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Itˆo Calculus and Derivative Pricing with Risk-Neutral

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Itˆo Calculus and Derivative Pricing with Risk-Neutral ITO^ CALCULUS AND DERIVATIVE PRICING WITH RISK-NEUTRAL MEASURE MAX CYTRYNBAUM Abstract. This paper will develop some of the fundamental results in the theory of Stochastic Differential Equations (SDE). After a brief review of stochastic processes and the It^oCalculus, we set our sights on some of the more advanced machinery needed to work with SDE. Chief among our results are the Feynman-Kac Theorem, which establishes a link between stochastic methods and the classic PDE approach, and the Girsanov theorem, which allows us to change the drift of an It^o diffusion by switching to an equivalent martingale measure. These results are also valuable on a practical level, and we will consider some of their applications to derivative pricing calculations in mathematical finance. Contents 1. Introduction and Motivation 1 2. It^oCalculus and Brownian Motion 1 3. Stochastic Differential Equations 5 4. It^oDiffusions 7 5. Feynman-Kac and PDE 11 6. The Girsanov Theorem 13 7. Risk-Neutral Measure and Black-Scholes 17 Acknowledgments 19 References 20 1. Introduction and Motivation In many applications, the evolution of a system with some fixed initial state is subject to random perturbations from the environment. For example, a stock price may rise in the long term yet be subject to seemingly random short-term fluctuations. Similarly, a small object in a liquid suspension may experience random impulses from collisions with surrounding water particles. This leads us to the consideration of a differential equation of the following form: dX t = b(t; X ) + σ(t; X ) · W (1.1) dt t t t where Wt represents \white noise". The way to formulate this precisely is through It^ocalculus and Brownian motion. We will briefly review some of the basic results from It^ocalculus. We assume the reader has a working knowledge of measure theoretic probability and is comfortable with the properties of conditional expectation. 2. Ito^ Calculus and Brownian Motion We will closely follow the exposition in [4]. Let (Ω; F;P ) be a probability space. Definition 2.1 (Filtration). We say that fFtgt≥0 is a filtration on the space (Ω; F;P ) if (1) Ft is sub σ-algebra of F for each t 1 2 MAX CYTRYNBAUM (2) Ft ⊂ Fs for all t ≤ s. n A measurable function f : [0; 1) × Ω ! R is said to be adapted to the filtration fFtgt≥0 if ! ! f(t; !) is an Ft measurable function for each t. The sub σ-algebra Ft represents the amount of information available to us at time t. Intuitively, this says that the value of the Ft-adapted random variable ft(!) can be determined by only using information in the σ-algebras up to Ft. This inability to see into the future is important in mathematical finance, where f(t; !) may represent some investment strategy and Ft may be the price information available up to time t. Thus, we cannot use future price information to determine our strategy. Definition 2.2 (Brownian Motion). We call a stochastic process B(t; !) = Bt(!) a version of Brownian motion starting at x 2 R when (1) B0 = x for all ! (2) For all times 0 ≤ t1 ≤ t2 ≤ · · · ≤ tn, B(t2) − B(t1);B(t2) − B(t3); ··· ;B(tn) − B(tn−1) are independent random variables (3) B(t + h) − B(t) is normally distributed with mean 0 and variance h for all t; h 2 [0; 1] (4) B(t) is P -a.s. t-continuous Similarly, we may define Bt 2 Rn to be a Brownian motion if B(t) = (B1(t);:::;Bn(t)) where each Bi(t) is a version of 1-dimensional brownian motion and all the Bi(t) are independent. For a construction of Brownian motion, we refer the reader to [5]. Definition 2.3 (Martingale). A stochastic process fMtg is said to be a martingale with respect to the filtration fFtgt≥0 if (1) fMtg is adapted to fFtgt≥0 (2) E[jMtj] < 1 for all t (3) For all 0 ≤ s < t and for a.s. ! E[MtjFs] = Ms A martingale neither increases nor decreases on average and can thus be thought of as a model of a fair game. From the above it is also clear that for any t we have E[Mt] = E[M0]. In what follows, we let fFtgt≥0 be the