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with imperfections’, involves a class of problems that we introduce in the next chapter. 15 Options in real markets In Chapter 14, we considered the option-pricing problem in ideal friction- less markets. Real markets are often eﬃcient, but they are never ideal. In this chapter, we discuss how the complexity of modeling ﬁnancial markets increases when we take into account aspects of real markets that are not formalized in the ideal model. These aspects are addressed in the literature as market microstructure [26] or market imperfections [127]. The terminology used in the economics literature suggests a clear parallel with similar scenarios observed in physical sciences. For example, it is much easier to construct a generalized description of the motion of a mechanical system in an idealized world without friction than in the real world. A similar situation is encountered when we compare equilibrium and non-equilibrium thermodynamics. In this chapter, we show that knowledge of the statistical properties of asset price dynamics is crucial for modeling real ﬁnancial markets. We also address some of the theoretical and practical problems that arise when we take market imperfections into account. 15.1 Discontinuous stock returns The existence of a portfolio containing both riskless and risky assets – replicating exactly the value of an option – is essential in determining the rational price of the option under the assumption that no arbitrage opportunities are present. Whether a portfolio is replicating or not depends on the statistical properties of the dynamics of the underlying asset. In the previous chapter, we saw that a replicating portfolio exists when the price of the underlying asset follows a geometric Brownian motion, but we also saw that this case cannot be generalized. For example, when the asset dynamic follows a jump-diﬀusion model [121], a simple replicating portfolio does not exist. A jump-diﬀusion model is a stochastic process composed of a diﬀusive 123 124 Options in real markets term (as in geometric Brownian motion) plus a second term describing jumps of random amplitudes occurring at random times. Roughly speaking, the presence of two independent sources of randomness in the asset price dynamics does not allow the building of a simple replicating portfolio.† It is not possible to obtain the rational price of an option just by assuming the absence of arbitrage opportunities. Other assumptions must be made concerning the risk aversion and price expectations of the traders. Taking a diﬀerent perspective, we can say that we need to know the statistical properties of a given asset’s dynamics before we can determine the rational price of an option issued on that asset. Discontinuity in the path of the asset’s price is only one of the ‘imperfections’ that can force us to look for less general option-pricing procedures. 15.2 Volatility in real markets Another ‘imperfection’ of real markets concerns the random character of the volatility of an asset price. The Black & Scholes option-pricing formula for an European option traded in an ideal market depends only on ﬁve parameters: (i) the stock price Y at time t, (ii) the strike price K , (iii) the interest rate r, (iv) the asset volatility rate σ, and (v) the maturity time T . Of these parameters, K and T are set by the kind of ﬁnancial contract issued, while Y and r are known from the market. Thus the only parameter that needs to be determined is the volatility rate σ. Note that the volatility rate needed in the Black & Scholes pricing formula is the volatility rate of the underlying security that will be observed in the future time interval spanning t = 0 and t = T . A similar statement can be made about the interest rate r, which may jump at future times. We know from the previous analysis that the volatility of security prices is a random process. Estimating volatility is not a straightforward procedure. 15.2.1 Historical volatility The ﬁrst approach is to determine the volatility from historical market data. Empirical tests show that such an estimate is aﬀected by the time interval used for the determination. One can argue that longer time intervals should provide better estimations. However, the local nonstationarity of the volatility versus time implies that unconditional volatility, estimated by using very long time periods, may be quite diﬀerent from the volatility observed in the lifetime of the option. † For a more rigorous discussion of this point, see [44, 70]. 15.2 Volatility in real markets 125 Fig. 15.1. Schematic illustration of the problems encountered in the determination of historical volatility. The nonstationary behavior of the volatility makes the deter- mination of the average volatility depend on the investigated period of time. Long periods of time are observed when the daily volatility is quite diﬀerent from the mean asymptotic value (solid line). An empirical rule states that the best estimate of volatility rate is obtained by considering historical data in a time interval t1 − t2 chosen to be as long as the time to maturity T of the option (Fig. 15.1). 15.2.2 Implied volatility A second, alternative approach to the determination of the volatility is to estimate the implied volatility σimp, which is determined starting from the options quoted in the market and using the Black & Scholes option-pricing formula (14.20). The implied volatility gives an indication about the level of volatility expected for the future by options traders. The value of σimp is obtained by using the market values of C(Y , t) and by solving numerically the equation C(Y ,T − t) = Y N(d1)−Ke−r(T−t)N(d2), (15.1) 126 Options in real markets Fig. 15.2. Schematic illustration of the implied volatility as a function of the diﬀer- ence between the strike price K and the stock price Y . The speciﬁc form shown is referred to as a volatility smile. where now the time is expressed in days from maturity, and d1 ≡ ln(Y /K) + (r + σ 2 imp/2)(T − t) σimp √ T − t , (15.2) and d2 ≡ d1 − σimp √ T − t. (15.3) In a Black & Scholes market, a determination of the implied volatility rate would give a constant value σ for options with diﬀerent strike prices and diﬀerent maturity. Moreover, the value of the implied volatility should coincide with the volatility obtained from historical data. In real markets, the two estimates, in general, do not coincide. Implied volatility provides a better estimate of σ. Empirical analysis shows that σimp is a function of the strike price and of the expiration date. Speciﬁcally, σimp is minimal when the strike price K is equal to the initial value of the stock price Y (‘at the money’), and increases for lower and higher strike prices. This phenomenon is often termed a ‘volatility smile’ (Fig. 15.2). The implied volatility increases when the maturity increases. These empirical ﬁndings conﬁrm that the Black & Scholes model relies on assumptions that are only partially veriﬁed in real ﬁnancial markets. When random volatility is present, it is generally not possible to determine the option price by simply assuming there are no arbitrage opportunities. In some models, for example, the market price of the volatility risk needs to be 15.4 Extension of the Black & Scholes model 127 speciﬁed before the partial diﬀerential equation of the option price can be obtained. 15.3 Hedging in real markets In idealized ﬁnancial markets, the strategy for perfectly hedging a portfolio consisting of both riskless and risky assets is known. In real markets, some facts make this strategy unrealistic: (i) the rebalancing of the hedged portfolio is not performed continuously; (ii) there are transaction costs in real markets; (iii) ﬁnancial assets are often traded in round lots of 100 and assume a degree of indivisibility. It has been shown that the presence of these unavoidable market imper- fections implies that a perfect hedging of