Options 101 – What is an option?

Welcome!

Introduction to Financial Options / Lecture 01

Agenda for today:

Administrative logistics
What is an option?
What kinds of options are there?
What are the parameters that define an option?

Learning Goals

Introduction to Financial Options / Lecture 01

At the end of this course, you will be able to:

Explain what financial options are and how they’re used.
Relate options to each other and find value bounds using arbitrage arguments.
Price vanilla options using both closed-form and numerical techniques.
Manage the risks of a portfolio of options.
Produce and use volatility models of options curves and surfaces.

About me and the course

Introduction to Financial Options / Lecture 01

I am an electrical engineer by training.
I began as a trader at Jane Street in 2008.
By 2011 I was the head of the options desk in London, running all EMEA options trading.
The content of this course is nearly identical to the intern and new-hire trainings that I conducted for years.

You will learn the same stuff.

Why should you care?

Introduction to Financial Options / Lecture 01

Options are everywhere. Options and option-like payoffs appear throughout finance and everyday life, as you’ll see later today.
The math is interesting. Options connect probability, calculus, optimization, and economics.
Your compensation may include them. You may need to value the stock options you receive at the high-flying startup you join or start someday.
They can become a career. You may even want to get into professional trading.

Lecture Sequence

Introduction to Financial Options / Lecture 01

What Is an Option?

Contracts, payoffs, and exercise rights

The Value of Optionality

Uncertainty, distributions, and time value

Put-Call Parity and Synthetic Equivalence

Synthetic positions and static arbitrage

Pricing by Replication

Binomial hedges, cash, and fair price

Black-Scholes

Dynamic hedging and the pricing argument

Implied Volatility

Turning option prices into vol quotes

Delta and Directional Hedging

Hedge ratios and changing exposure

Gamma, Theta, Vega, and P&L

Convexity, decay, and volatility risk

Volatility Curves and Surfaces

Smiles, skew, strike, and expiry

No-Arbitrage and Distributions

Bounds, convexity, and implied probabilities

Financial Time and Variance-Time

Trading time, calendar time, and events

Stochastic Volatility Models

Random volatility, jumps, and calibration

Evaluation Touchpoints

Introduction to Financial Options / Lecture 01

Assignments follow the lecture that supplies their core tools.

After Lecture 03

Assignment 1

Put-call parity

After Lecture 05

Computer Assignment 1

Write a fast options pricer

After Lecture 08

Assignment 2

Hedging an options portfolio

After Lecture 10

Assignment 3

Arbitrage-free volatility surface

After Lecture 12

Final Exam

Cumulative course assessment

You can use LLMs and one prompt to solve the assignments. It’s up to you whether to do this or not. I would suggest not.

Expectations and schedule

Introduction to Financial Options / Lecture 01

It is expected that students will have a basic understanding of single-variable calculus and basic programming skills in Python.
It is not expected that students will have prior experience with financial markets or trading.
We will meet once a week for lectures, and once again for supervisions.

Non-goals

Introduction to Financial Options / Lecture 01

We will not teach you profitable trading strategies.

Making money in financial markets is extraordinarily difficult.
Course sellers, TikTok influencers, Zlatan Ibrahimovic, none of them know how.
They make money off of you, not from their (supposed) trading.

Since this is a serious course, no more talk about “how to make money trading”.
If this disappoints you then this isn’t the course for you.
If you want to learn the basics of financial markets the way pros do, then it is.

Agreed?

Now let’s define an option.

But first: what’s a stock?

Introduction to Financial Options / Lecture 01

Assets

What the company owns

Liabilities

What the company owes

Equity

What belongs to shareholders

Equity is the residual claim: what remains for shareholders after liabilities are paid.

Private company

OpenAI

Shares are held by private investors.

Initial public offering

SpaceX, 2026

The company sells shares to public investors.

Public company

Google (Alphabet)

Those shares trade between investors in financial markets.

A stock is a tradable unit of equity ownership in a company.

Illustrative order book

Best market: 100.05 bid / 100.06 ask

Bid size	Price	Ask size
	100.08	860
	100.07	425
	100.06	160
125	100.05
380	100.04
910	100.03

Shorting stock

Introduction to Financial Options / Lecture 01

We need one more concept before we proceed:

How do you sell something you don’t own?

You borrow it first.

Borrow

Borrow one share from its owner.

Short

Sell the borrowed share.

Time passes

The share price can move.

Cover short

Buy one share back.

Return borrow

Give the share back to its owner.

An option is the right but not the obligation to {buy|sell}.

Introduction to Financial Options / Lecture 01

A call gives its holder the right to buy the underlying at a fixed price.

A put gives its holder the right to sell the underlying at a fixed price.

Note that we said a right.
The holder chooses whether to exercise the right or not.
These are bilateral contracts between two people.

Parameters of an option

Introduction to Financial Options / Lecture 01

Lots of words in the definition.

