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<p> </p><p>NTRODUCTION AND<br>PRELIMINARIES<br>1.1 Preliminaries<br>1.1.1 -algebra:<br>Let<br>be a non empty set, and a non empty collection of subsets of<br>.<br>Then is called a -algebra if the following properties hold:<br>(i)<br>2<br>(ii) If A 2 , then A0 2<br>(iii) If fAj : j 2 Jg , then<br>[<br>j2J<br>Aj 2<br>for any nite or innite countable subset of N.<br>1.1.2 Borel -algebra:<br>Let X be a non empty set and a topology on X i.e. is the collection<br>of subsets of X. Then ( ) is called the Borel -algebra of the topological<br>space (X; )<br>6<br>1.1.3 Probability Space:<br>Let<br>be a non-empty set and be a -algebra of subsets of<br>. Then the<br>pair (<br>,) is called a measurable space, and a member of is called a<br>measurable set. Let (<br>,) be a measurable space and be a real-valued<br>map on . Then is called a probability measure on (<br>,) if the following<br>properties hold:<br>I (A) 0; 8A 2<br>II (<br>)=1<br>III For fAngn2N with Aj Ak = ;, and i 6= j, then<br>(<br>[<br>n2N<br>An) =<br>X<br>n2N<br>(An)<br>i.e. is -additive (or countably additive).<br>Now if (<br>,) is a measurable space and is a probability on (<br>,), then the<br>triple (<br>; ; ) is called a probability space.<br>1.1.4 Measurable Map:<br>Let (<br>; ) and (ô€€€; ) be two measurable spaces. Then a map X :<br>ô€€€! ô€€€ is<br>called measurable if the set Xô€€€1(A) = f! 2<br>: X(!) 2 Ag is in whenever<br>A 2 : In particular, we take (ô€€€; ) to be (R; (R)) or (Rn; (Rn)) where<br>n 2 N and (R) is the Borel -algebra of R:<br>1.1.5 Random variables/vectors:<br>Let (<br>; ; ) be an arbitrary probability space and (Rn; (Rn)) be the n-<br>dimensional Borel measurable space. Then a measurable map X :<br>ô€€€! Rn<br>is called a random vector. If n = 1, then X is called a random variable.<br>We denote by L(<br>;Rn) the set of all Rn-valued random vectors on<br>, and<br>L1(<br>; ; ) the space of random variables.<br>1.1.6 Probability Distribution:<br>Let (<br>; ; ) be a probability space, (Rn; (Rn)) be the n-dimensional Borel<br>measurable space, and X :<br>ô€€€! Rn a random vector. Then the map<br>X : (Rn) ô€€€! [0; 1] dened by X(A) = (Xô€€€1(A));A 2 (Rn) is called<br>the probability distribution of X:<br>7<br>1.1.7 Mathematical Expectation:<br>Let (<br>,,) be a probability space. If X 2 L1(<br>; ; ), then<br>E(X) =<br>Z</p><p>X(!)d(!)<br>is called the mathematical expectation or expected value or mean of X:<br>1.1.8 Variance and Covariance of random variables:<br>Let (<br>; ; ) be a probability space and X an R-valued random variable on<br>,<br>such that X 2 L2(<br>; ; ). Then X is automatically in L1(<br>; ; ) (because<br>in general if p q, then Lq(<br>; ; ) Lp(<br>; ; ) for all p 2 [1;1) [ f1g:)<br>The variance of X is dened as<br>V ar(X) = E((X ô€€€ E(X))2):<br>The number X =<br>p<br>V ar(X) is called the standard deviation/error. Now<br>let X, Y 2 L2(<br>; ; ). Then the covariance of X and Y is given by:<br>Cov(X; Y ) = E((X ô€€€ E(X))(Y ô€€€ E(Y )))<br>1.1.9 Stochastic Process:<br>Let (<br>,,) be a probability space. A stochastic process X indexed by a<br>totally ordered set T (time), is a collection X = fX(t) : t 2 Tg, where each<br>X(t) or Xt is a random variable on<br>. We denote X(t) by Xt and write the<br>value of X(t) or Xt at ! 