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Processes A • X = X(t) { } is a of random variables. Stochastic Processes and X(t) (or X ) is a random variable for each time t and • t is usually called the state of the process at time t. A realization of X is called a sample path. • A sample path defines an ordinary function of t. •

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 396 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 398

Stochastic Processes (concluded) Of all the intellectual hurdles which the human mind If the times t form a countable set, X is called a • has confronted and has overcome in the last discrete-time stochastic process or a time series. fifteen hundred years, the one which seems to me In this case, subscripts rather than parentheses are to have been the most amazing in character and • usually employed, as in the most stupendous in the scope of its consequences is the one relating to X = X . { n } the problem of motion. If the times form a continuum, X is called a — Herbert Butterfield (1900–1979) • continuous-time stochastic process.

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 397 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 399 Random Walks

The binomial model is a in disguise. • Random Walk with Drift Consider a particle on the integer line, 0, 1, 2,... . • ± ± Xn = µ + Xn−1 + ξn. In each time step, it can make one move to the right • with probability p or one move to the left with ξn are independent and identically distributed with zero • probability 1 p. mean. − – This random walk is symmetric when p = 1/2. Drift µ is the expected change per period. • Connection with the BOPM: The particle’s position Note that this process is continuous in space. • • denotes the cumulative number of up moves minus that of down moves.

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 400 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 402

Martingalesa X(t), t 0 is a martingale if E[ X(t) ] < for • { ≥ } | | ∞ Position t 0 and ≥ 4 E[ X(t) | X(u), 0 ≤ u ≤ s ]= X(s), s ≤ t. (39) 2 In the discrete-time setting, a martingale means • Time 20 40 60 80 E[ X X ,X ,... ,X ]= X . (40) n+1 | 1 2 n n -2 X can be interpreted as a gambler’s fortune after the • n -4 nth gamble. -6 Identity (40) then says the expected fortune after the • -8 (n + 1)th gamble equals the fortune after the nth gamble regardless of what may have occurred before. aThe origin of the name is somewhat obscure.

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 401 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 403 Martingales (concluded) Still a Martingale? (continued) A martingale is therefore a notion of fair games. Well, no.a • • Apply the law of iterated conditional expectations to Consider this random walk with drift: • • both sides of Eq. (40) on p. 403 to yield Xi−1 + ξi, if i is even, Xi = E[ Xn ]= E[ X1 ] (41)  X , otherwise.  i−2 for all n. Above, ξ are random variables with zero mean. • n Similarly, E[ X(t)]= E[ X(0)] in the continuous-time a B89201033 • Contributed by Mr. Zhang, Ann-Sheng ( ) on April 13, case. 2005.

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 404 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 406

Still a Martingale? (concluded) It is not hard to see that Still a Martingale? • Xi−1, if i is even, Suppose we replace Eq. (40) on p. 403 with E[ Xi Xi−1 ]= • |  X , otherwise.  i−1 E[ Xn+1 Xn ]= Xn. | Hence it is a martingale by the “new” definition. • It also says past history cannot affect the future. • But • But is it equivalent to the original definition?a • Xi−1, if i is even, a Contributed by Mr. Hsieh, Chicheng (M9007304) on April 13, 2005. E[ Xi ... ,Xi−2,Xi−1 ]= |  X , otherwise.  i−2 Hence it is not a martingale by the original definition. •

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 405 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 407 Example Consider the stochastic process Probability Measure (continued) • n A stochastic process X(t), t 0 is a martingale with Z X , n 1 , • { ≥ } { n ≡ i ≥ } respect to information sets It if, for all t 0, i=1 { } ≥ X E[ X(t) ] < and | | ∞ where Xi are independent random variables with zero E[ X(u) I ]= X(t) mean. | t This process is a martingale because for all u > t. • E[ Z Z ,Z ,... ,Z ] The discrete-time version: For all n> 0, n+1 | 1 2 n • = E[ Zn + Xn+1 Z1,Z2,... ,Zn ] E[ X I ]= X , | n+1 | n n = E[ Zn Z1,Z2,... ,Zn ]+ E[ Xn+1 Z1,Z2,... ,Zn ] | | given the information sets In . { } = Zn + E[ Xn+1 ]= Zn.

