Lesson 19 of 22

Continuous-Time Models and Option Pricing

Advanced Topic – Optional. Simulate Brownian motion and GBM, then interpret Black–Scholes price and sensitivities under explicit assumptions.

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Advanced Topic – Optional

This chapter extends the core pathway. Complete Chapters 1–14 first, then use this material when your question and data justify the added model complexity.

Motivation

Why begin here

Continuous-time models connect random paths, risk-neutral valuation, and derivative payoffs while exposing discretization and model risk.

Why this matters

Simulate Brownian motion and GBM, then interpret Black–Scholes price and sensitivities under explicit assumptions.

Packages and data

Reproducible setup

Base stats; RQuantLib is optional for broader conventions.

Explanation 1

The live starter code is self-contained and uses only runner-compatible functions.

Explanation 2

Book-only outputs are labelled when the separate source dataset or advanced package was not supplied.

Explanation 3

R code, paths, function names, and formulas remain left-to-right in every locale.

Terminology

Core ideas and notation

Brownian motion

Continuous process with independent Gaussian increments.

Increment SD is sqrt(dt)

Risk-neutral measure

Pricing measure using risk-free drift under replication assumptions.

Not a physical forecast

Delta

Local option-price sensitivity to spot.

Not exercise probability

Notation and formulas

Key relationship

Call = S0 Phi(d1) - K exp(-rT) Phi(d2)

Black–Scholes maps inputs and strong assumptions to a price; payoff, price, probability, and sensitivity are distinct.

Step 1Define every variable and unit.
Step 2Check assumptions and information timing.
Step 3Calculate, verify, and interpret in context.

Worked example

Worked analysis

Scenario

What is the value and delta of an at-the-money one-year European call?

Variable: Spot 100, strike 100, rate 3%, volatility 20%, maturity one year.
Model choice: Black–Scholes maps inputs and strong assumptions to a price; payoff, price, probability, and sensitivity are distinct.
Calculation: Call price = $9.413; delta = 0.5987 per spot dollar under the stated assumptions.
Interpretation: Delta is a local derivative; the risk-neutral exercise probability Phi(d2) is different.
Reasonableness check: Constant volatility, continuous trading, frictionless markets, and lognormal dynamics are simplifications; this is education, not investment advice.

R check

S<-100; K<-100; r<-.03; sigma<-.20; T<-1
d1 <- (log(S/K)+(r+sigma^2/2)*T)/(sigma*sqrt(T))
d2 <- d1-sigma*sqrt(T)
call <- S*pnorm(d1)-K*exp(-r*T)*pnorm(d2)
print(c(call=call,delta=pnorm(d1),risk_neutral_exercise=pnorm(d2)))

Hints

Start by naming the question and unit.
Inspect intermediate values before the final result.
Explain what the calculation cannot establish.

Delta is a local derivative; the risk-neutral exercise probability Phi(d2) is different. Limitation: Constant volatility, continuous trading, frictionless markets, and lognormal dynamics are simplifications; this is education, not investment advice.

Visual

Call sensitivity

Illustrative call value rises with spot around a fixed strike.

R connection

Live, editable R practice

Run the self-contained starter code in the protected STATLAB R runner. The code prints an auditable result and avoids network or unrestricted file access.

Live R Lab

INTRO-R-CH19: Continuous-Time Models and Option Pricing

Run a self-contained example, verify its output, and explain one limitation for continuous-time models and option pricing.

Editable R code

Results

Ready to run

Ready to run. Results, errors, tables, plots, and fallback messages will appear here.

State the question, data, and units.

Run and inspect the printed output.

Interpret the result and one limitation.

Common mistake

Use volatility 0.20, not 20, and keep rate and maturity units consistent.

STATLAB Tip

Test limiting cases: price rises with spot and volatility, falls with strike, stays nonnegative, and approaches payoff near expiry.

Guided practice

Scenario

Calculate prices across three volatilities and maturities; explain why changes are not uniformly linear.

Hints

Reuse a small tested object.
Change one element at a time.
Finish with an interpretation and limitation.

Compare your result with the definition, units, and model assumptions—not only with a target number.

Exercises

Independent practice

Complete these without looking at the selected solutions. More than one defensible program may exist.

Explanation 1

Verify put–call parity.

Explanation 2

Estimate price with risk-neutral Monte Carlo.

Explanation 3

Approximate delta by centred difference.

Selected solutions

Check one defensible route

These are compact solution routes. Confirm dimensions, units, and any changed modelling choices.

Explanation 1

call-put = S-K*exp(-r*T)

Explanation 2

Simulate terminal GBM, discount pmax(ST-K,0), and report Monte Carlo error.

Explanation 3

[C(S+h)-C(S-h)]/(2h)

Chapter summary

What to carry forward

Simulate Brownian motion and GBM, then interpret Black–Scholes price and sensitivities under explicit assumptions.

Explanation 1

Brownian increments scale with sqrt(dt).

Explanation 2

GBM includes the Itô correction.

Explanation 3

Black–Scholes is risk-neutral and assumption-dependent.

Explanation 4

Greeks are local sensitivities.

Terminology

Chapter cheat sheet

cumsum(rnorm())

Brownian path

Use in Continuous-Time Models and Option Pricing.

exp()

GBM

Use in Continuous-Time Models and Option Pricing.

pnorm()

normal CDF

Use in Continuous-Time Models and Option Pricing.

pmax()

option payoff

Use in Continuous-Time Models and Option Pricing.

References and provenance

Black & Scholes (1973); Merton (1973); Safavi (2026), Chapter 19. Student notes: Mohammad Safavi, Ph.D., STATLAB Academy, Version 1.0.

Resource

Download the complete student notes

The authoritative 125-page English PDF accompanies this native lesson.

Download

Reflection

Pause and write

What assumption in continuous-time models and option pricing most affects the conclusion?

Name the assumption, evidence you would seek, and how the recommendation might change.

Exit check

Chapter knowledge check

Reach 70% to complete the chapter. Explanations appear after submission.

Checkpoint