APPLICATIONS OF STOCHASTIC CALCULUS TO FINANCIAL MODELINGmath.uchicago.edu/~may/REU2020/REUPapers/Rapport.pdf · 2020. 10. 22. · APPLICATIONS OF STOCHASTIC CALCULUS TO FINANCIAL

APPLICATIONS OF STOCHASTIC CALCULUS TO FINANCIAL

MODELING

BEN RAPPORT

Abstract. Following a brief discussion of measure theory and probability, weintroduce stochastic processes, specifically Brownian motion, as mathematicalmodels of randomness. Building upon this foundation, we derive fundamentalresults of Itô Calculus and use these results to solve Stochastic Di↵erentialEquations (SDEs). Alongside no-arbitrage arguments, we employ these toolsto reproduce the Black-Scholes-Merton (BSM) option pricing model and theMargrabe Formula. We conclude by discussing Monte Carlo methods as ameans of simulating stochastic processes.

Contents

1. Introduction 12. Review of Measure Theory and Probability 23. Stochastic Processes 44. Itô Calculus 55. Stochastic Di↵erential Equations 96. Applications to Financial Modeling 117. Stochastic Simulation 16Acknowledgments 19References 19Appendix 20

1. Introduction

Most real world processes contain elements of randomness. In specific contexts,the impact of this randomness can be disregarded because of its small relativemagnitude. However, in other contexts, this randomness is essential to describing aprocess’ evolution. These random, or stochastic, processes are di�cult to model us-ing traditional calculus techniques as most of these processes form non-di↵erentiablepaths. In an e↵ort to translate traditional calculus techniques and theory into thestochastic setting, Kiyosi Itô developed Itô Calculus to provide these random pro-cesses with a rigorous formalization. Itô Calculus retained many properties fromtraditional calculus while adding key new constructions, chief among them the ItôIntegral and Itô’s Formula. These two results, which produce the stochastic refor-mulations of integration and the chain rule respectively, have major applicationsthroughout applied mathematics.

Date: AUGUST 29, 2020.

1

2 BEN RAPPORT

In modern finance stochastic processes are used to model price movements ofsecurities in the stock market. One of the earliest pricing models, the BSM model,produces a PDE which describes how the value of an option changes over time inan arbitrage-free market. For European options, the solution to this di↵erentialequation can be written as a closed-form expression in which the characteristics ofthe underlying asset can be explicitly observed and interpreted. Many years later,William Margrabe built upon this model and derived a more general version of thisPDE that applies to a broader range of instruments and derivatives.

In this paper, we will begin with a brief review of measure theory and some basicprobabilistic concepts. We will then give a formalized introduction to stochasticprocesses and discuss some pertinent properties of Brownian motion. Using thisknowledge as our foundation, we will then begin our treatment of Itô Calculus inwhich we will derive the Itô Formula and explore stochastic di↵erential equations asa means to model financial instruments that depend on random price movements.Finally, we will consider the application of these models and review some numericalmethods for simulating the stochastic behavior of more complex derivatives.

2. Review of Measure Theory and Probability

To begin this paper, we introduce several measure-theoretic and probabilisticdefinitions and concepts for consideration – many of which can be found in [1]. Wedenote ⌦ to be the outcome space, the set of all possible outcomes, for a givenprocedure. For example, the outcome space for a singular die roll is given by⌦ = {1, 2, 3, 4, 5, 6}.

Definitions 2.1. Let ⌦ be given. Then a �-algebra F on ⌦ is a family of subsetsof ⌦ such that

(1) ; 2 F(2) F 2 F =) F C 2 F , where F C = ⌦ \ F is the compliment of F in ⌦

(3) A1, A2, ... 2 F =) A :=1Si=1

Ai 2 F

The pair (⌦,F) is called a measurable space. For any U =Si

Fi ⇢ ⌦, we define

HU :=\

{H;H is a �-algebra of ⌦ such that U ⇢ H}

to be the �-algebra generated by U . As is shown in [1], HU is also the smallest�-algebra which contains U . The Borel �-algebra B is the �-algebra generated bythe collection of all open subsets of Rn. A Borel set is an element B 2 B.

Definitions 2.2. We define a probability measure on (⌦,F) to be the functionP : F ! [0, 1] with the following properties:

(1) P (;) = 1� P (⌦) = 0

(2) if A1, A2, ... 2 F and Ai \Aj = ; for all i 6= j, then P (1Si=1

Ai) =1Pi=1

P (Ai)

A probability space is a triple (⌦,F , P ).

Definitions 2.3. Given a probability space (⌦,F , P ), a set F is F-measurable ifF 2 F . A function f : ⌦ ! Rn is said to be F-measurable if for all open setsU 2 Rn we have that

f�1(U) := {! 2 ⌦; f(!) 2 U} 2 F .

APPLICATIONS OF STOCHASTIC CALCULUS TO FINANCIAL MODELING 3

Definitions 2.4. A random variable on (⌦,F , P ) is an F-measurable functionX : ⌦ ! Rn. The �-algebra generated by the random variable X is given byHX = {X�1(B);B 2 B}. For any random variable, X, we define its distributionto be µX(B) :=P (X�1(B)) = P (X 2 B) where B is a Borel set.

Definitions 2.5. A collection of measurable sets {Fi} is independent if

P (F1 \ F2 \ ... \ Fn) = P (F1) · P (F2) · · · P (Fn).

Suppose X1, X2, ... , Xn are random variables on (⌦,F , P ). The collection of ran-dom variables {X1, X2, ... , Xn} is independent if the collection of their generated�-algebras {HX1 ,HX2 , ... ,HXn} is independent.

Remark 2.6. Any two elements of an independent set are independent. Thus, if{X1, X2, ... , Xn} is independent, then X1 and X2 are independent.

Definitions 2.7. Let X be a random variable which assumes values in Rn. IfZ

⌦

|X(!)|dP (!) < 1, then E[X] :=

Z

⌦

X(!)dP (!) =

Z

Rn

xdµX(x)

where E[X] is the expectation of X. If f : Rn ! R is B-measurable andZ

⌦

|f(X(!))|dP (!) < 1, then E[f(X)] :=

Z

⌦

f(X(!))dP (!) =

Z

Rn

f(x)dµX(x) .

If H ⇢ F is a �-algebra, then the conditional expectation of X given H, writtenas E[X|H], is a function from ⌦ to Rn satisfying the following two properties:

(1) E[X|H] is H measurable(2)

R

H

E[X|H] dP =R

H

XdP , for all H 2 H

The variance of X is defined as V ar[X] := E[X2]� [E[X]]2 = E[(X � E[X])2].