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 408 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 410

Probability Measure A martingale is defined with respect to a probability • measure, under which the expectation is taken. Probability Measure (concluded) – A probability measure assigns probabilities to states The above implies E[ X I ]= X for any m> 0 • n+m | n n of the world. by Eq. (15) on p. 137.

A martingale is also defined with respect to an – A typical In is the price information up to time n. • information set. – Then the above identity says the FVs of X will not – In the characterizations (39)–(40) on p. 403, the deviate systematically from today’s value given the information set contains the current and past values price history. of X by default. – But it needs not be so.

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 409 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 411 Example Martingale Pricing

Consider the stochastic process Zn nµ, n 1 . Recall that the price of a European option is the • { − ≥ } • – Z n X . expected discounted future payoff at expiration in a n ≡ i=1 i risk-neutral economy. – X1,X2P,... are independent random variables with mean µ. This principle can be generalized using the concept of • Now, martingale. • Recall the recursive valuation of European option via E[ Zn+1 (n + 1) µ X1,X2,...,Xn ] • − | = E[ Z X ,X ,... ,X ] (n + 1) µ C =[ pCu + (1 p) Cd ]/R. n+1 | 1 2 n − − = Z + µ (n + 1) µ n − – p is the risk-neutral probability. = Z nµ. n − – $1 grows to $R in a period.

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 412 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 414

Martingale Pricing (continued) Example (concluded) Let C(i) denote the value of the option at time i. • Define Consider the discount process • • In X1,X2,... ,Xn . ≡ { } C(i)/Ri, i = 0, 1,... ,n . Then { } • Z nµ, n 1 Then, { n − ≥ } • is a martingale with respect to I . C(i + 1) pCu + (1 p) Cd C { n } E C(i)= C = − = . Ri+1 Ri+1 Ri  

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 413 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 415 Martingale Pricing (continued) It is easy to show that Martingale Pricing (continued) • C(k) C Equation (43) holds for all assets, not just options. E C(i)= C = , i k. (42) • Rk Ri ≤   When interest rates are stochastic, the equation becomes • This formulation assumes: a C(i) C(k) • = Eπ , i k. (44) 1. The model is Markovian in that the distribution of M(i) i M(k) ≤   the future is determined by the present (time i ) and – M(j) is the balance in the money market account at not the past. time j using the rollover strategy with an initial 2. The payoff depends only on the terminal price of the investment of $1. underlying asset (Asian options do not qualify). – So it is called the bank account process. aContributed by Mr. Wang, Liang-Kai (Ph.D. student, ECE, Univer- B90902081 It says the discount process is a martingale under π. sity of Wisconsin-Madison) and Mr. Hsiao, Huan-Wen ( ) on • May 3, 2006.

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 416 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 418

Martingale Pricing (concluded) If interest rates are stochastic, then M(j) is a random • Martingale Pricing (continued) variable. In general, the discount process is a martingale in that • – M(0) = 1. C(k) C(i) – M(j) is known at time j 1. Eπ = , i k. (43) − i Rk Ri ≤   Identity (44) on p. 418 is the general formulation of π • – Ei is taken under the risk-neutral probability risk-neutral valuation. conditional on the price information up to time i. Theorem 14 A discrete-time model is arbitrage-free if and This risk-neutral probability is also called the EMM, or • only if there exists a probability measure such that the the equivalent martingale (probability) measure. discount process is a martingale. This probability measure is called the risk-neutral probability measure.

