Itˆo Calculus and Derivative Pricing with Risk-Neutral

ITOˆ 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 ItôCalculus, 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 Itô 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. ItôCalculus and Brownian Motion 1 3. Stochastic Differential Equations 5 4. ItôDiffusions 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 Itôcalculus and Brownian motion. We will briefly review some of the basic results from Itôcalculus. We assume the reader has a working knowledge of measure theoretic probability and is comfortable with the properties of conditional expectation.

2. Itoˆ 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 {Ft}t≥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, ∞) × Ω → R is said to be adapted to the ﬁltration {Ft}t≥0 if ω → f(t, ω) 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(ω) 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, ω) 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, ω) = Bt(ω) a version of Brownian motion starting at x ∈ R when (1) B0 = x for all ω (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 ∈ [0, ∞] (4) B(t) is P -a.s. t-continuous

Similarly, we may deﬁne Bt ∈ 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 {Mt} is said to be a martingale with respect to the ﬁltration {Ft}t≥0 if

(1) {Mt} is adapted to {Ft}t≥0 (2) E[|Mt|] < ∞ for all t (3) For all 0 ≤ s < t and for a.s. ω

E[Mt|Fs] = 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 {Ft}t≥0 be the ﬁltration generated by {Bt} (i.e. Ft is the σ-algebra generated by ω → B(t, ω) for each t.) Then using independence of increments and properties of conditional expectation, we calculate:

E[Bt|Fs] = E[Bt − Bs|Fs] + E[Bs|Fs] = 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, ω)dBt(ω) as a limit of sums of the form: X f(tj, ω)∆Btj (ω) j ITOˆ CALCULUS AND DERIVATIVE PRICING WITH RISK-NEUTRAL MEASURE 3

Intuitively, the increments f(tj, ω)∆Btj (ω) reflect the random change in Xt due to white noise. We now make this notion precise. Definition 2.4 (ItôIntegrable). We say that a function f(t, ω) is ItôIntegrable on [S, T ] if the following conditions are satisfied: (1) f(t, ω) is B ⊗ F measurable (2) f(t, ·) is Ft-measurable for each t Z T (3) E f 2(t, ω)dt < ∞ S In this case we write f ∈ V[S, T ], following the notation in [4]. We say that a function φ(t, ω) is X elementary if φ ∈ V[S, T ] and φ(t, ω) = φ(tj, ω)X[tj ,tj+1] where {tj} 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 (ItôIsometry). If φ is elementary and φ ∈ V[S, T ], then we have "Z T 2# Z T 2 E f(t, ω)dBt = E f (t, ω)dt S S Proof. 2 " 2#    Z T X E f(t, ω)dBt = E  f(tj, ω)∆Bj  S j   X = E  f(ti, ω)f(tj, ω)∆Bj∆Bi 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:     Z T X 2 2 X 2 X 2 2 E  ftj (∆Bj)  = ∆tjE[ftj ] = E  ftj ∆tj = E f (t, ω)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 Itôintegral to all of V[S, T ]. Specifically, given f ∈ V[S, T ] let {φn} be a sequence of elementary functions such that: Z T 2 E (f(t, ω) − φn(t, ω)) dt → 0 as n → ∞ (2.3) S

As the reader can check, Lemma 2.5 and equation (2.3) imply that {φn} 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, ω)dBt(ω) = lim φn(t, ω)dBt(ω) (2.4) S n→∞ 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 {φn}. For a closer look at the existence of such a sequence R {φn}, see [4]. We now provide a brief review of some of the properties of fdBt: + Proposition 2.6 (Properties of the ItˆoIntegral). Let [S, T ] ⊂ R and let f ∈ 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 equation (2.4). With a little more work we can show Theorem 2.7 (Martingale Property). Let f ∈ V[0,T ], 0 ≤ t ≤ T . R t (1) There exists a t-continuous version of 0 fdBs R t (2) Put Mt(ω) = 0 fdBs(ω). Then Mt is a {Ft}t≥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, ω) such that each vij(t, ω) ∈ V[S, T ]. Let Bt ∈ R . Then we define   T T v11 ··· v1m Z Z . . vdBt =  . .  dBt S S   vn1 ··· vnm to be the n × 1 vector with ith component m X Z T vik(t, ω)dBk(t, ω) k=1 S Definition 2.8 (ItôProcess). Let v ∈ V[0,T ] and let u(t, ω) be a measurable stochastic process adapted to the filtration {Ft}t≥0 and such that Z t P |u(s, ω)|ds < ∞ for all t ≥ 0 = 1 0

Then we say that X(t, ω) is ItôProcess if Xt has the form Z t Z t Xt = X0 + u(s, ω)ds + v(s, ω)dBs 0 0 This extends naturally to the n-dimensional case by considering expressions of the form Z t Z t X(t) = X(0) + u ds v dBs 0 0 where u is a 1 × n matrix and v is an n × m matrix with all components satisfying the conditions of Definition 2.8. This leads us to the following fundamental theorem, which shows that Itôprocesses are invariant under sufficiently smooth maps. This can be considered the stochastic version of the chain rule. ITOˆ CALCULUS AND DERIVATIVE PRICING WITH RISK-NEUTRAL MEASURE 5

Theorem 2.9 (ItôLemma). Let Z t Z t X(t) = X(0) + u(s, ω)ds + v(s, ω)dBs 0 0 n m 2 be an n-dimensional Itôprocess. Let f : [0, ∞) × R → R , where f is a C map. Define a new stochastic process Y (t) = f(t, Xt). Then Y(t) is an Itôprocess as above, and the ith coordinate is given by 2 ∂fi X ∂fi X ∂ fi dYi(t) = (t, Xt)dt + (t, Xt)dXk + (t, Xt)(dXk)(dXj) ∂t ∂xk ∂xk∂xj k k,j 2 where (dXi)(dXj) is calculated according to the rule (dt) = 0, (dt)(dBk) = (dBk)(dt) = 0 for all i, and (dBk)(dBj) = (dBj)(dBk) = δjkdt. We omit the proof (see page 46 of [4]) in order to start solving SDE as quickly as possible, merely noting that the second order term above, notably absent in the classic chain rule, comes from the fact that Brownian motion has non-zero quadratic variance.

