Portfolio Optimization

Portfolio Optimization

Gerhard-Wilhelm Weber1 Erik Kropat2 Zafer-Korcan Görgülü3

1Institute of Applied MathematicsMiddle East Technical University

Ankara, Turkey

2Department of MathematicsUniversity of Erlangen-Nuremberg

Erlangen, Germany

3University of the Federal Armed ForcesMunich, Germany

2008

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Outline I

1 The Mean-Variance Approach in a One-Period ModelIntroduction

2 The Continuous-Time Market ModelModeling the Security PricesExcursion 1: Brownian Motion and MartingalesContinuation: Modeling the Security pricesExcursion 2: The Ito IntegralExcursion 3: The Ito FormulaTrading Strategy and Wealth ProcessProperties of the Continuous-Time Market ModelExcursion 4: The Martingale Representation Theorem

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Outline II

3 Option PricingIntroductionExamplesThe Replication PrincipleArbitrage OpportunityContinuationPartial Differential Approach (PDA)Arbitrage & Option Pricing

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Outline III

4 Pricing of Exotic Options and Numerical AlgorithmsIntroductionExamplesExamplesEquivalent Martingale MeasureExotic Options with Explicit Pricing FormulaeWeak Convergence of Stochastic ProcessesMonte-Carlo SimulationApproximation via Binomial TreesThe Pathwise Binomial Approach of Rogers and Stapleton

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Outline IV

5 Optimal PortfoliosIntroduction and Formulation of the ProblemThe martingale methodOptimal Option PortfoliosExcursion 8: Stochastic ControlMaximize expected value in presence of quadratic control costsIntroductionPortfolio Optimization via Stochastic Control Method

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Outline

1 The Mean-Variance Approach in a One-Period Model

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Outline

1 The Mean-Variance Approach in a One-Period ModelIntroduction

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Introduction

MVA Based on H. MARKOWITZ

OPM • Decisions on investment strategies only at the beginning of theperiod

• Consequences of these decisions will be observed at the end of theperiod (−→ no action in between: static model )

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The one-period model

Market with d traded securitiesd different securities with positive prices p1, . . . , pd at time t = 0

Security prices P1(T ), . . . , Pd (T ) at final time t = T notforeseeable−→ modeled as non-negative random variables on probability space

(Ω,F ,P)

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Securities in a OPM

Returns of Securities

Ri(T ) := Pi (T )pi

(1 ≤ i ≤ d)

Estimated Means, Variances and Covariances

E (Ri(T )) = µi , Cov(Ri(T ), Rj(T )

)= σij (1 ≤ i ≤ d)

RemarkThe matrix

σ :=(σij)

i ,j∈1,...,d

is positive semi-definite as it is a variance-covariance matrix.

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Securities in a OPM

Each security perfectly divisable

Hold ϕi ∈ R shares of security i (1 ≤ i ≤ d)

Negative position (ϕi < 0 for some i) corresponds to a selling−→ Not allowed in OPM

−→ No negative positions: pi ≥ 0 (1 ≤ i ≤ d)−→ No transaction costs

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Budget equation and portfolio return

The Budget EquationInvestor with initial wealth x > 0 holds ϕi ≥ 0 shares of securityi with ∑

1≤i≤d

ϕi · pi = x

The Portfolio Vector π := (π1, . . . , πd)T

πi :=ϕi · pi

x(1 ≤ i ≤ d)

Portfolio Return

Rπ :=∑

1≤i≤d

πi · Ri(T ) = πT R

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Remarkπi . . . fraction of total wealth invested in security i

∑

1≤i≤d

πi =

∑1≤i≤d

ϕi · pi

x=

xx

= 1

Xπ(T ) . . . final wealth corresponding to x and π

Xπ(T ) =∑

1≤i≤d

ϕi · Pi(T )

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Remark (continued)Portfolio Return

Rπ =∑

1≤i≤d

πi · Ri(T ) =∑

1≤i≤d

ϕi · pi

x· Pi(T )

pi=

Xπ(T )

x

Portfolio Mean and Portfolio Variance

E (Rπ) =∑

1≤i≤d

πi · µi , Var (Rπ) =∑

1≤i ,j≤d

πi · σij · πj

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Selection of a portfolio–criterion

(i) Maximize mean return (choose security of highest mean return)−→ risky, big fluctuations of return

(ii) Minimize risk of fluction

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Selection of a portfolio–approach by Markowitz (MVA)

Balance Risk (Portfolio Variance) and Return (Portfolio Mean)(i) Maximize E (Rπ) under given upper bound c1 for Var (Rπ)

maxπ∈Rd

E (Rπ) subject to

πi ≥ 0 (1 ≤ i ≤ d)∑

1≤i≤d

πi = 1

Var (Rπ) ≤ c1

(ii) Minimize Var (Rπ) under given lower bound c2 for E (Rπ)

minπ∈Rd

Var (Rπ) subject to

πi ≥ 0 (1 ≤ i ≤ d)∑

1≤i≤d

πi = 1

E (Rπ) ≥ c2

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Solution methods

(i) Linear Optimization Problem with quadratic constraint−→ No standard algorithms, numerical inefficient

(ii) Quadratic Optimization Problem with positive semidefiniteobjective matrix σ

−→ efficient algorithms (i.e., GOLDFARB/IDNANI, GILL/MURRAY)Feasible region non-empty if c2 ≤ max

1≤i≤dµi

σ positive definite and feasible region non-empty−→ unique solution (even if one security riskless)

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Relations between the formulations (i) and (ii)

TheoremAssume:

σ positive definite

min1≤i≤d

µi ≤ c2 ≤ max1≤i≤d

µi c2 ∈ R+0

minπi≥0,

∑1≤i≤d πi=1

σ2(π) ≤ c1 ≤ maxπi≥0,

∑1≤i≤d πi=1

σ2(π) c1 ∈ R+0

Then

(1) π∗ solves (i) with Var(Rπ∗)

= c1 =⇒ π∗ solves (ii) withc2 := E

(Rπ∗)

(2) π solves (ii) with E(Rπ)

= c2 =⇒ π solves (i) withc1 := Var

(Rπ)

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The diversification effect–example

Holding different Securities reduces VarianceBoth security prices fluctuate

randomlyσ11, σ22 > 0

independentσ12 = σ21 = 0

Then for the Portfolio π =

(0.50.5

)we get

Var (Rπ) = Var (0.5 · R1 + 0.5 · R2) =σ11

4+

σ22

4

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The diversification effect–example

Holding different Securities reduces Variance

−→ If σ11 = σ22 then the Variance of Portfolio(

0.50.5

)is half as big

as that of(

10

)or(

01

)

−→ Reduction of Variance . . . Diversification Effect depends onnumber of traded securities

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Example

Mean-Variance CriterionInvesting into seemingly bad security can be optimal. Let be

µ =

(1

0.9

), σ =

(0.1 −0.1

−0.1 0.15

)

Formulation (ii) becomes (II)

minπ

Var (Rπ) = minπ

(0.1 · π2

1 + 0.15 · π22 − 0.2 · π1π2

)

subject to

π1, π2 ≥ 0π1 + π2 = 1

E (Rπ) = π1 + 0.9 · π2 ≥ 0.96

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Example

Consider Portfolios(

10

)and

(0.50.5

)(does not satify

expectation constraint)

Var(

R(1,0)T)

= 0.1 , E(

R(1,0)T)

= 1

Var(

R(0.5,0.5)T)

= 0.125 , E(

R(0.5,0.5)T)

= 0.95

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Example

Ignore expectation constraint and rememberπ1, π2 ≥ 0 π1 + π2 = 1. Hence

minπ

(0.1 · π2

1 + 0.15 · (1 − π1)2 − 0.2 · π1 · (1 − π1)

)

= minπ

(0.45 · π2

1 − 0.5 · π1 + 0.15)

−→ Minimizing Portfolio (No solution of (II) but better than(

0.50.5

))

π =19·(

54

)

−→ Portfolio Return Variance Var(

R( 59 ,

49 )

T)= 0.001

−→ Portfolio Return Mean E(

R( 59 ,

49 )

T)= 0.95

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Example

0.0 0.5 0.6 1.00.0

0.4

0.5

1.0

π1

π2

Pairs (π1, π2) satisfying expectation constraint are above thedotted line

Intersect with line π1 + π2 = 1−→ Feasible region of MeanVariance Problem (bold line)

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Example

0 0.5 0.6 1.0 1.50

0.05

0.1

0.15

π1

Var

Portfolio Return Variance (as function of π1 ) of all pairs satisfyingπ1 + π2 = 1

Minimum in feasible region π ∈ [0.6, 1] is attained at π = 0.6Optimal Portfolio in (II)

−→π

∗ =

(0.60.4

)with Var

(Rπ∗

)= 0.012 , E

(Rπ∗

)= 0.96

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Stock price model

OPMNo assumption on distribution of security returns

Solving MV Problem just needed expectations and covariances

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Stock price model

OPM with just one security (price p1 at time t = 0 )At time T security may have price d · p1 or u · p1

q : probability of decreasing by factor d1 − q : probability of increasing by factor u (u > d)

Mean and Variance of Return

E (R1(T )) = E(

P1(T )

p1

)= q · u + (1 − q) · d

Var (R1(T )) = Var(

P1(T )

p1

)= q · u2 + (1 − q) · d2

− (q · u + (1 − q) · d)2

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Stock price model

OPM with just one security (price p1 at time t = 0 )After n periods the security has price

P1(n · T ) = p1 · uXn · dn−Xn

with Xn ∼ B(n, p) number of up-movements of price inn periods

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Comments on MVA

Only trading at initial time t = 0

No reaction to current price changes possible( −→ static model )

Risk of investment only modeled by variance of return

Need of Continuous-Time Market ModelsDiscrete-time multi-period models(many periods −→ no fast algorithms)

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Outline

2 The Continuous-Time Market Model

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Outline


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Modeling the security prices

Market with d+1 securitiesd risky stocks withprices p1, p2, . . . , pd at time t = 0 andrandom prices P1(t), P2(t), . . . , Pd (t) at times t > 0

1 bond withprice p0 at time t = 0 anddeterministic price P0(t) at times t > 0.

Assume: Perfectly devisible securities, no transaction costs.

⇒ Modeling of the price development on the time interval [0, T ].

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The bond price

Assume: Continuous compounding of interest at

a constant rate r :

Bond price: P0(t) = p0 · er ·t for t ∈ [0, T ]

a non-constant, time-dependent and integrable rate r(t):

Bond price: P0(t) = p0 · e

t∫

0r(s) ds

for t ∈ [0, T ]

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The stock price

Stock price = random fluctuations around an intrinsic bond part

0 0.2 0.4 0.6 0.8 1

0.8

1

1.2

1.4

1.6

1.8

2

log-linear model for a stock price

ln(Pi(t)) = ln(pi) + bi · t + ”randomness”

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The stock price

Randomness is assumed

to have no tendency, i.e., E("randomness") = 0,

to be time-dependent,

to represent the sum of all deviations ofln(Pi(t)) from ln(pi) + bi · t on [0, T ],

∼ N (0, σ2t) for some σ > 0.

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The stock price

Deviation at time t

Y (t) := ln(Pi(t)) − ln(pi) − bi · t

withY (t) ∼ N (0, σ2t)

Properties:

E(Y (t)) = 0,

Y (t) is time-dependent.

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The stock price

Y (t) = Y (δ) + (Y (t) − Y (δ)), δ ∈ (0, t)

Distribution of the increments of the deviation Y (t) − Y (δ)

depends only on the time span t − δ

is independent of Y (s), s ≤ δ

=⇒ Y (t) − Y (δ) ∼ N(0, σ2(t − δ)

)

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The stock price

Existence and properties of the stochastic process

Y (t)t∈[0,∞)

will be studied in the excursion on the Brownian motion .

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Outline


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General assumptions

General assumptionsLet (Ω,F , P) be a complete probability space with sample space Ω,σ-field F and probability measure P.

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Filtration

DefinitionLet Ftt∈I be a family of sub-σ-fields of F , I be an ordered index setwith Fs ⊂ Ft for s < t , s, t ∈ I. The family Ftt∈I is called a filtration.

A filtration describes flow of information over time.

Ft models events observable up to time t .

If a random variable Xt is Ft -measurable, we are able todetermine its value from the information given at time t .

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Stochastic process

DefinitionA set (Xt , Ft)t∈I consisting of a filtration Ftt∈I and a family ofR

n-valued random variables Xtt∈I with Xt being Ft -measurable iscalled a stochastic process with filtration Ftt∈I .

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Remark

RemarkI = [0,∞) or I = [0, T ].

Canoncial filtration (natural filtration) of Xtt∈I :

Ft := FXt := σXs | s ≤ t , s ∈ I.

Notation: Xtt∈I = X (t)t∈I = X .

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Sample path

Sample pathFor fixed ω ∈ Ω the set

X .(ω) := Xt(ω)t∈I = X (t , ω)t∈I

is called a sample path or a realization of the stochastic process.

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Identification of stochastic processes

Can two stochastic processes be identified with each other?

DefinitionLet (Xt , Ft)t∈[0,∞) and (Yt , Gt)t∈[0,∞) be two stochastic processes.Y is a modification of X , if

Pω |Xt(ω) = Yt(ω) = 1 for all t ≥ 0.