2<br>by X(t; !) or Xt(!). Thus, a stochastic process<br>or random process is a collection of random variables, often used to represent<br>the evolution of some random value, or system overtime.<br>1.1.10 Brownian Motion:<br>The Brownian motion refers to the ceaseless, irregular random motion of<br>small particles immersed in a liquid or gas, as observed by R. Brown in<br>1827. The phenomena can be explained by the perpetual collisions of the<br>particles with the molecules of the surrounding medium. Mathematically,<br>let (<br>; ; ) be a probability space, and W = fW(t) 2 L(<br>;Rn) : t 2 Tg,<br>where T R+ = [0;1), be an Rn-valued stochastic process on<br>with the<br>8<br>following properties:<br>(i) W(0) = 0, almost surely.<br>(ii) W has continuous sample paths. i.e. If X is a stochastic process and<br>! 2<br>then the map t 7ô€€€! X(t; !) 2 Rn is called a sample path or trajectory<br>of X. Now if the map is continuous we say X has a continuous sample paths.<br>(iii) W(t)ô€€€W(s) is an N(0; (tô€€€s)T) random vector for all t &gt; s 0, where<br>T is the n n identity map.<br>(iv) W has a stochastically independent increments i.e. For every 0 &lt; t1 &lt;<br>t2 &lt; &lt; tk, the random vectors W(t1), W(t2) ô€€€W(t1); ;W(tk) ô€€€W(tkô€€€1)<br>are stochastically independent.<br>ThenW is called the standard n-dimensional Brownian motion or n-dimensional<br>Wiener process.<br>For the n-dimensional Brownian motion W(t) = (W1(t); ; Wn(t))<br>we have the following useful properties:<br>(I) E(Wj(t)) = 0, j = 1; 2; 3; ; n<br>(II) E(Wj(t)2) = t, j = 1; 2; 3; ; n<br>(III) E(Wj(t)Wk(s)) = min(t; s) for t, s 2 T.<br>To show the result in III above, we assume t &gt; s (without loss of<br>generality) and consider<br>E[Wj(t)Wk(s)] = E[(Wj(t) ô€€€Wk(s))Wk(s) +Wk(s)2]<br>= E[(Wj(t) ô€€€Wk(s))Wk(s)] + E[Wk(s)2]<br>(because E is linear). Then, since Wj(t)ô€€€Wk(s) and Wk(s) are independent<br>and both Wj(t) ô€€€Wk(s) and Wk(s) have zero mean, so<br>E[Wj(t)Wk(s)] = E[Wk(s)2] = s = min(t; s)<br>1.1.11 Filtrations and Filtered Probability space:<br>Let (<br>; ; ) be a probability space and consider F() = ft : t 2 Tg a family<br>of -algebras of with the following properties:<br>(i) For each t 2 T, t contains all the -null members of ,<br>(ii) s t whenever t s, s, t 2 T.<br>Then F() is called a Filtration of and (<br>; ; F(); ) is called a Filtered<br>Probability Space or Stochastic Basis.<br>9<br>1.1.12 Adaptedness:<br>A Stochastic process X = fX(t) 2 L(<br>;Rn) : t 2 Tg is said to be adapted<br>to the ltration F() = ft : t 2 Tg if X(t) is measurable with respect to<br>t for each t2T. It is plain that every stochastic process is adapted to its<br>natural ltration.<br>1.1.13 Conditional Expectation:<br>Let (<br>; ; ) be a probability space, X a real random variable in L1(<br>; ; )<br>and a -subalgebra of . Then the conditional expectation of X given<br>written E(X j ) is dened as any random variable Y such that:<br>(i) Y is measurable with respect to i.e. for any A 2 (R), the set Y ô€€€1(A) 2<br>.<br>(ii)<br>R<br>B X(!)d(!) =<br>R<br>B Y (!)d(!) for arbitrary B 2 :<br>A random variable Y which satises (i) and (ii) is called a version of E(X j ):<br>1.1.14 Martingales:<br>The term martingale has its origin in gambling. It refers to the gambling tac-<br>tic of doubling the stake when losing in order to recoup oneself. In the stud-<br>ies of stochastic processes, martingales are dened in relation to an adapted<br>stochastic process. Let X = fX(t) 2 L1(<br>; ; ) : t 2 Tg be a real-valued<br>stochastic process on a ltered probability space (<br>; ; F(); ). Then X is<br>called a<br>(i) Supermartingale if E(X(t) j s) X(s) almost surely whenever t<br>s.