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Lecture Notes for MA 5182:
Mathematical Methods in Finance

Professor Qiang Zhang
City University of Hong Kong

Reference book: Martin Baxter and Andrew Rennie, Financial Calculus, published by Cambridge University Press

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1. Contract:
An agreement to buy an underlying asset at a specific price on a specified date. specified price - strike price, exercise price specified date - maturity price
2. Options:
(i) Call option: a right for the holder of option to buy an underlying asset at a specified price on a specified date. (ii) Put option: a right for the holder of option to sell an underlying asset at a specified price on a specified date. long position: the person has the right to exercise or not. short position: the person has the obligation.

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3. Difference between the contracts and options:
(i) At the beginning of contract, the value of the contract is zero. This is due to the fact that one can adjust the strike price to make the contract value to be zero.
The option has nonzero value even at the beginning of the option.
(ii) Contract is a binding. It must be exercised on the maturity date. The option holder has the choice whether to exercise the option.
The main focus of the option pricing theory:
(i) pricing - to determine the value of the option.
(ii) hedging - to reduce risk.

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Types of the options:
European Options: They can only be exercised on the maturity date.
American Options: They can be exercised any time before and on the maturity date.

Coin f lipping game:
The game pays $1 if head is up, and nothing if tail is up. How much one should pay to play such a game?
Binary tree model for options:

V0 = e−rt[(1 − p)V+ + pV−]

Is this formula correct?

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No Arbitrage Opportunity(principle):
Arbitrage Opportunity: A strategy, that allows one to invest zero money, but has nonzero probability