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# lec_smry1 - Brief Summary of Math Finance 1 Fundamentals...

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Brief Summary of Math Finance 1 Fundamentals Stock markets around the world sell a great variety of different products. These include shares in companies, commodities, futures, currencies and options. We will consider a simplified market without commodities or futures and only one currency. A dominating factor in markets is the risk aversion of investors. This means that a key feature of any market model is the risk-free interest rate . We will always assume that cash H.2 can earn interest at some rate r without risk (many texts describe this by the purchase or sale of bonds ). For any sequence of cash flows x i at time t i the net present value of the cash flow is v ( x, t ) = summationdisplay i x i (1 + r ) t i when the times t i are measured in multiples of the compounding period . This becomes v ( x, t ) = i x i e rt i if interest is compounded continuously. The observed behaviour of stocks. Price changes of shares are largely unpredictable and the study of many sequences for many different shares has lead to the observation that the quantities S ( t + δt ) S ( t ) S ( t ) are approximately Normally distributed for a wide range of values of ∂t and that changes over non-overlapping time intervals appear to be independent. Portfolios We will describe our assets at any time as a portfolio . It consists of the shares, options and cash we have at that time and we suppose that any of these can be negative. For shares and options this can be achieved by selling short i.e. selling things you haven’t got! When you do this you will be required to buy the relevant stocks or shares at some future date at the market price to honour your short sale. Arbitrage When it is possible to assemble a portfolio that with certainty returns more than the risk free interest rate then we say that an arbitrage opportunity exists. Definition 1.1 A European call option ( put option ) gives its holder the right to buy H.1 from ( sell to ) the writer an asset at a nominated price at a specified future time (but the holder need not exercise the option). The nominated price is often called the exercise price or strike price while the specified time is the expiry date . A European call option returns max( S ( T ) K, 0) ( S ( T ) K ) + at the expiry date T . The main aim of the course this term is to explore the pricing of European call options in terms of the strike price K , expiry date T , the interest rate and any components of the stock price model which might be relevant. The essential aspect of pricing options is that they should not permit arbitrage opportunities.

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