Stochastic Calculus, Application of Real Analysis in Finance

, Application of Real Analysis in Finance Workshop for Young Mathematicians in Korea Seungkyu Lee Pohang University of Science and Technology August 4th, 2010

Contents 1 BINOMIAL ASSET PRICING MODEL

Contents 1 BINOMIAL ASSET PRICING MODEL 2

BINOMIAL ASSET PRICING MODEL Contents 1 BINOMIAL ASSET PRICING MODEL 2

BINOMIAL ASSET PRICING MODEL Binomial Asset Pricing Model Assumptions: Model stock prices in discrete time at each step, the stock price will change to one of two possible values begin with an initial positive stock price S 0 there are two positive numbers, d and u, with 0 < d < 1 < u such that at the next period, the stock price will be either ds 0 or us 0, d = 1/u.

BINOMIAL ASSET PRICING MODEL Binomial Asset Pricing Model Objectives: the definition of probability space and the concept of Arbitrage Pricing and its relation to Risk-Neutral Pricing is clearly illuminated Flip Coins Toss a coin and when we get a Head, the stock price moves up, but when we get a Tail, the price moves down. the price at time 1 by S 1 (H) = us 0 and S 1 (T) = ds 1

BINOMIAL ASSET PRICING MODEL Binomial Asset Pricing Model For the moment, let us assume that the third toss is the last one and denote by Ω = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} the set of all possible outcomes of the three tosses. The set Ω of all possible outcomes of a random experiment is called the sample space for the experiment, and the element ω of Ω are called sample points.

BINOMIAL ASSET PRICING MODEL Basic Probability Theory Probability space (Ω, F, P) like (R, B, µ), Lebesgue Measure Ω: the set of all possible realizations of the stochastic economy Ω = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} ω: a sample path, ω Ω F t : the sigma field of distinguishable events at time t F 2 = {, {HHH, HHT}, {HTH, HTT}, {THH, THT}, {TTH, TTT},..., Ω} P: a probability measure defined on the elements of F t

BINOMIAL ASSET PRICING MODEL Probability Spaces Definition If Ω is a given set, then a σ-algebra F on Ω is a family F of subsets of Ω with the following properties: F F F F C F, where F C = Ω \ F is the complement of F in Ω A 1, A 2,... F A := i=1 A i F The pair (Ω, F) is called a measurable space.

BINOMIAL ASSET PRICING MODEL Probability Spaces Definition A Probability measure P on a measurable space (Ω, F) is a function P : F [0, 1] such that P( ) = 0, P(Ω) = 1 if A 1, A 2,... F and {A i } i=1 is disjoint then ( ) P A i = P(A i ) i=1 i=1

BINOMIAL ASSET PRICING MODEL Random Variable Definition If (Ω, F, P) is a given probability space, then a function Y : Ω R n is called F-measurable if Y 1 (U) := {ω Ω; Y(ω) U} F for all open sets U R n (or, equivalently, for all Borel sets U R n ). In the following we let (Ω, F, P) denote a given complete probability space. A random variable X is an F-measurable function X : Ω R n.

BINOMIAL ASSET PRICING MODEL Arbitrage Price of Call Options A money market with interest rate r $1 invested in the money market $(1 + r) in the next period d < (1 + r) < u

BINOMIAL ASSET PRICING MODEL Arbitrage Price of Call Options A money market with interest rate r $1 invested in the money market $(1 + r) in the next period d < (1 + r) < u European call option with strike price K > 0 and expiration time 1 this option confers the right to buy the stock at time 1 for K dollars so is worth max[s 1 K, 0] denote V 1 (ω) = (S 1 (ω) K) + max[s 1 (ω) K] the value(payoff) at expiration. Compute the arbitrage price of the call option at time zero, V 0.

BINOMIAL ASSET PRICING MODEL Arbitrage Price of Call Options Suppose at time zero you sell the call option for V 0 dollars. ω 1 = H you should pay off (us 0 K) + ω 1 = T you should pay off (ds 0 K) + At time 0, we don t know the value of ω 1. Making replicating portfolio: V 0 0 S 0 dollars in the money market. 0 shares of stock the value of the portfolio is V 0 at time 0. the value should be (S 1 K) + at time 1.