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 417 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 419 Futures Price under the BOPM Futures prices form a martingale under the risk-neutral • Martingale Pricing and Numeraire (concluded) probability. Choose S as numeraire. – The expected futures price in the next period is • Martingale pricing says there exists a risk-neutral 1 d u 1 • p Fu + (1 p ) Fd = F − u + − d = F probability π under which the relative price of any asset f − f u d u d  − −  C is a martingale: (p. 380). C(i) C(k) = Eπ , i k. Can be generalized to S(i) i S(k) ≤ •   F = Eπ[ F ], i k, – S(j) denotes the price of S at time j. i i k ≤ So the discount process remains a martingale. where Fi is the futures price at time i. • It holds under stochastic interest rates. •

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 420 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 422

Martingale Pricing and Numeraire Example The martingale pricing formula (44) on p. 418 uses the Take the binomial model with two assets. • • money market account as numeraire.a In a period, asset one’s price can go from S to S or • 1 – It expresses the price of any asset relative to the S2. money market account. In a period, asset two’s price can go from P to P or • 1 The money market account is not the only choice for P2. • numeraire. Assume • S S S Suppose asset S’s value is positive at all times. 1 < < 2 • P1 P P2 a Leon Walras (1834–1910). to rule out arbitrage opportunities.

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 421 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 423 Example (continued) Example (concluded)

For any derivative security, let C1 be its price at time It is easy to verify that • • one if asset one’s price moves to S . 1 C C C = p 1 + (1 p) 2 . Let C be its price at time one if asset one’s price P P1 − P2 • 2 moves to S2. – Above, (S/P ) (S /P ) Replicate the derivative by solving p − 2 2 . • ≡ (S /P ) (S /P ) 1 1 − 2 2 αS1 + βP1 = C1, The derivative’s price using asset two as numeraire is • αS2 + βP2 = C2, thus a martingale under the risk-neutral probability p.

using α units of asset one and β units of asset two. The expected returns of the two assets are irrelevant. •

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 424 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 426

Brownian Motiona Brownian motion is a stochastic process X(t), t 0 • { ≥ } Example (continued) with the following properties. This yields 1. X(0) = 0, unless stated otherwise. • P C P C S C S C 2. for any 0 t0 < t1 < < tn, the random variables α = 2 1 − 1 2 and β = 2 1 − 1 2 . ≤ ··· P2S1 P1S2 S2P1 S1P2 − − X(t ) X(t − ) k − k 1 The derivative costs • for 1 k n are independent.b ≤ ≤ C = αS + βP 3. for 0 s < t, X(t) X(s) is normally distributed P S PS PS P S ≤ − = 2 − 2 C + 1 − 1 C . with mean µ(t s) and σ2(t s), where µ P S P S 1 P S P S 2 − − 2 1 − 1 2 2 1 − 1 2 and σ = 0 are real numbers. 6 aRobert Brown (1773–1858). bSo X(t) − X(s) is independent of X(r) for r ≤ s

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 425 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 427 Brownian Motion (concluded) Brownian Motion Is a Random Walk in Continuous Time Such a process will be called a (µ, σ) Brownian motion • with drift µ and variance σ2. Claim 1 A (µ, σ) Brownian motion is the limiting case of The existence and uniqueness of such a process is random walk. • a guaranteed by Wiener’s theorem. A particle moves ∆x to the left with probability 1 p. • − Although Brownian motion is a of t • It moves to the right with probability p after ∆t time. with probability one, it is almost nowhere differentiable. • Assume n t/∆t is an integer. The (0, 1) Brownian motion is also called the Wiener • ≡ • Its position at time t is process. • aNorbert Wiener (1894–1964). Y (t) ∆x (X + X + + X ) . ≡ 1 2 ··· n

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 428 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 430