DefinitionLet (Xt , Ft)t∈[0,∞) and (Yt , Gt)t∈[0,∞) be two stochastic processes.X and Y are indistinguishable, if

Pω |Xt(ω) = Yt(ω) for all t ∈ [0,∞) = 1.

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Identification of stochastic processes

RemarkX , Y indistinguishable ⇒ Y modification of X .

TheoremLet the stochastic process Y be a modification of X . If both processeshave continuous sample paths P-almost surely, then X and Y areindistinguishable.

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Brownian motion

DefinitionThe real-valued process Wtt≥0 with continuous sample paths and

i) W0 = 0 P-a.s.

ii) Wt − Ws ∼ N (0, t − s) for 0 ≤ s < t"stationary increments"

iii) Wt − Ws independent of Wu − Wr for 0 ≤ r ≤ u ≤ s < t"independent increments"

is called a one-dimensional Brownian motion.

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Brownian motion

RemarkBy an n-dimensional Brownian motion we mean the R

n-valued process

W (t) = (W1(t), . . . , Wn(t)),

with components Wi being independent one-dimensional Brownianmotions.

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Brownian motion and filtration

Brownian motion can be associated with

natural filtration

FWt := σWs |0 ≤ s ≤ t, t ∈ [0,∞)

P-augmentation of the natural filtration (Brownian filtration)

Ft := σFWt ∪ N |N ∈ F , P(N) = 0, t ∈ [0,∞)

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Requirement iii) of a Brownian motion with respect to a filtrationFtt≥0 is often stated as

iii)∗ Wt − Ws independent of Fs, 0 ≤ s < t .

Ftt≥0 natural filtration (Brownian filtration)

⇒ iii) and iii)∗ are equivalent.

Ftt≥0 arbitrary filtration

⇒ iii) and iii)∗ are usually not equivalent.

ConventionIf we consider a Brownian motion (Wt ,Ft)t≥0 with an arbitraryfiltration we implicitly assume iii)∗ to be fulfilled.

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Existence of the Brownian motion

How can we show the existence of a stochastic process satisfying therequirements of a Brownian motion?

Construction and existence proofs are long and technical.

Construction based on weak convergence and approximation byrandom walks [Billingsley 1968].

Wiener measure, Wiener process.

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TheoremThe Brownian filtration Ftt≥0 is right-continuous as well asleft-continuous, i.e., we have

Ft = Ft+ :=⋂

ε>0

Ft+ε and Ft = Ft− := σ

(⋃

s<t

Fs

).

DefinitionA filtration Gtt≥0 satifies the usual conditions, if it is right-continuousand G0 contains all P-null sets of F .

General assumption for this sectionLet Ftt≥0 be a filtration which satisfies the usual conditions.

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Martingales

DefinitionThe real-valued process (Xt ,Ft)t∈I with E |Xt | < ∞ for all t ∈ I(where I is an ordered index set), is called

a super-martingale, if for all s, t ∈ I with s ≤ t we have

E(Xt |Fs) ≤ Xs P-a.s. ,

a sub-martingale, if for all s, t ∈ I with s ≤ t we have

E(Xt |Fs) ≥ Xs P-a.s. ,

a martingale, if for all s, t ∈ I with s ≤ t we have

E(Xt |Fs) = Xs P-a.s. .

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Interpretation of the martingale concept

Example: Modeling games of chance

Xn: Wealth of a gambler after n-th participation in a fair game

Martingale condition: E(Xn+1|Fn) = Xn P-a.s.

⇒ "After the game the player is as rich as he was before"

favorable game = sub-martingalenon-favorable game = super-martingale

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Example: Tossing a fair coin

"Head": Gambler receives one dollar

"Tail": Gambler loses one dollar

⇒ Martingale

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TheoremA one-dimensional Brownian motion Wt is a martingale.

RemarkEach stochastic process with independent, centered increments isa martingale with respect to its natural filtration.

The Brownian motion with drift µ and volatility σ

Xt := µt + σWt , µ ∈ R, σ ∈ R

is a martingale if µ = 0, a super-martingale if µ ≤ 0 and asub-martingale if µ ≥ 0.

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Theorem(1) Let (Xt , Ft)t∈I be a martingale and ϕ : R → R be a convex

function with E |ϕ(Xt)| < ∞ for all t ∈ I. Then

(ϕ(Xt ),Ft)t∈I

is a sub-martingale.

(2) Let (Xt , Ft)t∈I be a sub-martingale and ϕ : R → R a convex,non-decreasing function with E |ϕ(Xt)| < ∞ for all t ∈ I. Then

(ϕ(Xt ),Ft)t∈I

is a sub-martingale.

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Remark(1) Typical applications are given by

ϕ(x) = x2, ϕ(x) = |x |.

(2) The theorem is also valid for d -dimensional vectors

X (t) = (X1(t), . . . , Xd(t))

of martingales and convex functions ϕ : Rd → R.

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Stopping time

DefinitionA stopping time with respect to a filtration Ftt∈[0,∞)

(or Fnn∈N) is an F-measurable random variable

τ : Ω → [0,∞] (or τ : Ω → N ∪ ∞)

with ω ∈ Ω | τ(ω) ≤ t ∈ Ft for all t ∈ [0,∞)(or ω ∈ Ω | τ(ω) ≤ n ∈ Fn for all n ∈ N).

TheoremIf τ1, τ2 are both stopping times then τ1 ∧ τ2 := minτ1, τ2 is also astopping time.

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The stopped process

The stopped processLet (Xt , Ft)t∈I be a stochastic process, let I be either N or [0,∞),and τ a stopping time. The stopped process Xt∧τt∈I is defined by

Xt∧τ (ω) :=

Xt(ω) if t ≤ τ(ω),

Xτ(ω)(ω) if t > τ(ω).

Example: Wealth of a gambler who participates in a sequence ofgames until he is either bankrupt or has reached a given level ofwealth.

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The stopped filtration

The stopped filtrationLet τ be a stopping time with respect to a filtration Ftt∈[0,∞).

σ-field of events determined prior to the stopping time τ

Fτ := A ∈ F |A ∩ τ ≤ t ∈ Ft for all t ∈ [0,∞)

Stopped filtrationFτ∧tt∈[0,∞).

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What will happen if we stop a martingale or a sub-martingale?

Theorem: Optional sampling

Let (Xt , Ft)t∈[0,∞) be a right-continuous sub-martingale (ormartingale). Let τ1, τ2 be stopping times with τ1 ≤ τ2. Then for allt ∈ [0,∞) we have

E(Xt∧τ2 | Ft∧τ1) ≥ Xt∧τ1 P-a.s.

(orE(Xt∧τ2 | Ft∧τ1) = Xt∧τ1 P-a.s.).

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CorollaryLet τ be a stopping time and (Xt , Ft)t∈[0,∞) a right-continuoussub-martingale (or martingale). Then the stopped process(Xt∧τ ,Ft)t∈[0,∞) is also a sub-martingale (or martingale).

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TheoremLet (Xt , Ft)t∈[0,∞) be a right-continuous process. Then Xt is amartingale if and only if for all bounded stopping times τ we have

EXτ = EX0.

→ Characterization of a martingale

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DefinitionLet (Xt , Ft)t∈[0,∞) be a stochastic process with X0 = 0. If there is anon-decreasing sequence τnn∈N of stopping times with

P(

limn→∞

τn = ∞)

= 1,

such that (X (n)

t := (Xt∧τn ,Ft)

)

t∈[0,∞)

is a martingale for all n ∈ N, then X is a local martingale. Thesequence τnn∈N is called a localizing sequence corresponding to X .

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Remark(1) Each martingale is a local martingale.

(2) A local martingale with continuous paths is calledcontinuous local martingale.

(3) There exist local martingales which are not martingales.

E(Xt) need not exist for a local martingale. However, theexpectation has to exist along the localizing sequence t ∧ τn.

The local martingale is a martingale on the random timeintervals [0, τn].

TheoremA non-negative local martingale is a super-martingale.

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Theorem: Doob’s inequalityLet Mtt≥0 be a martingale with right-continuous paths andE(M2

T ) < ∞ or all T > 0. Then, we have

E((

sup0≤t≤T

|Mt |)2)

≤ 4 · E(M2T ).

TheoremLet (Xt , Ft)t∈[0,∞) be a non-negative super-martingale withright-continuous paths. Then, for λ > 0 we obtain

λ · P

ω

∣∣∣∣ sup0≤s≤t

Xs(ω) ≥ λ

≤ E(X0).

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Outline


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Continuation: The stock price

log-linear model for a stock price

ln(Pi(t)) = ln(pi) + bi · t + ”randomness”

Brownian motion (Wt , Ft)t≥0 is the appropriate stochastic process tomodel the "randomness"

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Market with one stock and one bond (d=1)

ln(P1(t)) = ln(p1) + b1 · t + σ11Wt

P1(t) = p1 · exp(b1 · t + σ11Wt

)

Market with d stocks and one bond (d>1)

ln(Pi(t)) = ln(pi) + bi · t +

m∑

j=1

σijWj(t), i = 1, . . . , d

Pi(t) = pi · exp(

bi · t +

m∑

j=1

σijWj(t))

, i = 1, . . . , d

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Distribution of the logarithm of the stock prices

ln(Pi(t)) ∼ N(

ln(pi) + bi · t ,m∑

j=1

σ2ij · t)

⇒ Pi(t) is log-normally distributed.

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Lemma

Let bi := bi + 12

m∑

j=1

σ2ij for i = 1, . . . , d.

(1) E(Pi(t)) = pi · ebi t .

(2) Var(Pi(t)) = p2i · exp(2bi t) ·

(exp

( m∑

j=1

σ2ij t)− 1)

.

(3) Xt := a · exp( m∑

j=1

(cjWj(t) −

12

c2j t))

with a, cj ∈ R, j = 1, . . . , m

is a martingale.

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Interpretation of the stock price model

The stock price model

Pi(t) = pi · exp(bi t) · exp( m∑

j=1

[σijWj(t) −

12σ2

ij t])

,

Pi(0) = pi , i = 1, . . . , d .

The stock price is the product of

the mean stock price pi · exp(bi t) and

a martingale with expectation 1, namely

exp( m∑

j=1

[σijWj(t) −

12σ2

ij t])

which models the stock price around its mean value.

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Interpretation of the stock price model

Vector of mean rates of stock returns

b = (b1, . . . , bd )T

Volatility matrix

σ =

σ11 . . . σ1m...

. . ....

σd1 . . . σdm

Pi(t) is a geometric Brownian motion with drift bi and volatilityσi . = (σi1, . . . , σim)T .

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Summary: Stock prices

Bond price and stock prices

P0(t) = p0 · ertBond price

P0(0)= p0

Pi(t) = pi · exp(bi t) · exp( m∑

j=1

[σijWj(t) −

12

σ2ij t])

Stock prices

Pi(0) = pi , i = 1, . . . , d .

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Extension

Extension: Model with non-constant, time-dependent, and integrablerates of return bi(t) and volatilities σ(t).

Stock prices:

Pi(t) = pi · exp( t∫

0

(bi(s) − 1

2

m∑

j=1

σ2ij (s)

)ds)

· exp( m∑

j=1

t∫

0

σij(s) dWj(s)

)

Problem:t∫

0σij(s) dWj(s)

⇒ Stochastic integral (Ito integral)

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Outline


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The Ito integral

Is it possible to define the stochastic integral

t∫

0

Xs(ω) dWs(ω)

ω-wise in a reasonable way?

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The Ito integral

TheoremP-almost all paths of the Brownian motion Wtt∈[0,∞) are nowheredifferentiable.

⇒ A definition of the form

t∫

0

Xs(ω) dWs(ω) =

t∫

0

Xs(ω)dWs(ω)

dsds

is impossible.

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The Ito integral

TheoremWith the definition

Zn(ω) :=

2n∑

i=1

∣∣∣∣W i2n

(ω) − W i−12n

(ω)

∣∣∣∣, n ∈ N, ω ∈ Ω

we haveZn(ω)

n→∞−−−→ ∞ P-a.s. ,

i.e., the paths Wt(ω) of the Brownian motion admit infinite variation onthe interval [0, 1] P-almost surely.

The paths Wt(ω) of the Brownian motion have infinite variation on eachnon-empty finite interval [s1, s2] ⊂ [0,∞) P-almost surely.

80 / 477

General assumptions

General assumptions for this sectionLet (Ω,F , P) be a complete probability space equipped with a filtrationFtt satisfying the usual conditions. Further assume that on thisspace a Brownian motion (Wt ,Ft)t∈[0,∞) with respect to this filtrationis defined.

81 / 477

Simple process

DefinitionA stochastic process Xtt∈[0,T ] is called a simple process if there existreal numbers 0 = t0 < t1 < . . . < tp = T , p ∈ N, and bounded randomvariables Φi : Ω → R, i = 0, 1, . . . , p, with

Φ0 F0-measurable, Φi Fti−1 -measurable, i = 1, . . . , p

such that Xt(ω) has the representation

Xt(ω) = X (t , ω) = Φ0(ω) · 10(t) +

p∑

i=1

Φi(ω) · 1(ti−1,ti ](t)

for each ω ∈ Ω.

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Simple process

RemarkXt is Fti−1-measurable for all t ∈ (ti−1, ti ].