filtration generated by fBtg (i.e. Ft is the σ-algebra generated by ! ! B(t; !) for each t.) Then using independence of increments and properties of conditional expectation, we calculate: E[BtjFs] = E[Bt − BsjFs] + E[BsjFs] = E[B(t) − B(s)] + B(s) = B(s) so that Brownian motion is a martingale with respect to its own filtration. For more information on martingales in discrete-time, see [7]. We now attempt to find a precise interpretation of the SDE (1.1). To do this, we rewrite (1.1) in the more suggestive form: dXt = b(t; Xt)dt + σ(t; Xt)dBt (2.1) where we have interpreted white noise in terms of brownian motion as Wtdt = dBt, which leads to the tentative integral equation: Z Z Xt = b(t; Xt)dt + σ(t; Xt)dBt (2.2) Thinking in terms of Riemann-Stieltjes integration, it seems natural to interpret an integral of the R form f(t; !)dBt(!) as a limit of sums of the form: X f(tj;!)∆Btj (!) j ITO^ CALCULUS AND DERIVATIVE PRICING WITH RISK-NEUTRAL MEASURE 3 Intuitively, the increments f(tj;!)∆Btj (!) reflect the random change in Xt due to white noise. We now make this notion precise. Definition 2.4 (It^oIntegrable). We say that a function f(t; !) is It^oIntegrable on [S; T ] if the following conditions are satisfied: (1) f(t; !) is B ⊗ F measurable (2) f(t; ·) is Ft-measurable for each t Z T (3) E f 2(t; !)dt < 1 S In this case we write f 2 V[S; T ], following the notation in [4]. We say that a function φ(t; !) is X elementary if φ 2 V[S; T ] and φ(t; !) = φ(tj;!)X[tj ;tj+1] where ftjg is a partion of the interval j [S; T ]. We define: Z T X φ dBt := φ(tj)∆Bj S j where ∆Bj := B(tj+1) − B(tj). This yields the following Lemma 2.5 (It^oIsometry). If φ is elementary and φ 2 V[S; T ], then we have "Z T 2# Z T 2 E f(t; !)dBt = E f (t; !)dt S S Proof. 2 " 2# 20 1 3 Z T X E f(t; !)dBt = E 4@ f(tj;!)∆BjA 5 S j 2 3 X = E 4 f(ti;!)f(tj;!)∆Bj∆Bi5 i;j Now since the the ftj are Ftj adapted and by the independence of increments for Brownian motion, we get that ∆Bj is independent from ∆Biftj fti when i < j. Since E(∆Bj) = 0, E(ftj fti ∆Bj∆Bi) = 0 for i < j. Then the above simplifies to: 2 3 2 3 Z T X 2 2 X 2 X 2 2 E 4 ftj (∆Bj) 5 = ∆tjE[ftj ] = E 4 ftj ∆tj5 = E f (t; !)dt j j j S The first identity uses the fact the ∆Bj and ftj are independent and the result from definition (2.2) 2 that E(∆Bj ) = ∆tj. We can now use this result to extend the It^ointegral to all of V[S; T ]. Specifically, given f 2 V[S; T ] let fφng be a sequence of elementary functions such that: Z T 2 E (f(t; !) − φn(t; !)) dt ! 0 as n ! 1 (2.3) S As the reader can check, Lemma 2.5 and equation (2.3) imply that fφng is a Cauchy sequence in L2(P ), where P is the law of standard Brownian motion. Since L2(P ) is complete, we can define: Z T Z T f(t; !)dBt(!) = lim φn(t; !)dBt(!) (2.4) S n!1 S 4 MAX CYTRYNBAUM where the limit is taken in L2(P ). Standard analysis arguments can be used to show that this limit is independent of the chosen sequence fφng. For a closer look at the existence of such a sequence R fφng, see [4]. We now provide a brief review of some of the properties of fdBt: + Proposition 2.6 (Properties of the It^oIntegral). Let [S; T ] ⊂ R and let f 2 V[S; T ]. Z T (1) f ! fdBt is a linear operator on V[S; T ] S Z T (2) E fdBt = 0 S Z T (3) fdBt is FT -measurable S "Z T 2# Z T 2 (4) E fdBt = E f dt S S These all follow by proving the statement for elementary functions and taking limits using equa- tion (2.4). With a little more work we can show Theorem 2.7 (Martingale Property). Let f 2 V[0;T ], 0 ≤ t ≤ T . R t (1) There exists a t-continuous version of 0 fdBs R t (2) Put Mt(!) = 0 fdBs(!). Then Mt is a fFtgt≥0 martingale. Theorem 2.7(2) follows from the fact that Bt is a martingale with respect to its own filtration. We refer the reader to page 32 of [4] for the proof. The above extends naturally to n dimensions. Let n;m m V [S; T ] be the space of n × m matrices v(t; !) such that each vij(t; !) 2 V[S; T ]. Let Bt 2 R . Then we define 0 1 T T v11 ··· v1m Z Z . vdBt = B . C dBt S S @ A vn1 ··· vnm to be the n × 1 vector with ith component m X Z T vik(t; !)dBk(t; !) k=1 S Definition 2.8 (It^oProcess).
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