Proposition 2.8. For conciseness, we will now present several properties of ex-pectation and variance. Their proofs can be found in [2]. Suppose X is given asabove. Let a, b 2 R, H 2 F and Y : ⌦ ! Rn be another random variable such thatE[|Y |] < 1. We then observe

(1) E[aX + bY |H] = aE[X|H] + bE[Y |H](2) E[E[X|H]] = E[X](3) E[X|H] = X if X is H-measurable.(4) E[X|H] = E[X] if X and H are independent.(5) V ar[aX + b] = a2V ar[X]

Definition 2.9. The cumulative distribution function (CDF) of a random vari-able X is the function FX : Rn ! [0, 1] where FX(x) = P (X x) = µX( x). Arandom variable is said to be continuous if its CDF is continuous.

Definition 2.10. Let X be a continuous random variable X : ⌦ ! R. If thereexists a function f : R ! [0, 1] such that

FX(x) =

Zx

�1f(t)dt ,

then f(x) is the density function of X.

4 BEN RAPPORT

Definitions 2.11. A normally distributed random variable X : ⌦ ! R withE[X] = µ and Var[X] = �2 has cumulative distribution function

(2.12) FX(b) =

bZ

�1

1

�p2⇡

· e�(x�µ)2

2�2 dx .

For such a random variable we write X ⇠ N(µ,�2). A random variable Z is saidto have lognormal distribution if log(Z) ⇠ N(µ,�2). A random variable Y is saidto have standard normal distribution if Y ⇠ N(0, 1). The standard normal CDFand density function are given by

�(b) = FY (b) =

bZ

�1

1p2⇡

· e�x2

2 dx =

bZ

�1

f(x) dx .

3. Stochastic Processes

While it is possible to model randomness in higher dimensions, in this paper wewill focus exclusively on 1-D models, in particular 1-D Brownian motion.

Definition 3.1. A stochastic process in (⌦,F , P ) over an interval T is a param-eterized collection of random variables {Xt(!)}t2T where X : T ⇥ ⌦ ! R.

Definition 3.2. A filtration on (⌦,F) is an increasing sequence of �-algebrasK = {Ft}t�0 such that for all s 2 [0, t) we have that Fs ⇢ Ft ⇢ F .

Remark 3.3. Let K = {Ft}t�0 be a filtration and f : T ⇥ ⌦ ! R be a randomvariable on (⌦,F , P ). A process {ft(!)}t�0 is Ft-adapted if ft(!) is Ft-measurablefor every t � 0. We define the natural filtration of a process {Xt(!)}t�0 to be thecollection of �-algebras {Ft} generated by {Xt(!)}t�0. It follows that a stochasticprocess is adapted with respect to its natural filtration.

Definition 3.4. A martingale in (⌦,F , P ) is an Ft-adapted stochastic process{Mt}t�0 with the following properties:

(1) E[|Mt|] < 1 for all t � 0(2) E[Ms|Ft] = Mt for all s � t, which implies E[Ms �Mt|Ft] = 0

Remark 3.5. Consider a game whose value at time t is modeled by a martingale,Mt. Over any interval, the expected change in value, or return, of the game is 0. Assuch, the second martingale property is often refered to as the fair game property.

Definition 3.6. We define a Brownian motion about the origin with drift µ andvariance �2 to be the stochastic process {Bt(!)}t�0 for which

(1) B0 = 0(2) For s < t, the random variable Y (t,!) = Bt(!) � Bs(!) is normally dis-

tributed with E[Y ] = µ(t� s) and V ar[Y ] = �2(t� s)(3) If s < t, then the random variable Y = Bt(!) � Bs(!) is independent of

the natural filtration of {Bp(!)}s�p�0 (Independent Increments)(4) P [B : T ⇥ ! ! Bt(!) is continous with respect to t] = 1

A Brownian motion is standard if it has drift ⌘ 0 and variance ⌘ 1. From this defi-nition, it can be shown that a standard Brownian motion Bt is a Gaussian processand thus

(3.7) E[eiu(Bt�Bs)] = e�u2(t�s)

2 for u 2 R and i =p�1 .


Given its continuous nature, Brownian motion can be thought of as a random path.

Remark 3.8. Any Brownian motion Wt about the origin with drift µ and variance�2 may be described by Wt = µt + �Bt, where Bt is standard Brownian motion.Given this fact, Bt will be used to represent standard Brownian motion from thispoint onward.

Examples 3.9. Let (⌦,F , P ) be given. It follows from Proposition 2.8 that

E[(Bt �Bs)2] = E[(Bt �Bs)

2� E[(Bt �Bs)]

2] = V ar[(Bt �Bs)] = t� s .

Often we will compute E[(Btj+1 �Btj )2] which, as shown above, equals tj+1 � tj .

Moreover, for s � t > 0, we observe

E[Bs|Ft] = E[Bs �Bt|Ft] + E[Bt|Ft] = Bt.

Through manipulations of (2.12) it can be shown that E[|Bt|] < 1 for all t, at whichpoint we may conclude that standard Brownian motion is in fact a martingale. Astandard Brownian motion may thus be thought of as a fair game whose varianceover an interval T is exactly the length of that interval.

Proposition 3.10. E[(Bt�Bs)4] = 3(t�s)2. A proof can be found in the Appendix.

4. Itô Calculus

Before we begin our treatement of Itô Calculus, we must first establish a classof functions for which these concepts can be applied.

Definition 4.1. A function f(t,!) : [0,1)⇥ ⌦ ! R is nice on [S, T ) if

(1) (t,!) ! f(t,!) is B ⇥ ⌦-measurable (B := Borel � � algebra)(2) f(t,!) is Ft-adapted

(3) E[RT

S(f(t,!))2 dt] < 1

(4) P [RT

S(f(t,!))2 dt < 1] = 1, for any probability measure P

The last condition is stricter than necesary, but saves space. Additonally, a nicefunction is said to be elementary if it has the form �(t,!) =

Pjej(!) ·1{tj ,tj+1}(t).

Similarly,RT

S�(t,!) dBt(!) :=

Pjej(!) ·�Bj . The notation �Bj is used exten-

sively in this section and assumes the existence of a partition ⇡ = {to, .., tn}. Forany stochastic process we denote �Xj := Xtj+1 �Xtj , and �tj := tj+1 � tj .

Definition 4.2. Suppose u(t,!), v(t,!) are nice on [0,1). We define an Itô processto be the stochastic process {Xt}t�0 in (⌦,F , P ) which satisfies

(4.3) X0 2 R, dXt = u(t,!)dt+ v(t,!)dBt for t � 0 .