The paths X (., ω) of the simple process Xt are left-continuous stepfunctions with height Φi(ω) · 1(ti−1,ti ](t).

0 T0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

X(.,ω)

t1

t2

t3

83 / 477

Stochastic integral

DefinitionFor a simple process Xtt∈[0,T ] the stochastic integral I.(X ) fort ∈ (tk , tk+1] is defined according to

It(X ) :=

t∫

0

Xs dWs :=∑

1≤i≤k

Φi(Wti − Wti−1) + Φk+1(Wt − Wtk ),

or more generally for t ∈ [0, T ]:

It(X ) :=

t∫

0

Xs dWs :=∑

1≤i≤p

Φi(Wti∧t − Wti−1∧t).

84 / 477

Stochastic integral

Theorem: Elementary properties of the stochastic integral

Let X := Xtt∈[0,T ] be a simple process. Then we have

(1) It(X )t∈[0,T ] is a continuous martingale with respect to Ftt∈[0,T ].In particular, we have E(It(X )) = 0 for all t ∈ [0, T ].

(2) E( t∫

0Xs dWs

)2

= E( t∫

0X 2

s ds)

for t ∈ [0, T ].

(3) E(

sup0≤t≤T

∣∣∣∣t∫

0Xs dWs

∣∣∣∣)2

≤ 4 · E( T∫

0X 2

s ds)

.

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Stochastic integral

Remark(1) By (2) the stochastic integral is a square-integrable stochastic

process.

(2) For the simple process X ≡ 1 we obtain

t∫

0

1 dWs = Wt

and

E( t∫

0

dWs

)2

= E(W 2t ) = t =

t∫

0

ds.

86 / 477

Stochastic integral

Remark(1) Integrals with general boundaries:

T∫

t

Xs dWs :=

T∫

0

Xs dWs −t∫

0

Xs dWs for t ≤ T .

For t ≤ T , A ∈ Ft we have

T∫

0

1A(ω) · Xs(ω) · 1[t,T ](s) dWs = 1A(ω) ·T∫

t

Xs(ω) dWs.

(2) Let X , Y be simple processes, a, b ∈ R. Then we have

It(aX + bY ) = a · It(X ) + b · It(Y ) (linearity)

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Measurability

DefinitionA stochastic process (Xt , Gt)t∈[0,∞) is called measurable if themapping

[0,∞) × Ω → Rn

(s, ω) 7→ Xs(ω)

is B([0,∞)) ⊗F-B(Rn)-measurable.

RemarkMeasurability of the process X implies that X (., ω) isB([0,∞))-B(Rn)-measurable for a fixed ω ∈ Ω. Thus, for all t ∈ [0,∞),

i = 1, . . . , n, the integralt∫

0X 2

i (s) ds is defined.

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Measurability

DefinitionA stochastic process (Xt , Gt)t∈[0,∞) is called progressivelymeasurable if for all t ≥ 0 the mapping

[0, t] × Ω → Rn

(s, ω) 7→ Xs(ω)

is B([0, t]) ⊗ Gt -B(Rn)-measurable.

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Measurability

Remark(1) If the real-valued process (Xt , Gt)t∈[0,∞) is progressively

measurable and bounded, then for all t ∈ [0,∞) the integralt∫

0Xs ds is Gt -measurable.

(2) Every progressively measurable process is measurable.

(3) Each measurable process possesses a progressively measurablemodification.

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Measurability

TheoremIf all paths of the stochastic process (Xt , Gt)t∈[0,∞) areright-continuous (or left-continuous), then the process is progressivelymeasurable.

TheoremLet τ be a stopping time with respect to the filtration Gtt∈[0,∞). If thestochastic process (Xt , Gt)t∈[0,∞) is progressively measurable, thenso is the stopped process (Xt∧τ , Gt)t∈[0,∞). In particular, Xt∧τ is Gt

and Gt∧τ -measurable.

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Extension of the stochastic integral toL2[0, T ]-processes

Definition

L2[0, T ] := L2([0, T ],Ω,F , Ftt∈[0,T ], P

)

:=

(Xt ,Ft)t∈[0,T ] real-valued stochastic process

∣∣∣∣

Xtt progressively measurable, E( T∫

0

X 2t dt

)<∞

Norm on L2[0, T ]: ‖X‖2T := E

( T∫0

X 2t dt

).

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‖ · ‖2T L2-norm on the probability space(

[0, T ]× Ω, B([0, T ]) ⊗F , λ ⊗ P).

‖ · ‖2T semi-norm (‖X − Y‖2

T = 0 6⇒ X = Y ).

X equivalent to Y :⇔ X = Y a.s. λ ⊗ P.

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Ito isometryLet X be a simple process. The mapping X 7→ I.(X ) induces by

‖I.(X )‖2LT

:= E( T∫

0

Xs dWs

)2

= E( T∫

0

X 2s ds

)= ‖X‖2

T

a norm on the space of stochastic integrals.

⇒ I.(X ) linear, norm-preserving (= isometry)

⇒ I.(X ) Ito isometry

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Use processes X ∈ L2[0, T ] approximated by a sequence X (n) ofsimple processes.

I.(X (n)) is a Cauchy-sequence with respect to ‖ · ‖LT.

To show: I.(X (n)) is convergent, limit independent of X (n).

Denote limit by

I(X ) =

∫Xs dWs.

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X ∈ L2[0, T ]J(.) //_______ J(X ) ∈ MC

2

X (n)

‖·‖T

OO

I(.)// I(X (n))

‖·‖LT

OO

simple process stochastic integralfor simple processes

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Theorem

An arbitrary X ∈ L2[0, T ] can be approximated by a sequence ofsimple processes X (n).

More precisely: There exists a sequence X (n) of simple processes with

limn→∞

E

T∫

0

(Xs − X (n)

s

)2ds = 0.

97 / 477


LemmaLet (Xt , Gt)t∈[0,∞) be a martingale where the filtration Gtt∈[0,∞)

satisfies the usual conditions. Then the process Xt possesses aright-continuous modification (Yt , Gt)t∈[0,∞) such that(Yt , Gt)t∈[0,∞) is a martingale.

98 / 477


Construction of the Ito integral for processes in L2[0, T ]

There exists a unique linear mapping J from L2[0, T ] into the space ofcontinuous martingales on [0, T ] with respect to Ftt∈[0,T ] satisfying

(1) X = Xtt∈[0,T ] simple process⇒ P

(Jt(X ) = It(X ) for all t ∈ [0, T ]

)= 1

(2) E(

Jt(X )2)

= E( t∫

0X 2

s ds)

Ito isometry

Uniqueness: If J, J ′ satisfy (1) and (2), then for all X ∈ L2[0, T ] theprocesses J ′(X ) and J(X ) are indistinguishable.

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Definition

For X ∈ L2[0, T ] and J as before we define by

t∫

0

Xs dWs := Jt(X )

the stochastic integral (or Ito integral) of X with respect to W .

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Theorem: Special case of Doob’s inequality

Let X ∈ L2[0, T ]. Then we have

E(

sup0≤t≤T

∣∣∣∣

t∫

0

Xs dWs

∣∣∣∣)2

≤ 4 · E( T∫

0

X 2s ds

).

101 / 477


Multi-dimensional generalization of the stochastic integral

(W (t), Ft)t : m-dimensional Brownian motionwith W (t) = (W1(t), . . . , Wm(t))

(X (t), Ft)t : Rn,m-valued progressively measurable process with

Xij ∈ L2[0, T ].

Ito integral of X with respect to W :

t∫

0

X (s) dW (s) :=

m∑

j=1

t∫

0

X1j(s) dWj(s)

...m∑

j=1

t∫

0

Xnj(s) dWj(s)

, t ∈ [0, T ]

102 / 477

Further extension of the stochastic integral

Definition

H2[0, T ] := H2([0, T ], Ω, F , Ftt∈[0,T ], P

)

:=

(Xt ,Ft)t∈[0,T ] real-valued stochastic process

∣∣∣∣

Xtt progressively measurable,

T∫

0

X 2t dt < ∞ P-a.s.

103 / 477


Processes X ∈ H2[0, T ]

do not necessarily have a finite T -norm→ no approximation by simple processes as for

processes in L2[0, T ]

can be localized with suitable sequences of stopping times

Stopping times (with respect to Ftt ):

τn(ω) := T ∧ inf

0 ≤ t ≤ T

∣∣∣∣

t∫

0

X 2s (ω) ds ≥ n

, n ∈ N

Sequence of stopped processes:

X (n)t (ω) := Xt(ω) · 1τn(ω)≥t

⇒ X (n) ∈ L2[0, T ] ⇒ Stochastic integral already defined.

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Stochastic integral:

It(X ) := It(X (n)) for 0 ≤ t ≤ τn

Consistence property:

It(X ) = It(X (m)) for 0 ≤ t ≤ τn(≤ τm), m ≥ n

⇒ It(X ) well-defined for X ∈ H2[0, T ]

105 / 477


Stopping times:τn

n→∞−−−→ +∞ P-a.s.

⇒ It(X ) local martingale with localizing sequence τn.

⇒ Stochastic integral is linear and possesses continuous paths.

106 / 477

Outline


107 / 477

The Ito formula

General assumptions for this sectionLet (Ω,F , P) be a complete probability space equipped with a filtrationFtt satisfying the usual conditions. Further, assume that on thisspace a Brownian motion (Wt ,Ft)t∈[0,∞) with respect to this filtrationis defined.

108 / 477

The Ito formula

DefinitionLet (Wt ,Ft)t∈[0,∞) be an m-dimensional Brownian motion.

(1) (X (t),Ft )t∈[0,∞) is a real-valued Ito process if for all t ≥ 0 itadmits the representation

X (t) = X (0) +

t∫

0

K (s) ds +

t∫

0

H(s) dW (s)

= X (0) +

t∫

0

K (s) ds +

m∑

j=1

t∫

0

Hj(s) dWj(s) P-a.s.

109 / 477

The Ito formula

X (0) F0-measurable,K (t)t∈[0,∞), H(t)t∈[0,∞) progressively measurable with

t∫

0

|K (s)|ds < ∞,

t∫

0

H2i (s) ds < ∞ P-a.s.

for all t ≥ 0, i = 1, . . . , m.

(2) n-dimensional Ito process X = (X (1), . . . , X (n))= vector with components being real-valued Ito processes.

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The Ito formula

Remark

Hj ∈ H2[0, T ] for all T > 0.

The representation of an Ito process is unique up toindistinguishability of the representing integrands Kt , Ht .

Symbolic differential notation:

dXt = Kt dt + Ht dWt

111 / 477

The Ito formula

DefinitionLet X and Y be two real-valued Ito processes with

X (t) = X (0) +

t∫

0

K (s) ds +

t∫

0

H(s) dW (s),

Y (t) = Y (0) +

t∫

0

L(s) ds +

t∫

0

M(s) dW (s).

Quadratic covariation of X and Y :

〈X , Y 〉t :=

m∑

i=1

t∫

0

Hi(s) · Mi(s) ds.

112 / 477

The Ito formula

DefinitionQuadratic variation of X

〈X 〉t := 〈X , X 〉t .

NotationLet X be a real-valued Ito process, and Y a real-valued, progressivelymeasurable process. We set

t∫

0

Y (s) dX (s) :=

t∫

0

Y (s) · K (s) ds +

t∫

0

Y (s) · H(s) dW (s)

if all integrals on the right-hand side are defined.

113 / 477

The Ito formula

Theorem: One-dimensional Ito formulaLet Wt be a one-dimensional Brownian motion, and Xt a real-valued Itoprocess with

Xt = X0 +

t∫

0

Ks ds +

t∫

0

Hs dWs.

Let f ∈ C2(R). Then, for all t ≥ 0 we have

f (Xt) = f (X0) +

t∫

0

f ′(Xs) dXs +12

t∫

0

f ′′(Xs) d〈X 〉s

= f (X0) +

t∫

0

(f ′(Xs) · Ks +

12· f ′′(Xs) · H2

s

)ds +

t∫

0

f ′(Xs)Hs dWs

114 / 477

The Ito formula

RemarkThe Ito formula differs from the fundamental theorem of calculusby the additional term

12

t∫

0

f ′′(Xs) d〈X 〉s.

The quadratic variation 〈X 〉t is an Ito process.

Differential notation:

df (Xt) = f ′(Xt) dXt +12· f ′′(Xt) d〈X 〉t .

115 / 477

The Ito formula

LemmaLet X be a martingale with |Xs| ≤ C for all s ∈ [0, t] P-a.s.Let π = t0, t1, . . . , tm, t0 = 0, tm = t , be a partition of [0, t] with

‖π‖ := max1≤k≤m

|tk − tk−1|.

Then we have

(1) E( m∑

k=1

(Xtk − Xtk−1

)2)2

≤ 48 · C4

(2) X continuous ⇒ E( m∑

k=1

(Xtk − Xtk−1

)4)

‖π‖→0−−−−→ 0.

116 / 477

Some applications of Ito′s formula

Some applications of Ito′s formula I(1) Xt = t :

Representation:

Xt = 0 +

t∫

0

1 ds +

t∫

0

0 dWs.

For f ∈ C2(R) we have

f (t) = f (0) +

t∫

0

f ′(s) ds.

⇒ Fundamental theorem of calculus is a special case ofIto′s formula.