Brownian Motion as Limit of Random Walk Example (continued) If X(t), t 0 is the Wiener process, then (continued) • { ≥ } • X(t) X(s) N(0, t s). – Here − ∼ − A (µ, σ) Brownian motion Y = Y (t), t 0 can be • { ≥ } +1 ifthe ith move is to the right, expressed in terms of the Wiener process: Xi ≡  1 ifthe ith move is to the left.  − Y (t)= µt + σX(t). (45) – Xi are independent with Note that Y (t + s) Y (t) N(µs,σ2s). Prob[ Xi =1]= p = 1 Prob[ Xi = 1]. • − ∼ − − Recall E[ X ] = 2p 1 and Var[ X ] = 1 (2p 1)2. • i − i − −

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 429 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 431 Brownian Motion as Limit of Random Walk Geometric Brownian Motion (continued) Let X X(t), t 0 be a Brownian motion process. • ≡ { ≥ } Therefore, • The process • X(t) E[ Y (t)]= n(∆x)(2p 1), Y (t) e , t 0 , − { ≡ ≥ } Var[ Y (t)]= n(∆x)2 1 (2p 1)2 . is called geometric Brownian motion. − −   Suppose further that X isa (µ, σ) Brownian motion. With ∆x σ√∆t and p [1+(µ/σ)√∆t ]/2, • • ≡ ≡ X(t) N(µt, σ2t) with moment generating function E[ Y (t)] = nσ√∆t (µ/σ)√∆t = µt, • ∼ sX(t) s µts+(σ2ts2/2) Var[ Y (t)] = nσ2∆t 1 (µ/σ)2∆t σ2t, E e = E [ Y (t) ]= e − → h i as ∆t 0.   from Eq. (16) on p 139. →

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 432 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 434

Brownian Motion as Limit of Random Walk (concluded) Thus, Y (t), t 0 converges to a (µ, σ) Brownian • { ≥ } motion by the . Geometric Brownian Motion (continued) Brownian motion with zero drift is the limiting case of In particular, • • symmetric random walk by choosing µ = 0. 2 E[ Y (t)]= eµt+(σ t/2), Note that Var[ Y (t)]= E Y (t)2 E[ Y (t)]2 • − Var[ Y (t + ∆t) Y (t)] 2µt +σ2t  σ2t − = e e 1 . 2 2 − =Var[ ∆xXn+1 ]=(∆x) Var[ Xn+1 ] σ ∆t.   × → Similarity to the the BOPM: The p is identical to the • probability in Eq. (23) on p. 235 and ∆x = ln u.

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 433 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 435 Y(t) Geometric Brownian Motion (concluded) 6 Then • n 5 ln Yn = ln Xi 4 i=1 X 3 is a sum of independent, identically distributed random 2 variables. 1 Thus ln Y , n 0 is approximately Brownian motion. Time (t) n 0.2 0.4 0.6 0.8 1 • { ≥ } – And Y , n 0 is approximately geometric -1 { n ≥ } Brownian motion.

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 436 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 438

Geometric Brownian Motion (continued) It is useful for situations in which percentage changes • are independent and identically distributed. Let Y denote the stock price at time n and Y = 1. • n 0 Assume relative returns Continuous-Time Financial • Yi Xi ≡ Yi−1 are independent and identically distributed.

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 437 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 439 Stochastic (concluded) Typical requirements for X in financial applications are: • A proof is that which convinces a reasonable man; t 2 – Prob[ 0 X (s) ds < ]=1 for all t 0 or the a rigorous proof is that which convinces an t ∞ ≥ stronger E[ X2(s)] ds < . unreasonable man. R 0 ∞ – The information set at time t includes the history of — (1914–1984) R X and W up to that point in time.