Theorem 4.4 (The Itô Isometry for Elementary Functions). Let �(t,!) be abounded elementary function. Then

(4.5) E[(

ZT

S

�(t,!) dBt(!))2] = E[(

ZT

S

�(t,!)2 dt)] .

6 BEN RAPPORT

Proof. Observe, if i 6= j, then E[ei · ej ·�Bj ·�Bi] = E[ei · ej ·�Bi] ·E[�Bj ] = 0because Brownian motion has independent increments. Hence,

E[(

ZT

S

�(t,!) dBt(!))2] = E[(

X

j

ej(!) ·�Bj)(X

i

ei(!) ·�Bi)]

=X

i,j

E[ej · ei ·�Bj ·�Bi]

=X

j

E[e2j(!) · (�Bj)

2]

=X

j

E[e2j(!)] · (�tj)

= E[

ZT

S

�(t,!)2 dt] as needed. ⇤

We will now present the Itô Integral and give an outline of its proof, noting thesimilarities in construction to the Lebesgue and Riemann Integrals. The followingoutline derives from the proof given in [1].

Theorem 4.6 (The Itô Integral). Let f be a nice function on [S, T ). Then the ItôIntegral over that interval is

(4.7) I[f ](w) :=

ZT

S

f(t,!) dBt(!) = limn!1

ZT

S

�n(t,!) dBt(!)

where {�n} is a sequence of elementary functions for which

(4.8) E[

ZT

S

(f(t,!)� �n(t,!))2 dt] ! 0 as n ! 1 .

Proof. In the same way step functions are used to approximate bounded functionsin the derivation of the Riemann Integral, the Itô Integral uses elementary nicefunctions to approximate nice functions. In doing so, we will generate a sequenceof elementary functions which satisfy (4.8). The first step of the proof is to approx-imate nice functions using a sequence of bounded nice functions. Specifically, forany nice function f , we must produce a sequence {hn} of bounded nice functionssuch that

E[

ZT

S

(f � hn(t,!))2 dt] ! 0 as n ! 1 .

Next, we replicate this process to show that any bounded nice function can beapproximated by a sequence of bounded and continuous nice functions. Finally, wewould show that any bounded and continuous nice function can be approximatedby a sequence of elementary nice functions. It then follows that this sequence ofelementary nice functions {�n} yields the desired convergence in expectation. Wethus define the Itô Integral of f on (S, T ) as

I[f ](w) :=

ZT

S

f(t,!) dBt(!) = limn!1

ZT

S

�n(t,!) dBt(!) ,

where the convergence of the rightmost expression follows from the Itô Isometryand the convergence of Cauchy sequences. ⇤


A corollary to Theorem 4.4 found in [1] extends the Itô Isometry from elementaryfunctions to all nice functions and will be used in later theorems. Moreover, giventhis construction, it follows that the Itô process given by (4.3) is also described by

Xt = X0 +

Zt

0u(s,!)ds+

Zt

0v(s,!)dBs .

Lemma 4.9. Let f(s,!) be a nice function on [0, t) and define fj = f(sj ,!). Then

as �sj ! 0, we observeP

jfj · (�Bj)2 !

Pjfj ·�sj =

Rt

0 fj ds .

Proof. Following from analysis presented in the Itô Isometry, we find that if i 6= j,then ((�Bi)2 � �ti) and (fi)(fj)((�Bj)2 � �tj) are independent, and thus theexpectation of their product is 0. Following from the construction of the Itô Integral,we compute

E[(X

j

fj(�Bj)2�

X

j

fj�tj)2] = E[

X

j,i

fjfi((�Bj)2��tj)((�Bi)

2��ti)] .

Next, we expand the square, recall (3.10), use the independence argument presentedabove, and observe

E[X

j

f2j· ((�Bj)

2��tj)

2] = E[X

j

f2j· ((�Bj)

4� 2(�Bj)

2(�tj) + (�tj)2)]

=X

j

E[f2j] · E[(�Bj)

4� 2(�Bj)

2(�tj) + (�tj)2]

=X

j

E[f2j] · (3(�tj)

2� 2(�tj)

2 + (�tj)2)

=X

j

E[f2j] · (2(�tj)

2) ! 0

as �tj ! 0 given that f is nice. ⇤

Setting f ⌘ 1, this proof illustrates a fundamental construction of Itô Calculus:(dBt)2 = dt. Another important computation is revealed in the final line; as �tj !0, the summation

Pj(�tj)2 ! 0. Similarly, as we will show in (4.15) and (4.16),

as �tj ! 0, the summationP

j(�tj)(�Bj) ! 0. This is expressed in di↵erential

notation through the equations (dt)2 = 0 and (dt)(dBt) = 0.

Theorem 4.10 (Itô’s Formula). Let Xt be an Itô process. Suppose g(t, x) 2C2([0,1) ⇥ R). Then Yt = g(t,Xt) is also an Itô process which satisfies the fol-lowing Stochastic Di↵erential Equation

(4.11) dYt = @tg(t,Xt)dt+ @xg(t,Xt)dXt +1

2@xxg(t,Xt)(dXt)

2 .

Proof. By Taylor’s Theorem, we may rewrite Yt = g(t,Xt) as

(4.12) g(t,Xt) = g(0, X0) +X

j

8 BEN RAPPORT

where Rj is the remainder term of each polynomial. We then demonstrate thedesired result (4.11) by evaluating each summation separately. As �tj ! 0, weobserve that

(4.13)X

j


Recalling that (dXs)2 = u2(ds)2 + 2uv(dBs)(ds) + v2(dBs)2 = v2ds, we findZ

t

0d(g) =

Zt

0

@g

@sds+

Zt

0

@g

@xdXs +

1

2

Zt

0

@2g

@x2(dXs)

2 .

Therefore we concluded(Yt) = d(g(t,Xt))

=

✓@g(t,Xt)

@t+

v2(t,!)

2·@2g(t,Xt)

@x2

◆dt+

@g(t,Xt)

@xdXt

=@g(t,Xt)

@tdt+

@g(t,Xt)

@xdXt +

1

2

@2g(t,Xt)

@x2(dXt)

2 is an Itô process. ⇤

For our proof of the Margrabe Formula in the Applications to Financial Modelingsection, we use the 2-dimensional interpretation of the Itô Formula. The followingdefinition and theorem can be found in [1].

Definition 4.17. Let µi,�i be nice functions (i 2 (1, 2)). We define a 2-dimensionalItô process to be

dXt =

(dXt = µ1(t,!)dt+ �1(t,!)dBtdYt = µ2(t,!)dt+ �2(t,!)dWt

where Bt and Wt are both 1-dimensional standard Brownian motions.