117 / 477

Some applications of Ito′s formula II


(2) Xt = h(t) :

For h ∈ C1(R) Ito′s formula implies the chain rule

Xt = h(0) +

t∫

0

h′(s) ds +

t∫

0

0 dWs

⇒ (f h)(t) = (f h)(0) +

t∫

0

f ′(h(s)) · h′(s) ds.

118 / 477

Some applications of Ito′s formula III


(3) Xt = Wt , f (x) = x2 :

Due to

Wt = 0 +

t∫

0

0 ds +

t∫

0

1 dWs

we obtain

W 2t =

t∫

0

2 · Ws dWs +12·

t∫

0

2 ds = 2 ·t∫

0

Ws dWs + t

⇒ Additional term "t"(→ nonvanishing quadratic variation of Wt ).

119 / 477

The Ito formula

Theorem: Multi-dimensional Ito formula

X (t) =(X1(t), . . . , Xn(t)

)n-dimensional Ito process with

Xi(t) = Xi(0) +

t∫

0

Ki(s) ds +m∑

j=1

t∫

0

Hij(s) dWj(s), i = 1, . . . , n

where W (t) =(W1(t), . . . , Wm(t)

)is an m-dimensional Brownian motion.

Let f : [0,∞) × Rn → R be a C1,2-function. Then, we have

f (t , X1(t), . . . , Xn(t)) = f (0, X1(0), . . . , Xn(0))

+

t∫

0

ft(s, X1(s), . . . , Xn(s)) ds +

n∑

i=1

t∫

0

fxi (s, X1(s), . . . , Xn(s)) dXi(s)

+12·

n∑

i ,j=1

t∫

0

fxi xj (s, X1(s), . . . , Xn(s)) d〈Xi , Xj〉s.

120 / 477

Product rule or partial integration

Corollary: Product rule or partial integration

Let Xt , Yt be one-dimensional Ito processes with

Xt = X0 +

t∫

0

Ks ds +

t∫

0

Hs dWs,

Yt = Y0 +

t∫

0

µs ds +

t∫

0

σs dWs.

Then we have

Xt · Yt = X0 · Y0 +

t∫

0

Xs dYs +

t∫

0

Ys dXs +

t∫

0

d〈X , Y 〉s

= X0 · Y0 +

t∫

0

(Xsµs + YsKs + Hsσs

)ds +

t∫

0

(Xsσs + YsHs

)dWs.

121 / 477

The stock price equation

Simple continuous-time market model (1 bond, one stock).Stock price influenced by a one-dimensional Brownian motion

Price of the stock at time t :

P(t) = p · exp((

b − 12σ2)

t + σWt

)

Choose

Xt = 0 +

t∫

0

(b − 1

2σ2)

ds +

t∫

0

σ dWs, f (x) = p · ex

Ito formula implies

f (Xt) = p +

t∫

0

[f (Xs)(b − 1

2σ2) + 12 f (Xs) · σ2]ds +

t∫

0

f (Xs) · σ dWs

122 / 477



P(t) = p +

t∫

0

P(s) · b ds +

t∫

0

P(s) · σ dWs

RemarkThe stock price equation is valid for time-dependent b and σ, if

Xt =

t∫

0

(b(s) − 1

2σ2(s))

ds +

t∫

0

σ(s) dWs.

123 / 477


The stock price equation in differential form

dP(t) = P(t)(b dt + σ dWt

)

P(0) = p

124 / 477


Theorem: Variation of constantsLet (W (t), Ft)t∈[0,∞) be an m-dimensional Brownian motion.Let x ∈ R and A, a, Sj , σj be progressively measurable, real-valuedprocesses with

t∫

0

(|A(s)| + |a(s)|

)ds < ∞ for all t ≥ 0 P-a.s.

t∫

0

(S2

j (s) + σ2j (s)

)ds < ∞ for all t ≥ 0 P-a.s. .

125 / 477


Theorem: Variation of constantsThen the stochastic differential equation

dX (t) =(A(t) · X (t) + a(t)

)dt +

m∑

j=1

(Sj(t)X (t) + σj(t)

)dWj(t)

X (0) = x

possesses a unique solution with respect to λ ⊗ P :

X (t) = Z (t) ·(

x +

t∫

0

1Z (u)

(a(u) −

m∑

j=1

Sj(u)σj(u)

)du

+m∑

j=1

t∫

0

σj(u)

Z (u)dWj(u)

)

126 / 477


Theorem: Variation of constantsHereby is

Z (t) = exp( t∫

0

(A(u) − 1

2 · ‖S(u)‖2) du +

t∫

0

S(u) dW (u)

)

the unique solution of the homogeneous equation

dZ (t) = Z (t)(A(t) dt + S(t)T dW (t)

)

Z (0) = 1.

127 / 477


RemarkThe process (X (t), Ft)t∈[0,∞) solves the stochastic differentialequation in the sense that X (t) satisfies

X (t) = x +

t∫

0

(A(s) · X (s) + a(s)

)ds

+

m∑

j=1

t∫

0

(Sj(s) X (s) + σj(s)

)dWj(s)

for all t ≥ 0 P-almost surely.

128 / 477

Outline


129 / 477

General assumptions

General assumptions for this section(Ω,F , P) be a complete probability space,(W (t),Ft )t∈[0,∞) m-dimensional Brownian motion.

Dynamics of bond and stock prices:

P0(t) = p0 · exp( t∫

0

r(s) ds)

bond


0

(bi(s) − 1

2

m∑

j=1

σ2ij (s)

)ds

+m∑

j=1

t∫

0

σij(s) dWj(s)

)stock

for t ∈ [0, T ], T > 0, i = 1, . . . , d .

130 / 477

General assumptions (continued)

General assumptions for this section (continued)

r(t), b(t) = (b1(t), . . . , bd (t))T , σ(t) = (σij(t))ij

progressively measurable processes with respect to Ftt ,component-wise uniformly bounded in (t , ω).

σ(t)σ(t)T uniformly positive definite,i.e., it exists K > 0 with

xT σ(t)σ(t)T x ≥ KxT x

for all x ∈ Rd and all t ∈ [0, T ] P-a.s.

Deterministic rate of return r(t) is not requiredr(t) can be a stochastic process⇒ bond is no longer riskless.

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Bond and stock prices

Bond and stock prices are unique solutions of the stochasticdifferential equations

dP0(t) = P0(t) · r(t) dt bond

P0(t) = p0

dPi(t) = Pi(t)(

bi(t) dt +

m∑

j=1

σij(t) dWj(t))

, i = 1, . . . , d

Pi(0) = pi stock

⇒ Representations of prices as Ito processes

132 / 477

Possible actions of investors

(1) Investor can rebalance his holdings→ sell some securities→ invest in securities⇒ Portfolio process / trading strategy.

(2) Investor is allowed to consume parts of his wealth⇒ Consumption process.

133 / 477

Requirements on a market model

(1) Investor should not be able to foresee events→ no knowledge of future prices.

(2) Actions of a single investor have no impact on the stock prices(small investor hypothesis).

(3) Each investor has a fixed initial capital at time t = 0.

(4) Money which is not invested into stocks has to be invested inbonds.

(5) Investors act in a self-financing way(no secret source or sink for money).

134 / 477

Requirements on a market model

(6) Securities are perfectly divisible.

(7) Negative positions in securities are possiblebond → creditstock → we sold some stock short.

(8) No transaction costs.

135 / 477

Negative bond positions and credit interest rates

Negative bond positions and credit interest rates

Assume: Interest rate r(t) is constant

Negative bond position = it is possible to borrow money for thesame rate as we would get for investing in bonds.

Interest depends on the market situation ((t , ω) ∈ [0, T ] × Ω), butnot on positive or negative bond position.

136 / 477

Mathematical realizations of some requirements

Market with 1 bond and d stocks

Time t = 0: – Initial capital of investor: x > 0

– Buy a selection of securities

ϕ(0) =(ϕ0(0), ϕ1(0), . . . , ϕd(0)

)T

Time t > 0: – Trading strategy: ϕ(t)

(1) ⇒ trading strategy is progressively measurablewith respect to Ftt

Decisions on buying and selling are made on basis of informationavailable at time t (→ modelled by Ftt )

(5) ⇒ only self-financing trading strategies should be used.

137 / 477

Discrete-time example: self-financing strategy

Market with 1 riskless bond and 1 stock

Two-period model for time points t = 0, 1, 2.

Number of shares of bond and stock at time t :

(ϕ0(t), ϕ1(t))T ∈ R2

Consumption of investor at time t : C(t)

Wealth at time t : X (t)

Bond/stock prices at time t : P0(t), P1(t)

Initial conditions: C(0) = 0, X (0) = x

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t = 0Investor uses initial capital to buy shares of bond and stock

X (0) = x = ϕ0(0) · P0(0) + ϕ1(0) · P1(0).

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t = 1Security prices have changed, investor consumes parts of his wealth

Current wealth:

X (1) = ϕ0(0) · P0(1) + ϕ1(0) · P1(1) − C(1).

Total:

X (1) = x + ϕ0(0) ·(P0(1) − P0(0)

)+ ϕ1(0) ·

(P1(1) − P1(0)

)− C(1)

Wealth = initial wealth + gains/losses - consumption

Invest remaining capital at the market:

X (1) = ϕ0(1) · P0(1) + ϕ1(1) · P1(1).

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t = 2Invest remaining capital at the market

Wealth:X (2) = ϕ0(2) · P0(2) + ϕ1(2) · P1(2).

Wealth = total wealth of shares held

Total:

X (2) = x +2∑

i=1

[ϕ0(i − 1) · (P0(i) − P0(i − 1))

+ϕ1(i − 1) · (P1(i) − P1(i − 1))]

−2∑

i=1

C(i).

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Self-financing trading strategy:

wealth before rebalancing - consumption = wealth after rebalancing

Condition:

ϕ0(i) · P0(i) + ϕ1(i) · P1(i)

= ϕ0(i − 1) · P0(i) + ϕ1(i − 1) · P1(i) − C(i)

⇒ Useless in continuous-time setting

(securities can be traded at each time instant /"before" and "after" cannot be distinguished)

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Continuous-time setting

Wealth process corresponding to strategy ϕ(t):

X (t) = x +

t∫

0

ϕ0(s) dP0(s) +

t∫

0

ϕ1(s) dP1(s) −t∫

0

c(s) ds

⇒ Price processes are Ito processes.

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Trading strategy and wealth processes

Definition(1) A trading strategy ϕ with

ϕ(t) :=(ϕ0(t), ϕ1(t), . . . , ϕd (t)

)T

is an Rd+1-valued progressively measurable process with respect

to Ftt∈[0,T ] satisfying

T∫

0

|ϕ0(t)|dt < ∞ P-a.s.

d∑

j=1

T∫

0

(ϕi(t) · Pi(t)

)2 dt < ∞ P-a.s. for i = 1, . . . , d .

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DefinitionThe value

x :=

d∑

i=0

ϕi(0) · pi

is called initial value of ϕ.

(2) Let ϕ be a trading strategy with initial value x > 0.The process

X (t) :=d∑

i=0

ϕi(t)Pi (t)

is called wealth process corresponding to ϕ withinitial wealth x .

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Definition(3) A non-negative progressively measurable process c(t) with

respect to Ftt∈[0,T ] with

T∫

0

c(t) dt < ∞ P-a.s.

is called consumption (rate) process.

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DefinitionA pair (ϕ, c) consisting of a trading strategy ϕ and a consumption rateprocess c is called self-financing if the corresponding wealth processX (t) satisfies

X (t) = x +

d∑

i=0

t∫

0

ϕi(s) dPi(s) −t∫

0

c(s) ds P-a.s.

current wealth = initial wealth + gains/losses - consumption

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RemarkWe have

t∫

0

ϕ0(s) dP0(s) =

t∫

0

ϕ0(s) P0(s) r(s) ds

t∫

0

ϕi(s) dPi(s) =

t∫

0

ϕi(s) Pi(s) bi(s) ds

+

m∑

j=1

t∫

0

ϕi(s) Pi (s)σij(s) dWj(s), i = 1, . . . , d .

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Self-financing portfolio process

DefinitionLet (ϕ, c) be a self-financing pair consisting of a trading strategy and aconsumption process with corresponding wealth process X (t) > 0P-a.s. for all t ∈ [0, T ]. Then the R

d -valued process

π(t) =(π1(t), . . . , πd (t)

)T with πi(t) =ϕi(t) · Pi(t)

X (t)

is called a self-financing portfolio process corresponding to thepair (ϕ, c).

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Portfolio processes

Remark(1) The portfolio process denotes the fractions of total wealth invested

in the different stocks.

(2) The fraction of wealth invested in the bond is given by

(1 − π(t)T 1

)=

ϕ0(t) · P0(t)X (t)

, where 1 := (1, . . . , 1)T ∈ Rd .

(3) Given knowledge of wealth X (t) and prices Pi(t), it is possible foran investor to describe his activities via a self-financing pair (π, c).→ Portfolio process and trading strategy are

equivalent descriptions of the same action.

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The wealth equation

The wealth equation

dX (t) = [r(t) X (t) − c(t)] dt

+ X (t)π(t)T ((b(t) − r(t) 1) dt + σ(t) dW (t))

X (0) = x

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Alternative definition of a portfolio process

Definition

The progressively measurable Rd -valued process π(t) is called a

self-financing portfolio process corresponding to the consumptionprocess c(t) if the corresponding wealth equation possesses a uniquesolution X (t) = Xπ,c(t) with

T∫

0

(X (t) · πi(t)

)2 dt < ∞ P-a.s. for i = 1, . . . , d .

