Wiener process

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Mathematical Finance: Core Theory, Problems and Statistical Algorithms by Nikolai Dokuchaev

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Definition 4.5 We say that a continuous time process w(t) = (w1(t),…, wn(t)): [0, +∞)×Ω→Rn is a (standard) n-dimensional Wiener process if (i) wi(t) is a (one-dimensional) Wiener process for any i=1,…, n; (ii) the processes {wi(t)} are mutually independent. Remark 4.6 Let be a matrix such that Then the process is also said to be a Wiener process (but not a standard Wiener process, since it has correlated components). We shall omit the word ‘standard’ below; all Wiener processes in this book are assumed to be standard. For simplicity, one can assume for the first reading that n=1, and all processes used in this chapter are one-dimensional. After that, one can read this chapter again taking into account the general case. Proposition 4.7 A Wiener process is a Markov process. be the filtration Proof. We consider an n-dimensional Wiener process w(t). Let generated by w(t). We have that w(t+τ)=w(t+τ)−w(t)+w(t).

© 2007 Nikolai Dokuchaev 4 Basics of Ito calculus and stochastic analysis This chapter introduces the stochastic integral, stochastic differential equations, and core results of Ito calculus. 4.1 Wiener process (Brownian motion) Let T>0 be given, Definition 4.1 We say that a continuous time random process w(t) is a (onedimensional) Wiener process (or Brownian motion) if (i) w(0)=0; (ii) w(t) is Gaussian with Ew(t)=0, Ew(t)2=t, i.e., w(t) is distributed as N(0, t); (iii) w(t+τ)−w(t) does not depend on {w(s), s≤t} for all t≥0, τ>0. Theorem 4.2 (N. Wiener). There exists a probability space such that there exists a pathwise continuous process with these properties. This is why we call it the Wiener process. The corresponding set Ω in Wiener’s proof of this theorem is the set C(0, T). Remember that C(0, T) denotes the set of all continuous functions f:[0, T]→R. Corollary 4.3 Let ∆t>0, Corollary 4.4 then Var ∆w=∆t. This can be interpreted as This means that a Wiener process cannot have pathwise differentiable trajectories.

Clearly, w(s) is measurable and does not depend on For any bounded measurable function F:Rnk→R, we have that for some measurable functions F1:Rnk→R and F2:Rn→R. It follows that w(s) is a Markov process. Proposition 4.8 Let be a filtration such that an n-dimensional Wiener process w(t) and w(t+τ)−w(t) does not depend on Then w(t) is a martingale with is adapted to respect to Proof. We have that w(t+τ)=w(t+τ)−w(t)+w(t). Hence since w(t+τ)−w(t) does not depend on © 2007 Nikolai Dokuchaev Therefore, the martingale property holds. Basics of Ito Calculus and Stochastic Analysis 51 Corollary 4.9 A Wiener process w(t) is a martingale. (In other words, if is the ) filtration generated by w(t), then w(t) is a martingale with respect to Up to the end of this chapter, we assume that we are given an n-dimensional Wiener process w(t) and the filtration such as described in Proposition 4.8. One may assume that this filtration is generated by the process (w(t), η(t)), where η(S) is a process where T>0 is given deterministic independent from w(·).

Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues by Alain Ruttiens

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The drift can be defined as the instantaneous expected value of change in (t) per time unit and the volatility is the instantaneous STD of change in (t) per time unit. It follows that the expected value and variance of the general Wiener process are Finally, applying the Eqs. 8.3–8.5 to (t), we obtain: (8.9) 8.4 THE ITÔ PROCESS The most generalized form of the general Wiener process is called the Itô process, where the drift and volatility are functions of the stochastic variable and of the time: (8.10) So that the expected value and the variance of the Itô process are also functions of (t) and t. 8.5 APPLICATION OF THE GENERAL WIENER PROCESS General Wiener processes4 are widely used to describe the behavior of financial products. In practice, we actually model the returns (rather than the prices) of financial time series by a general Wiener process. Long time series can indeed present large prices variations, while returns are more stable over time, as can be viewed in the example next.

With respect to the expected value of the product of (t) at two different points of time t1 and t2, (8.5) Finally, the product of two different standard Wiener processes Z1 and Z2, is not random: (8.6) where ρ1, 2 is the correlation coefficient between the two processes. These relationships constitute the core of the stochastic calculus, together with the more general hypothesis that (8.7) as a reasonable assertion. A more general diffusion process is: 8.3 THE GENERAL WIENER PROCESS This process describes a random variable combining a deterministic process – the μdt term – with a standard Wiener process in d (8.8) In the general Wiener process, the μ and σ coefficients are posited constant and are called the drift and the volatility of the process. The drift can be defined as the instantaneous expected value of change in (t) per time unit and the volatility is the instantaneous STD of change in (t) per time unit.

Any other value for μ implies some degree of risk aversion, which can be defined as (8.15) This measure recalls – despite somewhat different notations – the price of risk measure within the framework of the CAPM (cf. Chapter 4, Section 4.3.4, Eq. 4.4). 8.9 NOTION OF MARTINGALE The geometric Wiener process applied on the returns of a stock price S (Eq. 8.1), has been built by using a physical probability measure, given the μ drift, associated with the stochastic standard Wiener process dZ. By assuming μ = r, this equation can be rewritten with the risk neutral probability measure, called Q. Defining dZQ as we obtain (8.16) that is, a geometric Wiener process involving a standard Wiener process under Q8. Integrating Eq. 8.16 in the same manner as in Section 8.7, instead of obtaining (Eq. 8.14) we obtain a similar relationship, but with r instead of μ: Therefore, by integration, the realization of S(t) on T will involve a (log-normal) risk neutral probability distribution, centered on the objective, “neutral” value of r, instead of on an arbitrary value μ.

Tools for Computational Finance by Rüdiger Seydel

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For a given realization of the Wiener process Wt we obtain as solution a trajectory (sample path) Xt . For another realization of the Wiener process the same theoretical solution (3.2) takes other values. If a Wiener process Wt is given, we call a solution Xt of the SDE a strong solution. In this sense the solution in (3.2) is a strong solution. If one is free to select a Wiener process, then a solution of the SDE is called weak solution. For a weak solution, only the distribution of X is of interest, not its path. Assuming an identical sample path of a Wiener process for the SDE and for the numerical approximation, a pathwise comparison of the trajectories 3.1 Approximation Error 93 Xt of (3.2) and y from (3.1) is possible for tj . For example, for tm = T the absolute error for a given Wiener process is |XT − yT |.