Theorem 4.18. Let dXt be a 2-dimensional Itô process as defined above and letg(t,Xt) = g(t,Xt, Yt) be a C2 map g : [0,1]⇥R2 ! R. Then Yt = g(t,Xt) is alsoan Itô process which satisfies

(4.19) dYt = @tg(t,Xt)dt+ @xg(t,Xt)dXt + @yg(t,Xt)dYt +

1

2[@xxg(t,Xt)(dXt)

2 + 2@xyg(t,Xt)(dXt)(dYt) + @yyg(t,Xt)(dYt)2] .

5. Stochastic Differential Equations

While we will not discuss di↵erential equations in depth, we present some exam-ples of common SDEs in mathematical finance and their solutions. We will focusprimarily on Itô processes.

Example 5.1. Let Xt be an Itô process satisfying

dXt = µtXtdt+ �tdBt where µt = µ(t,Xt), �t = �(t,Xt) 2 C2([0,1)⇥R)

To solve this SDE we multiply both sides of the equality by an integrating factor:

e�µttdXt = e�µtt(µtXtdt+ �tdBt) .

Thus, letting g(t,Xt) = e�µttXt = Yt, we apply Itô’s Formula to obtain

d(Yt) = �µte�µttXtdt+ e

�µttdXt + 0 · (dXt)2 .

Substituting dXt into d(Yt), we find

d(Yt) = �e�µttµtXtdt+ e

�µttµtXtdt+ e�µtt�tdBt .

Therefore d(e�µttXt) = e�µtt�tdBt, and thusRT

0 d(e�µttXt) =

RT

0 e�µtt�tdBt.

Solving this integral we find e�µTTXT �X0 =RT

0 e�µtt�tdBt and equivalently

XT = X0eµTT +

ZT

0e(µTT�µtt)�tdBt .

10 BEN RAPPORT

The above example is a variation of the famous Ornstein-Uhlenbleck equation.Altering this equation slightly, we get the di↵erential equation for Geometric Brown-ian motion (GBM). Because GBM is a strictly positive stochastic processes (Xt > 0for all t assuming X0 > 0) and has lognormal distribution, it is used extensively tomodel financial market behaviors. We show the derivation of its solution below.

Example 5.2. Let Xt be a stochastic process satisfying the following SDE:

dXt = Xt[µtdt+ �tdBt] µt = µ(t,Xt), �t = �(t,Xt) 2 C2([0,1)⇥R)

We solve this SDE via careful selection of g. Let g(t,Xt) = ln(Xt) = Yt . Hence,

d(Yt) =1

XtdXt �

1

2(Xt)2(dXt)

2

= µtdt+ �tdBt �1

2[µtdt+ �tdBt]

2

= µtdt+ �tdBt �1

2[(µt)

2(dt)2 + 2µt�t(dt)(dBt) + �t2(dBt)

2]

= (µt �1

2�t

2)dt+ �tdBt .

Integrating both sides of this expression we observe

ln(XT /Xt) =

ZT

t

d(Ys)

=

ZT

t

(µs �1

2�s

2)ds+

ZT

t

�sdBs

Consequently, we derive XT = XteR Tt (µs�

12�s

2)ds+R Tt �sdBs . If µ and � are constants

in R, then we define the stochastic process {XT }T�t to be a Geometric Brownianmotion with drift µ and variance �2. For such a Brownian motion we claim

XT = Xte[(T�t)(µ� 12�

2)+�(BT�Bt)]

= Xte[(T�t)(µ� 12�

2)+�pT�t·Y ]

(5.3)

where Y ⇠ N(0, 1). As such, we compute the conditional expectation

E[XT |Ft] = E[Xte[(T�t)(µ� 12�

2)+�pT�t·Y ]

|Ft]

= Xte(T�t)(µ� 12�

2)· E[e�

pT�t·Y

|Ft]

= Xte(T�t)(µ� 12�

2)· e

12�

2(T�t)

= Xteµ(T�t) .

(5.4)

Lastly, let Z = (T � t)(µ � 12�2) + �

pT � t · Y = log(XT

Xt). We then assert that

E[Z] = (µ� �2

2 )(T � t) and V ar[Z] = �2(T � t). It follows that

Z = log(XTXt

) ⇠ N((µ��2

2)(T � t),�

p

T � t) ;

hence, we claim that XT has lognormal distribution.


6. Applications to Financial Modeling

The stochastic modeling techniques developed in the previous sections are readilyapplied to finance. A security x is an object which has value V (x) = X and canbe bought and sold. An asset is a security whose change in value can be describedusing SDEs. A derivative of an asset is a security whose value is a function of timeand the asset’s value. A market, M , is the set of all securities. The risk-free rateis a security whose value Rt is a function of time and the other securities in themarket X̄t. Formally, the risk free rate is modeled by R0 = 1, dRt = r(t,Xt)Rtdt

and equivalently RT = Rt · eR Tt r(s,Xs) ds for T > t � 0.

A portfolio is an allocation in a market given by ✓ = {ai}, where ai 2 R. In thisway, a portfolio can be thought of as a quantity of each security in a market. Astrategy is an algorithm which buys and sells securities in a market, thus generatinga collection of portfolios {✓t}. The value of a portfolio at time t, is the sum of thevalues of its securities and is denoted V (✓t) = ✓t·Xt =

PaitXi

twhereXt = (Rt, X̄t).

The return of a portfolio over an interval [t, T ] is given by V (✓T )� V (✓t).A strategy inM is an arbitrage when P [V (✓T ) > V (✓t)(1+

RTRt

)] = 1. Conversely,a market is arbitrage-free when no strategy can be an arbitrage. In this way, we canthink of an arbitrage-free market as a market in which the risk-free rate o↵ers thebest guaranteed return. Finally, a portfolio is self-financing if dV (✓t) =

PiaitdXi

t.

Definition 6.1. A trading space is a pair (M , ✓) for which

(1) The market M is arbitrage-free and includes the risk-free rate.(2) Every portfolio ✓ is self-financing.(3) Securities in the market can be bought and sold at any time (Continuity).(4) Asset values in the market are modeled by GBM.

Remark. The value of an asset in a trading space can also be thought of as its price.For such an asset Xt, we describe the instantaneous change of its price as dXt.

One common example of a financial derivative is an option. Given an asset xwith value Xt, an x-option grants its owner the ability to buy 1 unit of x at aspecific point in time, or over a specific time interval, for a fixed price K.1 Noticethat an option grants its owner the right to buy the underlying asset, but does notobligate its owner to buy the asset – the owner of the option may choose NOT tobuy the asset. A European option is an option wherein the owner of the option hasthe right to buy the underlying asset only at a specific point in time T .2

Suppose we want to sell an option, but do not want to be exposed to risk. Wemust derive a strategy such that at time T , the value of our portfolio is guaranteedto be greater than or equal to the value of the option which we sell. This strategyis derived via the Black-Scholes-Merton equation and is shown below.