BINOMIAL ASSET PRICING MODEL Arbitrage Price of Call Options Thus, V 1 (H) = 0 S 1 (H) + (1 + r)(v 0 0 S 0 ) (1) V 1 (T) = 0 S 1 (T) + (1 + r)(v 0 0 S 0 ) (2)

BINOMIAL ASSET PRICING MODEL Arbitrage Price of Call Options Thus, V 1 (H) = 0 S 1 (H) + (1 + r)(v 0 0 S 0 ) (1) V 1 (T) = 0 S 1 (T) + (1 + r)(v 0 0 S 0 ) (2) Subtracting (2) from (1), we obtain V 1 (H) V 1 (T) = 0 (S 1 (H) S 1 (T)), (3) so that 0 = V 1(H) V 1 (T) S 1 (H) S 1 (T) (4)

BINOMIAL ASSET PRICING MODEL Arbitrage Price of Call Options Substitute (4) into either (1) or (2) and solve for V 0 ; V 0 = 1 [ 1 + r d 1 + r u d V 1(H) + ] u (1 + r) V 1 (T) u d (5)

BINOMIAL ASSET PRICING MODEL Arbitrage Price of Call Options Substitute (4) into either (1) or (2) and solve for V 0 ; V 0 = 1 [ 1 + r d 1 + r u d V 1(H) + ] u (1 + r) V 1 (T) u d (5) Simply, where V 0 = 1 1 + r [ pv 1(H) + qv 1 (T)] (6) p = 1 + r d u (1 + r), q = = 1 p u d u d We can regard them as probabilities of H and T, respectively. They are the risk-neutral probabilities.

Contents 1 BINOMIAL ASSET PRICING MODEL 2

Problem - The Movement of Stock Price A risky investment(e.g. a stock), where the price X(t) per unit at time t satisfies a stochastic differential equation: dx dt = b(t, X t) + σ(t, X t ) noise, (7) where b and σ are some given functions.

dx dt = b(t, X t) + σ(t, X t )W t

dx dt = b(t, X t) + σ(t, X t )W t Based on many situations, one is led to assume that the noise, W t, has these properties: t 1 t 2 W t1 and W t2 are independent. {W t } is stationary, i.e. the (joint)distribution of {W t1 +t,..., W tk +t} does not depend on t. E[W t ] = 0 for all t. Let 0 = t 0 < t 1 < < t m = t and consider a discrete version of (7): X k+1 X k = b(t k, X k ) t k + σ(t k, X k )W k t k, (8) where X j = X(t j ), W k = W tk, t k = t k+1 t k.

dx dt = b(t, X t) + σ(t, X t )W t Replace W k k by V k = V tk+1 V tk, where {V t } t 0 is some suitable stochastic process.

dx dt = b(t, X t) + σ(t, X t )W t Replace W k k by V k = V tk+1 V tk, where {V t } t 0 is some suitable stochastic process. It can be easily proved that {V t } t 0 satisfies the four properties which define the standard Brownian Motion, {B t } t 0 : B 0 = 0. The increments of B t are independent; i.e. for any finite set of times 0 t 1 < t 2 < < t n < T the random variables B t2 B t1, B t3 B t2,..., B tn B tn 1 are independent. For any 0 s t < T the increment B t B s has the Gaussian distribution with mean 0 and variance t s. For all w in a set of probability one, B t (w) is a continuous function of t.

dx dt = b(t, X t) + σ(t, X t )W t Thus we put V t = B t and obtain from (8): k 1 k 1 X k = X 0 + b(t j, X j ) t j + σ(t j, X j ) B j. (9) j=0 j=0

dx dt = b(t, X t) + σ(t, X t )W t Thus we put V t = B t and obtain from (8): k 1 k 1 X k = X 0 + b(t j, X j ) t j + σ(t j, X j ) B j. (9) j=0 When t j 0, by applying the usual integration notation, we should obtain X t = X 0 + t 0 b(s, X s )ds + j=0 t 0 σ(s, X s )db s (10) Now, in the remainder of this chapter we will prove the existence, in a certain sense, of t 0 f(s, ω)db s (ω) where B t (ω) is 1-dim l Brownian motion.