The pursuit of mathematics is a – But it contains nothing about the evolution of X or divine madness of the human spirit. W after t (nonanticipating, so to speak). — Alfred North Whitehead (1861–1947), – The future cannot influence the present. Science and the Modern World X(s), 0 s t is independent of • { ≤ ≤ } W (t + u) W (t),u> 0 . { − }

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 440 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 442

Stochastic Integrals Use W W (t), t 0 to denote the Wiener process. • ≡ { ≥ } Ito The goal is to develop integrals of X from a class of • a A theory of stochastic integration. stochastic processes, • t As with calculus, it starts with step functions. • It(X) X dW, t 0. ≡ 0 ≥ A stochastic process X(t) is simple if there exist Z • { } 0= t0 < t1 < such that It(X) is a random variable called the stochastic integral ··· • of X with respect to W . X(t)= X(t − ) for t [ t − , t ), k = 1, 2,... k 1 ∈ k 1 k The stochastic process I (X), t 0 will be denoted • { t ≥ } for any realization (see figure next page). by X dW .

aIto (1915–).R

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 441 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 443 Ito Integral (continued) Xa t f Let X = X(t), t 0 be a general stochastic process. • { ≥ } Then there exists a random variable I (X), unique • t almost certainly, such that It(Xn) converges in

probability to It(X) for each sequence of simple

stochastic processes X1,X2,... such that Xn converges in probability to X. If X is continuous with probability one, then I (X ) t • t n t0 t1 t2 t3 t4 t5 converges in probability to It(X) as δ max ≤ ≤ (t t − ) goes to zero. n ≡ 1 k n k − k 1

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 444 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 446

Ito Integral (concluded) Ito Integral (continued) It is a fundamental fact that X dW is continuous The Ito integral of a simple process is defined as • • almost surely. R n−1 I (X) X(t )[ W (t ) W (t )], (46) The following theorem says the Ito integral is a t ≡ k k+1 − k • k=0 martingale. X where tn = t. A corollary is the mean value formula • – The integrand X is evaluated at tk, not tk+1. b E X dW = 0. Define the Ito integral of more general processes as a • " Za # limiting random variable of the Ito integral of simple stochastic processes. Theorem 15 The Ito integral X dW is a martingale. R

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 445 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 447 Discrete Approximation Recall Eq. (46) on p. 445. • The following simple stochastic process X(t) can be • { } X X used in place of X to approximate the stochastic Y$ t b X$ integral 0 X dW , R X(s) X(t − ) for s [ t − , t ), k = 1, 2,... ,n. ≡ k 1 ∈ k 1 k Noteb the nonanticipating feature of X. t t • – The information up to time s, (a) (b) b X(t),W (t), 0 t s , { ≤ ≤ } cannot determineb the future evolution of X or W .

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 448 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 450

Ito Process Discrete Approximation (concluded) The stochastic process X = X , t 0 that solves • { t ≥ } Suppose we defined the stochastic integral as • t t X = X + a(X ,s) ds + b(X ,s) dW , t 0 n−1 t 0 s s s ≥ Z0 Z0 X(tk+1)[ W (tk+1) W (tk)]. − is called an Ito process. kX=0 – X is a scalar starting point. Then we would be using the following different simple 0 • – a(X , t): t 0 and b(X , t): t 0 are stochastic process in the approximation, { t ≥ } { t ≥ } stochastic processes satisfying certain regularity Y (s) X(t ) for s [ t − , t ), k = 1, 2,... ,n. ≡ k ∈ k 1 k conditions.

Thisb clearly anticipates the future evolution of X. The terms a(Xt, t) and b(Xt, t) are the drift and the • • diffusion, respectively.

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 449 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 451 Ito Process (continued) Euler Approximation A shorthanda is the following stochastic differential The following approximation follows from Eq. (48), • • equation for the Ito differential dXt, X(tn+1)

dXt = a(Xt, t) dt + b(Xt, t) dWt. (47) =X(t )+ a(X(t ), t ) ∆t + b(X(t ), t ) ∆W (t ), b n n n n n n (49) – Or simply dXt = at dt + bt dWt. b b b where tn n∆t. This is Brownian motion with an instantaneous drift at ≡ • 2 and an instantaneous variance bt . It is called the Euler or Euler-Maruyama method. • X is a martingale if the drift at is zero by Theorem 15 Under mild conditions, X(t ) converges to X(t ). • • n n (p. 447). Recall that ∆W (tn) should be interpreted as a • b Paul Langevin (1904). W (t ) W (t ) instead of W (t ) W (t − ). n+1 − n n − n 1