1 The derivative refered to as an option throughout this paper is more specifically known asa call-option. Likewise, a put-option grants its owner the ability to sell 1 unit of a security at aspecific point in time, or over a specific time interval, for a fixed price K.

2 Options are often used in the financial services industry as insurance contracts. For example,suppose that one asset made up a large percentage of our portfolio. This being the case, we mightwant to purchase a put-option so that if the asset were to dramatically decrease in value, we wouldstill retain a baseline value of K.

12 BEN RAPPORT

Theorem 6.2 (Black-Scholes-Merton Equation). Let T = (M, ✓) be a trading spacewith an asset y, whose price is given by {St}. Suppose f(t, St) 2 T is the price of ay-option whose value at time T is given by f(T, ST ) = F (ST ) = max(ST �K, O),for K 2 R fixed. Then for x = x(t) = St

(6.3) @tf(t, x) = r(t, x)(f(t, x)� x@xf(t, x))�(�(t, x))2x2

2@xxf(t, x) .

Proof. The BSM strategy {✓t} first sells an option at time 0 for a price f(0, S0).Simultaneously, the strategy creates a portfolio, ✓0, allocated between the risk-freerate and the asset y, such that V (✓0) = f(0, S0) = a10R0 + a

20S0. For t 2 [0, T ], the

strategy generates {✓t} such that the value of the portfolio at time T satisfies

(6.4) P [V (✓T ) � F (ST )] = 1 .

The above inequality asserts that at time T the BSM strategy produces a portfolio✓T with value greater than or equal to the value of the sold option almost surely.Due to the arbitrage-free nature of a trading space, we refine (6.4) as follows.For the sake of contradiction, suppose that there exists a strategy {✓⇤

t} given by

✓⇤t= (b1

t, b2

t, 0) such that P [V (✓T ) > F (ST )] = 1. Now consider the strategy {✓0t}

given by ✓0t= (b1

t, b2

t,�1). We would then observe

P [V (✓0T) = V (✓T )� F (ST ) > f(0, S0)� V (✓0) = V (✓

00) = 0] = 1

implying that P [V (✓0T)� V (✓00) > V (✓

00)(1 +RT ) = 0] = 1 .

This result would indicate that {✓0t} is an arbitrage, contradicting the arbitrage-free

nature of a trading space. Hence, we assert that for t 2 [0, T ] the strategy mustgenerate {✓t} such that

(6.5) P [V (✓T ) = F (ST )] = 1 where ✓t = (a1t, a2

t, 0) for t 2 [0, T ] .

Next, we will use a similar arbitrage argument to extend this result, in turn provingthat V (✓t) = f(t, St) for all t 2 (0, T ). For the sake of contradiction, suppose thatV (✓t) > f(t, St) for some fixed set of points t 2 E = {t0, t1, ...} ✓ [0, T ]. In thiscase, the strategy defined by

✓0t=

((a1

t, a2

t,�1) if t t0

(�a1t+ [V (✓t0)� f(t0, St0)],�a

2t, 1) if t > t0

is an arbitrage as

P [V (✓0T) = F (ST )� V (✓T ) + � = � > 0 = V (✓0)� f(0, S0) = V (✓

00) = 0] = 1

where � = [V (✓t0) � f(t0, St0)] > 0. Therefore, E must be empty as there cannotexist any points t for which V (✓t) ⇧ f(t, St). A similar argument, employing thestrategy defined by

✓0t=

((�a1

t,�a2

t, 1) if t t0

(a1t+ [f(t0, St0)� V (✓t0)], a

2t,�1) if t > t0

can be made to show V (✓t0) ⌅ f(t0, St0). Therefore, by trichotomy, we find thatf(t, St) = V (✓t) for all t 2 [0, T ], which yields the crucial equality

(6.6) dV (✓t) = d(f(t, St)) for all t 2 [0, T ) .


Let St be a GBM represented by dSt = St[µtdt+ �tdBt] where µt = µ(t,Xt), �t =�(t,Xt) 2 C2([0,1) ⇥ R). As every portfolio in a trading space is self-financing,we assert dV (✓t) = a1dRt + a2dSt. By (6.6) we claim

d(V (✓t)) = d(f(t, St)) ,

and thus a1dRt + a2dSt = @tf(t, St)dt+ @xf(t, St)dSt +1

2@xxf(t, St)(dSt)

2 .

Noticing, (dSt)2 = [St(µtdt+ �tdBt)]2 = S2t �2tdt we proceed to show

a1dRt + [a2 � @xf(t, St)] · St[µtdt+ �tdBt] = @tf(t, St)dt+1

2@xxf(t, St)S

2t�2tdt .

Recall, that we wanted the BSM strategy to be riskless, i.e. have a guarenteedreturn. To do so, we must constantly update our portfolio such that the randomterm in the above di↵erential equation is always eliminated. Setting a2 = @xf(t, St),

and consequentially a1 =f(t,St)�@xf(t,St)

Rt, we compute

[f(t, St)� @xf(t, St)]r(t, St)dt = @tf(t, St)dt+1

2@xxf(t,Xt)S

2t�2dt

and @tf(t, St) = [f(t, St)� @xf(t, St)]r(t, St)��2tS2t

2@xxf(t,Xt)

which is the Black-Scholes-Merton Equation. ⇤

The PDE derived in this theorem yields the risk-free pricing of a y-option withstrike price K and expiration T . We consider this pricing schema to be risk-freebecause we can construct a portfolio of the risk-free rate and the underlying stock,for which we can entirely eliminate randomnness. Even though this schema is risk-free, the y-option itself need not be risk-free – the y-option will almost surely nothave a guaranteed value at time T . However, the y-option does have an expectedvalue at time T . We can use this expected value to derive the closed form risk-neutral solution to the BSM model, wherein the expected return of the y-optionequals the return of the risk-free rate.

Corollary 6.7. Suppose r(t, St) and �(t, St) are constant. Then the risk-neutralprice of the y-option described above is

(6.8) f(t, St) = St�(d1)� e�r(T�t)K�(d2)

where K 2 R is from 6.2 and d1, d2 are given by

d1 =ln(St

K) + (r + �

2

2 )(T � t)

�pT � t

and d2 =ln(St

K) + (r � �

2

2 )(T � t)

�pT � t

.

Proof. We begin with a risk-neutral argument. If we desire the price of our y-option to be risk-neutral, then the expected return of the y-option should be equalto the expected return of an initial portfolio of f(t,St)

Rtunits of the risk free rate.