Markov process: Only the present value of Xt is relevant for its future motion. That is, the past history is fully reﬂected in the present value.4 An example of a process that is both Gaussian and Markov, is the Wiener process. 4 This assumption together with the assumption of an immediate reaction of the market to arriving informations are called hypothesis of the eﬃcient market [Bo98]. 26 Chapter 1 Modeling Tools for Financial Options 11500 11000 10500 10000 9500 9000 8500 8000 7500 7000 0 50 100 150 200 250 300 350 400 450 500 Fig. 1.14. The Dow at 500 trading days from September 8, 1997 through August 31, 1999 1.6.1 Wiener Process Deﬁnition 1.7 (Wiener process, Brownian motion) A Wiener process (or Brownian motion; notation Wt or W ) is a timecontinuous process with the properties (a) W0 = 0 (with probability one) (b) Wt ∼ N (0, t) for all t ≥ 0.

The relation (1.21b) is also known as 1.6 Stochastic Processes E((∆Wt )2 ) = ∆t . 27 (1.21c) The independence of the increments according to Deﬁnition 1.7(c) implies for tj+1 > tj the independence of Wtj and (Wtj+1 − Wtj ), but not of Wtj+1 and (Wtj+1 − Wtj ). Wiener processes are examples of martingales — there is no drift. Discrete-Time Model Let ∆t > 0 be a constant time increment. For the discrete instances tj := j∆t the value Wt can be written as a sum of increments ∆Wk , Wj∆t = j Wk∆t − W(k−1)∆t . k=1 =:∆Wk The ∆Wk are independent and because of (1.21) normally distributed with Var(∆Wk ) = ∆t. Increments ∆W with such a distribution can be calculated from standard normally distributed random numbers Z. The implication √ Z ∼ N (0, 1) =⇒ Z · ∆t ∼ N (0, ∆t) leads to the discrete model of a Wiener process √ ∆Wk = Z ∆t for Z ∼ N (0, 1) for each k . (1.22) We summarize the numerical simulation of a Wiener process as follows: Algorithm 1.8 (simulation of a Wiener process) Start: t0 = 0, W0 = 0; ∆t loop j = 1, 2, ... : tj = tj−1 + ∆t draw Z ∼ N (0, 1) √ Wj = Wj−1 + Z ∆t The drawing of Z —that is, the calculation of Z ∼ N (0, 1)— will be explained in Chapter 2.

Analysis of Financial Time Series by Ruey S. Tsay

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In the literature, some authors use x(t) instead of xt to emphasize that t is continuous. However, we use the same notation xt , but call it a continuous-time stochastic process. 6.2.1 The Wiener Process In a discrete-time econometric model, we assume that the shocks form a white noise process, which is not predictable. What is the counterpart of shocks in a continuoustime model? The answer is the increments of a Wiener process, which is also known as a standard Brownian motion. There are many ways to define a Wiener process {wt }. We use a simple approach that focuses on the small change wt = wt+ t − wt associated with a small increment t in time. A continuous-time stochastic process {wt } is a Wiener process if it satisfies √ 1. wt = t, where is a standard normal random variable, and 2. wt is independent of w j for all j ≤ t. The second condition is a Markov property saying that conditional on the present value wt , any past information of the process, w j with j < t, is irrelevant to the future wt+ with > 0.

Because i ’s are independent, we have E(wt − w0 ) = 0, Var(wt − w0 ) = T t = T t = t. i=1 Thus, the increment in wt from time 0 to time t is normally distributed with mean zero and variance t. To put it formally, for a Wiener process wt , we have that wt − w0 ∼ N (0, t). This says that the variance of a Wiener process increases linearly with the length of time interval. CONTINUOUS - TIME MODELS -0.4 w -1.5 -1.0 -0.5 0.0 w 0.0 0.2 0.4 0.6 0.5 224 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.6 0.8 1.0 time -0.6 -0.4 w -0.2 0.2 w 0.0 0.2 0.4 time 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 time time Figure 6.1. Four simulated Wiener processes. Figure 6.1 shows four simulated Wiener processes on the unit time interval [0, 1]. They are obtained by using a simple version of the Donsker’s Theorem in the statistical literature with n = 3000; see Donsker (1951) or Billingsley (1968).

This is the purpose of discussing Ito’s calculus in the next section. 6.2.2 Generalized Wiener Processes The Wiener process is a special stochastic process with zero drift and variance proportional to the length of time interval. This means that the rate of change in expectation is zero and the rate of change in variance is 1. In practice, the mean and variance of a stochastic process can evolve over time in a more complicated manner. Hence, further generalization of stochastic process is needed. To this end, we consider the generalized Wiener process in which the expectation has a drift rate µ and the rate of variance change is σ 2 . Denote such a process by xt and use the notation dy for a small change in the variable y. Then the model for xt is d xt = µ dt + σ dwt , (6.1) where wt is a Wiener process. If we consider a discretized version of Eq. (6.1), then √ xt − x0 = µt + σ t for increment from 0 to t.

Mathematics for Finance: An Introduction to Financial Engineering by Marek Capinski, Tomasz Zastawniak

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As a result, the American call is equivalent to its European counterpart. 8.12 The ex-dividend stock prices are n 0 1 11.20 2 12.32 < / S(n) ex-div 12.00 10.64 \ 10.34 9.40 < 8.93 294 Mathematics for Finance The corresponding European and American put prices will be n 0 1 2.53 2.80 P E (n) P A (n) 2 1.68 1.68 < / 3.36 3.36 \ 3.66 3.66 3.42 3.69 4.33 4.60 < 5.07 5.07 At time 1 the payoﬀ of the American put option in both the up and down states will be higher than the value of holding the option to expiry, so the option should be exercised in these states (indicated by bold ﬁgures). 1 2 8.13 Take b such that S(0)eσb+ru− 2 σ u = a and put V (t) = W (t)+ m − r + 12 σ 2 σt for any t ≥ 0, which is a Wiener process under P∗ . In particular, V (u) is normally distributed under P∗ with mean 0 and variance u. The right-hand side of (8.8) is therefore equal to 1 2 E∗ e−ru S(u)1S(u)<a = S(0)E∗ eσV (u)− 2 σ u 1V (u)<b \$ b 1 2 x2 1 = S(0) eσx− 2 σ u √ e− 2u dx 2πt −∞ \$ b (x−σu)2 1 √ = S(0) e− 2u dx. 2πt −∞ Now observe that, since V (t) is a Wiener process under P∗ , the random variables V (u) and V (t) − V (u) are independent and normally distributed with mean 0 and variance u and t − u, respectively. As a result, their joint density 2 2 − y −x 1 e 2(t−u) 2u 2πt . This enables us to compute the left-hand side of (8.8), 1 2 E∗ e−rt S(t)1S(u)<a = S(0)E∗ eσV (t)− 2 σ u 1V (u)<b 1 2 = S(0)E∗ eσ(V (t)−V (u))+σV (u)− 2 σ u 1V (u)<b \$ b \$ ∞ 2 y2 1 2 1 − 2(t−u) −x 2u dy = S(0) eσy+σx− 2 σ t dx e 2πt −∞ −∞ \$ b \$ ∞ 2 (x−σu)2 1 − (y−σ(t−u)) − 2u 2(t−u) = S(0) dy dx e −∞ −∞ 2πt \$ b (x−σu)2 1 √ = S(0) e− 2u dx 2πt −∞ is We can now see that the two sides of (8.8) are equal to one another.