Proposition 2.10 (Integration by Parts). Let Xt and Yt be 1-dimensional stochastic processes. Then d(XtYt) = XtdYt + YtdXt + dXt · dYt (2.5) so that Z t Z t Z t XsdYs = XtYt − X0Y0 − YsdXs − (dXs)(dYs) 0 0 0 2 Proof. Let g : R → R, g(x1, x2) = x1x2. Then g is smooth so applying the Itˆolemma, ∂g ∂g 1 ∂2g d(g(Xt,Yt)) = (Xt,Yt) + (Xt,Yt) + 2 · (Xt,Yt) ∂x1 ∂x2 2 ∂x1∂x2 = YtdXt + XtdYt + (dXt)(dYt) Then the integral equation above follows immediately from equation (2.5). 3. Stochastic Differential Equations In this section, we illustrate the power of Itˆo’sLemma by explicitly solving an elementary SDE. We will then prove an important existence and uniqueness result for well-behaved SDE. Let us consider a model for population growth in a stochastic environment.

Example 3.1 (Geometric Brownian Motion). Put θt = α + β · Wt where Wt is white noise and deﬁne dXt = θtXtdt This seems a reasonable model for a growing population subject to random environmental shocks. Using the Brownian motion interpretation, we can rewrite this as

dXt = αXtdt + βXtdBt (3.1) Expecting some sort of exponential behavior, we apply the ItˆoLemma to the C2 function f(x) = ln(x). Using Theorem 2.9, we get 1 d(f(X )) = f 0(X )dX + f 00(X )(dX )2 t t t 2 t t 1 1 1 2 2 = dXt − 2 (β Xt dt) Xt 2 Xt 1 β2 = dXt − dt Xt 2 6 MAX CYTRYNBAUM

Combining this with equation 3.1, we get that β2 d(ln(X )) = α − dt + βdB t 2 t so that 2 Xt β ln = α − t + βBt X0 2 or β2 X = X exp α − t + βB t 0 2 t β2 From the law of iterated logarithm for Brownian motion (see [5]), we see that if α > 2 , Xt(ω) → ∞ as t → ∞ for a.s. ω, while for lower values of α the stochastic term takes over. Given empirically determined coefficients α and β, when do we expect the population to reach a certain size N? We will return to this question later. Using the ItôLemma, a wide range of classic ODE methods such as matrix exponentiation and integrating factors can be applied to suitable SDE. See the exercises in Chapter 5 of [4] for a wealth of examples. Now we turn to the interesting question of existence and uniqueness of solutions to SDE. Lemma 3.2 (Gronwall Inequality). Let v(t) be a non-negative, Borel function such that for some constants C, A, we have Z t v(t) ≤ C + A v(s)ds 0 then we have v(t) ≤ C exp(At) n n Theorem 3.3 (Existence and Uniqueness of SDE). Let T ≥ 0 and b(·, ·) : [0,T ] × R → R , n n×m σ(·, ·) : [0,T ] × R → R be measurable functions such that we have |b(t, x)| + |σ(t, x)| ≤ C(1 + |x|) (3.2) n P for some constant C and for all t ∈ [0,T ], x ∈ R (where |σ| = |σij|). Assume also that |b(t, x) − b(t, y)| + |σ(t, x) − σ(t, y)| ≤ D|x − y| (3.3) n for some constant D and all t ∈ [0,T ], x, y ∈ R . Let Z be a random variable with finite second (m) moment such that Z is independent of F∞ := σ(Ft : t ≥ 0), where Ft is generated by Bt. Then the SDE dXt = b(t, Xt) + σ(t, Xt) X0 = Z(ω) Z has a unique t-continuous solution that is adapted to Ft := σ(Ft, σ(Z)) and Z T 2 E |Xt| dt < ∞ (3.4) 0 ˜ ˜ R t ˜ R t ˜ Proof. Uniqueness: Suppose there exists another solution Xt = Z + 0 b(s, Xs)ds + 0 σ(s, Xs)ds. put f(t, ω) = b(t, Xt) − b(t, X˜t) and put g(t, ω) = σ(t, Xt) − σ(t, X˜t). Then we calculate " Z t Z t 2# 2 E(|Xt − X˜t| ) = E Z − Z˜ + fds + gdBs 0 0 "Z t 2# "Z t 2# h 2i ≤ 3E (Z − Z˜) + 3E fds + 3E gdBs 0 0 h i Z t Z t ≤ 3E (Z − Z˜)2 + 3tE f 2ds + 3E g2ds 0 0 ITOˆ CALCULUS AND DERIVATIVE PRICING WITH RISK-NEUTRAL MEASURE 7

The above squares should be interpreted in the matrix sense. The ﬁrst inequality uses (x+y+z)2 ≤ 3x2 + 3y2 + 3z2 and the 2nd inequality uses the Itˆoisometry. Then equation (3.3) implies that the above is Z t Z t h 2i 2 2 2 2 ≤ 3E (Z − Z˜) + 3tD E |Xs − X˜s| ds + 3D E |Xs − X˜s| ds 0 0 Z t h 2i 2 h 2i = 3E (Z − Z˜) + 3D (t + 1) E |Xs − X˜s| ds 0 Condition (3.4) allows us to use Fubini in the second line. Put B = 3D2(T + 1), A = 3E[|Z − Z˜|2]. 2 Then with v(t) = E[|Xt − X˜t| ], we have that Z t v(t) ≤ A + B v(s)ds 0 Then Gronwall implies that v(t) ≤ A exp(Bt)