Construction of the Itô Integral It is reasonable to start with a definition for a simple class of functions f and then extend by some approximation procedure. Thus, let us first assume that f has the form φ(t, ω) = j 0 e j (ω) X [j 2 n,(j+1)2 n )(t), (11) where X denotes the characteristic (indicator) function.

Construction of the Itô Integral In general, as we did in Real Analysis, it is natural to approximate a given function f(t, ω) by f(t j, ω) X [tj,t j+1 )(t) j where the points t j belong to the intervals [t j, t j+1 ], specifically in Itô Integral, t j = t j, and then define t 0 f(s, ω)db s(ω).

Construction of the Itô Integral Definition The Itô integral of f is defined by t 0 f(s, ω)db s (ω) = lim n t 0 φ n (s, ω)db s (ω) (limit in L 2 (P)) where {φ n } is a sequence of elementary functions such that (13) [ t ] E (f(s, ω) φ n (s, ω)) 2 dt 0 as n. (14) 0 Note that such a sequence {φ n } satisfying (14) exists.

Example (proof) t 0 B s db s = 1 2 B2 t 1 2 t.

Example t 0 B s db s = 1 2 B2 t 1 2 t. (proof) Put φ n (s, ω) = B j (ω) X [tj,t j+1 )(s), where B j = B tj. Then

Example t 0 B s db s = 1 2 B2 t 1 2 t. (proof) Put φ n (s, ω) = B j (ω) X [tj,t j+1 )(s), where B j = B tj. Then [ t ] E (φ n B s ) 2 ds 0 = E j = j tj+1 t j tj+1 t j (B j B s ) 2 ds (s t j )ds = 1 2 (t j+1 t j ) 2 0 as t j 0

Example t 0 B s db s = lim t j 0 t 0 φ n db s = lim B j B j. t j 0 j

Example Now t 0 B s db s = lim t j 0 t 0 φ n db s = lim B j B j. t j 0 (B 2 j ) = B2 j+1 B2 j = (B j+1 B j ) 2 + 2B j (B j+1 B j ) = ( B j ) 2 + 2B j B j, j

Example Now t 0 B s db s = lim t j 0 t 0 φ n db s = lim B j B j. t j 0 (B 2 j ) = B2 j+1 B2 j = (B j+1 B j ) 2 + 2B j (B j+1 B j ) = ( B j ) 2 + 2B j B j, j and therefore, B 2 t = j (B 2 j ) = j ( B j ) 2 + 2 j B j B j

Example Now t 0 B s db s = lim t j 0 t 0 φ n db s = lim B j B j. t j 0 (B 2 j ) = B2 j+1 B2 j = (B j+1 B j ) 2 + 2B j (B j+1 B j ) = ( B j ) 2 + 2B j B j, j and therefore, B 2 t = j (B 2 j ) = j ( B j ) 2 + 2 j B j B j or B j B j = 1 2 B2 t 1 ( B 2 j ) 2. j j

The Itô Formula The previous example illustrates that the basic definition of Itô integrals is not very useful when we try to evaluate a given integral as ordinary Riemann integrals without the fundamental theorem of calculus plus the chain rule in the explicit calculations. It turns out that it is possible to establish an Itô integral version of the chain rule, called the Itô formula.

The Itô Formula Theorem Let X t be an Itô process given by dx t = udt + vdb t. Let g(t, x) C 2 ([0, ) R). Then Y t = g(t, X t ) is again an Itô process, and dy t = g t (t, X t)dt + g x (t, X t)dx t + 1 2 g 2 x 2 (t, X t) (dx t ) 2, (15) where (dx t ) 2 = (dx t ) (dx t ) is computed according to the rules dt dt = dt db t = db t dt = 0, db t db t = dt.

Example, Again Calculate the integral I = t 0 B s db s

Example, Again Calculate the integral I = t 0 B s db s Choose X t = B t and g(t, x) = 1 2 x2. Then, Y t = 1 2 B2 t.

Example, Again Calculate the integral I = t 0 B s db s Choose X t = B t and g(t, x) = 2 1x2. Then, Y t = 1 2 B2 t. By Itô s formula, ( ) 1 d 2 B2 t = B t db t + 1 2 dt. In other words, 1 2 B2 t = t 0 B s db s + 1 2 t.