توضیح 1
The live starter code is self-contained and uses only runner-compatible functions.
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درس 19 از 22
Advanced Topic – Optional. Simulate Brownian motion and GBM, then interpret Black–Scholes price and sensitivities under explicit assumptions.
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
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
Base stats; RQuantLib is optional for broader conventions.
توضیح 1
The live starter code is self-contained and uses only runner-compatible functions.
توضیح 2
Book-only outputs are labelled when the separate source dataset or advanced package was not supplied.
توضیح 3
R code, paths, function names, and formulas remain left-to-right in every locale.
اصطلاحات
Continuous process with independent Gaussian increments.
Increment SD is sqrt(dt)
Pricing measure using risk-free drift under replication assumptions.
Not a physical forecast
Local option-price sensitivity to spot.
Not exercise probability
نمادگذاری و فرمولها
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.
مثال حلشده
سناریو
What is the value and delta of an at-the-money one-year European call?
بررسی با R
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)))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
Illustrative call value rises with spot around a fixed strike.
ارتباط با R
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
Run a self-contained example, verify its output, and explain one limitation for continuous-time models and option pricing.
Ready to run
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.
تمرین هدایتشده
سناریو
Calculate prices across three volatilities and maturities; explain why changes are not uniformly linear.
Compare your result with the definition, units, and model assumptions—not only with a target number.
Exercises
Complete these without looking at the selected solutions. More than one defensible program may exist.
توضیح 1
Verify put–call parity.
توضیح 2
Estimate price with risk-neutral Monte Carlo.
توضیح 3
Approximate delta by centred difference.
Selected solutions
These are compact solution routes. Confirm dimensions, units, and any changed modelling choices.
توضیح 1
call-put = S-K*exp(-r*T)
توضیح 2
Simulate terminal GBM, discount pmax(ST-K,0), and report Monte Carlo error.
توضیح 3
[C(S+h)-C(S-h)]/(2h)
Chapter summary
Simulate Brownian motion and GBM, then interpret Black–Scholes price and sensitivities under explicit assumptions.
توضیح 1
Brownian increments scale with sqrt(dt).
توضیح 2
GBM includes the Itô correction.
توضیح 3
Black–Scholes is risk-neutral and assumption-dependent.
توضیح 4
Greeks are local sensitivities.
اصطلاحات
Brownian path
Use in Continuous-Time Models and Option Pricing.
GBM
Use in Continuous-Time Models and Option Pricing.
normal CDF
Use in Continuous-Time Models and Option Pricing.
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.
منبع
The authoritative 125-page English PDF accompanies this native lesson.
دریافتبازاندیشی
Name the assumption, evidence you would seek, and how the recommendation might change.
ارزیابی پایانی