Set Z = Z˜, then we have that v(t) = 0 so that for a.s. ω, |Xt − X˜t| = 0. We get that ˜ P [|Xt − Xt| = 0 for all t ∈ Q ∩ [0,T ]] = 1

By the assumed t-continuity of solutions, we ﬁnd that t → |Xt − X˜t| is continuous, so that we actually have Xt = X˜t for all t ∈ [0,T ] for a.s. ω, which completes the proof of uniqueness. The proof of existence uses Picard iterations and is very similar to the proof for ODE. Because of its length, it is ommitted. See [4, p.70]. The solution given by the previous theorem is called a strong solution because it is constructed with respect to a speciﬁc version of Brownian motion Bt. By a weak solution, we mean any pair of processes (Xct, Bct) that satisfy the equation

Xct = b(t, Xct)dt + σ(t, Xct)dBct (3.5) An SDE has a strongly unique solution if any two solutions are equal for a.s. (t, ω). An SDE is said to have a weakly unique solution if any two solutions have the same law. This distinction is subtle but will be necessary for our proof of the Girsanov theorem. What we require is the following: Lemma 3.4. Let b and σ be as in the previous theorem. Then any solution to equation (3.5) is weakly unique.

4. Itoˆ Diffusions An Itôdiffusion is a model of a well-behaved stochastic process that shares some fundamental characteristics with nice processes like n-dimensional Brownian motion. Specifically,

n Definition 4.1 (ItôDiffusion). We say a stochastic process Xt ∈ R is a an Itôdiffusion if it satisfies a stochastic differential equation

dXt = b(Xt)dt + σ(Xt)dBt n n n n×m where b : R → R and σ : R → R are functions satisfying the conditions of Theorem 3.3. Since we have no time argument, this just means that n |b(x) − b(y)| + |σ(x) − σ(y)| ≤ D|x − y| for all x, y ∈ R Theorem 3.3 shows that the above is well-deﬁned. The next concept is fundamental to the study of both martingales and stochastic processes. 8 MAX CYTRYNBAUM

Definition 4.2 (stopping time). Let {Mt}t≥0 be a filtration on the probability space (Ω, M,P ) and let τ :Ω → R. We say that τ is a stopping time w.r.t the filtration {Mt}t≥0 if for each t we have

{ω : τ(ω) ≤ t} ∈ Mt Sometimes we will write {τ ≤ t} for {ω : τ(ω) ≤ t}. The intuition behind the above definition is that one can know whether or not a stopping time has passed at time t without looking into the future (only using the information in Mt). Thus if Xt represents the value of a portfolio, and {Mt}t≥0 is the filtration generated by Xt, then sup{t : Xt > 20} would not be a stopping time while inf{t : Xt > 20} would be. The reader can show that any deterministic time t is trivially a stopping time w.r.t. any filtration.

Proposition 4.3. Let τ1 and τ2 both be stopping times w.r.t. {Mt}. Then the following random variables are also stopping times w.r.t. {Mt}

(1) τ1 ∧ τ2 (2) τ1 ∨ τ2

Here, (τ1 ∧τ2)(ω) = τ1(ω)∧τ2(ω) := min(τ1(ω), τ2(ω)). The maximum in (2) is interpreted likewise. Moreover, if τn is a sequence of stopping times such that lim τn = τ and such that, for each ω, n→∞ τn(ω) ≤ τ(ω) for large n, then τ is a stopping time w.r.t the same ﬁltration.

Proof. For (1) and (2) Note that {τ1 ∧ τ2 ≤ t} = {τ1 ≤ t} ∪ {τ2 ≤ t} ∈ Mt and that {τ1 ∨ τ2 ≤ t} = {τ1 ≤ t} ∩ {τ2 ≤ t} ∈ Mt. For the last statement, the reader can check that in this case we have ∞ [ \ {τ ≤ t} = {τm ≤ t} ∈ Mt n=1 m≥n

Deﬁnition 4.4. Let {Mt} be a ﬁltration on Ω and let M∞ be the smallest σ-algebra containing all the Mt. By Mτ we mean the σ-algebra of all sets in M ∈ M∞ such that

M ∩ {τ ≤ t} ∈ Mt

Intuitively, Mτ is the σ-algebra of all events that occured before the stopping time τ. In the s,x n s,x following, Xt will mean a diffusion started at a point x ∈ R and started at time s. E will s,x denote expectation w.r.t. the law afforded by Xt . One of the properties that Itôdiffusions share x X(t) with Brownian motion is their “forgetfulness”. Formally, E [X(t + h)|Ft] = E [Xh]. In fact, more is true:

x n Theorem 4.5 (Strong Markov Property). Let Xt ∈ R be an Itôdiffusion such that X0 = x and (m) let τ be a stopping time w.r.t. the filtration {Ft }t≥0. Then given any bounded Borel function n n f : R → R , x Xτ (ω) E [f(Xτ(ω)+h)|Fτ ] = E [f(Xh)] The proof involves a lot of technical bookkeeping, so we omit it (see page 117 of [4]). Let M∞ = σ(Mt : t ≥ 0), where Mt is the σ-algebra generated by Xt. If we define the operator θt so that θt(Xs) = Xs+t. It can be shown that the strong Markov property extends to M∞ so that for f a bounded, M∞ measurable function:

x (m) Xτ (ω) E [θτ f|Fτ ] = E [f] (4.1) The next lemma is fundamental: ITOˆ CALCULUS AND DERIVATIVE PRICING WITH RISK-NEUTRAL MEASURE 9

n Lemma 4.6. Let Yt be an Itˆoprocess in R such that Z t Z t Yt = x + u(s, ω)ds + v(s, ω)dBs 0 0 (m) 2 n Let τ be a stopping time w.r.t. the ﬁltration {Ft }t≥0. Let f ∈ C0 (R ) and let u and v above be bounded on the set of (s, ω) such that Y (t, ω) ∈ supp(f). Also, suppose Ex[τ] < ∞. Then we get     Z τ X ∂f 1 X ∂2f Ex(f(Y )) = f(x) + Ex (Y )u (s, ω) + (Y )(vvT ) (s, ω) ds (4.2) τ   ∂x s i 2 ∂x ∂x s ij   0 i i i,j i j

