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2.4.4 Arbitrage Pricing Theory Empirical examinations of the CAPM showed significant deficiencies in its ability to forecast and alleviate risk. These studies led to the development of the arbitrage pricing theory (APT), first introduced by Ross37 and further developed by other scientists. APT is based on the empirical observation that different instruments have simultaneous and homogeneous development ranges. The theory implicitly assumes that the returns are linked to a certain number of factors which influence the instrument prices. The part explained by these factors is assigned to the systematic factors, whereas the nonexplainable part of the return (and thus the risk) is assigned to specific factors. In theory, the factors are uncorrelated, as empirical examination supports correlated factors. Such correlated factors have to be transformed into an observation-equivalent model with uncorrelated factors. The factors cannot be observed and have to be examined empirically. A critical difference between CAPM and APT is that APT is an equilibrium theory, based on the arbitrage condition. As long as it is possible with intensive research to find factors that systematically impact the return of a position, it is possible to do arbitrage based on this superior knowledge.
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Modern portfolio theory is not based solely on return calculations. Risk and risk management become increasingly important. As the portfolio theory shows, despite diversification, an investor is still exposed to systematic risk. With the development of portfolio and position insurance, an approach has been created to hedge (insure) against unwanted moves of the underlying position. The theoretical framework introduced a range of applications, such as replication of indices, dynamic insurance, leveraging, immunization, structured products, etc. To understand the current state of option pricing, the different approaches, and the critics, it is necessary to summarize the development of, and approaches to, modern option-valuation theory. Valuation and pricing of income streams is one of the central problems of finance. The issue seems straightforward conceptually, as it amounts to identifying the amount and the timing of the cash flows expected from holding the claims and then discounting them back to the present. Valuation of a European-style call option requires that the mean of the call option s payout distribution on the expiration date be estimated, and the discount rate be applied to the option s expected terminal payout. The first documented attempt to value a call option occurred near the turn of the twentieth century. Bachelier wrote in his 1900 thesis that the call option can be valued under the assumption that the underlying claim follows an arithmetic Brownian motion.38 Sprenkle and Samuelson used a geometric Brownian motion in their attempt to value options.39 As the underlying asset prices have multiplicative, rather than additive (as with the arithmetic motion) fluctuations, the asset price distribution at the expiration date is lognormal, rather then normal. Sprenkle and Samuelson s research set the stage, but there was still a problem. Specifically, for implementation of their approach, the risk-adjusted rates of price appreciation for both the asset and the option are required. Precise estimation was the problem, which was made more difficult as the option s return depends on the asset s return, and the passage of time. The breakthrough came in 1973 with Black, Scholes, and Merton.40 They showed that as long as a risk-free hedge may be formed between the option and its underlying asset, the value of an option relative to the asset will be the same for all investors, regardless of their risk preferences. The argument of the risk-free hedge is convincing, because in equilibrium, no arbitrage opportunities can exist, and any arbitrage opportunity is obvious for all market participants and will be eliminated. If the observed price of the call is above (or below) its theoretical price, risk-free arbitrage profits are possible by selling the call and buying (or selling) a portfolio consisting of a long position in a half unit of the asset, and a short position in the other half in risk-free bonds. In equilibrium, no arbitrage opportunities can exist, and any arbitrage opportunity can exist.
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2.4.5.1 Analytical Formulas The option valuation theory goes beyond the mathematical part of the formula. The economic insight is that if a risk-free hedge between the option and its underlying asset my be formed, risk-neutral valuation may be applied. The Black-Scholes model follows the work of Sprenkle and Samuelson. In a risk-neutral market, all assets (and options) have an expected rate of return equal to the risk-free interest rate. Not all assets have the same expected rate of price appreciation. Some assets, such as bonds, have coupons, and equities have dividends. If the asset s income is modeled as a constant and continuous proportion of the asset price, the expected rate of price appreciation on the asset equals the interest rate less the cash disbursement rate. The Black-Scholes formula covers a wide range of underlying assets. The distinction between the valuation problems described as follows rests in the asset s risk-neutral price appreciation parameter:
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Non-dividend-paying stock options. The best-known option valuation problem is that of valuing options on non-dividendpaying stocks. This is, in fact, the valuation problem addressed by Black and Scholes in 1973.41 With no dividends paid on the underlying stock, the expected price appreciation rate of the stock equals the risk-free rate of interest, and the call option valuation equation becomes the familiar Black-Scholes formula. Constant-dividend-yield stock options. Merton generalized stock option valuation in 1973 by assuming that stocks pay dividends at a constant, continuous dividend yield.42 Futures options. Black valued options on futures in 1976.43 In a risk-neutral world with constant interest rates, the expected rate of price appreciation on a futures contract is zero, because it involves no cash outlay. Futures-style futures options. Following the work of Black, Asay valued futures-style futures options.44 Such options, traded on various exchanges, have the distinguishing feature that the option premium is not paid up front. Instead, the option position is marked to market in the same manner as the underlying futures contract. Foreign currency options. Garman and Kohlhagen valued options on foreign currency in 1983.45 The expected rate of price appreciation of a foreign currency equals the domestic rate of interest less the foreign interest. Dynamic portfolio insurance. Dynamic replication is at the heart of one of the most popular financial products of the 1980s dynamic portfolio insurance. Because long-term index put options were not traded at the time, stock portfolio managers had
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