<br>(ii) Submartingale if E(X(t) j s) X(s) almost surely whenever t s.<br>(iii) martingale if X is both a submartingale and a supermartingale i.e. If<br>E(X(t)j s) = X(s) almost surely whenever t s.<br>1.1.15 Ito Calculus:<br>Let (<br>; ; F(); ) be a ltered probability space and W a Brownian motion<br>relative to this space. We dene an integral of the form<br>W(f; t) =<br>Z t<br>0<br>f(s)dW(s); t 2 R+<br>10<br>where f belongs to some class of stochastic processes adapted to (<br>; ; F(; ).<br>1.1.16 Quadratic Variation:<br>Let X be a stochastic process on a ltered probability space (<br>; ; F(); ):<br>Then the quadratic variation of X on [0; t], t &gt; 0, is the stochastic process<br>hXi dened by<br>hXi(t) = lim<br>jPj! 0<br>Xnô€€€1<br>j=0<br>jX(tj+1) ô€€€ X(tj))j2<br>where P = ft; t1; ; tng is any partition of [0; t] i.e. 0 = t1 &lt; t2 &lt; &lt; tn = t<br>and jPj = max0jnô€€€1jtj+1 ô€€€ tj j<br>If X is a dierentiable stochastic process, then hXi=0.<br>1.1.17 Stochastic Dierential Equations:<br>These are equations of the form<br>dX(t) = g(t;X(t))dt + f(t;X(t))dW(t)<br>with initial condition X(t) = x<br>1.1.18 Ito Formula and Lemma:<br>Let (<br>; ; F(); ) be a ltered probability space, X an adapted stochas-<br>tic process on (<br>; ; F(); ) whose quadratic variation is hXi and U 2<br>C1;2([0; 1] R).<br>Then,<br>U(t;X(t)) = U(s;X(s)) +<br>Z t<br>s<br>@U<br>@t<br>(;X( ))ds +<br>Z t<br>s<br>@U<br>@x<br>(;X( ))dX( )<br>+<br>1<br>2<br>Z t<br>s<br>@2U<br>@x2 (;X( ))dhXi( )<br>which may be written as<br>11<br>dU(t; x) =<br>@U<br>@t<br>(t;X(t))dt +<br>@U<br>@x<br>(t;X(t))dX(t)<br>+<br>1<br>2<br>@2U<br>@x2 (t;X(t))dhXi(t)<br>The equation above is normally referred to as the Ito formula. If X<br>satsies the stochastic dierential equation (SDE)<br>dX(t) = g(t;X(t))dt + f(t;X(t))dW(t)<br>X(t) = x;<br>then<br>dU(t;X(t)) = gu(t;X(t))dt + fu(t;X(t))dW(t)<br>U(t;X(t)) = U(t; x)<br>where<br>gu(t; x) =<br>@U<br>@t<br>(t; x) + g(t; x)<br>@U<br>@x<br>(t; x) +<br>1<br>2<br>(f(t; x))2 @2U<br>@x2 (t; x);<br>fu(t; x) = f(t; x)<br>@U<br>@x<br>(t; x)<br>We obtain a particular case of the Ito formula called the Ito lemma, if we<br>take X = W, where g 0 and f 1 on T R. Then<br>dU(t;W(t)) = [<br>@U<br>@t<br>(t;W(t)) +<br>1<br>2<br>@2U<br>@x2 (t;W(t))]dt +<br>@U<br>@x<br>(t;W(t))dW(t)<br>The equation above is referred to as the Ito lemma.<br>1.1.19 Risk-neutral Probabilities:<br>These are probabilities for future outcomes adjusted for risk, which are then<br>used to compute expected asset values. The benet of this risk-neutral pric-<br>ing approach is that once the risk-neutral probabilities are calculated, they<br>can be used to price every asset based on its expected payo. These the-<br>oretical risk-neutral probabilities dier from actual real world probabilities;<br>if the latter were used, expected values of each security would need to be<br>adjusted for its individual risk prole. A key assumption in computing risk-<br>neutral probabilities is the absence of arbitrage. The concept of risk-neutral<br>probabilities is widely used in pricing derivatives.