payoffs, scrambled Sobol points give near-\(O(1/N)\) accuracy and an error estimate; make them a default.
Remember these are model prices: they inherit every assumption of the SDE (constant \(\sigma\), no jumps, frictionless trading). Calibrate and stress-test before trusting a number.

Knowledge Check

Q1 Medium

The Monte Carlo price of a European option is estimated as:

\(\tfrac1N\sum g(S_T^{(i)})\) with \(S_T\) drifting at \(\mu\)

\(e^{-rT}\tfrac1N\sum g(S_T^{(i)})\) with \(S_T\) drifting at \(r\) under \(\mathbb{Q}\)

\(e^{-\mu T}\tfrac1N\sum g(S_T^{(i)})\)

\(\max_i g(S_T^{(i)})\)

Q2 Hard

Which Greek estimator best handles a discontinuous (digital) payoff?

Pathwise method

Likelihood-ratio method

Forward finite difference with fresh random numbers

None can

Q3 Medium

Quasi-Monte Carlo can improve the error rate from \(O(N^{-1/2})\) toward \(O(N^{-1})\) primarily because:

It uses more random seeds

Its low-discrepancy points fill space more evenly (small discrepancy)

It discards the payoff variance

It uses the physical measure

Practical Exercise

You must price a barrier option (knocks out if \(S\) ever exceeds \(B\)) and estimate its delta. (a) Why is a closed form unlikely and Monte Carlo natural? (b) Which delta estimator would you choose and why? (c) Name one variance-reduction technique well-suited to this payoff.

▶ Show full solution

(a) The payoff depends on the entire path (the running maximum), so simulate paths and apply the knock-out rule; closed forms exist only under idealized continuous monitoring, and real contracts monitor discretely, which breaks them.

(b) The payoff is discontinuous at the barrier, so the pathwise method fails there; use the likelihood-ratio method (or a smoothed/conditional-expectation variant), which differentiates the density and tolerates the jump.

(c) Importance sampling that tilts the drift toward (or away from) the barrier concentrates paths in the region that determines knock-out, sharply cutting variance; a vanilla-option control variate is another strong choice.

After the reveal, answer for yourself: How would discrete vs continuous barrier monitoring change your simulated price, and in which direction?

Lesson Summary

A Monte Carlo price is a discounted risk-neutral expectation, \(V_0=e^{-rT}\E_{\mathbb{Q}}[g(S_T)]\), estimated with a confidence interval; the drift is \(r\), never the physical \(\mu\). Greeks come from finite differences (with common random numbers), the low-variance pathwise method (smooth payoffs), or the likelihood-ratio method (discontinuous payoffs). Quasi-Monte Carlo replaces random points with low-discrepancy sequences for near-\(O(1/N)\) accuracy on smooth, low-dimensional problems. All such prices are model-dependent and must be calibrated and stress-tested.

Retrieval Practice

Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.

▶ Show retrieval prompts & answers

Q: Write the Monte Carlo option-pricing formula and name the measure.
A: \(V_0=e^{-rT}\E_{\mathbb{Q}}[g(S_T)]\), an expectation under the risk-neutral measure \(\mathbb{Q}\) (drift \(r\)), estimated by averaging discounted payoffs with a \(1/\sqrt N\) error bar.

Q: Contrast the pathwise and likelihood-ratio Greek estimators.
A: Pathwise differentiates the payoff along each path (low variance, needs a differentiable payoff); likelihood-ratio differentiates the density (handles discontinuous payoffs, higher variance).

Completion Checklist

I can explain the core ideas in my own words
I worked the derivations/examples by hand
I completed the interactive workbench(es)
I passed the knowledge check

Confidence / mastery rating

Personal notes

Source References

This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.

Monte Carlo Methods in Financial Engineering (Paul Glasserman, 2004) foundational - Ch. 5-7 - Ch. 5–7: estimating sensitivities (pathwise, likelihood ratio), and quasi-Monte Carlo with low-discrepancy sequences.
Stochastic Calculus for Finance II (Steven Shreve, 2004) foundational - Ch. 5-6 - Ch. 5–6: risk-neutral valuation \(V_0=e^{-rT}\E_{\mathbb{Q}}[g(S_T)]\) underpinning the simulation.

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