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0809Karlsen - Karlsen K.H 2010 “Pricing of financial...

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PRICING OF FINANCIAL DERIVATIVES KENNETH H. KARLSEN 1. Introduction A financial derivative, for example an option, is an instrument (contract) whose value depends on the values of some underlying variables, where the underlying can be a commodity, an interest rate, stock, a stock index, a currency, to mention just a few examples. The financial derivatives market is enormous and is regularly reported to be worth $500 trillion. Derivatives provide an efficient way to reduce risk related to changes in the value of the underlying variable. Once a financial derivative is defined the first question is the following: “what is the fair price that the seller of the derivative should charge to the buyer?” Eventually this question is answered, at least for options, by the so-called Black and Scholes formula. The purpose of this short note is to display in very simplified contexts the main underlying argument leading up to the fair price of a derivative, namely that markets eliminate any opportunity for risk-free profits (the principle of no arbitrage). 2. Forward (futures) contracts Forward contracts amount to the simplest example of a financial derivative. A forward contract is an agreement between two parties in which one party agrees to buy or sell an asset at a fixed time in the future for a particular price that they agree upon today. The price that the underlying asset is bought or sold for is called the delivery price. This price must be “fair”, i.e., it must be chosen so that the value of the contract to both parties is zero at the onset. This is the principle of “no arbitrage” (arbitrage means taking advantage of price differences in two markets). Let us illustrate by an example. Let us assume that I enter into a forward in which I agree to sell 100 ounces of gold at $300 per ounce in one year’s time. The payoff from this contract depends on the actual price of gold on the delivery date. If the price of a gold is $350 at the end of the year, I make a loss on the transaction because I have to buy 100 ounces of gold at $350 per ounce to meet my obligation to deliver the gold (assuming I do not own the gold myself). On delivery I receive $300 per ounce, a total loss of $5000 plus the delivery costs. On the other hand if the gold price went down to $250 per ounce, I make a profit of $5000. In general, if we denote the delivery price by K and S is the current price of gold at the time of delivery (or any other underlying asset) the payoff is K - S per ounce of gold if I have agreed to sell gold at a fixed price, in which case I have taken a “short position” in gold. Similarly, the payoff is S - K per ounce of gold if I have agreed to buy gold at a fixed price, in which case I have taken a “long position” in gold, cf. Figure 1 (Top-left) for (short and long) payoff diagrams.
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