N tN As N → ∞, we have tN → t and N tN → ∞, so that wN (tN ) → W (t) √ in distribution, where W (t) = tX. The last equality means that W (t) is normally distributed with mean 0 and variance t. This argument, based on the Central Limit Theorem, works for any single ﬁxed time t > 0. It is possible to extend the result to obtain a limit for all times t ≥ 0 simultaneously, but this is beyond the scope of this book. The limit W (t) is called the Wiener process (or Brownian motion). It inherits many of the properties of the random walk, for example: 1. W (0) = 0, which corresponds to wN (0) = 0. 2. E(W (t)) = 0, corresponding to E(wN (t)) = 0 (see the solution of Exercise 3.25). 3. Var(W (t)) = t, with the discrete counterpart Var(wN (t)) = t (see the solution of Exercise 3.25). 4. The increments W (t3 ) − W (t2 ) and W (t2 ) − W (t1 ) are independent for 0 ≤ t1 ≤ t2 ≤ t3 ; so are the increments wN (t3 ) − wN (t2 ) and wN (t2 ) − wN (t1 ). 2 See, for example, Capiński and Zastawniak (2001). 70 Mathematics for Finance 5.

Our treatment of continuous time is a compromise lacking full mathematical rigour, which would require a systematic study of Stochastic Calculus, a topic 186 Mathematics for Finance treated in detail in more advanced texts. In place of this, we shall exploit an analogy with the discrete time case. As a starting point we take the continuous time model of stock prices developed in Chapter 3 as a limit of suitably scaled binomial models with time steps going to zero. In the resulting continuous time model the stock price is given by (8.5) S(t) = S(0)emt+σW (t) , where W (t) is the standard Wiener process (Brownian motion), see Section 3.3.2. This means, in particular, that S(t) has the log normal distribution. Consider a European option on the stock expiring at time T with payoﬀ f (S(T )). As in the discrete-time case, see Theorem 8.4, the time 0 price D(0) of the option ought to be equal to the expectation of the discounted payoﬀ e−rT f (S(T )), (8.6) D(0) = E∗ e−rT f (S(T )) , under a risk-neutral probability P∗ that turns the discounted stock price process e−rt S(t) into a martingale.

Handbook of Modeling High-Frequency Data in Finance by Frederi G. Viens, Maria C. Mariani, Ionut Florescu

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S0 Indeed, assumption (1) is equivalent to ask that the increment Xt+h − Xt of the process X over [t, t + h] is Gaussian with mean μh and variance σ 2 h. Assumption (2) simply means that the increments of X over disjoint periods of time are independent. Finally, the last condition is tantamount to asking that X has continuous paths. Note that we can represent a general geometric Brownian motion in the form St = S0 eσ Wt +μt , where (Wt )t≥0 is the Wiener process. In the context of the above Black–Scholes model, a Wiener process can be deﬁned as the log return process of a price process satisfying the Black–Scholes conditions (1)–(3) with μ = 0 and σ 2 = 1. As it turns out, assumptions (1)–(3) above are all controversial and believed not to hold true especially at the intraday level (see Cont (2001) for a concise description of the most important features of ﬁnancial data). The empirical distributions of log returns exhibit much heavier tails and higher kurtosis than a Gaussian distribution does and this phenomenon is accentuated when the frequency of returns increases.

In the following section, we concentrate on two important and popular types of exponential Lévy models. 1.2.2 VARIANCE-GAMMA AND NORMAL INVERSE GAUSSIAN MODELS The VG and NIG Lévy models were proposed in Carr et al. (1998) and BarndorffNielsen (1998), respectively, to describe the log return process Xt := log St /S0 of a ﬁnancial asset. Both models can be seen as a Wiener process with drift that is time-deformed by an independent random clock. That is, (Xt ) has the representation Xt = σ W (τ (t)) + θτ (t) + bt, (1.1) where σ > 0, θ, b ∈ R are given constants, W is Wiener process, and τ is a suitable independent subordinator (nondecreasing Lévy process) such that Eτ (t) = t, and Var(τ (t)) = κt. In the VG model, τ (t) is Gamma distributed with scale parameter β := κ and shape parameter α := t/κ, while in the NIG model τ (t) follows an inverse Gaussian distribution with mean μ = 1 and shape parameter λ = 1/(tκ).

LANCETTE Department of Statistics, Purdue University, West Lafayette, IN KISEOP LEE Department of Mathematics, University of Louisville, Louisville, KY; Graduate Department of Financial Engineering, Ajou University, Suwon, South Korea YA N H U I M I Department of Statistics, Purdue University, West Lafayette, IN 1.1 Introduction Driven by the necessity to incorporate the observed stylized features of asset prices, continuous-time stochastic modeling has taken a predominant role in the ﬁnancial literature over the past two decades. Most of the proposed models are particular cases of a stochastic volatility component driven by a Wiener process superposed with a pure-jump component accounting for the Handbook of Modeling High-Frequency Data in Finance, First Edition. Edited by Frederi G. Viens, Maria C. Mariani, and Ionuţ Florescu. © 2012 John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc. 3 4 CHAPTER 1 Estimation of NIG and VG Models discrete arrival of major inﬂuential information. Accurate approximation of the complex phenomenon of trading is certainly attained with such a general model.