Mathematically, we assert that

(6.9)f(t, St)

Rt·RT � f(t, St) = E[F (ST )� f(t, St)|Ft].

As r(t, St) is constant,RTRt

is Ft-measurable and thus we simplify (6.9) to be

(6.10) f(t, St)er(T�t) = E[F (ST )|Ft] .

14 BEN RAPPORT

Recalling (5.4), another risk-neutrality argument can be made to show

(6.11) Z = e(T�t)(µ�12�

2)+�pT�t·Y = e(T�t)(r�

12�

2)+�pT�t·Y .

Using this result, we compute

E[F (ST )|Ft] =

Z 1

�1max(Ste

z�K, 0)f(z)dz

=

Zln( KSt

)

�1max(Ste

z�K, 0)f(z)dz +

Z 1

ln( KSt)max(Ste

z�K, 0)f(z)dz

=

Z 1

ln( KSt)Ste

z�Kf(z)dz

= St

Z 1

ln( KSt)ezf(z)dz �K

Z 1

ln( KSt)f(z)dz

= St

Z 1

ln( KSt)e(T�t)(µ�

12�

2)+�pT�t·ye�y

2

dy �K

Z 1

ln( KSt)e�z

2

dz

which following several lines of computation simplifies to be

= Ster(T�t)�

ln(St

K) + (r + �

2

2 )(T � t)

�pT � t

!�K�

ln(St

K) + (r � �

2

2 )(T � t)

�pT � t

!.

Finally, we isolate f(t, St) to derive the closed form BSM pricing equation

f(t, St) = St�(d1)� e�r(T�t)K�(d2). ⇤

Before continuing, it is worth discussing some assumptions and limitations of theBlack-Scholes-Merton model, the majority of which derive from the constructionof the trading space. Firstly, the BSM model assumes that securities are boughtand sold continously. In reality, there are specific intervals in which markets areopen and securities can be exchanged – after normal market hours, securities arestill bought and sold, but there is a signifcant decrease in the volume of exchangedsecurities. This leads to a drastically di↵erent market environment. Additionally,arbitrage strategies exist in realistic markets, but are almost never present for morethan a microsecond. This fact combined with our assumption of continous tradingleads to a certain tension in our result: if we assume the continuous exchangeof assets, then in a more realistic model there could be instantaneous arbitragestrategies.3

Lastly, our assumption that assets follow GBM with constant drift and varianceis an over-generalization of market behaviors. It is very rare for an asset to haveconstant variance, and even rarer for its drift to remain constant. However, BSMis simply a model, and should not be viewed as an all encompassing formula. BSMis a framework.

Now we will consider a derivative, whose payo↵ does not depend on a singularasset, but instead on two assets. One such example is a spread-option. An x/y-spread-option grants its owner the ability to exchange 1 unit of an asset x for 1 unitof an asset y at a specific point in time. Like an plain y-option, a spread-optiongrants its owner the right to exchange the assets, but does not obligate the owner

3 One could construct a heuristic measure-theoretic argument that these moments of inconsis-tency are so minute that we may treat them as a set of measure zero, but this lacks formalization.


to do so. Unlike a y-option, a spread-option does not have a strike price, its payo↵depends solely on the price of the two underlying assets at time T . The formulageneralizing BSM to this environment is known as the Margrabe Formula, and isshown below

Corollary 6.12 (The Margrabe Formula). Let T = (M, ✓) be a trading space withtwo assets x, y, whose prices satisfy

dXt = Xt[µ1dt+ �1dWt] and dYt = Yt[µ2dt+ �2dBt] where µ1,�1, µ2,�2 2 R

and Wt and Bt are independent standard Brownian motions. Suppose thatf(t,Xt, Yt) 2 T is the price of a x/y-spread-option whose value at time T is givenby f(T,XT , YT ) = F (XT , YT ) = max(XT � YT , O). Then the risk-neutral price ofthe x/y-spread-option described above is

(6.13) f(t,Xt, Yt) = Xt�(q1)� Yt�(q2)

where q1 and q2 are given by

q1 =ln(S1

S2) + �

2

2 (T � t)

�pT � t

and q2 =ln(S1

S2)� �

2

2 (T � t)

�pT � t

, for � =q

�21 + �22 .

Proof. The proof of the Margrabe formula is a direct extension of the BSM equation.Consider a self-financing portfolio

(6.14) V (✓t) = a1tRt + a

2tXt + a

3tYt such that V (✓T ) = F (XT , YT ) .

Using the same type of arbitrage arguments as those found in BSM, we have that

V (✓t) = f(t,Xt, Yt) for all t 2 [0, T ]

which yields the crucial equality:

dV (✓t) = d(f(t,Xt, Yt)) .

Following the same procedure as in Theorem 6.2, this time using the 2-d Itô For-mula, we find that

(6.15) a1tdRt + a

2tdXt + a

3tdYt = @tf(t,Xt)dt+ @xf(t,Xt)dXt + @yf(t,Xt)dYt

+1

2[@xxf(t,Xt)(dXt)

2 + 2@xyf(t,Xt)(dXt)(dYt) + @yyf(t,Xt)(dYt)2] .

Using analogous riskless arguments from Theorem 6.2, we set

a2 = @xf(t,Xt), a3 = @yf(t,Xt) and a1 =f(t,Xt)� @xf(t,Xt)� @yf(t,Xt)

Rt.

Recalling that dRt = r(t,Xt)Rtdt, we assert

a1tdRt = r(t,Xt)(f(t,Xt)� @xf(t,Xt)� @yf(t,Xt))dt .

We then reduce (6.15) to be

(6.16) r(t,Xt)(f(t,Xt)� @xf(t,Xt)� @yf(t,Xt))dt

= @tf(t,Xt)dt+1

2[@xxf(t,Xt)(dXt)

2+2@xyf(t,Xt)(dXt)(dYt)+@yyf(t,Xt)(dYt)2] .

Next, we look to expand (dXt)2, (dXt)(dYt), and (dYt)2. We observe that

(dXt)2 = [Xt(µ1dt+ �1dWt)]

2

= X2t(µ21(dt)

2 + 2µ1�1dtdWt + �21(dWt)

2)

= X2t�21dt.

16 BEN RAPPORT

Symmetrically, we also find that (dYt)2 = Y 2t �22dt. Recalling that Wt and Bt are

independent, we compute

(dXt)(dYt) = [Xt(µ1dt+ �1dWt)][Yt(µ2dt+ �2dBt)]

= XtYt[µ1µ2(dt)2 + µ1�2dtdBt + µ2�1dWtdt+ �1�2(dWt)(dBt)]

= 0 .