P 2 Proof. Note that dYi(t) = uidt + k vikdBk. Applying the Itôlemma to the C function f, we find that X ∂f 1 X ∂2f d(f(Y )) = f(Y ) + dY + (dY )(dY ) 0 ∂x i 2 ∂x ∂x i j i i i,j i j X ∂f X ∂f 1 X ∂2f X = f(x) + uidt + vikdBk + vikvjkdt ∂xi ∂xi 2 ∂xi∂xj i i,k i,j k P where we have used that (dYi)(dYj) = k vikvjkdt from the Itôlemma. Note that this can be T rewritten as (vv )ij. Writing this differential equation in integral form, replacing t with the stopping time τ and taking an expectation, we get that   Z τ ∂f X ∂2f Ex[f(Y )] = f(x) + E (Y )u (s, ω) + (Y )(vvT ) (s, ω)ds τ  ∂x s i ∂x ∂x s ij  0 i i,j i j   Z τ X ∂f + E  (Ys)vik(s, ω)dBk ∂xi 0 i,k Therefore it will suffice to prove that the last expectation is 0. Let g be a bounded borel function n on R s.t. |g| < K. then apparently g ∈ V[0,T ] for all T , so with k > 0 we calculate Z τ∧k Z k x x E g(Ys)dBs = E X{s≤τ}g(Ys)dBs = 0 (4.3) 0 0 because if τ is a stopping time then the product of g and the indicator X{s≤τ} is in V[0,T ], so we can apply Proposition 2.6(2). Now note that "Z τ Z τ∧k 2# Z τ x x 2 E g(Ys)dBs − g(Ys)dBs = E g(Ys) ds 0 0 τ∧k ≤ K2Ex[τ − τ ∧ k] and since Ex[τ] < ∞ we get that this tends to 0 as k → ∞. Note that the equality above uses the Itôisometry. Then we have that Z τ∧k Z τ x x E g(Ys)dBs → E g(Ys)dBs 0 0 in L2(P x) as k → ∞. Then equation (4.3) implies that the limit is identically 0, which completes the proof of the lemma. In many applications, we will wish to associate to a second-order partial differential operator to each Itôdiffusion. This operator encodes a wealth of information about the process Xt and 10 MAX CYTRYNBAUM will allow us to forge a direct link between stochastic theory and PDE through the Feynman-Kac formula.

Definition 4.7 (Infinitesimal Generator). Let Xt be an Itôdiffusion. We define the infinitesimal generator: Ex(f(X )) − f(x) Af(x) = lim t t↓0 t n We let DA denote the set of functions f for which the above limit exists at all x ∈ R . n 2 n Theorem 4.8. Let Xt = b(Xt)dt + σ(Xt)dBt be an Itôdiffusion in R . If f ∈ C0 (R ), then n f ∈ DA and for all x ∈ R we have that X ∂f 1 X ∂2f Af(x) = b (x) + (σσT ) (x) i ∂x 2 ∂x ∂x ij i i i,j i j

Proof. Apply Lemma 4.6 to the Itôdiffusion Xt with τ = t. Using bounded convergence to pass the limit through the expectation, the theorem is a simple consequence of the fundamental theorem of calculus. Thinking of the operator A as a derivative of sorts, the following important theorem can be seen as a generalization of the fundamental theorem of calculus: 2 n Theorem 4.9 (Dynkin’s Formula). Let f ∈ C0 (R ) and let Xt be an Itôdiffusion as above. Assume (m) also that τ is a stopping time w.r.t. the filtration {Ft }t≥0. Then we have Z τ x x E [f(Xτ )] = f(x) + E Af(Xs)ds 0 Proof. This follows immediately from Theorem 4.8 and the above lemma. Example 4.10 (Population Growth and Hitting Times). In Section 3, we considered a stochastic model of population growth and produced the equation β2 X = x exp α − t + βB (4.4) t 2 t β2 β2 If α > 2 , Xt → ∞ for a.s. ω. If α < 2 , Xt → 0 for a.s. ω. Given coefficients α and β, we ask β2 (i) for α > 2 when do we expect the population to reach a certain size R? β2 (ii) for α < 2 , do we expect the population to ever reach size R? γ For fixed γ ∈ R we can use Theorem 4.8 to compute Xt’s infinitesimal generator for f(x) = x . γ 2 By letting f go to 0 in a smooth way, we may assume that x ∈ C0 (R). Then since dXt = αXtdt + βXtdBt, we get 1 α2(γ − 1) Af(x) = βxf 0(x) + f 00(x)α2x2 = γxγ β + (4.5) 2 2

2α γ1 If we put γ1 = 1 − β2 and f(x) = x then from equation (4.5) we see that Af(x) = 0. We deﬁne 1 τn = inf{t > 0 : Xt 6∈ [ n ,R]. This is known as a hitting time, and it can be shown that the hitting time for any Borel set is a stopping time w.r.t. {Ft}. Let g(x) = ln(x) for x < ln(R) and let g be 0 outside some compact set containing [0, ln(R)] (i.e. let g go to 0 in some smooth way). Then 1 2 Theorem 4.8 implies that Ag(x) = α − 2 β . Applying the Dynkin formula to g with the stopping time τn ∧ k we get Z τn∧k 2 x x β x E [g(Xτn∧k)] = ln(x) + E Ag(Xs)ds = ln(x) + (α − )E [τn ∧ k] 0 2 ITOˆ CALCULUS AND DERIVATIVE PRICING WITH RISK-NEUTRAL MEASURE 11

