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160a_finance

# 160a_finance - Winter 2010 PSTAT 160a Introduction to...

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Winter 2010 PSTAT 160a Introduction to Discrete time Math Finance I. Introduction This is a very brief introduction to one of the main problem in Finance, namely Option Pricing. This problem is interesting because it is a typical example of risk management, so it is also closely related to insurance. To understand what this is about, let’s look at 2 examples: Example 1: Imagine that GM, a US car maker, is introducing a new model in the European market. The company has to decide how much to charge for the car, given production costs and currency exchange rates. However, cars are going to be sold in e Euro in future, and the costs are fixed in \$ . Exchange rates fluctuate, and the company can’t be overly conservative if it wants their selling price to be competitive. Suppose now the rate is \$1 for e 1 and the car price is set for \$10,000= e 10,000. Suppose the Euro decrease by 20% in the future so the rate is then \$.8 for e 1. The car price is still e 10,000 but GM gets only \$ 8,000 and that amount may not cover the manufacturing costs. Example 2: A power plant in California is producing electricity from natural gas. Gas is bought on a monthly basis, and its price fluctuates randomly. Let us first assume that the company is selling its electricity at a price fixed at the beginning of the year. If gas price goes down, the company is quite happy, but if it goes up, it may lose a lot of money as a large part of its cost comes from gas, and its revenues are fixed. The company may want to protect itself from this risk which is similar as in our first example. Typically, a significant portion of the electricity is sold on short term contracts. In that case, the company may want to be protected against large increase in the difference between gas price and electricity price. II. Definitions from Finance An asset is a product that can be traded on the market. It includes: Bonds with interest rate r . It could be a government bond or a bank saving account. In both cases, \$1 invested yields \$ e rT after T unit of time. For simplicity, we consider the market to be completely symmetric, so that anyone can borrow with this same rate r . The discounted value at time 0 of an asset worth \$1 at time T is then \$ e - rT . Stocks: This includes shares of a company, commodities such as gas or electricity, currencies, etc... We will denote stock prices at time n by S n . Contingent claims: (or just claim for short) also called derivatives . They are contracts on some underlying assets, and of course the holder has to pay a price for this contract. One of the main problem in finance is to find this price. Options: It is a special class of contingent claims that gives the right to exercise the contract. A call option give the right to buy the underlined asset at a fixed strike price K , at or before the maturity time T , a put option gives the right to sell.

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