Monte Carlo Simulation and Finance by Don L. McLeish

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For a more formal definition, as well as an explanation of how we should interpret the integral, see the appendix. This process Zs is, both in discrete and continuous time, a martingale. MODELS IN CONTINUOUS TIME 67 Wiener Process 3 2.5 2 W(t) 1.5 1 0.5 0 -0.5 -1 0 1 2 3 4 5 t 6 7 8 9 Figure 2.6: A sample path of the Wiener process Models in Continuous Time We begin with some oversimplified rules of stochastic calculus which can be omitted by those with a background in Brownian motion and diﬀusion. First, we define a stochastic process Wt called the standard Brownian motion or Wiener process having the following properties; 1. For each h > 0, the increment W (t+h)−W (t) has a N (0, h) distribution and is independent of all preceding increments W (u) − W (v), t > u > v > 0. 2. W (0 ) = 0 .

Does your price oﬀer an arbitrage to another trader? What is the risk-neutral measure for this bond? 5. Which would you prefer, a gift of \$100 or a 50-50 chance of making \$200? A fine of \$100 or a 50-50 chance of losing \$200? Are your preferences self-consistent and consistent with the principle that individuals are riskaverse? PROBLEMS 93 6. Compute the stochastic diﬀerential dXt (assuming Wt is a Wiener process) when (a) Xt = exp(rt) Rt (b) Xt = 0 h(t)dWt (c) Xt = X0 exp{at + bWt } (d) Xt = exp(Yt ) where dYt = µdt + σdWt . 7. Show that if Xt is a geometric Brownian motion, so is Xtβ for any real number β. 8. Suppose a stock price follows a geometric Brownian motion process dSt = µSt dt + σSt dWt Find the diﬀusion equation satisfied by the processes (a) f (St ) = Stn ,(b) log(St ), (c) 1/St . Find a combination of the processes St and 1/St that does not depend on the drift parameter µ.

Consider approximating an integral of the form RT 0 g(t)dWt ≈ P g(t){W (t+ h) − W (t)} where g(t) is a non-random function and the sum is over values of t = nh, n = 0, 1, 2, ...T /h − 1. Show by considering the distribution 94 CHAPTER 2. SOME BASIC THEORY OF FINANCE of the sum and taking limits that the random variable normal distribution and find its mean and variance. RT 0 g(t)dWt has a 12. Consider two geometric Brownian motion processes Xt and Yt both driven by the same Wiener process dXt = aXt dt + bXt dWt dYt = µYt dt + σYt dWt . Derive a stochastic diﬀerential equation for the ratio Zt = Xt /Yt . Suppose for example that Xt models the price of a commodity in \$C and Yt is the exchange rate (\$C/\$U S) at time t. Then what is the process Zt ? Repeat in the more realistic situation in which (1) dXt = aXt dt + bXt dWt (2) dYt = µYt dt + σYt dWt (1) (2) and Wt , Wt are correlated Brownian motion processes with correlation ρ. 13.

Frequently Asked Questions in Quantitative Finance by Paul Wilmott

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See Richardson (1922). Richardson later worked on the mathematics for the causes of war. 1923 Wiener Norbert Wiener developed a rigorous theory for Brownian motion, the mathematics of which was to become a necessary modelling device for quantitative finance decades later. The starting point for almost all financial models, the first equation written down in most technical papers, includes the Wiener process as the representation for randomness in asset prices. See Wiener (1923). 1950s Samuelson The 1970 Nobel Laureate in Economics, Paul Samuelson, was responsible for setting the tone for subsequent generations of economists. Samuelson ‘mathematized’ both macro and micro economics. He rediscovered Bachelier’s thesis and laid the foundations for later option pricing theories. His approach to derivative pricing was via expectations, real as opposed to the much later risk-neutral ones.

Kiyosi Itô showed the relationship between a stochastic differential equation for some independent variable and the stochastic differential equation for a function of that variable. One of the starting points for classical derivatives theory is the lognormal stochastic differential equation for the evolution of an asset. Itô’s lemma tells us the stochastic differential equation for the value of an option on that asset. In mathematical terms, if we have a Wiener process X with increments dX that are normally distributed with mean zero and variance dt then the increment of a function F(X) is given by This is a very loose definition of Itô’s lemma but will suffice. See Itô (1951). 1952 Markowitz Harry Markowitz was the first to propose a modern quantitative methodology for portfolio selection. This required knowledge of assets’ volatilities and the correlation between assets.

Example The obvious example concerns the random walkdS = µS dt + σ S dX commonly used to model an equity price or exchange rate, S. What is the stochastic differential equation for the logarithm of S, lnS? The answer is Long Answer Let’s begin by stating the theorem. Given a random variable y satisfying the stochastic differential equationdy = a(y, t) dt + b(y, t) dX , where dX is a Wiener process, and a function f(y, t) that is differentiable with respect to t and twice differentiable with respect to y, then f satisfies the following stochastic differential equation Itô’s lemma is to stochastic variables what Taylor series is to deterministic. You can think of it as a way of expanding functions in a series in dt, just like Taylor series. If it helps to think of it this way then you must remember the simple rules of thumb as follows. 1.

Market Risk Analysis, Quantitative Methods in Finance by Carol Alexander

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Further detail on unit root tests is given in Section II.5.3. I.3.7.2 Mean Reverting Processes and Random Walks in Continuous Time A continuous time stochastic process has dynamics that are represented by a stochastic differential equation. There are two parts to this representation: the first term defines the deterministic part and the second term defines the stochastic part. A Wiener process, also called a Brownian motion, describes the stochastic part when the process does not jump. Thus a Wiener process is a continuous process with stationary, independent normally distributed increments. The increments are normally distributed so we often use any of the notations Wt Bt or Zt for such a process. The increments of the process are the total change in the process over an infinitesimally small time period; these are denoted dWt dBt or dZt, with EdW = 0 and VdW = dt Now we are ready to write the equation for the dynamics of a continuous time stochastic process X t as the following SDE: dXt = dt + dZt (I.3.141) where is called the drift of the process and is called the process volatility.

So if the growth rate is a constant , we can write dSt = St (I.1.29) dt Now by the chain rule, dSt d ln St d ln St dSt = = St−1 dt dSt dt dt Thus an equivalent form of (I.1.29) is d ln St = dt (I.1.30) St = S0 exp t (I.1.31) Integrating (I.1.30) gives the solution Hence, the asset price path would be an exponential if there were no uncertainty about the future price. 22 Quantitative Methods in Finance However, there is uncertainty about the price of the asset in the future, and to model this we add a stochastic differential term dWt to (I.1.29) or to (I.1.30). The process Wt is called a Wiener process, also called a Brownian motion. It is a continuous process that has independent increments dWt and each increment has a normal distribution with mean 0 and variance dt.19 On adding uncertainty to the exponential price path (I.1.31) the price process (I.1.29) becomes dSt = dt + dWt (I.1.32) St This is an example of a diffusion process. Since the left-hand side has the proportional change in the price at time t, rather than the absolute change, we call (I.1.32) geometric Brownian motion.