<br>12<br>1.1.20 Log-normal Distribution:<br>A random variable X is said to have a lognormal distribution if its logarithm<br>has a normal distribution. i.e. ln(X) N(; ), meaning logrithim of X is<br>distributed normal with mean and variance .<br>1.1.21 Bivariate Normal Density Function:<br>The bivariate normal density function is given by:<br>f(x; y) =<br>1<br>2<br>p<br>1 ô€€€ 2<br>exp[ô€€€<br>x2 ô€€€ 2xy + y2<br>2(1 ô€€€ 2)<br>1.1.22 Cumulative Bivariate Normal Distribution Func-<br>tion:<br>The standardised cumulative normal distribution function returns the prob-<br>ability that one random variable is less than “a”, and that a second random<br>variable is less than “b” when the correlation between the two variables is<br>and is given by:<br>M(a; b; ) =<br>1<br>2<br>p<br>1 ô€€€ 2<br>Z a<br>ô€€€1<br>Z b<br>ô€€€1<br>exp[ô€€€<br>x2 ô€€€ 2xy + y2<br>2(1 ô€€€ 2)<br>]dxdy<br>1.1.23 Markov Process:<br>A Markov process is a stochastic process satisfying a certain property, called<br>the Markov property. Let (<br>; ; ) be a probability space with a ltration<br>F() = ft : t 2 Tg for some totally ordered set T, and let (S; ) be a mea-<br>surable space. An s-valued stochastic process X = fXt : t 2 Tg adapted<br>to the ltration is said to posses the Markov property with respect to the<br>lteration F() if, for each A 2 and s; t 2 T with s &lt; t,<br>P(Xt 2 Ajs) = P(Xt 2 AjXs)<br>A Markov process is a stochastic process which satises the Markov prop-<br>erty with respect to its natural ltration.<br>13<br>1.1.24 Backward Kolmogorov Equation:<br>The Kolmogorov backward equation (diusion) is a partial dierential eqau-<br>tion (PDE) that arises in the theory of continuous-time Markov processess.<br>Assume that the system state X(t) evolves according to the stochastic dif-<br>ferential erquation<br>dXt = (Xt; t)dt + (Xt; t)dW(t)<br>then the Kolmogorov backward equation is as follows<br>ô€€€<br>@<br>@t<br>p(x; t) = (x; t)<br>@<br>@x<br>+<br>1<br>2<br>2(x; t)<br>@2<br>@x2 p(x; t)<br>for t s, subject to the nal condition p(x; s) = us(x): This can be derived<br>using Ito’s lemma on p(x; t) and setting the dt term equal to zero.<br>1.1.25 Forkker-Planck Equation:<br>The Fokker-Planck equation describes the time evolution of the of the veloc-<br>ity of a particle, and can be generalised to other observables as well. It is<br>also known as the Kolmogorov forward equation (diusion). In one spatial<br>dimension X, for an Ito process given by the stochastic dierential equation<br>dXt = (Xt; t)dt +<br>p<br>2D(Xt; t)dWt<br>with drift (Xt; t) and diusion coecient D(Xt; t); the Fokker-Planck<br>equation for the probability density f(x; t) of the random variable Xt is<br>@<br>@t<br>f(x; t) = ô€€€<br>@<br>@x<br>[(x; t)f(x; t)] +<br>@2<br>@x2 [D(x; t)f(x; t)]<br>The Fokker-Planck also exist in many dimensions, but we are going to restrict<br>ourselves to one dimension only.<br>1.1.26 Diusion Process:<br>A diusion process is a solution to a stochastic dierential equation. It<br>is a continuous-time Markov process with almost surely continuous sample<br>paths. Mathematically, it is a Markov process with continuous sample paths<br>for which the Kolmogorov forward equation is the Forkker-Planck equation.<br>Brownian motion, re ected Brownian motion and Ornstein-Uhlenbeck pro-<br>cesses are examoles of diusion process.