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Admissibility

DefinitionA self-financing pair (ϕ, c) or (π, c) consisting of a trading strategy ϕ ora portfolio process π and a consumption process c will be calledadmissible for the initial wealth x > 0, if the corresponding wealthprocess satisfies

X (t) ≥ 0 P-a.s. for all t ∈ [0, T ].

The set of admissible pairs will be denoted by A(x).

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An example

Portfolio process:π(t) ≡ π ∈ R

d constant

Consumption rate:c(t) = γ · X (t), γ > 0

Wealth process corresponding to (π, c) :

X (t)

Investor rebalances his holdings in such a way that the fractions ofwealth invested in the different stocks and in the bond remainconstant over time.

Consumption rate is proportional to the current wealth of theinvestor.

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An example

Wealth equation:

dX (t) = [r(t) − γ] X (t) dt

+ X (t)πT ((b(t) − r(t) 1) dt + σ(t) dW (t))

X (0) = 0

Wealth process:

X (t) = x · exp( t∫

0

[r(s) − γ + πT (b(s) − r(s) · 1

)− 1

2‖πT σ(s)‖2

]ds

+

t∫

0

πT σ(s) dW (s)

)

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Outline


156 / 477

Properties of the continuous-time market model

Assumptions:

Dimension of the underlying Brownian motion= number of stocks

Past and present prices are the only sources of information for theinvestors⇒ Choose Brownian filtration Ftt∈[0,T ]

Aim: Final wealths X (T ) when starting with initial capital of x .

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General assumption / notation

General assumption for this section

d = m

Notation

γ(t) := exp(−

t∫

0

r(s) ds)

θ(t) := σ−1(t)(b(t) − r(t) 1

)

Z (t) := exp(−

t∫

0

θ(s)T dW (s) − 12

t∫

0

‖θ(s)‖2 ds)

H(t) := γ(t) · Z (t)

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Properties of the continuous-time market model

b, r uniformly boundedσσT uniformly positive definite⇒ ‖θ(t)‖2 uniformly bounded

Interpretation of θ(t): Relative risk premium for stock investment.

Process H(t) is important for option pricing .

H(t) is positive, continuous, and progressively measurable withrespect to Ftt∈[0,T ].

H(t) is the unique solution of the SDE

dH(t) = −H(t)(r(t) dt + θ(t)T dW (t)

)

H(0) = 1.

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Completeness of the market

Theorem: Completeness of the market(1) Let the self-financing pair (π, c) consisting of a portfolio process π

and a consumption process c be admissible for an initial wealth ofx ≥ 0, i.e.,

(π, c) ∈ A(x).

Then the corresponding wealth process X (t) satisfies

E(

H(t) X (t) +

t∫

0

H(s)c(s) ds)

≤ x for all t ∈ [0, T ].

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Theorem: Completeness of the market(2) Let B ≥ 0 be an FT -measurable random variable and c(t) a

consumption process satisfying

x := E(

H(T ) B +

T∫

0

H(s)c(s) ds)

< ∞.

Then there exists a portfolio process π(t) with (π, c) ∈ A(x) andthe corresponding wealth process X (t) satisfies

X (T ) = B P-a.s.

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H(t) can be regarded as the appropriate discounting process thatdetermines the initial wealth at time t = 0

E( T∫

0

H(s) · c(s) ds)

+ E(H(T ) · B)

which is necessary to attain future aims.

(1) puts bounds on the desires of an investor given his initialcapital x ≥ 0.

(2) proves that future aims which are feasible in the sense of part(1) can be realized.

(2) says that each desired final wealth in t = T can be attainedexactly via trading according to an appropriate self-financing pair(π, c) if one possesses sufficient initial capital(completeness/complete model) .

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Remark1/H(t) is the wealth process corresponding to the pair

(π(t), c(t)

)=(σ−1(t)T θ(t), 0

)

with initial wealth x := 1/H(0) = 1

and final wealth B:= 1/H(T ).

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Outline


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Excursion 4: The martingale representation theorem

General assumptions(Ω,F , P) complete probability space.(Wt ,Ft)t∈[0,∞) m-dimensional Brownian motion.Ftt Brownian filtration.

DefinitionA real-valued martingale (Mt ,Ft)t∈[0,T ] with respect to the Brownianfiltration Ftt is called a Brownian martingale.

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The martingale representation theorem

Martingale representation theorem

Let (Mt ,Ft)t∈[0,T ] be a square-integrable Brownian martingale, i.e.,

EM2t < ∞ for all t ∈ [0, T ].

Then there exists a progressively measurable Rm-valued process Ψ(t)

with

E( T∫

0

‖Ψ(t)‖2 dt)

< ∞

and

Mt = M0 +

t∫

0

Ψ(s)T dW (s) P-a.s. .

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CorollaryLet (Mt ,Ft)t∈[0,T ] be a local martingale with respect to the Brownianfiltration Ftt . Then there exists a progressively measurableR

m-valued process Ψ(t) withT∫

0

‖Ψ(t)‖2 dt < ∞

and

Mt = M0 +

t∫

0

Ψ(s)T dW (s) P-a.s.

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RemarkEach local martingale with respect to the Brownian filtration canbe represented as an Ito process.

Each Brownian martingale can be represented as an Ito process.

⇒ Quadratic variation and quadratic covariation are defined.

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Outline

3 Option Pricing

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Outline


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Introduction

Option derivative security (i.e., from underlying assets)

call buy fixed amount of asset at fixed time in future forfixed strike price

put sell fixed amount of asset for strike price

American Option Sell/buy asset during timespan of contract

European Option Act at maturity ( =⇒ expiry )

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Outline


172 / 477

European call

European callRight to buy security at time t = T for strike price K > 0fixed at time t = 0

P(T ) > K Gain (P(T ) − K )+

P(T ) ≤ K No gain (Holder will buy in market)

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European put

European putRight to sell security at time t = T for price K > 0 fixed at time t = 0

P(T ) ≥ K No gain (Holder will sell in market)

P(T ) < K Gain (K − P(T ))+

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The payoff diagram

The payoff diagramGraph of final gain (through the option) as function of the stock price attime T

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Payoff diagram of a call

P1(T)

(P1(T)−K)+

K

K0

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Payoff diagram of a put

P1(T)

(K−P1(T))+

K

K0

177 / 477

The sense of options

Hedge against price fluctuations of underlying asset

Bound risks of future cash flows

Risks of option speculationHigh losses not unusual

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Outline


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Option pricing

Option PricingBond Interest Rate (BIR) r constant

Fixed deterministic payment of B at time T has present value of

e−rT · B

Sum to be invested at t = 0 to obtain amount of B at maturity.

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Option pricing

Option Pricing

BIR r(t) time dependent and random variableExpected value

E(

e−

T∫

0r(s) ds

· B)

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Examples of option pricing

Pricing in discrete-time

Market with bond ( P0(t) ) and stock ( P1(t) ) and BIRr = 0 , European Call with strike K = 1

P0(0) = 1 , P (P0(T ) = 1) = 1

P1(0) = 1 ,

P (P1(T ) = 3) = aP (P1(T ) = 0.5) = 1 − a

(a ∈ (0, 1))

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E((P1(T ) − K )+

)= (3 − 1) · a + 0 · (1 − a)

= 2a

Parameter a unkown

Final payment of option can be obtained by following aself-financing trading strategy in stock and bond( −→ Replication Principle )

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Determine (ϕ0(0), ϕ1(0)) such that

X (T ) = ϕ0(0) · P0(T ) + ϕ1(0) · P1(T )

= (P1(T ) − K )+

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Option Price

p = ϕ0(0) · P0(0) + ϕ1(0) · P1(0)

equals capital in t = 0 to buy replication strategy (ϕ0(0), ϕ1(0))

For all other choices riskless gains are possible without initialcapital −→ arbitrage opportunities

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Proof– Part 1(i) Be option price p < p. Then buy option for p and sell

(ϕ0(0), ϕ1(0)) for p(Hold position (−ϕ0(0),−ϕ1(0)) )

=⇒ Initial gain of p − p without own capital

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Proof– Part 2(ii) Be option price p > p. Then sell call and hold position

(ϕ0(0), ϕ1(0)) for p

=⇒ Initial gain of p − p without own capital

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Our example yields the system of equations

ϕ0(0) · 1 + ϕ1(0) · 3 = 2

ϕ0(0) · 1 + ϕ1(0) · 0.5 = 0

with unique solution

(ϕ0(0), ϕ1(0)) = (−0.4, 0.8)

=⇒ Option price p = −0.4 · 1 + 0.8 · 1 = 0.4

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Option price independent of unknown probability a

No arbitrage opportunities in market

Calculated call price p equals expected discounted terminalpayment of call ⇐⇒ a = 0.2−→ P1(t) is martingale

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General assumptions

Assumptions for this sectionSelf-financing pair (π, c) ∈ A(x) admissible for initial capital x ≥ 0

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Outline


191 / 477

Arbitrage opportunity

DefinitionBe

(ϕ, c) a self-financing and admissible pair

ϕ a trading strategy

c a consumption process

(ϕ, c) is called arbitrage opportunity ⇐⇒ corresponding wealthprocess satisfies

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Definition continued

X (0) = 0 , X (T ) ≥ 0 P-a.s.

P (X (T ) > 0) > 0 or P

T∫

0

c(t) dt > 0

> 0

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CorollaryIn the complete continuous-time market model there is no arbitrageopportunity.

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Contingent claim

DefinitionA contingent claim (g, B) consists of an Ftt -progressivelymeasurable payout rate process g(t), t ∈ [0, T ], g(t) ≥ 0, and aFT -measurable terminal payment B ≥ 0 at T with

E

T∫

0

g(t) dt + B

µ < ∞

for some µ > 1.

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Replication strategy

Definition(π, c) is called replication strategy for the contingent claim (g, B) if

g(t) = c(t) P-a.s. ∀ t ∈ [0, T ]

X (T ) = B P-a.s.

where X (t) is wealth process corresponding to (π, c).

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Set of replication strategies of price x

Definition

D(x) := D (x ; (g, B))

:= (π, c) ∈ A(x)| (π, c) replication

strategy for (g, B)

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Fair price

DefinitionThe fair price of the contingent claim (g, B) is defined as

p := inf p | D(p) 6= ∅ .

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Fair price of contingent claim (g, B)

TheoremThe fair price of the contingent claim (g, B) is given by

p = E

H(T )B +

T∫

0

H(t)g(t) dt

< ∞,

and there exists a unique (with respect to P ⊗ λ) replication strategy(π, c

)∈ D

(p).

Its wealth process X (t) (also called the valuation process of (g, B)) isgiven by

X (t) =1

H(t)E

H(T )B +

T∫

t

H(s)g(s) ds

∣∣∣∣ Ft

.

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The valuation process

RemarkThe above theorem gives us the fair price of the contingent claim(g, B) at time t as this price p(t) has to coincide with X (t). Otherwise,there would be arbitrage opportunities in the market consisting ofstock, bond, and contingent claim.

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Black-Scholes formula

TheoremConsider a market model with just one stock and a bond with constantmarket coefficients, i.e.,

d = m = 1

andr(t) ≡ r , b(t) ≡ b , σ(t) ≡ σ > 0

for all t ∈ [0, T ], T > 0, r , b, σ ∈ R.

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(i) The price XC(t) of a European call with strike price K > 0 andmaturity T is given by

XC(t) = P1(t)Φ(d1(t)) − Ke−r(T−t)Φ(d2(t))

d1(t) =ln(

P1(t)K

)+ (r + 0.5σ2)(T − t)

σ√

T − t

d2(t) =ln(

P1(t)K

)+ (r − 0.5σ2)(T − t)

σ√

T − t

= d1(t) − σ√

T − t

where Φ is the distribution function of the standard normaldistribution.

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(ii) Price XP(t) of European put with strike K > 0

XP(t) = Ke−r(T−t)Φ(−d2(t)) − P1(t)Φ(−d1(t)).

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The change of measure

W Q(t) := W (t) + θt =⇒

p = XC(0) = EQ

(e−rT (P1(T ) − K )+

)

where EQ(·) denotes expected value with respect to measure Q givenby Radon-Nikodym derivative

dQdP

= e−0.5θ2T−θW (T ).

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General assumptions of this section

General Assumptions of this SectionBe (X (t),Ft )t≥0 an m-dimensional progressively measurableprocess, Ft the Brownian filtration with

t∫

0

X 2i (s) ds < ∞ P-a.s. for all t ≥ 0, i = 1, . . . , m.

Let further

Z (t , X ) := exp(−

m∑

i=1

t∫

0

Xi(s) dWi(s) − 12

t∫

0

‖X (s)‖2 ds)

.

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Consequences

The argument in Z (t , X ) is an Ito process and we have

Z (t , X ) = 1 −m∑

i=1

t∫

0

Z (s, X )Xi(s) dWi(s).

Z (t , X ) is a continuous local martingale with Z (0, X ) = 1.

