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a portfolio is not guaranteed in a real market, even if one assumes that the asset dynamics are well described by a geometric Brownian motion [58]. When we consider real markets, the complexity of the modeling grows, the number of assumptions increases, and the generality of the solutions diminishes. 15.4 Extension of the Black & Scholes model It is a common approach in science to use a model system to understand the basic aspect of a scientiﬁc problem. The idealized model is not able to describe all the occurrences observed in real systems, but is able to describe those that are most essential. As soon as the validity of the idealized model is assessed, extensions and generalizations of the model are attempted in order to better describe the real system under consideration. Some extensions do not change the nature of the solutions obtained using the model, but others do. The Black & Scholes model is one of the more successful idealized models currently in use. Since its introduction in 1973, a large amount of literature dealing with the extension of the Black & Scholes model has appeared. These extensions aim to relax assumptions that may not be realistic for real ﬁnancial markets. Examples include • option pricing with stochastic interest rate [4, 120]; • option pricing with a jump-diﬀusion/pure-jump stochastic process of stock price [13, 121]; • option pricing with a stochastic volatility [71, 72]; and • option pricing with non-Gaussian distributions of log prices [7, 21] and with a truncated Le´vy distribution [118]. 128 Options in real markets We will brieﬂy comment on general equations describing the time evolution of stock price and volatility [12] that is much more general than the Black & Scholes assumption of geometric Brownian motion. Our aim is to show how the complexity of equations increases when one or several of the Black & Scholes assumptions are relaxed. These general equations are dY (t) Y (t) = [r(t)− λµJ]dt+ σ(t)dWY (t) + J(t)dq(t) (15.4) and dσ2(t) = [θv −Kvσ2(t)]dt+ σvσ(t)dWv(t), (15.5) while the Black & Scholes assumption of geometric Brownian motion is, from (14.5), dY (t) Y (t) = µdt+ σdW (t); σ = const. (15.6) Here r(t) is the instantaneous spot interest rate, λ the frequency of jumps per year, σ2(t) the diﬀusion component of return variance, WY (t) and Wv(t) standard Wiener processes with covariance cov[dWY (t), dWv(t)] = ρdt, J(t) the percentage jump size with unconditional mean µJ , q(t) a Poisson process with intensity λ, and Kv, θv and σv parameters of the diﬀusion component of return variance σ2(t). It is worth pointing out that the increase in complexity is not only technical, but also conceptual. This is the case because the process is so general that it is no longer possible to build a simple replicating portfolio, or to perfectly hedge an ‘optimal’ portfolio. The elegance of the Black & Scholes solution is lost in real markets. 15.5 Summary Complete knowledge of statistical properties of asset return dynamics is essential for fundamental and applied reasons. Such knowledge is crucial for the building and testing of a statistical model of a ﬁnancial market. In spite of more than 50 years of eﬀort, this goal has not yet been achieved. The practical relevance of the resolution of the problem of the statistical properties of asset return dynamics is related to the optimal resolution of the rational pricing of an option. This is a ﬁnancial activity that is extremely important in present-day ﬁnancial markets. We saw that the dynamical properties of asset return dynamics – such as the continuous or discontinuous nature of its changes, the random character of its volatility, 15.5 Summary 129 and the knowledge of the pdf function of asset returns – need to be known in order to adequately pose, and possibly solve, the option-pricing problem. Statistical and theoretical physicists can contribute to the resolution of these scientiﬁc problems by sharing – with researchers in the other disciplines involved – the background in critical phenomena, disordered systems, scaling, and universality that has been developed over the last 30 years. Appendix A: Notation guide Chapter 1 x income of a given individual y number of people having income x or greater ν exponent of Pareto law xi random variable Sn sum of n random variables P (x) probability density function of the random variable x α index of the Le´vy stable distribution d dimension of a chaotic attractor ∼ symbol to denote asymptotic equality Chapter 2 t time Yt price of a ﬁnancial asset at time t E{x} expected value of the variable x E{x|y1, y2, y3, . . .} expected value of x conditional on the occurrence of y1, y2, y3, . . . K(n) bit length of the shortest computer program able to print a given string of length n Chapter 3 xi random variable n number of random variables Sn sum of n random variables E{f(x)} average value of f(x) δij Kronecker delta ∆t time step 130 Appendix A: Notation guide 131 Chapter 3 (cont.) x(n∆t) sum of n random variables, each one occurring after a time step ∆t s2 second moment of a dichotomic variable xi D diﬀusion constant⊗ convolution symbol σn standard deviation of Sn Ui truncated random variable � small number x˜ scaled variable P (x) probability density function P˜ (x˜) scaled probability density function π pi Fn(S) distribution function of a scaled S˜(n) Φ(S) distribution function of a Gaussian process Qj(S) polynomial encountered in convergence studies ri third moment of the absolute value of xi s2n sum of n variances σ 2 i Chapter 4 P (x) probability density function ϕ(q) characteristic function F[f(x)] operator indicating the Fourier transform of f(x) F(q) Fourier transform of f(x) ϕn(q) characteristic function of random variable Sn PL(x) symmetric Le´vy stable distribution γ scale factor of the Le´vy distribution µ average value of a random variable β asymmetry parameter of Le´vy distribution Γ(x) Gamma function S˜n scaled variable P˜ (S˜n) scaled probability density function ϕk(q) characteristic function of elementary random variable concurring to an inﬁnitely divisible random variable - length k integer number Z(t) price change at time t 132 Appendix A: Notation guide Chapter 5 Y (t) price of a ﬁnancial asset at time t Z(t) price change at time t ZD(t) deﬂated or discounted price change D(t) deﬂating or discounting time series R(t) return at time t S(t) successive diﬀerences of the natural logarithm of price Chapters 6 and 7 E{f(x)} expected value of f(x) f(x, t) probability density of observing x at time t f(x1, x2; t1, t2) joint probability density of observing x1 at time t1 and x2 at time t2 f(x1; t1|x2; t2) conditional probability density of observing x2 at t2 after observing x1 at t1 µ average value of the random process R(t1, t2) autocorrelation function τ ≡ t2 − t1 time lag C(t1, t2) autocovariance τc characteristic time τ0 time scale ν exponent η exponent τ∗ typical time σ2 variance f frequency S(f) power spectrum σ(t) standard deviation at diﬀerent time horizons Chapters 8 and 9 Y (t) price of a ﬁnancial asset at time t S(t) diﬀerence of the logarithm of price x random variable yi random variables Z random variable P (Z) probability density function n, k, m integers Cn constant Γ(x) Gamma function Ω(t) directing process Appendix A: Notation guide 133 Chapters 8 and 9 (cont.) PL(x) Le´vy distribution - truncation length PG(x) Gaussian distribution α index of the Le´vy distribution γ scale factor of the Le´vy distribution c constant n× number of i.i.d. variables needed to observe a crossover between Le´vy and Gaussian regimes S˜ scaled variable P˜ (S˜) scaled probability density function ϕ(q) characteristic function Z(t) index changes σ standard deviation gi normalized diﬀerence of the logarithm of price of company i F(g) cumulative distribution σi volatility (standard deviation) of company i Chapter