High School Mathematics Extensions/Financial Options

High School Mathematics Extensions

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Supplementary Chapters — Primes and Modular Arithmetic — Logic

Mathematical Proofs — Set Theory and Infinite Processes — Counting and Generating Functions — Discrete Probability

Matrices — Further Modular Arithmetic — Mathematical Programming — Markov Chains

Binary tree option pricing

Introduction

We have all heard of at least one stock exchange. NASDAQ, Dow Jones, FTSE and Hang Sheng. Less well-known, but more useful to many people, are the futures exchanges. A stock exchange allows stock brokers, also known as investment advisors, to trade company stocks, while futures exchanges allow more exotic derivatives to be traded. For example, financial options, which is also the focus of this chapter.

An option is a contract that gives the holder the choice to buy (or sell) a certain good in some time in the future for a certain price. What are options for? Initially, they are used to protect against risk. But they are also used to take advantage foreseeable opportunities, like what Thales has done¹.

Thales, the great Greek philosopher, was credited with the first recorded use of an option in the western world. A popular anecdote suggests, in one particular year while still in winter, he forecasted a great harvest of olives in the coming year. He had next to no money, so he purchased the option for the use of all the olive presses in his area. Naturally, when the time to harvest came everyone wanted to use the presses he had optioned! Needless to say he made a lot of money out of it.

Basics

An option is a contract of choice. You can choose whether to exercise the option or not.

If you own an option that states

You may purchase 1 kg of sugar from Shop A tomorrow for $2

suppose tomorrow the market price of sugar is $3, you would want to exercise the option i.e. buy the sugar for $2. Then you would sell it for $3 on the market and make $1 in the process. But if the market price for 1 kg of sugar is $1, then you would choose not to exercise the option, because it's cheaper on the market.

Let us be a little bit more formal about what an option is. In particular there are two types of option:

Call Option

A call option is a contract that gives the owner the option to buy an 'underlying stock' at the 'strike price', on the 'expiry date'.

Put Option

A put option is a contract that gives the owner the option to sell an 'underlying stock' at the 'strike price', on the 'expiry date'.

In the above example, the 'underlying stock' was sugar and the 'strike price' $2 and 'expiry date' is tomorrow.

We shall represent an option like below

{C or P, $amount, # periods to expiry}

. For example

{C,$3,1}

represents a call option with strike price for some unspecified underlying stock expiring in one time-unit's time. A time-unit here may be a year, a month, one day or one hour. The important point is the mathematics we will present later does not really depend on what this time-unit is. Also, we need not specify the underlying stock either. Another example

{P,$100,2}

represents a put option with strike price $100 for some unspecified underlying stock expiring in 2 time-units' time.

Now that we have a basic idea of what an option is, we can start to imagine a market place where options are traded. We assume that such a market exists. Also we assume that there is no fee of any kind to participate in a trade. Such a market is called a frictionless market. Of course, a market place where the underlying stock is traded is also assumed to exist.

Info -- American or European

Actually there are two major types of options: American or European. An European option allows you to exercise the option only on the 'expiry date'; while the American version allows you to exercise the option at any time prior to the 'expiry date'. We shall only discuss European options in this chapter.

Arbitrage

Another very important concept is arbitrage. In short, an arbitrage is a way to make money out of nothing. We assume that there is no free-lunch in this world, in other words our market is arbitrage-free. We will show an example of how to perform an arbitrage later on in the chapter.

The real meat of this chapter is the technique used to price the options. In simple terms, we have an option, how much should it be? From this angle, we will see that the arbitrage-free requirement is a very strong one, in that it basically dictates what the price of the option should be.

Option's value on expiry

Pricing the option is about how much is it worth now. Of course the present value of an option depends on its possible future values. Therefore it is vital to understand how much the option is worth at expiry, when it is time to choose whether to exercise the option or not. For example, consider the option

{C,$2,1}

it is the call (buy) option that expires in one week's time (or day or year or whatever period it is supposed to be). How much should the option be if the market price of the underlying stock on expiry is $3? What if the market price is $1?

It is sensible to say the option has a value of $1 if the market price (for the underlying stock) is $3, and the option should be worthless ($0) if the market price is $1.

Why do we say it is sensible to price the option as above? It is because we assume the market is arbitrage-free. Also in a market, we assume

• there is a bank that's willing to lend you money
• if you repay the bank in the same day you borrowed, no fee will be charged.

With those assumptions, we show that if you price the option any differently, someone can make money without using any of his/her own money. For example, suppose on expiry, the market price for the underlying stock is $3 and you decide to sell the option for $0.7 (not $1 as is sensible). An intelligent buyer would do the following:

Action	Money	Balance
Borrow $2.7	+$2.7	$2.7
Purchase your option for $0.7	-$0.7	$2
Purchase sugar for $2 with option	-$2	$0
Sell 1kg of sugar for $3 in market	+$3	$3
Repay bank $2.7	-$2.7	$0.3

He/she made $0.3 and at no time did he/she use his/her own money (i.e. balance never less than zero)! This is a free lunch, which is contrary to the assumption of a arbitrage-free market!