(independent and identically distributed) variables central limit theorem 121 error process 148 financial modelling 186 GEV distribution 101 regression 148, 157, 175 stable distribution 106 stochastic process 134–5 Implicit function 185 Implied volatility 194, 196, 200–1 Implied volatility surface 200–1 Incremental change 31 Indefinite integral 15 Independent events 74 Independent and identically distributed (i.i.d.) variables central limit theorem 121 error process 148 financial modelling 186 GEV distribution 101 regression 148, 157, 175 stable distribution 106 stochastic process 134–5 284 Index Independent variable 72, 143 random 109–10, 115, 140 Index tracking regression model 182–3 Indicator function 6 Indices, laws 8 Indifference curves 248–9 Inequality constraint, minimum variance portfolio 245–6 Inference 72, 118–29, 141 central limit theorem 120–1 confidence intervals 72, 118–24 critical values 118–20, 122–3, 129 hypothesis tests 124–5 means 125–7 non-parametric tests 127–9 quantiles 118–20 variance 126–7 Inflexion points 14, 35 Information matrix 133, 203 Information ratio 257, 259 Instability, finite difference approximation 209–10 Integrated process, discrete time 134–6 Integration 3, 15–16, 35 Intensity, Poisson distribution 88 Interest rate 34, 171–3 Interest rate sensitivity 34 Interpolation 186, 193–200, 223 cubic spline 197–200 currency option 195–7 linear/bilinear 193–5 polynomial 195–7 Intrinsic value of option 215 Inverse function 6–7, 35 Inverse matrix 41, 43–4, 133 Investment bank 225 Investment 2, 256–7 Investor risk tolerance 230–1, 237 Irrational numbers 7 Isoquants 248 Iteration 186–93, 223 bisection method 187–8 gradient method 191–3 Newton–Raphson method 188–91 Itô’s lemma 138–9, 219 iTraxx Europe credit spread index 172 Jacobian matrix 202 Jarque–Bera normality test Jensen’s alpha 257–8 158 Joint density function 114–15 Joint distribution function 114–15 Joint probability 73 Jumps, Poisson process 139 Kappa indices 263–5 Kernel 106–7 Kolmogorov–Smirnoff test 128 Kuhn–Tucker conditions 30 Kurtosis 81–3, 94–6, 205–6 Lagrange multiplier (LM) test 124, 167 Lagrange multiplier 29–30, 244 Lagrangian function 29–30 Lattice 186, 210–16, 223 Laws of indices 8 Least squares OLS estimation 143–4, 146–50, 153–61, 163, 170–1, 176 problems 201–2 weighted 179 Leptokurtic density 82–3 Levenberg–Marquardt algorithm 202 Lévy distribution 105 Likelihood function 72, 130–31 MLE 72, 130–34, 141, 202–3 optimization 202–3 ratio test 124, 167 Linear function 4–5 Linear interpolation 193–5 Linear portfolios 33, 35 correlation matrix 55–60 covariance matrix 55–61 matrix algebra 55–61 P&L 57–8 returns 25, 56–8 volatility 57–8 Linear regression 143–84 Linear restrictions, hypothesis tests 165–6 Linear transformation 48 Linear utility function 233 LM (Lagrange multiplier) 29–30, 124, 167, 244 Local maxima 14, 28–9 Local minima 14, 28–9 Logarithmic utility function 232 Logarithm, natural 1, 9, 34–5 Log likelihood 131–2 Lognormal distribution 93–4, 213–14, 218–20 Log returns 16, 19–25 Index Long portfolio 3, 17, 238–40 Long-short portfolio 17, 20–1 Low discrepancy sequences 217 Lower triangular square matrix 62, 64 LR (likelihood ratio) test 124, 167 LU decomposition, matrix 63–4 Marginal densities 108–9 Marginal distributions 108–9 Marginal probability 73–4 Marginal utility 229–30 Market behaviour 180–1 Market beta 250 Market equilibrium 252 Market maker 2 Market microstructure 180 Market portfolio 250–1 Market risk premium, CAPM 253 Markets complete 212 regime-specific behaviour 96–7 Markowitz, Harry 226, 238, 266 Markowitz problem 200–1, 226, 244–5 Matrix algebra 37–70 application 38–47 decomposition 61–4, 70 definite matrix 37, 46–7, 54, 58–9, 70 determinant 41–3, 47 eigenvalues/vectors 37–8, 48–54, 59–61, 70 functions of several variables 27–31 general linear model 161–2 hypothesis testing 165–6 invariant 62 inverse 41, 43–4 law 39–40 linear portfolio 55–61 OLS estimation 159–61 PCA 64–70 product 39–40 quadratic form 37, 45–6, 54 regression 159–61, 165–6 simultaneous equation 44–5 singular matrix 40–1 terminology 38–9 Maxima 14, 28–31, 35 Maximum likelihood estimation (MLE) 72, 130–4, 141, 202–3 Mean confidence interval 123 Mean excess loss 104 Mean reverting process 136–7 Mean 78–9, 125–6, 127, 133–4 285 Mean square error 201 Mean–variance analysis 238 Mean–variance criterion, utility theory 234–7 Minima 14, 28–31, 35 Minimum variance portfolio 3, 240–7 Mixture distribution 94–7, 116–17, 203–6 MLE (maximum likelihood estimation) 72, 130–4, 141, 202–3 Modified duration 2 Modified Newton method 192–3 Moments probability distribution 78–83, 140 sample 82–3 Sharpe ratio 260–3 Monotonic function 13–14, 35 Monte Carlo simulation 129, 217–22 correlated simulation 220–2 empirical distribution 217–18 random numbers 217 time series of asset prices 218–20 Multicollinearity 170–3, 184 Multiple restrictions, hypothesis testing 166–7 Multivariate distributions 107–18, 140–1 bivariate 108–9, 116–17 bivariate normal mixture 116–17 continuous 114 correlation 111–14 covariance 110–2 independent random variables 109–10, 114 normal 115–17, 220–2 Student t 117–18 Multivariate linear regression 158–75 BHP Billiton Ltd 162–5, 169–70, 174–5 confidence interval 167–70 general linear model 161–2 hypothesis testing 163–6 matrix notation 159–61 multicollinearity 170–3, 184 multiple regression in Excel 163–4 OLS estimation 159–61 orthogonal regression 173–5 prediction 169–70 simple linear model 159–61 Multivariate Taylor expansion 34 Mutually exclusive events 73 Natural logarithm 9, 34–5 Natural spline 198 Negative definite matrix 46–7, 54 Newey–West standard error 176 286 Index Newton–Raphson iteration 188–91 Newton’s method 192 No arbitrage 2, 179–80, 211–12 Non-linear function 1–2 Non-linear hypothesis 167 Non-linear portfolio 33, 35 Non-parametric test 127–9 Normal