Therefore, (6.16) simplifies to be

(6.17) r(t,Xt)(f(t,Xt)� @xf(t,Xt)� @yf(t,Xt))

= @tf(t,Xt) +1

2[@xxf(t,Xt)X

2t�21 + @yyf(t,Xt)Y

2t�22 ] .

We claim that V (✓T ) = F (XT , YT ) = max(XT � YT , 0) = YTmax(XTYT

� 1, 0).Suppose that we standardized our market such that all asset values are measuredin terms of Yt – the new numeraire.

4 By asserting Yt as the numeraire, the spread-option thus becomes an x-option with K = 1 and r = 0. Hence, we apply the BSMmodel to Zt =

XtYt

, noting Zt is a GBM with drift µ1 and variance �21 + �22 . As

such, we obtain the equivalences

(6.18)f(t,Xt, Yt)

Yt= f(t, Zt) and

f(T,XT , YT )

YT= F (ZT ) = max(ZT � 1, 0) .

Applying the BSM schema to Zt, (6.17) reduces to

(6.19) 0 = @tf(t, Zt) +1

2Z2t@zzf(t, Zt)(�

21 + �

22) .

Next we use Corollary 6.7 and observe

f(t, Zt) = Zt�

0

@ ln(S1S2

) + �21+�

22

2 (T � t)p(�21 + �

22)(T � t)

1

A�1e�0·(T�t)�

0

@ ln(S1S2

)� �21+�

22

2 (T � t)p(�21 + �

22)(T � t)

1

A .

Isolating f(t,Xt, Yt), we conclude

⇤(6.20) f(t,Xt, Yt) = Xt�(q1)� Yt�(q2) .

7. Stochastic Simulation

While it is sometimes possible to model stochastic process with closed formequations, more often than not, doing so is computationally intensive and ine�cient.As a result, approximation schema known as Monte Carlo methods have becomeimportant tools for simulating Itô calculus in applied settings. In a Monte Carlomethod, a sample value of an asset at time T is computed recursively from an initialcondition via a step formula. The expected value of a process is hence computedas the average of a large number of sample values.

We will present two common Monte Carlo methods, the Euler method and theMilstein method, and use them both to simulate models of complex financial deriva-tives.

4 Often this procedure of standardizing a market with respect to Yt is formalized via appli-cations of the Girsanov Theorem and the Radon-Nikodym Theorem, which provide a foundationfor the change in measure induced by a change in the numeraire. While a more heuristic repre-sentation is shown here, formal proofs of both theorems and their application can be found in [2]and [7].


Construction 7.1 (Euler Method). Let x be an asset whose price Xt is modeledby (4.3) and let �t be the step size of the method. Given X0 = x0 2 R+ we definethe step formula of the Euler Method of Xt to be

X̂t+1 = X̂t + u(t, Ŝt)�t+ v(t, X̂t)�Bt where �Bt ⇠ N(0,�t) .

In the Euler Method, we see that the step formula follows almost directly fromthe asset value’s SDE. This is one of the most fundamental Monte Carlo meth-ods because of its simplistic formula and computation. As with any Monte Carlomethod, as the number of sample values is increased, the approximation error de-creases. Similarly, as we decrease the method’s step size, our approximation errordecreases. The Milstein Method, presented below, is a refinement of the Eulermethod and derives from the Itô Formula.

Construction 7.2 (Milstein Method). Let x be an asset whose price Xt is alsomodeled by (4.3). Given X0 = x0 2 R+ we define the step formula of the MilsteinMethod of Xt to be

X̂t+1 = X̂t + u(t, X̂t)�t+ v(t, X̂t)�Bt +1

2v(t, X̂t)@xv(t, X̂t) · ((�Bt)

2��t)

where �Bt ⇠ N(0,�t) and �t is the step size of the method.

We now present a number of simulations using these Monte Carlo methods. Foreach of the following examples our step size will be equal to 1100 .

Example 7.3. Let x be an asset whose price Xt is modeled by

X0 = 200, dXt =1000X̂t

· cos( t2000 )dt+ log(X̂2t+ t)dBt for t 2 [0, 10) .

The step formula of the Milstein Method of Xt is given by

X̂t+1 = X̂t+10

X̂t· cos(

t

2000)+ log(X̂2

t+ t)�Bt+

log(X̂2t+ t) · 2X̂t

X̂2t+ t

· ((�Bt)2�

1

100)

where �Bt ⇠ N(0,1

100 ). A plot of 100 sample paths generated by the MilsteinMethod is shown below.

From the sample paths generated in this schema, we approximate the expectedvalue of this security at time T = 10 to be 189.66 with a variance of 10.99.

18 BEN RAPPORT

However, 100 sample paths is a relatively small sample size, which in turnproduces an inexact estimate.

The purpose of the above model is not to represent any specific real life assetclass, but rather to show the computational e�ciency of Monte Carlo methods inhandling complex stochastic structures. Next, we take a basic GBM and evaluatehow close the Monte Carlo method approximation comes to the theoretic valuederived in (5.5).

Example 7.4. Let x be an asset whose price Xt is modeled by

X0 = 100, dXt = .03Xtdt+ 1.4XtdBt for t 2 [0, 10) .

A plot of 400 samples generated by the Milstein Method is shown below.

From the sample paths generated in this schema, we approximate the expectedvalue of this asset at time t = 10 to be 134.17 with a variance of 1062.07. In (5.5),we derived the expected value of a GBM with mean µ and � to be X0eTµ. Assuch, we compute E[Xt] = 134.99, which is incredibly close to our simulated valueof 134.17. Next, we use Monte-Carlo methods to model a spread-option derivativesimilar to the one discussed in our presentation of the Margrabe formula.

Example 7.5. Let x and y be assets whose prices are modeled by X0 = 300,dXt = .03Xtdt + .4XtdBt for t 2 [0, 5) and Y0 = 200, dYt = .1Ytdt + .8YtdBtfor t 2 [0, 5), respectively. Lastly, let z be a derivative whose value at time T , isgiven as F (ZT ) = max(XT �YT , 0). Using Euler methods, we can approximate theexpected value of ZT by simulating pairs of the assets Xt and Yt. A plot of 1000sample paths of Zt is shown below.


From the sample paths generated in this schema, we approximate the expectedvalue of this derivative at time T = 10 to be 39.27 with a variance of 1917.65.While the initial value of this derivative at time t is exactly 100, by time T wesee that its expected value is significantly less than 100. Again, it is importantto recall that a real life spread-option strategy using this model would run sev-eral thousand simulations so that the approximated price could converge to thetheoretical expected value.