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Girsanov’s theorem

Girsanov’s theoremBe process Z (t , X ) a martingale and define process(

W Q(t),Ft)

t≥0 by

W Qi (t) := Wi(t) +

t∫

0

Xi(s) ds (1 ≤ i ≤ m, t ≥ 0)

Then, for each fixed T ∈ [0,∞) the process(

W Q(t),Ft)

t∈[0,T ]is

an m-dimensional Brownian motion on (Ω,FT , QT ) with probabilitymeasure

QT (A) := E (1A · Z (T , X )) ∀ A ∈ FT

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The Novikov condition

Z (t , X ) martingale −→ apply Girsanov’s theorem

Sufficient condition is the Novikov condition:

E(

e0.5

t∫

0‖X(s)‖2 ds

)< ∞

208 / 477

The Novikov condition

Proposition

If we haveT∫0

‖X (s)‖2 ds < K for some constant K > 0, then

Z (t , X ) is a martingale.

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Outline


210 / 477

Girsanov’s theorem & Option Pricing

LemmaBe Q a probability measure which is equivalent to P restricted onFT . Then the density process Dtt∈[0,T ] defined by

Dt :=dQdP

∣∣∣∣Ft

, t ∈ [0, T ]

satisfies: (Dt ,Ft)t∈[0,T ] is a positive Brownian martingale withrespect to P satisfying

Dt = 1 +

∫ t

0Ψ(s)T dW (s)

for a progressively measurable d -dimensional process Ψ with

T∫

0

‖Ψ(s)‖2 ds < ∞ P-a.s.

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Uniqueness of the equivalent martingale measure

TheoremIn the complete market model QT is the unique equivalent martingalemeasure on Ftt∈[0,T ] for the price processes Pi(t), 0 ≤ i ≤ d .

RemarkThe existence of an equivalent martingale measure implies theabsence of arbitrage opportunities in the market.

The absence of arbitrage implies the existence of an equivalentmartingale measure.

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Option pricing & equivalent martingale measure

CorollaryBe (g, B) a contingent claim such that g(s) is uniformly bounded on[0, T ]. Then its price process X (t) satisfies

X (t) = EQ

e

−T∫t

r(s) dsB +

T∫

t

e−∫ s

t r(u) dug(s) ds

∣∣∣∣Ft

for 0 ≤ t ≤ T with EQ = EQ,T .

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Independence of the option price

In the case of g ≡ 0 we have

p = EQ

e

−T∫

0r(s) ds

B

Thus, p equals the natural price with respect to a new (uniquelydetermined) probability measure Q.

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European digital call

If the shock price P1(t) exceeds a certain boundary K at time T , theowner of the call is paid B∗, otherwise he gets nothing.

Choose B∗ = 1

Final payment B = 1P1(T )≥KBlack-Scholes model (d = m = 1, b, r , σ constant, σ > 0) and lastCorollary yields

X(t) = EQ

(e−r(T−t) · 1P1(T )≥K |Ft

)

= e−r(T−t) · Q (P1(T ) ≥ K |P1(t) ) .

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For fixed t we have P1(T ) ≥ K ⇐⇒

W Q(T ) − W Q(t) ≥ln(

KP1(t)

)−(

r − σ2

2

)(T − t)

σ︸︷︷︸:=K

As W Q(T ) − W Q(t) is normally distributed with expectation 0 andvariance T − t , we obtain

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X (t) = e−r(T−t)∫ ∞

K

1√2π(T − t)

e− x2

2(T−t) dx

= e−r(T−t) Φ

ln(

P1(t)K

)+(

r − σ2

2

)(T − t)

σ√

T − t

217 / 477

Outline


218 / 477

Option pricing by PDA

General Assumptions nowConsider a Black-Scholes model, i.e.,

d = m = 1

andb, r , σ constant with σ > 0.

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Proposition-Part ILet there exist a polynomially bounded solution

f : [0, T ]× (0,∞)d −→ R, i.e.

max0≤t≤T

|f (t , p)| ≤ M(

1 + ‖p‖k)

for a fixed M > 0, k ∈ N, p ∈ (0,∞)d , for the Cauchy problem

ft +12

∑

1≤i ,j≤d

aijpipj fpi pj +∑

1≤i≤d

rpi fpi − rf = 0

on [0, T ) × Rd ,

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Proposition–Part II

f (T , p1, . . . , pd ) = g(p1, . . . , pd ) for p ∈ Rd

such that f is continuous and f ∈ C1,2([0, T ] × (0,∞)d

). Further, let

EQ(g(P1(T ), . . . , Pd (T ))) < ∞,

where EQ = EQ,T . Then the price XB(t) of the contingent claimB = g(P1(T ), . . . , Pd(T )) in the d -dimensional Black-Scholes model isgiven by

XB(t) = f (t , P1(t), . . . , Pd (t)).

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Proposition–Part IIIFurther for 1 ≤ i ≤ d ,

Ψi(t) = fpi (t , P1(t), . . . , Pd (t))

Ψ0(t) =

f (t , P1(t), . . . , Pd (t)) − ∑1≤i≤d

Ψi(t)Pi(t)

P0(t)

is a replication strategy for B.

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SDE

Definition–Part IIf on (Ω,F , P) there exists a d -dimensional continuous process(X (t),Ft )t≥0 with X (0) = x , x ∈ R

d fixed,

Xi(t) = xi +

t∫

0

bi(s, X (s)) ds +∑

0≤j≤m

t∫

0

σij(s, X (s)) dWj(s)

P-a.s. for all t ≥ 0, 1 ≤ i ≤ d , satisfying

t∫

0

|bi(s, X (s))| +

∑

1≤j≤m

σ2ij (s, X (s))

ds < ∞

P-a.s. for all t ≥ 0, 1 ≤ i ≤ d ,

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SDE

Definition–Part IIthen X (t) is called a strong solution of the stochastic differentialequation (SDE)

dX (t) = b(t , X (t)) dt + σ(t , X (t)) dW (t)

X (0) = x

whereb : [0,∞) × R

d −→ Rd , σ : [0,∞) × R

d −→ Rd ,m

are given functions (m dimension of Brownian motion).

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Existence and uniqueness of solutions of SDEs

Theorem–Part ILet the coefficients b(t , x), σ(t , x) of the SDE be continuous functionswith

‖b(t , x) − b(t , y)‖ + ‖σ(t , x) − σ(t , y)‖ ≤ K‖x − y‖‖b(t , x)‖2 + ‖σ(t , x)‖2 ≤ K 2

(1 + ‖x‖2

)

for all t ≥ 0, x , y ∈ Rd and a constant K > 0.

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Theorem–Part IIThen there exists a continuous, strong solution (X (t),Ft )t≥0 of theSDE with

E(‖X (t)‖2

)≤ C

(1 + ‖x‖2

)eCT ∀ t ∈ [0, T ]

for some constant C = C(K , T ) and T > 0. Further, X (t) is unique upto indistinguishability.

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Proof planUniqueness

Existence – some estimates

Existence – convergence of the iteration

Solution property

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LemmaThe solution X of the SDE satisfies for m ≥ 1 and fixed T > 0

E(

max0≤s≤t

‖X (s)‖2m)

≤ C(

1 + ‖x‖2m)

eCt

for all t ∈ [0, T ] and suitable constant C = C(T , K , m, d).

Notation

Solution of the SDE with X (t) = x denote X t,s(s)

E(. . . X t,s(s) . . .) = E t,s(. . . X (s) . . .)

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SDEs

DefinitionBe X (t) unique solution of the SDE under the introduced conditions.For f : R

d −→ Rd , f ∈ C2

(R

d), the operator At defined by

(At f )(x) :=12

∑

1≤i≤d

∑

1≤k≤d

aik (t , x)∂2f

∂xi∂xk(x)

+∑

1≤i≤d

bi(t , x)∂f∂xi

(x)

withaik (t , x) :=

∑

1≤j≤m

σij(t , x)σkj (t , x)

is called the characteristic operator corresponding to X (t).

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The Cauchy problem

Description of the Cauchy problem IBe T > 0 fixed. Consider the following Cauchy problem correspondingto operator At :Find a function v(t , x) : [0, T ]× R

d −→ R with

−vt + kv = Atv + g on [0, T ] × Rd

v(T , x) = f (x) for x ∈ Rd

where

f : Rd −→ R , g : [0, T ] × R

d −→ R , k : [0, T ] × Rd −→ (0,∞).

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The Cauchy problem

Description of the Cauchy problem IITo ensure the uniqueness of a solution we require that v obeys apolynomial growth condition:

max0≤t≤T

|v(t , x)| ≤ M(

1 + ‖x‖2µ)

with M > 0 , µ ≥ 1.

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The Cauchy problem

Description of the Cauchy problem IIIUsually we assume that for suitable constants L, λ the functions f , g, kare continuous with

|f (x)| ≤ L(

1 + ‖x‖2λ)

, L > 0, λ ≥ 1 or f (x) ≥ 0

|g(t , x)| ≤ L(

1 + ‖x‖2λ)

, L > 0, λ ≥ 1 or g(t , x) ≥ 0.

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The Feynman-Kac representation

Theorem–Part ILet the inequalities for f and g be satisfied. Let furtherv(t , x) : [0, T ] × R

d −→ R be continuous solution of the Cauchyproblem with v ∈ C1,2([0, T ) × R

d). Denote by At the characteristicoperator corresponding to the unique solution X (t) of the SDE withcontinuous coefficients b, σ with

bi(t , x), σ(t , x) : [0,∞) × Rd −→ R

for 1 ≤ i ≤ d , 1 ≤ j ≤ m.

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The Feynman-Kac representation

Theorem–Part IIIf v(t , x) satisfies the polynomial growth condition we have therepresentation

v(t , x) = E t,x

f (X (T )) e

−T∫t

k(θ,X(θ)) dθ

+

+

T∫

t

g(s, X (s)) e−

s∫t

k(θ,X(θ)) dθ

ds

.

In particular, v(t , x) is unique solution.

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Outline


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General assumptions

General AssumptionsConsider market with d + 1 traded securities with positive pricesP0(t), . . . , Pd (t). Prices shall be Ito processes with respect to anm-dimensional Brownian motion (Wt ,Ft)t∈[0,∞) with m ≥ d whereFtt∈[0,∞) is the Brownian filtration.

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General assumptions

Notations

As trading strategy ϕ(t) = (ϕ0(t), . . . , ϕd (t))T , t ≥ 0, define a(d + 1)-dimensional progressively measurable process such that thestochastic integrals

T∫

0

ϕi(s) dPi(s) ,

T∫

0

ϕi(s) dPi(s), 0 ≤ i ≤ d

exist for all T ≥ 0 where

Pi(t) :=Pi(t)P0(t)

denotes the discounted price process.

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General assumptions

NotationsWealth process X (t) corresponding to the trading strategy ϕ(t) and theself-financing condition are defined by

X (t) =∑

0≤i≤d

ϕi(t)Pi(t)

= x +∑

0≤i≤d

t∫

0

ϕi(s) dPi(s)

P-a.s., for all t ≥ 0.

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Equivalent martingale measures

DefinitionA probability measure Q defined on (Ω,FT ) equivalent to P (P and Qhave same zero sets) is called an equivalent martingale measure forP0(t), . . . , Pd (t) if the discounted prices

Pi(t) =Pi(t)P0(t)

(1 ≤ i ≤ m), t ∈ [0, T ]

are martingales with respect to Q.

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Equivalent martingale measures

TheoremAll martingale measures Q for P0(t), . . . , Pd (t) equivalent to P can beobtained from P by a Girsanov transformation with an m-dimensionalprogressively measurable stochastic process (θ(t),Ft )t≥0 where forall t ≥ 0 we have

t∫

0

θ2i (s) ds < ∞ P-a.s. (1 ≤ i ≤ m)

and where Z (t , θ) is a martingale with respect to P.In particular, Q is given as

Q(A) := QT (A) := E(1A · Z (T , θ)) ∀ A ∈ FT .

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Equivalent martingale measures =⇒ no arbitrage

TheoremIf there exists an equivalent martingale measure then the market givenby the price process P0(t), . . . , Pd(t) contains no arbitrage opportunity.

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Definition(i) A contingent claim B is a non-negative FT -measureable random

variable with

EQ

(1

P0(T )B)

< ∞

for all equivalent martingale measures Q.

(ii) B is called attainable if there exists an admissible trading strategyϕ(t) with wealth process X (t) and

B = X (T ) P-a.s.

such that X (t) = X (t)/P0(t) is martingale with respect to someequivalent martingale measure Q.

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TheoremThe security market under examination is complete (i.e. eachcontingent claim is attainable) if and only if there exists a uniqueequivalent martingale measure Q.

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Option pricing in incomplete markets

DefinitionA market in which not every contingent claim is attainable is calledincomplete.

Possible reasons for incomplete markets can be

trading constraints to invest into particular stock.

additional random fluctuations in the market coefficients.

In an incomplete market, typically the σ-algebra FT is bigger than theone generated by final wealths produced by admissible tradingstrategies

X (T ) = x +∑

0≤i≤d

T∫

0

ϕi (s) dPi(s).

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Prices of attainable contingent claims

TheoremThe unique price process X ∗(t) of an attainable contingent claim B isgiven by

X ∗(t) = EQ

(P0(t)P0(T )

B

∣∣∣∣Ft

)(t ∈ [0, T ]),

where Q is an equivalent martingale measure.