Exercises

1. In an arbitrage-free market, consider an option T = {C,$100,1}.

i) How much should the option be on expiry if the price of the underlying stock is $90.

ii) What if the underlying stock costs $110 on expiry.

iii} $100?

2. Consider an option T = {C,$10,1}.

i) On expiry, would you consider buying the option if it was for sell for $2 if the underlying stock costs $12?

ii) What if the underlying stock costs $13.

3. Consider the put option T = {P,$2,1}. On expiry the underlying stock costs $1. Jenny owns T, she decides on the following actions

Borrow $1

Purchase the underlying stock from the market for $1

Exercise the option i.e. sell the stock for $2

Repay $1

Did she do the right thing?

4. In an arbitrage-free market, consider the put option T = {P,$2,1}.

i) On expiry, how much should the option cost if the underlying stock costs $1?

ii) $3?

5. Consider the put option T = {P,$2,1}. On expiry the underlying stock costs $1. And the option T is on sale for $0.5. Jenny immediately sees an arbitrage opportunity. Detail the actions she should take to capitalise on the arbitrage opportunity. (Hint: imitate the Action, Money, Balance table )

Pricing an option

Consider this hypothetical situation where a company, MassiveSoft, is in negotiation to merge with another company, Pears. The share price of MassiveSoft currently stands at $7. If the negotiation is successful, the share price will rise to $11; otherwise it will fall to $5. Experts predict the probability of a success is 90%. Consider a call option that lets you buy 1000 shares of MassiveSoft at $8 when the negotiation is finalised. How much should the option be?

Since the market is arbitrage-free, the value of the option at expiry is already determined. Of course

if the negotiation is successful, the option is valued at (11 - 8) × 1000 = $3000

otherwise, the option should be worthless ($0)

the above are the only correct values of the option at expiry or people can "rip you off".

Let x be the price of the option at present, we can use the following diagrams to illustrate the situation,

	$\nearrow$	$\$3000\!$
$\$x\!$
	$\searrow$	$\$0\!$

the diagram shows that the current price of the option should be $x, and if the negotiation is successful, it will be worth $3000, otherwise it is worthless. In similar fashion, the following diagram shows the value of the company stock now, and in the future

	$\nearrow$	$\$11\!$
$\$7\!$
	$\searrow$	$\$5\!$

You may have notice that we didn't put down the probability of success or failure. Interestingly (and counter-intuitively), they don't matter! Again, the arbitrage-free principle dictates that what we have in the two diagrams above are sufficient for us to price the option!

How?

What is the option? It is the contract that gives you the option to buy ... Wait, wait, wait. Think of it from another angle

it is a tradable object that is worth $3000 if the negotiation is successful, and $0 if otherwise

This is the main idea behind how to price the option. The option must be the same price as another object that goes up to $3000 or down to $0 depending on the success of the negotiation. Hopefully, this object is something we know the price of. This idea is called constructing a replicating portfolio.

A portfolio is a collection of tradable things. We want to construct a portfolio that behaves in the same way as the option. It turns out that we can construct a portfolio that behaves in the way as the option by using only two things. They are

1. MassiveSoft shares
2. and money

let's assume that money is tradable in the sense that you can buy a dollar with a dollar. This concept may seem very unintuitive at first. However let's proceed with the mathematics, suppose this portfolio consists of y units of MassiveSoft shares and z units of money. If the negotiation is successful, then each share will be worth $11, and the whole portfolio should be worth $3000, as it behaves in the same way as the option, so we have the following

$11y+z=3000\!$

but if the negotiation is unsuccessful then the portfolio is worthless ($0) and MassiveSoft share prices will fall to $5, giving

$5y+z=0\!$

we can easily solve the above simultaneous equations. We get

$6y=3000\!$

and so

$y=500$ and $z=-\$2500$

So this portfolio consists of 500 MassiveSoft shares and -$2500. But what is -$2500? This can be understood as an obligation to pay back some money (e.g. from borrowings) on the expiry date of the option. So the portfolio we constructed can be thought of as

500 MassiveSoft shares and an obligation to pay $2500

Now, 500 MassiveSoft shares costs $7 × 500 = 3500, so the option should be priced as 3500 - 2500 = $1000.

Let's price a few more options.

...

The famous mathematician John Nash, as portrayed in the movie A Beautiful Mind, did some pioneering work in portfolio theory with equivalent functions.

Call-Put parity

...more to come

Reference

1. A Brief History of Options

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