MATH4210 Financial Mathematics (2015-2016)
Tutorial 1
Goal of this course
1. Learn some basic concepts and techniques in Stochastic calculus (such as Itô’s Lemma)
2. Learn how to use Mathematics (Stochastic calculus) to price derivatives such as options
Derivatives
Derivatives are financial instrucments whose values depend on the values of other assets, which we call the underlying
assets, e.g., stocks, foreign currencies, etc..
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Forward/Future contracts
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Swaps
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Foreign currency options
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Stock index options
Derivatives
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 Interest rate options
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Options(European, American, . . . )
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Commodity options (Silver options)
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Stock options
Hedging using forward/future contracts
Suppose Mr Wong is performing a comedy show. One of his fans Jeffery gave him $ 1000 and ask for two tickets of
his comedy show in the next year.
Dayo Wong Chi-Wah stand-up comedy
What they are thinking is
1. Jeffery: It may be even more difficult to buy the tickets in the next year. The price of the tickets may rise as
well.
2. Mr Wong: I may not be attractive in the next year and the show may not be able to orgainze successfully.
Mr Wong accepted the $1000 and promised to give him two tickets IF he can really organize such show. In such
case, a forward contract is made. Mr Wong has an obligation to fulfill this forward/future contract.
Suppose we are in 2017 now, if Mr Wong is really on show,
1. The ticket price is more than $500, Jeffery is happy with the decision of accepting the contract.
2. The ticket price is less than $500, Jeffery may regret this hedging.
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Hedging using options
Options give you the right to buy/sell an asset in the future, i.e., Jeffery can choose whether to buy the two tickets
from Mr Wong with $500 or NOT.
1. The ticket price is more than $500. Jeffery wants to execite this right and buy two tickets.
2. The ticket price is less than $500. Jeffery can forget about this option and buy the two tickets directly in the
market.
Remark: Obviously Jeffery gets more freedom this time. Therefore Jeffery needs to spend MONEY to do the
hedging with options! The problem of determining how much an option costs is known as option pricing.
Lots of special terms
1. Derivatives: Holder/Buyer/Long position, Writer/Seller/Short position
2. Options: Put, Call, Expiration/Maturity date, Exercise/Strike price, European, American
The meaning of these terms...
1. Holder: One who owns an option contract. The option holder has the right, but not the obligation to buy/sell
the underlying asset.
2. Writer: One who originally sells an option contract. The option writer takes on an obligation to buy/sell the
underlying asset at the discretion of the option holder.
3. European option: Can only be exercised at the maturity/Expiration date.
4. American option: Can be exercised any time up to and including the maturity/Expiration date.
5. Put option: Give the holder the right to sell (or put) a fixed number of underlying assets at a fixed price
(Strike price) on or before the expiration date (European/American).
6. Call option: Give the holder the right (but not the obligation) to buy the underlying asset at a fixed price
(Strike price) on or before the expiration date (European/American).
Common notation in this course
1. S(t): the price of a stock at time t
2. T : The expiration date
3. E (or K): Strike price
4. c(t): the price of European call option at time t
5. p(t): the price of European put option at time t
6. C(t): the price of American call option at time t
7. P (t): the price of American put option at time t
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