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Option price and equivalent martingale measure

TheoremLet Q be an equivalent martingale measure to P. Let B be an arbitrarynot necessary attainable contingent claim. If we choose

X QB (t) := EQ

(P0(t)P0(T )

B

∣∣∣∣Ft

)

as price of the contingent claim then the extended security marketconsisting of d + 1 securities and the contingent claim contains noarbitrage opportunity.

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Market numeraire and numeraire invariance

DefinitionLet (Y (t),Ft )t∈[0,T ] be a strictly positive Ito process and discountprocess. Then we call such a discount process a numeraire.

Questions:

Does change of numeraire (i.e., the choice of a numerairedifferent from P0(t)) affect the option price or its calculation?

Does there exist a numeraire such that the option price is given asexpected value of final payment B with respect to originalmeasure P when discounted by this numeraire?

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Market numeraire and numeraire invariance

General AssumptionsConsider the complete market model with d = m and

1H(t)

= exp( t∫

0

(r(s) +

12‖θ(s)‖2

)ds +

t∫

0

θ(s)T dW (s)

).

By using the product rule and the SDE of the stock prices, we canverify that H(t) · Pi(t) are P-martingales for 0 ≤ i ≤ d .

1H(t) forms the wealth process corresponding to the admissible

pair (π, c) = (σ(t)−1(b(t) − r(t) 1), 0) ∈ A(1).

This numeraire can be replicated at market by trading in suitableway (market numeraire/numeraire portfolio).

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Option price and equivalent martingale measure

Theorem

In complete market 1H(t) is the unique numeraire such that the

corresponding discounted price processes H(t) Pi(t), 0 ≤ i ≤ d , aremartingales with respect to P.

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Numeraire invariance in the complete market

TheoremConsider the complete market model with constant market coefficientsr(t) ≡ r , b(t) ≡ b, σ(t) ≡ σ > 0. Then we have

(i) The process ZY (t) = H(t) Y π(t) is P-martingale for all constantportfolio processes π(t) ≡ π, where Y π(t) is wealth processcorresponding to π.

(ii) The corresponding probability measure QY with ZY (T ) = dQYdP is

unique equivalent martingale measure for price processesdiscounted by Y π(t).

(iii) Fair price p of a contingent claim B with E(Bµ) < ∞ for someµ > 1 is given as

p = E(H(T )B) = E(

1Yπ(T )

B)

if Y π(t) is numeraire of part (i)/(ii).

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Outline

4 Pricing of Exotic Options and Numerical Algorithms

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Outline


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General assumptions

General assumptionsConsider a Black-Scholes model with d = m and constantcoefficients b, r , σ with σ > 0 (or σ regular if d > 1).

All options are assumed to be of European type.

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Outline


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Exotic options

Options on Minimum/Maximum of the Stock PriceEuropean Call on Maximum given by the terminal payment

B =

(max

0≤t≤TP1(t) − K

)+

.

Barrier OptionsThey have a zero value if stock price exceeds certain barrier orhave a positive payment at time T if a certain barrier is reached.

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Outline


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Some examples for exotic options

Down-and-out Call

B = (P1(T ) − K1)+ 1

min0≤t≤T

P1(t)>K2

Down-and-in Call

B = (P1(T ) − K1)+ 1

min0≤t≤T

P1(t)≤K2

Double-barrier Call

B = (P1(T ) − K1)+ 1

min0≤t≤T

P1(t)>K2 , max0≤t≤T

P1(t)<K3

with K2 < K1 < K3.

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Outline


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Price of European option

The price of a European option is given by

p = EQ

(e−rT B

),

where Q is unique equivalent martingale measure andEQ its expectation.

AssumeStock prices are given as solutions of the SDE (1 ≤ i ≤ d)

dPi(t) = Pi(t)

r dt +

∑

1≤j≤d

σij dWj(t)

.

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Outline


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Path independent options on one stock

Binary Option–Part ITerminal payment in T of binary option with bound K given by

BCalld = 1P1(T )>K , BPut

d = 1P1(T )<K

if the final price P1(T ) exceeds K (in the case of a call) or issmaller than K (in the case of a put).

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Binary Option–Part IIPricing of these digital options

X Calld (t) = e−t(T−t)Φ

(d2(t)

),

X Putd (t) = e−t(T−t)Φ

(−d2(t)

)

with

d2(t) =ln(

P1(t)K

)+(

r − σ2

2

)(T − t)

σ√

T − t,

where Φ is distribution function of standard normal distribution.

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Paylater Options–Part IFinal payoffs

BCallPL =

(P1(T ) −

(K + DCall

))· 1P1(T )≥ K

BPutPL =

((K − DPut

)− P1(T )

)· 1P1(T )≤K

where DCall , DPut determined in such a way that prices of paylateroptions are zero at initial time.

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Paylater Options–Part IIFinal payoffs decomposed

BCallPL = BCall − DCall BCall

d

BPutPL = BPut − DPut BPut

d .

WithX Call

PL (0) = 0 = X PutPL (0)

DCall =X Call(0)

X Calld (0)

, DPut =X Put(0)

X Putd (0)

we get

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Paylater Options–Part III

X CallPL = P1(t)Φ(d1(t)) −

p1Φ(d1(0))

Φ(d2(0))Φ(d2(t)) ert

X PutPL = −P1(t)Φ(−d1(t)) +

p1Φ(−d1(0))

Φ(−d2(0))Φ(−d2(t)) ert

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Proposition–Part I(i) Given K > 0, maturity T1 and strike K1 there exists a unique

p∗ > 0 for T ≤ T1 such that for P1(T ) = p∗ we have

X Call(T ) = X Call(T , p∗) = K .

(ii) With

g1(t) =ln(

P1(t)p∗

)+(

r + σ2

2

)(T − t)

σ√

T − t,

g2(t) = g1(t) − σ√

T − t ,

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h1(t) =ln(

P1(t)K1

)+(

r + σ2

2

)(T1 − t)

σ√

T1 − t,

h2(t) = h1(t) − σ√

T1 − t ,

we get the price of a call on a call

X CCcom(t) = P1(t)Φ

(ρ1)(g1(t), h1(t))

− K1e−r(T1−t)Φ(ρ1)(g2(t), h2(t))

− Ke−r(T−t)Φ(g2(t))

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Proposition–Part III

for t ∈ [0, T ], where Φ(ρ)(x , y) is distribution function of bivariatestandard normal distribution with correlation coefficient ρ and with

ρ1 :=

√T − tT1 − t

,

(XY

)∼ N

((00

),

(1 ρ1

ρ1 1

)).

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Lemma–Part IIf X and Y independent random variables with

X ∼ N (µ, σ2) , Y ∼ N (0, 1)

then for x , α, β ∈ R , α > 0 we have

∞∫

x

ϕµ,σ2(x) Φ(αx + β) dx = P(X ≥ x , Y ≤ αX + β)

= P(X ≥ x , Z ≤ β),

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Lemma–Part IIwhere

(X , Z ) ∼ N((

µ−αµ

),

(σ2 −ασ2

−ασ2 1 + α2σ2

))

=⇒ ϕµ,σ2 is density function of the normal distribution with mean µ

and variance σ2.

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Options on more than one underlying Stock

Indexed OptionsConsider 2-dimensional Black-Scholes model with given stock prices

dPi(t) = Pi (bi dt + σi1 dW1(t) + σi2 dW2(t)) ,

Pi(0) = pi (i = 1, 2)

Be a1, a2 ∈ R+. An indexed option with parameters a1, a2 is then given

by final payment

Bind = (a1P1(T ) − a2P2(T ))+ .

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Options on minimum/maximum of 2 stocks

Call on minimum/maximum

BCallmin =

(min

(P1(T ), P2(T )

)− K

)+

BCallmax =

(max

(P1(T ), P2(T )

)− K

)+

Put on minimum/maximum

BPutmin =

(K − min

(P1(T ), P2(T )

))+

BPutmax =

(K − max

(P1(T ), P2(T )

))+

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Proposition–Part IThe prices of the minimum/maximum options are given by

X Callmin (0) = p1Φ

(ρ) (d1, d3) + p2Φ(˜ρ) (d2, d4)

−Ke−rT Φ(ρ)(

d1 − σ1

√T , d2 − σ2

√T)

,

X Putmin (0) = X Call

min (0) + Ke−rT − p1Φ (d3) − p2Φ (d4) ,

X Callmax (0) = X Call

(1) (0) + X Call(2) (0) − X Call

min (0),

X Putmax (0) = X Put

(1) (0) + X Put(2) (0) − X Put

min (0),

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where X Call(i) , X Put

(i) denote prices of ordinary European calls/puts onstock i with strike K (i = 1, 2) and

σi :=√

σ2i1 + σ2

i2 (i = 1, 2),

ρ :=σ11σ21 + σ12σ22

σ1σ2,

ρ :=ρ σ2 − σ1

σ,

˜ρ :=ρσ1 − σ2

σ,

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Proposition–Part III

di :=ln(pi

K

)+(r + 0.5σ2

i

)T

σi√

T,

di+2 :=(−1)i ln

(p1p2

)− 0.5σ2 T

σ√

T(i = 1, 2).

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Path dependent options

One-sided barrier optionsOwner receives final payoff of European call/put if stock price does notexceed/does exceed given barrier before time T . Look atdown-and-out call and down-and-in call:

BCalldo = (P1(T ) − K )+ · 1P1(t)>b ∀ t∈[0,T ],

BCalldi = (P1(T ) − K )+ · 1∃ t∈[0,T ]: P1(t)≤b.

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Path dependent options

LemmaBe M(t) := max0≤s≤t W (s) running maximum of 1-dimensionalBrownian motion W (t). Then for x ≥ 0, x ≥ w , we have

(i) P (W (t) ≤ w , M(t) < x) = Φ(

w√t

)− 1 + Φ

(2x−w√

t

)

(ii) For µ ∈ R be W (t) := W (t) + µ · t , M(t) := max0≤s≤t W (s).Thus we have

P(

W (t) ≤ w , M(t) < x)

= Φ

(w − µt√

t

)− e2µ xΦ

(w − 2x − µt√

t

)

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Outline


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General assumptions

General assumptions for this sectionWe consider the probability space

(Ω, F , P) = (C[0, 1], B(C[0, 1]), P),

i.e., the space of continuous, real-valued functions on [0,1] equippedwith the corresponding Borel σ-field and a probability measure P.

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Stochastic process with distribution P

The function valued random variable X on (Ω, F , P) given by

X (ω) := ω, ω ∈ C[0, 1]

defines a real-valued stochastic process with distribution P.

Value of the process at time t ∈ [0, 1]:

X (t , ω) := πt X (ω) := ω(t).

→ projection on the "t-th coordinate" of ω.

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Notion of convergence

The notion of convergence of stochastic process Xn via the usualweak convergence of random variables

Xn(t)n→∞−−−→ X (t) for all t ∈ [0, 1] in distribution

is too weak.

⇒ Consider weak convergence of probability measureson metric spaces

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Weak convergence

DefinitionLet (S,B(S)) be a metric space with metric ρ and the Borel-σ-fieldB(S) over S. Let further P, Pn, n ∈ N, be probability measures on(S,B(S)). Then we say that the sequence Pn converges weakly (orconverges in distribution) to P if for every continuous and boundedreal-valued function f on S we have

∫

S

f dPnn→∞−−−→

∫

S

f dP.

Special case: Weak convergence of stochastic processeswith continuous paths.

Remark: (C[0, 1], B(C[0, 1])) is a metric space with the metric

ρ(x , y) = sup0≤t≤1

|x(t) − y(t)|.282 / 477

Weak convergence

DefinitionThe sequence of continuous stochastic processes Xn(t)t∈[0,1]

converges weakly (or in distribution) to X if for all f ∈ C(C[0, 1], R) wehave

Ef (Xn)n→∞−−−→ Ef (X ).

Remark: C(C[0, 1], R) is the space of the uniformly continuous,bounded functionals on C[0, 1].

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Weak convergence

Stochastic process Xn is defined on (C[0, 1], B(C[0, 1]), Pn).

Process X is defined on (C[0, 1], B(C[0, 1]), P).

P, Pn are probability measures on (C[0, 1], B(C[0, 1])).

Ef (Xn) =∫

f (Xn) dP =∫

f dPnn→∞−−−→

∫f dP =

∫f (X ) dP = Ef (X ).

⇒ The weak convergence of stochastic processesis represented as the weak convergence of theprobability measures Pn → P.

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Weak convergence

TheoremLet P, Pn, n ∈ N, be probability measures on the metric space(S,B(S)) endowed with the metric ρ. Further, let h : S → S′ be ameasurable mapping into a metric space S′ with metric ρ′ andBorel-σ-field B(S′). If for the set Dh of points of discontinuity of h wehave

P(Dh) = 0,

then we get

Pnn→∞−−−→ P in distribution ⇒ Pn · h−1 n→∞−−−→ P · h−1 in distribution.

⇒ Weak convergence is preserved under continuous mappings.

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Weak convergence

CorollaryIf the sequence Xn of continuous stochastic processes convergesweakly to the continuous process X, then for every fixed t ∈ [0, 1] therandom variables Xn(t) converge in distribution to X (t).