confidence interval 119–20 Normal distribution 90–2 Jarque–Bera test 158 log likelihood 131–2 mixtures 94–7, 140–1, 203–6 multivariate 115–16, 220–2 standard 218–19 Normalized eigenvector 51–3 Normalized Student t distribution 99 Normal mixture distribution 94–7, 116–17, 140–1 EM algorithm 203–6 kurtosis 95–6 probabilities of variable 96–7 variance 94–6 Null hypothesis 124 Numerical methods 185–223 binomial lattice 210–6 inter/extrapolation 193–200 iteration 186–93 Objective function 29, 188 Offer price 2 Oil index, Amex 162–3, 169–70, 174 OLS (ordinary least squares) estimation 143–4, 146–50 autocorrelation 176 BHP Billiton Ltd case study 163 heteroscedasticity 176 matrix notation 159–61 multicollinearity 170–1 properties of estimator 155–8 regression in Excel 153–5 Omega statistic 263–5 One-sided confidence interval 119–20 Opportunity set 246–7, 251 Optimization 29–31, 200–6, 223 EM algorithm 203–6 least squares problems 201–2 likelihood methods 202–3 numerical methods 200–5 portfolio allocation 3, 181 Options 1–2 American 1, 215–16 Bermudan 1 call 1, 6 currency 195–7 European 1–2, 195–6, 212–13, 215–16 finite difference approximation 206–10 pay-off 6 plain vanilla 2 put 1 Ordinary least squares (OLS) estimation 143–4, 146–50 autocorrelation 176 BHP Billiton Ltd case study 163 heteroscedasticity 176 matrix notation 159–61 multicollinearity 170–1 properties of estimators 155–8 regression in Excel 153–5 Orthogonal matrix 53–4 Orthogonal regression 173–5 Orthogonal vector 39 Orthonormal matrix 53 Orthonormal vector 53 Out-of-sample testing 183 P&L (profit and loss) 3, 19 backtesting 183 continuous time 19 discrete time 19 financial returns 16, 19 volatility 57–8 Pairs trading 183 Parabola 4 Parameter notation 79–80 Pareto distribution 101, 103–5 Parsimonious regression model 153 Partial derivative 27–8, 35 Partial differential equation 2, 208–10 Pay-off, option 6 PCA (principal component analysis) 38, 64–70 definition 65–6 European equity indices 67–9 multicollinearity 171 representation 66–7 Peaks-over-threshold model 103–4 Percentage returns 16, 19–20, 58 Percentile 83–5, 195 Performance measures, RAPMs 256–65 Period log returns 23–5 Pi 7 Index Piecewise polynomial interpolation 197 Plain vanilla option 2 Points of inflexion 14, 35 Poisson distribution 87–9 Poisson process 88, 139 Polynomial interpolation 195–7 Population mean 123 Portfolio allocation 237–49, 266 diversification 238–40 efficient frontier 246–9, 251 Markowitz problem 244–5 minimum variance portfolio 240–7 optimal allocation 3, 181, 247–9 Portfolio holdings 17–18, 25–6 Portfolio mathematics 225–67 asset pricing theory 250–55 portfolio allocation 237–49, 266 RAPMs 256–67 utility theory 226–37, 266 Portfolios bond portfolio 37 delta-hedged 208 linear 25, 33, 35, 55–61 minimum variance 3, 240–7 non-linear 33, 35 rebalancing 17–18, 26, 248–9 returns 17–18, 20–1, 91–2 risk factors 33 risk free 211–12 stock portfolio 37 Portfolio volatility 3 Portfolio weights 3, 17, 25–6 Positive definite matrices 37, 46–7, 70 correlation matrix 58–9 covariance matrix 58–9 eigenvalues/vectors 54 stationary point 28–9 Posterior probability 74 Post-sample prediction 183 Power series expansion 9 Power utility functions 232–3 Prediction 169–70, 183 Price discovery 180 Prices ask price 2 asset price evolution 87 bid price 2 equity 172 generating time series 218–20 lognormal asset prices 213–14 market microstructure 180 offer price 2 stochastic process 137–9 Pricing arbitrage pricing theory 257 asset pricing theory 179–80, 250–55 European option 212–13 no arbitrage 211–13 Principal cofactors, determinants 41 Principal component analysis (PCA) 38, 64–70 definition 65–6 European equity index 67–9 multicollinearity 171 representation 66–7 Principal minors, determinants 41 Principle of portfolio diversification 240 Prior probability 74 Probability and statistics 71–141 basic concepts 72–85 inference 118–29 laws of probability 73–5 MLE 130–4 multivariate distributions 107–18 stochastic processes 134–9 univariate distribution 85–107 Profit and loss (P&L) 3, 19 backtesting 183 continuous time 19 discrete time 19 financial returns 16, 19 volatility 57–8 Prompt futures 194 Pseudo-random numbers 217 Put option 1, 212–13, 215–16 Quadratic convergence 188–9, 192 Quadratic form 37, 45–6, 54 Quadratic function 4–5, 233 Quantiles 83–5, 118–20, 195 Quartiles 83–5 Quasi-random numbers 217 Random numbers 89, 217 Random variables 71 density/distribution function 75 i.i.d. 101, 106, 121, 135, 148, 157, 175 independent 109–10, 116, 140–1 OLS estimators 155 sampling 79–80 Random walks 134–7 Ranking investments 256 287 288 Index RAPMs (risk adjusted performance measures) 256–67 CAPM 257–8 kappa indices 263–5 omega statistic 263–5 Sharpe ratio 250–1, 252, 257–63, 267 Sortino ratio 263–5 Realization, random variable 75 Realized variance 182 Rebalancing of portfolio 17–18, 26, 248–9 Recombining tree 210 Regime-specific market behaviour 96–7, 117 Regression 143–84 autocorrelation 175–9, 184 financial applications 179–83 heteroscedasticity 175–9, 184 linear 143–84 multivariate linear 158–75 OLS estimator properties 155–8 simple linear model 144–55 Relative frequency 77–8 Relative risk tolerance 231 Representation, PCA 66–7 Residuals 145–6, 157, 175–8 Residual sum of squares (RSS) 146, 148–50, 159–62 Resolution techniques 185–6 Restrictions, hypothesis testing 165–7 Returns 2–3, 16–26 absolute 58 active 92, 256 CAPM 253–4 compounding 22–3 continuous time 16–17 correlated simulations 220 discrete time 16–17, 22–5 equity index 96–7 geometric Brownian motion 21–2 linear portfolio 25, 56–8 log returns 16, 19–25 long-short portfolio 20–1 multivariate normal distribution 115–16 normal probability 91–2 P&L 19 percentage 16, 19–20, 59–61 period log 23–5 portfolio holdings/weights 17–18 risk free 2 sources 25–6 stochastic process 137–9 Ridge estimator, OLS 