Acknowledgments

I would like to thank DeVon Ingram for his guidance throughout the programand for helping me understand the intricacies of measure theory. I would like tothank Sebastián Muñoz for first introducing me to Itô calculus during our springDRP. I would like to thank Greg Lawler for his insightful lecture series on randomwalks and stochastic systems. I would like to thank Peter May for making thisprogram not only possible, but exciting despite less than ideal conditions. Lastly,thank you to Mom, Dad, E and M – I love you guys.

References

[1] Øksendal, B. (1995). Stochastic Di↵erential Equations, An Introduction with Applications.http://th.if.uj.edu.pl/ gudowska/dydaktyka/Oksendal.pdf

[2] Evans, L. (2013). An Introduction to Stochastic Di↵erential Equations.[3] Lawler, G. F. (2014). Stochastic Calculus: An Introduction with Applications.

http://www.math.uchicago.edu/ lawler/finbook.pdf[4] Bass, R. (2013). Real Analysis graduate students. http://bass.math.uconn.edu/rags010213.pdf[5] Hull, J. C. (1989). Options, Futures, and Other Derivatives. Pearson.[6] Suresh, E. (2019). Stochastic Calculus Applied to Arbitrage-Free Options Pricing.

http://math.uchicago.edu/ may/REU2019/REUPapers/Suresh.pdf[7] Margrabe, W. (1978). The Value of an Option to Exchange One Asset for Another.

Journal of F inance. http://www.jstor.com/stable/2326358

20 BEN RAPPORT

Appendix

A. Proof for Proposition 3.10

Proof. Let Wt = Bt �Bs ⇠ N(0, t� s). Using the Taylor expansion for ex we find

E[eiu(Bt�Bs)] = E[eiu(Wt)]

= E[1X

n=0

(iuWt)n

n!]

=1X

n=0

E[(iuWt)n

n!]

= E[1] + E[iuWt] + E[�u2W 2

t

2] + E[

�iu3W 3t

3!] + E[

u4W 4t

4!]

E[eiu(Bt�Bs)] = 1 + uE[iWt]�u2

2E[W 2

t]�

u3

3!E[iW 3

t] +

u4

4!E[W 4

t] + ...

Recall, by (3.6) we have E[eiu(Bt�Bs)] = e�u2(t�s)

2 . Expanding the right hand sideof the equality we find

e�u2(t�s)

2 = 1�u2(t� s)

2+

u4(t� s)2

2 · 4�

u6(t� s)3

3! · 8+

u8(t� s)4

4! · 16+ ...

Setting the even powered u terms equal we find

(�1)ku2kE[W 2k

t]

(2k)!= (�1)k

u2k · (t� s)k

k! · 2kfor k 2 N

Thus, E[W 2kt

] =(2k)!

2k · k!· (t� s)k for k 2 N

Therefore, E[(Bt �Bs)4] = E[W 4

t] = 3(t� s)2. ⇤

B. Coding for Section 7 can be found on the following pages.

Stochastic Simulation Code

August 28, 2020

[1]: import numpy as np

import math

import random

from matplotlib import pyplot as plt

from IPython.display import clear_output

PI = 3.1415926

e = 2.71828

[2]: def random_normal(sigma):

mu = 0

r = np.random.normal(mu, sigma)

return r

print(random_normal(.01))

-0.0009863635516575134

[56]: #EXAMPLE 1#the u, v and v_prime functions are input manually...

def u_fcn(t,x,mu,sigma):

value = 1000*np.cos(t/2000)/x

return value

def v_fcn(t,x,mu,sigma):

value = np.log(x*x+t)

return value

def v_prime(t,x,mu,sigma):

value = 2*x/(x*x+t) #= v_prime = (d/dx)(v_fcn)return value

testing=[]

def average(lst):

return sum(lst)/len(lst)

1

def milstein_method(mu, sigma, start, step, end):

prices = [start]

step_count = int(end/step)

time = 0

for i in range(step_count):

current = prices[-1]

time+=step_count/100

#notice here we divide by 100 so that the time value remains appropriately scaledrand = random_normal(step)

new = current + u_fcn(time,current,mu,sigma)*step +�

,!v_fcn(time,current,mu,sigma)*rand +\

.

,!5*v_fcn(time,current,mu,sigma)*v_prime(time,current,mu,sigma)*(rand**2-step)

prices.append(new)

testing.append(current)

plt.plot(prices)

plt.figure(figsize=(10,5),dpi=300)

plt.xlabel('Time (in centiseconds)')

plt.ylabel('Value of Security')

plt.title('Value of Exotic Security over time')

for i in range(100):

milstein_method(mu = .1, sigma = 10, start = 200, step = .01, end = 10)

print('the simulated mean at t=10 is ' + str(average(testing)))

print('the simulated variance at t=10 is ' + str(np.var(testing)))

print('the simulated volatility at t=10 is ' + str(np.std(testing)))

plt.savefig("REUfig1.png")

the simulated mean at t=10 is 189.65625938675225

the simulated variance at t=10 is 10.989436532745883

the simulated volatility at t=10 is 3.315031905238

2

[48]: #EXAMPLE 2

testing=[]

def euler_method(mu, sigma, start, step, end):

prices = [start]

step_count = int(end / step)


current = prices[-1]

rand = random_normal(step)

new = current + current*mu*step + current*sigma*rand

prices.append(new)

testing.append(current)

plt.plot(prices)




plt.title('Value of Security over time')


euler_method(mu = .03, sigma = .7, start = 100, step = .01, end = 10)





3




[49]: 100*e**(10*.03)

[49]: 134.98585351801717

[51]: #EXAMPLE 3testing = []

def spreadoption_euler_method(mu1, sigma1, start1, mu2, sigma2, start2, step,�

,!end):

prices1 = [start1]

prices2= [start2]

prices3 = [start1-start2]

step_count = int(end / step)


current1 = prices1[-1]

rand1 = random_normal(step)

new1 = current1 + current1*mu1*step + current1*sigma1*rand1

prices1.append(new1)

current2 = prices2[-1]

rand2 = random_normal(step)

new2 = current2 + current2*mu2*step + current2*sigma2*rand2


4

delta = new1 - new2

new3 = max(delta,0)


plt.plot(prices3)

testing.append(prices3[-1])




plt.title('Value of Spread-Option over time')


spreadoption_euler_method(mu1 = 0.03, sigma1 = 0.4, start1 = 300, mu2 = .1,�

,!sigma2 = 0.8, start2 = 200, step = .01, end = 5)








5

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APPLICATIONS OF STOCHASTIC CALCULUS TO FINANCIAL MODELINGmath.uchicago.edu/~may/REU2020/REUPapers/Rapport.pdf · 2020. 10. 22. · APPLICATIONS OF STOCHASTIC CALCULUS TO FINANCIAL