Since g is bounded, we can apply bounded convergence on both sides and let k → ∞. Since x τn ∧ k → τn pointwise for a.s. ω, the above implies that E [τn] < ∞. This allows us to apply γ Dynkin to the function f(x) = x 1 with the stopping time τn. Since Af = 0,

1− 2α x β2 E [f(Xτn )] = f(x) + 0 = x for all n (4.6) 1 If we let (1 − pn) be the probability that Xt exits the bottom of the interval [ n ,R] before it exits at R, (4.6) becomes 1 γ1 (1 − p ) + p (R)γ1 = xγ1 n n n so that γ1 x γ1 1 p = − (1 − p ) n R n R · n γ 1 2 x 1 Since α < β implies that γ1 > 0, we see that lim pn = . Let En be the event that Xτ = R. 2 n R n Then En ⊂ En+1 [ A := {ω : Xt(ω) = R for some t > 0} = En n So from basic probability, ! γ x x [ x 1 P (A) = P En = lim pn = n R n β2 where x is the initial population x = X0, which answers question (i). Now suppose that α > 2 . For (ii), if we apply the Dynkin formula with f(x) = ln(x) as above, similar arguments can be used x to show that E [τR], the expected time for a population with initial size Xt to reach population R, is given by ln R Ex[τ ] = x R 1 2 α − 2 β We have given the above proof in great detail to illustrate some important techniques, such as trun- cation of stopping times and limiting procedures with bounded convergence. For more information on this model, see Exercise 7.9 of [4]. Using the A operator, we will now prove an interesting connection between classic PDE theory and stochastic processes.

5. Feynman-Kac and PDE n 2 n Theorem 5.1 (Feynman-Kac). Let Xt ∈ R be an Itôdiffusion with generator A. Let f ∈ C0 (R ) n and q ∈ C(R ) such that q is lower bounded. Set Z t x v(t, x) = E exp − q(Xs)ds f(Xt) 0 Then we have that (i) ∂v = Av − qv (5.1) ∂t v(0, x) = f(x) (5.2) 1,2 n n (ii) If there is another function g(t, x) ∈ C (R×R ) such that g is bounded on K×R for every compact K ⊂ R, then if g satisfies (5.1) and (5.2) above we must have g(t, x) = v(t, x). 12 MAX CYTRYNBAUM

Z t R t Proof. We let Zt denote the stochastic process exp − q(Xs)ds . We set Nt = − 0 q(Xs)ds 0 and h(x) = exp(x). Then using the Itôlemma, we see that 1 d(h(N )) = h(N )dN + h(N )(dN )2 = −h(N )q(X )dt t t t 2 t t t t so that we have dZt = −Ztq(Xt)dt (5.3) Note that d(f(Xt)) can be calculated using Lemma 4.6. Using Proposition 2.10, we see that if Yt := f(Xt), d(ZtYt) = ZtdYt + YtdZt + (dZt)(dYt) = ZtdYt + YtdZt because dZt = −Ztq(Xt)dt implies that (dZt)(dYt) = 0 by the Itôlemma. Then ZtYt is an Itô process. By the above assumptions, ZtYt is bounded for all ω, so using Fubini’s theorem, equation x (4.2) immediately implies that E (f(Xt)Zt) is differentiable. Then we directly calculate Av using the limit definition: 1 1 Ex[v(t, X ) − v(t, x)] = Ex[EXs [Z f(X )]] − Ex[f(X )Z ] s s s t t t t Z t 1 x x x = E [E [f(Xt+s) exp − q(Xs+r)dr |Fs] − E [f(Xt)Zt|Fs]] s 0 The second equality follows from the Markov property and properties of conditional expectation. We can rewrite this as Z s 1 x x x = E E f(Xt+s)Zt+s exp q(Xr)dr − E [f(Xt)Zt]|Fs s 0 Z s 1 x 1 x = E [Zt+sf(Xt+s) − Ztf(Xt)] + E f(Xt+s)Zt+s exp q(Xr)dr − 1 s s 0 A calculation similar to the one that produced (5.3) shows that we can rewrite Z s Z s exp q(Xr)dr = Zrq(Xr)dr + 1 0 0 Z s Since the integrand is continuous for a.s. ω, it follows that exp q(Xr)dr − 1 is differen- 0 tiable, and its derivative at t = 0 is Z0q(X0) = q(x). It follows from the lower boundedness of q x that for each t, Ztf(Xt) is bounded for a.s. ω. Then since we already showed that E [f(Xt)Zt] is differentiable with respect to t, we can apply bounded convergence to show that 1 ∂ Ex[Z f(X ) − Z f(X )] → Ex[f(X )Z ] as s → 0 s t+s t+s t t ∂t t t and Z s 1 x E f(Xt+s)Zt+s exp q(Xr)dr − 1 → q(x)v(t, x) as s → 0 s 0 ∂v Then we have shown that Av = ∂t − qv. For (ii), we refer the reader to p.142 of [4] ∂v In the special case that q = 0 and with v(t, x) as above, we find that ∂t = Av, which is known as the Kolmogorov Backward Equation. To illustrate the theorem’s power, we will consider two examples of deterministic PDE solved with stochastic methods. ITOˆ CALCULUS AND DERIVATIVE PRICING WITH RISK-NEUTRAL MEASURE 13