171 Risk active risk 256 diversifiable risk 181 portfolio 56–7 systematic risk 181, 250, 252 Risk adjusted performance measure (RAPM) 256–67 CAPM 257–8, 266 kappa indices 263–5 omega statistic 263–5 Sharpe ratio 251, 252, 257–63, 267 Sortino ratio 263–5 Risk averse investor 248 Risk aversion coefficients 231–4, 237 Risk factor sensitivities 33 Risk free investment 2 Risk free portfolio 211 Risk free returns 2 Risk loving investors 248–9 Risk neutral valuation 211–12 Risk preference 229–30 Risk reversal 195–7 Risk tolerance 230–1, 237 Robustness 171 Roots 3–9, 187 RSS (residual sum of squares) 146, 148–50, 159–62 S&P 100 index 242–4 S&P 500 index 204–5 Saddle point 14, 28 Sample 76–8, 82–3 Sampling distribution 140 Sampling random variable 79–80 Scalar product 39 Scaling law 106 Scatter plot 112–13, 144–5 SDE (stochastic differential equation) 136 Security market line (SML) 253–4 Self-financing portfolio 18 Sensitivities 1–2, 33–4 Sharpe ratio 257–63, 267 autocorrelation adjusted 259–62 CML 251, 252 generalized 262–3 higher moment adjusted 260–2 making decision 258 stochastic dominance 258–9 Sharpe, William 250 Short portfolio 3, 17 22, 134, Index Short sales 245–7 Short-term hedging 182 Significance level 124 Similarity transform 62 Similar matrices 62 Simple linear regression 144–55 ANOVA and goodness of fit 149–50 error process 148–9 Excel OLS estimation 153–5 hypothesis tests 151–2 matrix notation 159–61 OLS estimation 146–50 reporting estimated model 152–3 Simulation 186, 217–22 Simultaneous equations 44–5 Singular matrix 40–1 Skewness 81–3, 205–6 Smile fitting 196–7 SML (security market line) 253–4 Solver, Excel 186, 190–1, 246 Sortino ratio 263–5 Spectral decomposition 60–1, 70 Spline interpolation 197–200 Square matrix 38, 40–2, 61–4 Square-root-of-time scaling rule 106 Stable distribution 105–6 Standard deviation 80, 121 Standard error 80, 169 central limit theorem 121 mean/variance 133–4 regression 148–9 White’s robust 176 Standard error of the prediction 169 Standardized Student t distribution 99–100 Standard normal distribution 90, 218–19 Standard normal transformation 90 Standard uniform distribution 89 Stationary point 14–15, 28–31, 35 Stationary stochastic process 111–12, 134–6 Stationary time series 64–5 Statistical arbitrage strategy 182–3 Statistical bootstrap 218 Statistics and probability 71–141 basic concepts 72–85 inference 118–29 law of probability 73–5 MLE 130–4 multivariate distribution 107–18 stochastic process 134–9 univariate distribution 85–107 Step length 192 Stochastic differential equation (SDE) 22, 134, 136 Stochastic dominance 227, 258–9 Stochastic process 72, 134–9, 141 asset price/returns 137–9 integrated 134–6 mean reverting 136–7 Poisson process 139 random walks 136–7 stationary 111–12, 134–6 Stock portfolio 37 Straddle 195–6 Strangle 195–7 Strictly monotonic function 13–14, 35 Strict stochastic dominance 258 Structural break 175 Student t distribution 97–100, 140 confidence intervals 122–3 critical values 122–3 equality of means/variances 127 MLE 132 multivariate 117–18 regression 151–3, 165, 167–8 simulation 220–2 Sum of squared residual, OLS 146 Symmetric matrix 38, 47, 52–4, 61 Systematic risk 181, 250, 252 Tail index 102, 104 Taylor expansion 2–3, 31–4, 36 applications 33–4 approximation 31–4, 36 definition 32–3 multivariate 34 risk factor sensitivities 33 Theory of asset pricing 179–80, 250–55 Tic-by-tic data 180 Time series asset prices/returns 137–9, 218–20 lognormal asset prices 218–20 PCA 64–5 Poisson process 88 regression 144 stochastic process 134–9 Tobin’s separation theorem 250 Tolerance levels, iteration 188 Tolerance of risk 230–1, 237 Total derivative 31 Total sum of square (TSS) 149, 159–62 289 290 Index Total variation, PCA 66 Tower law for expectations 79 Traces of matrix 62 Tradable asset 1 Trading, regression model 182–3 Transition probability 211–13 Transitive preferences 226 Transposes of matrix 38 Trees 186, 209–11 Treynor ratio 257, 259 TSS (total sum of squares) 149, 159–62 Two-sided confidence interval 119–21 Unbiased estimation 79, 81, 156–7 Uncertainty 71 Unconstrained optimization 29 Undiversifiable risk 252 Uniform distribution 89 Unit matrix 40–1 Unit vector 46 Univariate distribution 85–107, 140 binomial 85–7, 212–13 exponential 87–9 generalized Pareto 101, 103–5 GEV 101–3 kernel 106–7 lognormal 93–4, 213–14, 218–20 normal 90–7, 115–16, 131–2, 140, 157–8, 203–6, 217–22 normal mixture 94–7, 140, 203–6 Poisson 87–9 sampling 100–1 stable 105–6 Student t 97–100, 122–3, 126, 132–3, 140–1, 151–3, 165–8, 220–2 uniform 89 Upper triangular square matrix 62, 64 Utility theory 226–37, 266 mean–variance criterion 234–7 properties 226–9 risk aversion coefficient 231–4, 237 risk preference 229–30 risk tolerance 230–1, 237 Value at risk (VaR) 104–6, 185, 194 Vanna–volga interpolation method 196 Variance ANOVA 143–4, 149–50, 154, 159–60, 164–5 confidence interval 123–4 forecasting 182 minimum variance portfolio 3, 240–7 mixture distribution 94–6 MLE 133 normal mixture distribution 95–6 portfolio volatility 3 probability distribution 79–81 realized 182 tests on variance 126–7 utility theory 234–7 VaR (value at risk) 104–6, 185, 194 Vector notation, functions of several variables 28 Vectors 28, 37–9, 48–54, 59–61, 70 Venn diagram 74–5 Volatility equity 3, 172–3 implied volatility 194, 196–7, 200–1 interpolation 194, 196–7 linear portfolio 57–8 long-only portfolio 238–40 minimum variance portfolio 240–4 portfolio variance 3 Volpi, Leonardo 70 Vstoxx index 172 Waiting time, Poisson process 88–9 Wald test 124, 167 Weakly stationary process 135 Weak stochastic dominance 258–9 Weibull distribution 103 Weighted least squares 179 Weights, portfolio 3, 17, 25–6 White’s heteroscedasticity test 177–8 White’s robust standard errors 176 Wiener process 22, 136 Yield 1, 197–200 Zero matrix 39 Z test 126