<br>14<br>1.2 Introduction<br>An option is a nancial instrument that species a contract between two<br>parties for a future transaction at a reference price. This transaction can be<br>to buy or sell an underlying assets such as stocks, bonds, an interest rate<br>e.t.c. The option holder has the right but not the obligation to carry out the<br>specic transaction (i.e. to buy if it is a call option” or to sell if it is a put<br>option”) at or by a specied date (reference time).<br>A European option give the holder the right but not the obligation to buy,<br>(if it is a call) or to sell (if it is a put), an underlying asset on the specied<br>time or maturity date at the specied price. While an American option, give<br>the holder the right but not the obligation to buy, or sell an underlying asset<br>on or prior to the specied time or maturity date at the specied price.<br>A compound option is an option on an option. Hence, the compound op-<br>tion, or the mother option gives the holder the right but not the obligation<br>to buy, or sell another underlying option, the daughter option; for a certain<br>strike price K1 at a specied time T1. The daughter option then gives the<br>holder another right to buy or sell a nancial asset for another strike price<br>K2 at a later point in time T2. So, a compound option has two strike prices,<br>and two expiration dates. Also, Compound options are very frequently en-<br>countered in capital budgeting problems when projects require sequential<br>decisions. For example, when dealing with development projects, the initial<br>development expense allows one later to make a decision to wait or, to engage<br>in further development expenses eventually leading to a nal capital invest-<br>ment project. All R&amp;D expenditures involve a sequence of decisions. In the<br>mining and extraction industries, one conducts geological surveys that will<br>lead to the opening of a mine, or to the decision to drill. Then, the owner<br>of the mine, or the drilling platform can any day stop operations, and begin<br>them again later. An investment in the production of a movie, might lead to<br>sequels. The value of a sequel is the value of a compound option.<br>This project is divided into ve chapters; chapter one is the preliminar-<br>ies and introduction, chapter two is the Literature Review. Chapter three<br>will consist of nancial derivatives and compound options, where we’ll give<br>a full explanation of what compound option is all about. As in the case<br>of pricing and valuation of other nancial instruments (bonds or stocks) or<br>derivatives (futures or swaps), options too can be priced to avoid underesti-<br>mates or overestimates of the prices. As such, option pricing theory is one of<br>15<br>the cornerstones, and most successful theory in nance and economics as de-<br>scribed by Ross. Therefore, chapter four will deal with pricing of compound<br>options, where we are going to give some methods that are used in pricing<br>compound options, which is the main work of the project. Black-Scholes<br>formula for pricing compound options, forward valuation of compound op-<br>tions will also be discussed, where we use the Forkker-Planck equation and<br>backward Kolmogorov equation to obtain the formula for pricing compound<br>options. We will also discuss the binomial lattice model or binomial tree<br>model for pricing sequential compound options. Finally, chapter ve will<br>deal with applications.<br>16</p> <br><p></p>