What do they mean?

Underlying: One share of XYZ
Strike price, K: €100 per share
Expiry, T: Three months from today
Exercise style: European: at expiry only; American: any time before expiry
Settlement type (what changes hands): Physical: the actual underlying; Cash: A cash amount equivalent to the value

The holder chooses.

The writer must deliver if exercised.

At expiry, take the better of two choices.

Introduction to Financial Options / Lecture 01

C_T = \max(S_T-K,\,0)

Exercise

Buy at K, receive a share worth S_T.

Value: S_T-K

Let it expire

If buying is unattractive, do nothing.

Value: 0

With K=100: a share price of €112 gives a €12 payoff.

S_T: share price at expiry. Payoff excludes the premium paid.

Calls and puts respond in opposite directions.

Introduction to Financial Options / Lecture 01

Call · right to buy

C_T = \max(S_T-K,\,0)

Put · right to sell

P_T = \max(K-S_T,\,0)

Long positions · € per share · strike €100 · payoffs at expiry, before premium

Better terms cannot cost less

Introduction to Financial Options / Lecture 01

Calls

If K_L<K_H:

C(K_L) \ge C(K_H)

The lower strike buys the same asset for less.

Puts

If K_L<K_H:

P(K_L) \le P(K_H)

The higher strike sells the same asset for more.

If prices violate this ordering, buy the cheaper dominant option and sell the more expensive dominated option.

Same underlying, expiry, exercise style and settlement terms

A positive payoff can still mean a loss.

Introduction to Financial Options / Lecture 01

Long call · € per share · hold to expiry · ignore financing and transaction costs

Would you exercise, and did you make money?

Introduction to Financial Options / Lecture 01

You paid €6 for a call with strike €100. At expiry, the share is worth €103.

Think it through

Is exercise economically worthwhile?
What is the payoff?
What is the profit?

Yes. €3. −€3.

Exercise value: 103-100=3.

Profit: 3-6=-3.

Exercise recovers value even though the trade lost money.

€ per share · ignore financing and transaction costs

What about the other side?

Introduction to Financial Options / Lecture 01

So far, we have talked about holding an option. But every option holder bought the contract from someone else.

Long · holder

Pays the premium
Receives a right
Chooses whether to exercise

Short · writer

Receives the premium
Accepts an obligation
Must perform if assigned

Every long option is matched by a short option.

Before premiums and fees, their payoffs are equal and opposite.

Short calls and puts respond in opposite directions.

Introduction to Financial Options / Lecture 01

Short call · writer

C_T^{\text{short}} = -\max(S_T-K,\,0)

Short put · writer

P_T^{\text{short}} = -\max(K-S_T,\,0)

Short positions · € per share · strike €100 · payoffs at expiry, before premium

What happens when you’re short an option?

Introduction to Financial Options / Lecture 01

Assignment connects the holder’s exercise decision to a writer’s obligation.

Exercise

The long holder exercises. At expiry, an automatic procedure may exercise the option.

Assignment

The clearing system assigns an open short position; the broker allocates it to a short account.

Fulfilment

Short call: sell or deliver at K.

Short put: buy at K.

Once assigned, the writer cannot decline. Cash-settled options fulfil the obligation with a cash payment instead.

The history of options

Introduction to Financial Options / Lecture 01

1600s · Amsterdam

Puts and calls traded on shares of the Dutch East India Company.

1688 · Confusion of Confusions

Joseph de la Vega described premiums, exercise, puts and calls.

1900 · Louis Bachelier

Used Brownian motion to derive mathematical option prices in Théorie de la spéculation.

1973 · Chicago

CBOE and its clearing corporation opened a standardized listed market.

1973 · Black–Scholes–Merton

Connected option value to no-arbitrage and dynamic hedging.

2004 · VIX futures

Volatility becomes an asset class.

Amsterdam Stock Exchange, 1612 · Claes Jansz. Visscher · public domain

Options everywhere

Introduction to Financial Options / Lecture 01

Many ordinary contracts have an option-like payoff: pay something now to preserve a choice later.

Insurance

Pay a premium; claim only if something bad happens. Put-like protection.

Refundable booking

Pay more today; keep the right to cancel and recover value later.

Fixed-rate mortgage

The borrower can often prepay or refinance when rates fall. A call on the debt.

Reservation deposit

Pay a small amount now; decide later whether to complete a much larger purchase.

The option may be hidden inside another product. The logic is the same: right, not obligation.

Summary

Introduction to Financial Options / Lecture 01

An option gives its holder a right and its writer an obligation.
A call is the right to buy; a put is the right to sell.
Exercise depends on the payoff; profit also includes the premium.
If exercised, an open short position can be assigned and must perform.
Options have centuries of history.
Option-like choices are everywhere today.

Next lecture: how can we determine what an option is worth?