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 452 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 454

Ito Process (concluded) More Discrete Approximations dW is normally distributed with mean zero and • Under fairly loose regularity conditions, approximation variance dt. • (49) on p. 454 can be replaced by An equivalent form to Eq. (47) is • X(tn+1) dXt = at dt + bt√dt ξ, (48) =X(t )+ a(X(t ), t ) ∆t + b(X(t ), t )√∆tY (t ). b n n n n n n where ξ N(0, 1). ∼ – Y b(t0),Y (t1),...b are independentb and identically This formulation makes it easy to derive Monte Carlo • distributed with zero mean and unit variance. simulation algorithms.

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 453 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 455 More Discrete Approximations (concluded) A simpler discrete approximation scheme: • Trading and the Ito Integral (concluded) X(tn+1) The equivalent Ito integral, • =X(t )+ a(X(t ), t ) ∆t + b(X(t ), t )√∆tξ. b n n n n n T T T GT (φ) φt dSt = φtµt dt + φtσt dWt, – Prob[ ξ = 1 ] = Prob[ ξ = 1]=1/2. ≡ 0 0 0 b b − b Z Z Z – Note that E[ ξ ] = 0 and Var[ ξ ]=1. measures the gains realized by the trading strategy over the period [0,T ]. This clearly defines a binomial model. • As ∆t goes to zero, X converges to X. • b

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 456 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 458

Ito’s Lemma Trading and the Ito Integral A smooth function of an Ito process is itself an Ito process. Consider an Ito process dS = µ dt + σ dW . • t t t t Theorem 16 Suppose f : R R is twice continuously – S is the vector of security prices at time t. → t differentiable and dX = at dt + bt dW . Then f(X) is the Let φ be a trading strategy denoting the quantity of Ito process, • t each type of security held at time t. f(Xt) t t Hence the stochastic process φ St is the value of the • t = f(X )+ f ′(X ) a ds + f ′(X ) b dW portfolio φ at time t. 0 s s s s t Z0 Z0 t 1 ′′ 2 φt dSt φt(µt dt + σt dWt) represents the change in the + f (X ) b ds • ≡ 2 s s value from security price changes occurring at time t. Z0 for t 0. ≥

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 457 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 459 Ito’s Lemma (continued) Ito’s Lemma (continued) In differential form, Ito’s lemma becomes Theorem 17 (Higher-Dimensional Ito’s Lemma) Let • W1,W2,... ,Wn be independent Wiener processes and ′ ′ 1 ′′ 2 df(X)= f (X) a dt + f (X) bdW + f (X) b dt. X (X ,X ,... ,X ) be a vector process. Suppose 2 ≡ 1 2 m (50) f : Rm R is twice continuously differentiable and X is → i an Ito process with dX = a dt + n b dW . Then Compared with calculus, the interesting part is the third i i j=1 ij j • df(X) is an Ito process with the differential, term on the right-hand side. P m 1 m m A convenient formulation of Ito’s lemma is df(X)= f (X) dX + f (X) dX dX , • i i 2 ik i k 1 i=1 i=1 k=1 df(X)= f ′(X) dX + f ′′(X)(dX)2. X X X 2 where f ∂f/∂x and f ∂2f/∂x ∂x . i ≡ i ik ≡ i k

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 460 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 462

Ito’s Lemma (continued) Ito’s Lemma (continued) We are supposed to multiply out • The multiplication table for Theorem 17 is (dX)2 =(a dt + bdW )2 symbolically according to • dWi dt dW dt × × dWk δik dt 0 dW dt 0 dt 0 0 dt 0 0 in which – The (dW )2 = dt entry is justified by a known result. 1 if i = k, δ = This form is easy to remember because of its similarity ik  • 0 otherwise. to the Taylor expansion.  