2 n Example 5.2 (Cauchy Problem). Let φ ∈ C0 (R ). We seek a bounded solution g of the initial value problem ∂g(t, x) 1 = ∆ g(t, x) (5.4) ∂t 2 x g(0, x) = φ(x) (5.5) n 1 Let Xt = Bt ∈ R . Clearly dBt = IndBt, so using theorem (3.8) we get that Af = 2 ∆f for any 2 n x 2 n f ∈ C0 (R ). Let v(t, x) = E [φ(Bt)]. Then since φ ∈ C0 (R ), we can apply Fubini to show that Z 2 x 1 |x − y| v(t, x) = E [φ(Bt)] = φ(y) n exp dy Rn (2πt) 2 2t 2 n Then since φ is bounded, it is clear that for each t, we have x → v(t, x) ∈ C0 (R ). Therefore, for 1 each t we have that Avt(x) = 2 ∆vt(x). Applying the Kolmogorov backward equation, we conclude that v(t, x) is a solution to (5.4) and (5.5) and is bounded because φ is. Example 5.3. In this example, we will give an explicit formula for the solution u(t, x) to the initial value problem ∂u 1 = λu + ∆u ∂t 2 u(0, x) = f(x)

n 2 n where x ∈ R , λ is a constant and f ∈ C0 (R ) is given. Set Z t x v(t, x) = E exp −λds f(Bt) 0 2 n i.e. set q(x) = −λ in the Feynman-Kac theorem. Then if we can prove that v(t, x) ∈ C0 (R ), Feynman-Kac immediately implies that v(t, x) is a solution to the above initial value problem (we 1 2 showed above that for Brownian motion Ag = 2 ∆g for g ∈ C0 (R).) Using the density of Bt, we compute the expectation: Z 2 x exp(−λt) |x − y| v(t, x) = E [exp(−λt)f(Bt)] = n f(y) exp dy (5.6) Rn (2πt) 2 2t 2 n whence it is clear that x → v(t, x) ∈ C0 (R ) for each t. Then v(t, x) is indeed a solution to the above differential equation and is given explicitly by equation (5.6). The next result tells us that if we change the drift coefficient of an Itôdiffusion, the law of the new process is absolutely continuous w.r.t. the original process. The Radon-Nikodym derivative provided by the theorem is important for doing computations with risk-neutral measure in mathematical finance.

6. The Girsanov Theorem First we prove a general result from probability theory: Theorem 6.1 (Bayes’ Rule). Let (Ω, N ) be a measurable space equipped with a measure µ. Let f ∈ L1(µ) and put dν = fdµ. Let H be a sub σ-algebra of N . Then if X is an r.v. such that Z |X(ω)|f(ω)dµ(ω) < ∞ Ω we get that Eν[X|H] · Eµ[f|H] = Eµ[fX|H] (6.1) 14 MAX CYTRYNBAUM

Proof. Using the elementary properties of conditional expectation, it will suﬃce to show that the left and right hand side of (6.1) have the same expectation over any set H ∈ H: Z Z Z Z Eµ[Xf|H]dµ = Xfdµ = Xdν = Eν[X|H]dν (6.2) H H H H where the second equality uses the L1(µ) condition above. Note that Z Eν[X|H]dν = Eµ[Eν[X|H]fXH ] = Eµ[Eµ[Eν[X|H]fXH |H]] (6.3) H

n Theorem 6.2 (LévyCharacterization of Brownian Motion). Let X(t) ∈ R be a stochastic process on the probability space (Ω, H,P ). Then the following are equvalent −1 (i) X(t) is a Brownian motion w.r.t. the probability measure P (i.e. P ◦ Xt is the law of n Brownian motion on R ). (ii) (a) X(t) is a martingale w.r.t. its own filtration under the measure P (i.e. EP [Mt|Ms] = Ms) (b) Xi(t)Xj(t) − δijt is a martingale w.r.t. its own filtration under the measure P We refer the reader to Peres (2010) for more information. We may now state the main theorem of this section.

n Theorem 6.3 (Girsanov Theorem). Let X(t) ∈ R be an Itˆoprocess such that X0 = 0 and for ﬁxed T with 0 ≤ t ≤ T ≤ ∞

dXt = a(t, ω)dt + dBt n where Bt ∈ R is standard Brownian motion w.r.t. the measure P. Define Z t Z t 1 2 Mt = exp − a(s, ω)dBs − a (s, ω)ds 0 2 0 (m) and suppose that Mt is a martingale w.r.t. the filtration {Ft } generated by Bt under the measure (m) P . Define the measure dQ = MT dP . Then Q is a probability measure on FT and Xt is n- dimensional Brownian motion w.r.t. Q.

(m) Proof. It is easy to show that Q is a probability measure on FT :

Q(Ω) = EP [MT ] = EP [M0] = 1 (m) (m) We note that on Ft , we actually have dQ = MtdP . More precisely, let f be a bounded Ft - measurable function. Then we get

(m) (m) EQ[f] = EP [fMT ] = EP [EP [fMT |Ft ]] = EP [fEP [MT |Ft ]] = EP [fMt] ITOˆ CALCULUS AND DERIVATIVE PRICING WITH RISK-NEUTRAL MEASURE 15

1 2 In context of the above definition of Mt, let dNt = −adBt − 2 a dt. Applying the Itôlemma to the function g(x) = exp(x), we find that 1 d(M ) = d(exp(N )) = exp(N )dN + exp(N )(dN )2 t t t t 2 t t ! X 1 X 1 X = M −a dB (t) − a2dt + M a2dt t i i 2 i 2 t i i i i ! X = −Mt aidBi(t) i

Put Zt = MtXt. From (2.10), we have that

dZi(t) = MtdXi(t) + Xi(t)dMt + (dMt)(dXi(t))     X X = Mt(aidt + dBi(t)) + Xi(t)Mt  −ajdBj(t) + Bi(t)Mt  −ajdBj(t) j j    X = Mt dBi(t) − Xi(t)  ajdBj(t) (6.4) j

n In matrix notation, (6.4) can be written as MtVtdBt, where Vt ∈ R is deﬁned by

Vj(t) = (δij − Xj(t)aj(t)) (m) By Theorem 2.7, we have shown that MtXi(t) is a martingale w.r.t. {Ft } under the measure P . Using Bayes’ Theorem, we have that for s < t ≤ T ,

E [X (t)M |F (m)] X (s)M E [X (t)|F (m)] = P i t s = i s = X (s) Q i s (m) M i EP [Mt|Fs ] s (m) So that Xi(t) is a martingale w.r.t. {Ft } under the measure Q. The proof that XiXj − δijt is also a martingale is similar. By the L´evycharacterization of Brownian motion, this completes the proof.