Asset and Risk Management: Risk Oriented Finance by Louis Esch, Robert Kieffer, Thierry Lopez

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If we then wish to move to continuous modelling while retaining the same variability per (X)2 unit of time, that is, with: = 1, for example, we obtain the stochastic process t (n) wt = limn→∞ Zt . This is a standard Brownian motion (also known as a Wiener process). It is clear that this stochastic process wt , deﬁned on R+ , is such that w0 = 0, that wt has independent and stationary increments, and that in view of the √ central limit theorem wt is distributed according to a normal law with parameters (0; t). It can be shown that the paths of a Wiener process are continuous everywhere, but cannot generally be differentiated. In fact √ wt ε t ε = =√ t t t where, ε is a standard normal r.v. 356 Asset and Risk Management 2.3.2.3 Itô process If a more developed model is required, wt can be multiplied by a constant in order to produce variability per time unit (X)2 /t different from 1 or to add a constant to it in order to obtain a non-zero mean: Xt = X0 + b · wt This type of model is not greatly effective because of the great variability √ of the development in the short term, the standard deviation of Xt being equal7 to b t.

When HS cannot rely on the normality hypothesis, this method of working is incorrect50 and the previous technique should be applied. • For MC and for this method only, it is possible to generate not only a future price value but a path of prices for the calculation horizon. We now explain this last case a little further, where, for example, the price evolution of an equity is represented by geometric Brownian motion (see Section 3.4.2): St+dt − St = St · (ER · dt + σR · dwt ) where the Wiener process (dwt ) obeys a law with a zero expectation and a variance equal to dt. If one considers a normal random variable ε with zero expectation and variance of 1, we can write: √ St+dt − St = St · (ER · dt + σR · ε dt) Simulation of a sequence of independent values for ε using the Monte Carlo method allows the variations St+dt − St to be obtained, and therefore, on the basis of the last price observed S0 , allows the path of the equity’s future price to be generated for a number of dates equal to the number of ε values simulated.51 7.5.1.2 Models used (1) The valuation models play an important part in the VC and MC methods.

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Any dimension, well, up to 160, given that that’s all the primitive polynomials I have. I had meanwhile learned from some presentation slides by Mark Broadie and JWPR007-Lindsey May 18, 2007 21:24 Peter Jäckel 171 Paul Glasserman that it is always a good idea to order dimensions by importance when using low discrepancy numbers, and duly pay attention to this fact. Also I had learned from them that the Brownian bridge, for discretized Wiener process path construction, helps to tickle (nearly) the maximum benefit out of low-discrepancy numbers, while not having to use a full matrix multiplication for each path (as one would with the theoretically superior spectral construction method), which could dominate the computational effort. Alright then, I say to myself, there are a couple of minor caveats to heed when using Sobol’ numbers, but that’s fair enough, given that they are the equivalent to nitromethane-fuelled engines in Indy car races: you just ought to be a bit more careful than with ordinary engines!

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We won’t go into all the details as to exactly what (,ℑW,℘W ) represents but you can think of the probability measure, ℘W, which is called Wiener measure, to be deﬁned in terms of the transition density function p(T,y;t,x) for =T–t, OPTION PRICING IN CONTINUOUS TIME 541 p(T ,y;t,x ) = p( ,x,y ) ⎛ 1 ⎞ −(y −x )2 / 2 =⎜ ⎟e ⎝ 2 ⎠ Norbert Wiener gave the ﬁrst rigorous mathematical construction (existence proof) for ABM and, because of this, it is sometimes called the Wiener process. It has the following properties, 1. W0=0 (starts at 0). 2. For every set of times t0=0<t1<t2<…tn–1<tn the increments (changes) Wt1–Wt0,Wt2–Wt1,…,Wtn–Wtn-1 are independent (independent increments). 3. For any times s and t with 0 s<t, the random variable Wt()–Ws() is normally distributed with mean E(Wt ()–Ws())=0 and variance Var(Wt ()–Ws())=t–s (normally distributed increments). 4. Almost all the sample paths of {Wt()}t0 are continuous (continuous sample paths).

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And in 1920, after briefly work- ing as a journalist for the Boston Herald to tide himself over between jobs, he joined the mathematics faculty at MIT. It was not a prestigious appointment. MIT's transformation still lay in the fu- ture, and the mathematics department existed mainly to teach math to the engi- neering students. The school wasn't oriented toward research at all. However, no one seems to have informed Wiener of that fact, and his mathematical output soon became legendary. The Wiener measure, the Wiener process, the Wiener- Hopf equations, the Paly-Wiener theorems, the Wiener extrapolation of linear times series, generalized harmonic analysis-he saw mathematics everywhere he looked. He also made significant contributions to quantum theory as it devel- oped in the 1920s and 1930s. Moreover, he did all this in a style that left his more conventional colleagues shaking their heads. Instead of treating mathematics as a formal exercise in the manipulation of symbols, Wiener worked by intuition, often groping his way toward a solution by trying to envision some physical model of the problem.