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 461 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 463 Geometric Brownian Motion Ito’s Lemma (continued) Consider the geometric Brownian motion process • Y (t) eX(t) Theorem 18 (Alternative Ito’s Lemma) Let ≡ W1,W2,... ,Wm be Wiener processes and – X(t) isa (µ, σ) Brownian motion. X (X1,X2,... ,Xm) be a vector process. Suppose – Hence dX = µ dt + σdW by Eq. (45) on p. 429. ≡ m f : R R is twice continuously differentiable and Xi is → As ∂Y/∂X = Y and ∂2Y/∂X2 = Y , Ito’s formula (50) an Ito process with dXi = ai dt + bi dWi. Then df(X) is the • on p. 460 implies following Ito process, 2 m 1 m m dY = Y dX + (1/2) Y (dX) df(X)= f (X) dX + f (X) dX dX . i i 2 ik i k = Y (µ dt + σdW )+(1/2) Y (µ dt + σdW )2 i=1 i=1 X X kX=1 = Y (µ dt + σdW )+(1/2) Y σ2 dt.

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 464 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 466

Ito’s Lemma (concluded) The multiplication table for Theorem 18 is Geometric Brownian Motion (concluded) • dW dt Hence × i • dY = µ + σ2/2 dt + σ dW. dWk ρik dt 0 Y dt 0 0 The annualized instantaneous rate of return is µ + σ2/2 • Here, ρ denotes the correlation between dW and not µ. • ik i dWk.

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 465 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 467 Product of Geometric Brownian Motion Processes

Let •

dY/Y = a dt + bdWY , Product of Geometric Brownian Motion Processes

dZ/Z = f dt + gdWZ . (concluded) ln U is Brownian motion with a mean equal to the sum Consider the Ito process U YZ. • • ≡ of the means of ln Y and ln Z. Apply Ito’s lemma (Theorem 18 on p. 464): • This holds even if Y and Z are correlated. dU = Z dY + Y dZ + dY dZ • Finally, ln Y and ln Z have correlation ρ. = ZY (adt + b dWY )+ Y Z(f dt + g dWZ ) •

+Y Z(adt + b dWY )(f dt + g dWZ )

= U(a + f + bgρ) dt + UbdWY + UgdWZ .

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 468 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 470

Product of Geometric Brownian Motion Processes (continued) Quotients of Geometric Brownian Motion Processes The product of two (or more) correlated geometric Suppose Y and Z are drawn from p. 468. • • Brownian motion processes thus remains geometric Let U Y/Z. Brownian motion. • ≡ We now show that Note that • • dU 2 =(a f + g bgρ) dt + bdWY gdWZ . Y = exp a b2/2 dt + bdW , U − − − − Y (51) 2 Z = exp f g /2 dt + gdWZ , − 2 2 Keep in mind that dWY and dWZ have correlation ρ. U = exp  a + f b + g /2 dt + bdWY + gdWZ . • −    

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 469 c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 471 Quotients of Geometric Brownian Motion Processes (concluded) The multidimensional Ito’s lemma (Theorem 18 on • p. 464) can be employed to show that

dU 2 2 3 2 = (1/Z) dY − (Y/Z ) dZ − (1/Z ) dY dZ +(Y/Z )(dZ) 2 = (1/Z)(aY dt + bY dWY ) − (Y/Z )(fZ dt + gZ dWZ ) 2 3 2 2 −(1/Z )(bgY Zρdt)+(Y/Z )(g Z dt)

= U(adt + b dWY ) − U(f dt + g dWZ ) 2 −U(bgρdt)+ U(g dt) 2 = U(a − f + g − bgρ) dt + UbdWY − UgdWZ .

c 2006 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 472