Exponential martingales such as Mt above are actually a commonly used tool in stochastic calculus. The following theorem is useful

Theorem 6.4 (Novikov Condition). Let Mt be defined as in the previous theorem. A sufficient (m) condition for Mt to be a martingale w.r.t. {Ft } under the measure P is that Z t E exp a2(s, ω)ds < ∞ 0 For the proof, see, for instance, Karatzas and Shreve (1991). Any continuous, deterministic function trivially satisfies the Novikov condition.

n Theorem 6.5 (Girsanov II). Let Xt ∈ R be an Itˆoprocess

dXt = β(t, ω)dt + θ(t, ω)dBt m n n×m where Bt ∈ R . Now suppose that we can ﬁnd α(t, ω) ∈ V [0,T ] and u(t, ω) ∈ V [0,T ] such that θ(t, ω)u(t, ω) = β(t, ω) − α(t, ω) 16 MAX CYTRYNBAUM

Deﬁne Mt as in the previous theorem with u(s, ω) in place of a(s, ω) and assume that this is a (m) (m) martingale w.r.t. {Ft } under the measure P . Deﬁne dQ = MT dP on FT . Then the process Z t Bct := u(s, ω)ds + Bt (6.4) 0 is standard Brownian motion w.r.t. Q, and we can write

dX(t) = α(t, ω)dt + θ(t, ω)dBct (6.5) Proof. In view of Theorem 6.3, it suﬃces to prove that (6.5) holds. This is clear in view of (6.4):

dX(t) = β(t, ω)dt + θ(t, ω)dBt

= β(t, ω)dt + θ(t, ω)(dBct − u(t, ω)dt)

= β(t, ω)dt + θ(t, ω)dBct − (α(t, ω) − β(t, ω))dt

= α(t, ω)dt + θ(t, ω)dBct The final version applies specifically to Itôdiffusions and contains a helpful uniqueness statement: n Theorem 6.6 (Girsanov III). Let Xt ∈ R be an Itôdiffusion and define Yt such that

dXt = b(Xt)dt + σ(Xt)dBt

Yt = (b(Xt) + γ(t, ω))dt + σ(Xt)dBt n n m where Bt ∈ R and γ ∈ V [0,T ]. Suppose we can ﬁnd u(t, ω) ∈ V [0,T ] such that σ(Yt)u(t, ω) = (m) γ(t, ω). Deﬁne Mt, Q, and Bct as in the last theorem, where Mt is a martingale w.r.t. {FT } under the measure P . Then we get that

dYt = b(Yt)dt + σ(Yt)dBct n x x x Also, with x ∈ R a starting point for both diffusions, the Q law of Yt is equal to the P law of x Xt . Proof. This follows by applying Girsanov II with appropriate choices of α and β. The statement about equivalence of laws is an immediate consequence of weak uniqueness of solutions to SDE Lemma 3.4, which from Theorem 3.3 we can clearly apply to the above diffusions. The theorem essentially states that we can change the drift coefficient of an Itôdiffusion and it will keep the same law up to a change of measure. (m) Remark 6.7. With Q, P , and MT as above, we clearly have that Q P . Suppose that A ∈ FT and note that if Q(A) = 0 then EP [MT XA] = 0 so that MT XA = 0 for P -a.s. ω. Since MT > 0 for P -a.s. ω, we have that XA = 0 for P -a.s. ω. Then immediately P (A) = 0. This shows that P Q so that P ∼ Q. Because of this, Q is sometimes called an equivalent martingale measure. Example 6.8. Let 0 1 3 dX(t) = dt + dB 1 −1 −2 t If we set 1 3 0 u(t, ω) = −1 −2 1 −3 we get that u = . Since u trivially satisfies the Novikov condition, and −1 −3 dB := + dB ct −1 t ITOˆ CALCULUS AND DERIVATIVE PRICING WITH RISK-NEUTRAL MEASURE 17 is standard Brownian motion w.r.t. the measure dQ = MT dP , where MT is the exponential martingale defined as in the previous theorems. Girsanov also implies that we can rewrite X(t) as " # 1 3 dB (t) X(t) = c1 −1 −2 dBc2(t)

7. Risk-Neutral Measure and Black-Scholes In this section, we will apply some of the previous results to asset pricing theory and mathematical finance. We will see that stochastic processes provide a natural framework for the analysis of derivative securities. Our discussion is brief and informal. For a comprehensive introduction, see [2]. Let (Ω, N ,P ) be a probability space. In this context, we will model risky assets as random variables on Ω. Consider the European call: at t = 0 an investor purchases the right to buy a certain security at time T > 0 for a specified price K. Since the buyer is not obligated to exercise this right if the price at time T if XT (ω) < K, the payoff at time T is given by:

max(XT (ω) − K, 0) (7.1) We let X(t, ω) be an Itˆoprocess representing the value of the security at time t. How should we price such a contingent claim with payoﬀ at time T given by (7.1)? One such procedure, known as pricing by arbitrage, takes as given a collection of primitive securities (bonds, currencies, etc...) with known price processes that can be used to price the claim. More precisely:

Definition 7.1 (Replicable Claims). A contingent claim CT with payoff at maturity given by CT (ω) is said to be replicable if (i) There exists a portfolio of primitive securities with price process S(t, ω) such that CT (ω) = S(T, ω) a.s. (ii) The portfolio is self-financing i.e. there are no net cash infusions into the portfolio between t = 0 and maturity. A simple economic argument can be used to show that in the absence of arbitrage the contingent claim’s price is uniquely determined. In this situation, the claim is said to be priced by arbitrage.A market in which all contingent claims can be priced by arbitrage is said to be complete. Using this approach to price derivatives has the drawback of being computationally inefficient as a different portfolio must be constructed to price each contingent claim. Luckily, we have another method: Definition 7.2 (Risk-Neutral Measure). A measure Q on the space (Ω, N ,P ) is said to be Risk- Neutral if (i) Q ∼ P i.e. Q P and P Q (ii) Any price process X(t, ω) is a martingale w.r.t. its own filtration under the measure Q. To explain the second condition, let us suppose that the market has some risk free security such as a bond. By way of normalization, we discount every price process by the bond’s price process. If we wish to create a “risk-neutral” measure and avoid arbitrage opportunities, the expected change in value of every asset should be the same. Because the bond’s discounted expected value change is 0, this must hold for all other assets, which is reflected in condition (ii). Definition 7.3 (Risk-Neutral Price). Let X(t, ω) be the price process of an asset in a complete, arbitrage free market. Let Q be a risk-neutral measure for the associated probability space. Given a contingent claim CT on the security Xt we define the risk-neutral price F (CT ) by X F (C ) = E T T Q V (T ) where V (t) is the price process of a fixed risk-free asset. 18 MAX CYTRYNBAUM

Note that the existence of such a measure Q is a consequence of a result known as the Funda- mental Theorem of Asset Pricing. This brings us to a crucial result, the proof of which can be found in any text on mathematical ﬁnance:

Theorem 7.4. Let CT be a replicable claim in a complete, arbitrage free market. Then the arbitrage free price (using a replicating portfolio of primitive securities) and the risk-neutral price F (CT ) are the same. We will see that the Girsanov theorem gives us the tools to compute a claim’s risk-neutral price. Example 7.5 (Black-Scholes Formula). Consider the Black-Scholes model where a certain security’s price process Xt obeys geometric Brownian motion:

dXt = αXtdt + βXtdBt where α and β are constants. We think of αXt as the mean rate of change in price and βXt as the asset’s uncertainty. Let V (t) be the price process of a bond, which we assume is a risk-free security. dV (t) = rV (t)dt Then the discounted price process for X is Z = Xt . We can apply the Itˆolemma to the function t t Vt g(x , x ) = x1 to conclude that 1 2 x2

dXt dVt dZt = − 2 Xt Vt Vt = αZtdt + βZtdBt − rZtdt

= (α − r)Ztdt + βZtdBt Let us use the Girsanov Theorem to get rid of the drift coefficient of the above diffusion. We need to find a u(t, ω) such that βZtu(t, ω) = (α − r)Zt α−r and clearly u = β will work. This is a constant, so it trivially satisfies the Novikov condition whence by the Girsanov Theorem we get that α − r Bct = t + Bt β is standard Brownian motion w.r.t. the measure Q, where dQ = MT dP and Mt is as in the Girsanov Theorem. Then the discounted price process Zt has the representation dZt = βZtdBct. Using our formula for the solution to the geometric Brownian motion SDE, this is just 1 2 Zt = Z0 exp − β + βBct (7.2) 2

From Theorem 2.7, it is clear that Zt is a martingale w.r.t. its own filtration under the measure Q. From Remark 6.7, we also have that Q ∼ P . Then we have shown that Q is a risk-neutral measure according to Definition 7.2. By Theorem 7.4, we now know how to price replicable claims on the security Xt. We once again consider the European claim described above. According to the theorem, the expectation to be calculated is 1 EQ[max(XT − K, 0)] = EQ[max(ZT − exp(−rT )K, 0] VT by evaluating the ODE for the bond price. From equation (7.2) and using the fact that Bct is standard Brownian motion w.r.t. the measure Q, the above expectation is given by Z 2 1 2 1 −y max Z0 exp − β T + βy − exp(−rT )K, 0 √ exp dy (7.3) R 2 2πT 2T A trivial computation shows that 1 Z exp − β2T + βy − exp(−rT )K ≥ 0 iff y ≥ a 0 2 where a is defined by 1 Z 1 a = − ln 0 + β2 − r T β K 2 Thus, we may rewrite eqn. (7.3) as Z ∞ Z 1 y2 Z ∞ K y2 √ 0 exp − β2T + βy − dy − exp(−rT ) √ exp dy (7.4) a 2πT 2 2T a 2πT 2T 1 2 1 2 y2 where we have merged the exponentials. Notice that − 2T (y−βT ) = − 2T β T +βy− 2T . Removing the constants from the integrals, the above are just integrals of the density functions for two normal random variables with distribution N(βT, T ) and N(0,T ) respectively. We can thus rewrite equation (7.4) in terms of the CDF of the standard normal distribution as

X0Φ(−w1) − K exp(−rT )Φ(−w2) (7.5) where w = a−√βT and w = √a , and we have used the fact that Z = X , the initial price of the 1 T 2 T 0 0 security. Then Theorem 7.4 implies that the arbitrage-free price of the European call described above is given by F (CT ) = X0Φ(−w1) − K exp(−rT )Φ(−w2) Acknowledgments I would like to thank Peter May for organizing this REU, which has been both productive and enjoyable. I also want to thank my mentors Marcelo Alvisio, Andrew Lawrie, and Casey Rodriguez for their enthusiasm to meet with me every week and for the mathematical insights they shared with me during the course of the REU. 20 MAX CYTRYNBAUM

References [1] Kai Lai Chung, Lectures from Markov Processes to Brownian Motion, Springer, 1982 [2] Darrell Duffie Dynamic Asset Pricing Theory Princeton University Press 2001 [3] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, Wiley-Interscience, Second Edition, 1999 [4] Bernt Øksendal, Stochastic Differential Equations, Wiley, Sixth Edition, 2007 [5] Peter Mörtersand Yuval Peres, Brownian Motion, Cambridge University Press, 2010 [6] Rangarajan Sundaram “Equivalent Martingale Measures and Risk-Neutral Pricing: An Expository Note” Journal of Derivatives 1997 [7] David Williams, Probability with Martingales, Cambridge University Press, 1991