Black-Scholes Equation Derivation

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

Which factor contributed to the initial skepticism surrounding Carlos Juan Finlay's theory on yellow fever transmission?

It took many years before the scientific community took his theory seriously. (correct)
His lack of formal medical training limited his credibility among his peers.
Finlay's research was primarily focused on finding a cure, not identifying the cause.
The scientific community was already convinced that yellow fever was caused by poor sanitation.

What aspect of yellow fever epidemics did Carlos Juan Finlay observe that led him to his hypothesis?

The disease primarily affected individuals with compromised immune systems.
Aedes aegypti mosquitoes were often present in houses during outbreaks. (correct)
The cyclical pattern of outbreaks corresponded with specific weather patterns.
Contaminated water sources correlated with higher infection rates.

How did researchers on the US Army Yellow Fever Board validate Carlos Juan Finlay's hypothesis?

By administering a newly developed vaccine to a large population.
By isolating and identifying the virus responsible for yellow fever.
By demonstrating mosquito transmission through self-experimentation. (correct)
By statistically correlating mosquito populations with the incidence of yellow fever.

What was August Weismann's primary contribution to evolutionary biology?

Providing evidence that acquired characteristics are not inherited. (A) Signup and view all the answers

What experimental method did August Weismann employ to investigate inheritance?

He cut off the tails of mice for five generations. (C) Signup and view all the answers

According to August Weismann, what is the exclusive mechanism of inheritance?

Inheritance via sperm and eggs. (A) Signup and view all the answers

How did Weismann's work on heredity relate to Darwin's theory of evolution?

It provided a mechanism for how traits are passed down, which is a requirement for natural selection to operate. (D) Signup and view all the answers

How did August Weismann’s experiment with mice contribute to the understanding of acquired characteristics?

It helped prove that acquired characteristics, such as a cut tail, are NOT inherited. (B) Signup and view all the answers

What connection did Carlos Juan Finlay observe that led to his yellow fever transmission hypothesis?

The presence of Aedes aegypti mosquitos in houses during epidemics. (D) Signup and view all the answers

Why is August Weismann considered a significant evolutionary biologist after Darwin?

For providing evidence that traits acquired during an organism's life cannot be inherited. (B) Signup and view all the answers

Flashcards

Who was Carlos Juan Finlay?

Cuban physician who first suggested in 1881 that yellow fever is transmitted by mosquitoes.

What is Aedes aegypti?

A mosquito species often present in houses during yellow fever epidemics.

What did August Weismann prove?

Inheritance occurs only via sperm and eggs, not from body cells.

Who was August Weismann?

German biologist regarded as the most significant evolutionary biologist after Darwin.

Signup and view all the flashcards

What did Weismann's mice experiment demonstrate?

Acquired characteristics cannot be inherited

Signup and view all the flashcards

Study Notes

Derivation of Black-Scholes Equation

This equation is used to determine the fair price of a European-style option.

Assumptions

Stock prices follow a geometric Brownian motion, where dS = μSdt + σSdWt.
- μ represents the drift.
- σ represents the volatility.
- dWt represents the Wiener process.
Short selling is allowed, and assets can be divided into smaller units.
No dividends are paid out during the option's life.
There are no opportunities for arbitrage (risk-free profit).
Trading can happen at any point in time (continuous).
The risk-free interest rate (r) remains constant for all maturities

Portfolio Setup

A portfolio (Π) is created, consisting of:
- Selling one call option (-1 call option)
- Buying Δ shares of the underlying asset $\qquad \Pi = -C + \Delta S$
Value of the portfolio changes over a short time period (dt) is defined by: $\qquad d\Pi = -dC + \Delta dS$

Ito's Lemma

Ito's Lemma is used to describe the change in call option value (dC) based on the changes in the stock price (dS) and time (dt).
For a function C(S, t): $\qquad dC = \frac{\partial C}{\partial t} dt + \frac{\partial C}{\partial S} dS + \frac{1}{2} \frac{\partial^2 C}{\partial S^2} (dS)^2$
Given that $dS = \mu S dt + \sigma S dW_t$, it follows that $(dS)^2 = \sigma^2 S^2 dt$
Substituting into the $dC$ equation: $\qquad dC = \frac{\partial C}{\partial t} dt + \frac{\partial C}{\partial S} (\mu S dt + \sigma S dW_t) + \frac{1}{2} \frac{\partial^2 C}{\partial S^2} \sigma^2 S^2 dt$

Substituting $dC$ into $d\Pi$

$dC$ is substituted into the portfolio change equation ($d\Pi$) to get: $\qquad d\Pi = -\left( \frac{\partial C}{\partial t} dt + \frac{\partial C}{\partial S} (\mu S dt + \sigma S dW_t) + \frac{1}{2} \frac{\partial^2 C}{\partial S^2} \sigma^2 S^2 dt \right) + \Delta (\mu S dt + \sigma S dW_t)$

Risk Elimination

By setting $\Delta = \frac{\partial C}{\partial S}$, the risk (dWt term) is eliminated from the portfolio: $\qquad d\Pi = -\frac{\partial C}{\partial t} dt - \frac{1}{2} \frac{\partial^2 C}{\partial S^2} \sigma^2 S^2 dt$

Arbitrage Argument

With no risk, the portfolio should yield the risk-free rate: $\qquad d\Pi = r \Pi dt$
Substituting $\Pi = -C + \frac{\partial C}{\partial S}S$ into the equation yields: $\qquad -\frac{\partial C}{\partial t} dt - \frac{1}{2} \frac{\partial^2 C}{\partial S^2} \sigma^2 S^2 dt = r \left( -C + \frac{\partial C}{\partial S}S \right) dt$

Black-Scholes Equation

After rearranging the terms, the Black-Scholes equation is derived: $\qquad \frac{\partial C}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} + rS \frac{\partial C}{\partial S} - rC = 0$