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Weak convergence

RemarkDefine the projections

πt1,...,tk : C[0, 1] → Rk

byπt1,...,tk (ω) = (ω(t1), . . . , ω(tk ))

for 0 ≤ t1 < . . . < tk ≤ 1. Then, we have

Xnn→∞−−−→ X in distribution

⇒ (Xn(t1), . . . , Xn(tk ))n→∞−−−→ (X (t1), . . . , X (tk )) in distribution.

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Weak convergence

RemarkWeak convergence of stochastic processes implies convergenceof the finite-dimensional distributions.

Convergence of all finite-dimensional distributions

Pn · π−1t1,...,tk

does not in general imply convergence of the distributions Pn ofthe corresponding processes.

If the sequence of the Pn is relatively compact (i.e., eachsubsequence contains a weakly convergent subsequence), thenthe convergence of the finite-dimensional subsequences impliesweak convergence of the Pn.

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Approximation of the one-dimensional Brownianmotion

Algorithm

(1) Choose a sequence ξnn∈N of i.i.d. random variables of a simpleform with

E(ξi) = 0, Var(ξi) = σ2 < ∞and set

S0 := 0, Sn :=n∑

i=1

ξi .

Example: ξi = Yi − q with Yi ∼ B(1, q).

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Algorithm(2) By means of linear interpolation construct a stochastic process

Xn(t) with continuous paths of that sequence

Xn(t , ω) =1

σ√

nS[nt](ω) + (nt − [nt])

1σ√

nξ[nt]+1(ω)

for t ∈ [0, 1], n ∈ N, i.e., we have

Xn( k

n , ω)

=1

σ√

nSk(ω),

and for t ∈(k

n , k+1n

)we obtain Xn(t) by linear interpolation.

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Algorithm(3) The finite-dimensional distributions of Xn converge in distribution

to that of a Brownian motion.

• From [ns]n

n→∞−−−→ s and the central limit theorem, we obtain

1σ√

nS[ns]

n→∞−−−→ W (s) in distribution.

• Chebychev’s inequality yields∣∣∣∣Xn(s) − 1

σ√

nS[ns]

∣∣∣∣ ≤1

σ√

n|ξ[ns]+1| n→∞−−−→ 0 in probability.

Hence,Xn(s)

n→∞−−−→ W (s) in distribution.

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Algorithm• Due to the independence of the ξi and the theorem of Slut-

sky this results in(

1σ√

nS[ns],

1σ√

n

(S[nt] − S[ns]

)) n→∞−−−→ (Ws, Wt − Ws)

for s < t . From this we get

(Xn(s), Xn(t) − Xn(s))n→∞−−−→ (Ws, Wt − Ws) in distribution.

Slutsky’s theorem implies

(Xn(s), Xn(t))n→∞−−−→ (Ws, Wt) in distribution.

Analogously: Convergence of finite tuples ofXn(ti)-components.

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Algorithm(4) Show that the sequence of the distributions Pn on

(C[0, 1], B(C[0, 1])) corresponding to Xn is relatively compact.

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Donsker’s theorem

Donsker’s theoremLet ξnn∈N be an i.i.d. sequence with

E(ξi) = 0 and 0 < Var(ξi) = σ2 < ∞.

Then, the sequence Xn of stochastic processes defined by

Xn(t , ω) =1

σ√

nS[nt](ω) + (nt − [nt])

1σ√

nξ[nt]+1(ω), t ∈ [0, 1], n ∈ N

converges weakly to the one-dimensional Brownian motionW (t), t ∈ [0, 1].

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Donsker’s theorem

RemarkThe convergence assertion and the limiting distribution areindependent of the exact choice of ξi

(Donsker’s invariance principle).

Donsker’s theorem can be viewed as the "process version" of thecentral limit theorem.

Donsker’s theorem can be assumed valid for arbitrary intervals[0, T ].

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Donsker’s theorem for triangular schemes

Donsker’s theorem for triangular schemesThe random variables ξn1, . . . , ξnkn

, n ∈ N, kn ∈ N, are assumed to bei.i.d. with E(ξn1) = 0 and 0 < Var(ξn1) = σ2

n1≤ c, where c > 0. Let

Sni:= ξn1 + . . . + ξni , 1 ≤ i ≤ kn,

s2ni

:= σ2n1

+ . . . + σ2ni

= i · σ2n1

,

s2n := s2

nkn= kn · σ2

n1.

Define the process Xn(t), t ∈ [0, 1], by

Xn(0) := 0,

Xn

(s2

ni/s2n

):= Sni/sn , i = 1, . . . , kn,

and via linear interpolation on the intervals[s2

ni−1/s2

n, s2ni

/s2n

].

If kn → ∞ and sn → ∞ for n → ∞, then Xn converges weakly to theBrownian motion W .

296 / 477

Donsker’s theorem

RemarkWe have

E(h(Xn))n→∞−−−→ E(h(X ))

for continuous and bounded functionals h : C[0, T ] → R.

Not sufficient for practical applications.

If in the Black-Scholes model we approximate the Brownianmotion by a sequence of processes Xn, Donsker’s theorem wouldnot directly imply

E(

eb·T+σ·Xn(T ))

n→∞−−−→ E(

eb·T+σ·W (T ))

as the exponential functional is not bounded.

Additionally, we need the uniform integrability of the sequenceexp(σXn(t)).

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Weak convergence

TheoremLet the sequence of random variables Xnn∈N be uniformly integrable.Assume further that we have

Xnn→∞−−−→ X in distribution.

Then this impliesE(Xn)

n→∞−−−→ E(X ).

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Outline


299 / 477

The basic idea

Description of the idea

The basis of Monte-Carlo simulation is the strong law of largenumbers

⇒ Arithmetic mean of independent, identically distributed ran-dom variables converges towards their mean almost surely.

Computing an option price = Computing the discountedexpectation (with respect to the equivalent martingale measure) ofthe payoff B.

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The basic idea

Algorithm: Determine the option price via Monte-Carlosimulation(1) Simulate n independent realizations Bi of the final payoff B.

(2) Choose(

1n

n∑

i=1

Bi

)· e−rT as an approximation for the option price

EQ(e−rT B).

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Simulation of the payoff B

Assume that the payment B is a functional of the price processP1(t), t ∈ [0, T ].

Simulate a path P1(t) of the stock price process with respect tothe equivalent martingale measure Q.

A path can only be simulated approximately (it is given by infinitelymany values).

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Procedure(1) Divide the interval [0, T ] into N ≫ 1 equidistant parts.

(2) Generate N independent random numbers Yi which areN (0, 1)-distributed.

(3) From those, simulate an (approximate) path W (t) of the Brownianmotion on [0, T ]:

W (0) = 0,

W(j · T

N

)= W

((j − 1) · T

N

)+√

TN · Yj , j = 1, . . . , N,

W (t) = W((j − 1) · T

N

)

+ (t − (j − 1) · TN ) · N

T ·[W(j · T

N

)− W

((j − 1) · T

N

)],

for t ∈[(j − 1) · T

N , j · TN

]

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Procedure(4) Use this to generate an (approximate) path of the price process

P1(t):

P1(t) = p1 · e(

r−12 σ2)

t · eσ·W (t), t ∈ [0, T ].

(5) With this simulated path of the price process compute an estimatefor the payoff B.

Example (European call): Bi = (P1(T ) − K )+.

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RemarkFor the practical realization of the computation of Bi in step 5 it oftenproves to be more convenient to do the interpolation in step 4 and notin step 3:

P1(0) = p1,

P1(j · T

N

)= P1

((j − 1) · T

N

)· e(

r−12 σ2) T

N · eσ·√

TN ·Yj

, j = 1, . . . , N,

P1(t) = P1((j − 1) · T

N

)

+ (t − (j − 1) · TN ) · N

T ·[P1(j · T

N

)− P1

((j − 1) · T

N

)],

for t ∈[(j − 1) · T

N , j · TN

].

For large N the differences between both methods are negligible.

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Convergence of the method

Let P(N)1 (t), t ∈ [0, T ] be the approximate price process.

If B is a continuous and bounded functional on C([0, T ]) then byDonsker’s theorem we have the convergence

EQ

(B(

P(N)1 (t), t ∈ [0, T ]

))N→∞−−−−→ EQ

(B(P1(t), t ∈ [0, T ])

).

If B is a continuous functional on C([0, T ]), then the convergenceof the option price is guaranteed if the family

B(

P(N)1 (t), t ∈ [0, T ]

) ∣∣∣∣N ∈ N

is uniformly integrable.

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Convergence of the method

Showing the uniformly integrability can be tricky for specificchoices of B. Then for a given N the expected value

EQ

(B(

P(N)1 (t), t ∈ [0, T ]

))

will be approximated by the arithmetic mean

1n

n∑

i=1

B(

P(N)1,i (t), t ∈ [0, T ]

)

by the strong law of large numbers. Here, P(N)1,i (t) are different

paths which have been generated according to the aboveprocedure.

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Advantages/Disadvantages of the Monte-Carlo method

Advantages

Easy to implement.

Reasonable number generators are available.

Every exotic option can be approximated.

Refinements to obtain faster convergence are available.

Disadvantages

Method is time-consuming (n and N have to be very large).

Frequently n and N have to be so large that the whole reservoir ofpseudo-random numbers is used and an already used sequenceof pseudo-random numbers has to be reused (→ independenceassumption of the different simulations is no longer true).

Strong dependence of the method on the quality of the randomnumber generator.

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Outline


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The basic idea

Description of the idea

Monte-Carlo simulation relies on the strong law of large numbers.

The approximation method via binomial trees can be motivated bythe central limit theorem.

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Cox-Ross-Rubinstein model

Stock price process: P(n)1 (i), i = 0, 1, . . . , n

Possible paths given by binomial tree→ price process starts in p at time t = 0→ in each node the price P(n)

1 (i) can increaseby the factor u with probability q orby the factor d with probability (1 − q) (where d < u).

The probability of a price increase by the factor u and the possiblevalues of the relative price change

P(n)1 (i)

P(n)1 (i − 1)

are the same in each node.The factors u and d satisfy

d < er∆t < u with ∆t :=Tn

.

→ avoid riskless gains311 / 477

Binomial tree

t = 0 1 · T/n 2 · T/n T ”time”

unpppppppppppppppp

NNNNNNNNNNNNNN

u2p un−1dp

up

mmmmmmmmmmmmmmmm

QQQQQQQQQQQQQQQQ

p

nnnnnnnnnnnnnnnnn

PPPPPPPPPPPPPPPPP udp

dp

mmmmmmmmmmmmmmmm

QQQQQQQQQQQQQQQQ

d2p udn−1ppppppppppppppp

NNNNNNNNNNNNNNN

dnp ”price”

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Number of "up-moves" of P(n)1 (i): Xn

Properties:Xn ∼ B(n, q)

P(n)1 (n) = p · uXn · dn−Xn = p · eXn·ln(u/d)+n·ln(d).

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Special case:

q =12

, b ∈ R

and

u = eb∆t+σ√

∆t , d = eb∆t−σ√

∆t

b =12

ln(u) + ln(d)

∆t, σ =

12

ln(u) − ln(d)

∆t

Convergence:

P(n)1 (n) = p · exp

(b · T + σ

√T(

2Xn − n√n

))

n→∞−−−→ p · exp(

b · T + σ · W (T )

)= P1(T ) in distribution

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Note that2Xn − n√

n

has zero mean and variance 1.2Xn is the sum of n independent, double Bernoulli variables.Important: Convergence of the discrete to the continuous priceprocess.Discounted expected payoff of the option in the discrete modelcan be computed easily.With increasing degree of fineness of the time discretization, thediscounted expected final payment in the discrete-time model willconverge to that of the continuous-time model if the family

Bn := B(

P(n)1 (i), i = 0, 1, . . . , n

)

is uniformly integrable.315 / 477

Approximation via binomial trees

Algorithm(1) For n ≫ 1 set up a suitable binomial tree for the price process

P(n)1 (i) in discrete time.

(2) Compute the discounted expected payoff E (n)(e−rtBn

)in the

discrete-time model as an approximation for EQ(e−rtB

).

Remark:

The choice of n (i.e., the fineness of the (space and) time discretization)is the essential factor for the accuracy of the approximation. Therefore,the algorithm is performed iteratively for different (increasing) values ofn and will be stopped if the sequence of approximations for the optionprice converges.

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Choice of parameters in the binomial trees

Independence of the identically distributed increments

P(n)1 (i)

P(n)1 (i − 1)

in the binomial tree implies weak convergence of

P(n)

1 (i), i = 0, 1, . . . , n

to

P1(t), t ∈ [0, T ]

if the first two moments of the logarithm of increments

ln(

P(n)1 (i)

P(n)1 (i − 1)

)and ln

(P1(i · T

n )

P1((i − 1) · T

n

))

of the discrete and continuous price processes coincide at times i · Tn .

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If we define a continuous process P(s,n)1 (t) via linear interpolation

betweenln(

P(n)1 (i − 1)

)and ln

(P(n)

1 (i))

i.e.,

ln(

P(s,n)1 (t)

)= ln

(P(n)

1 (i − 1))

+(t − (i − 1) · T

n

)· n

T

·[ln(

P(s,n)1 (i)

)− ln

(P(n)

1 (i − 1))]

for t ∈[(i − 1) · T

n , i · Tn

],

then this continuous process converges weakly to the stochastic priceprocess P1(t) if the moment conditions hold.

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Consider the price process with respect to the equivalentmartingale measure Q and assume that

P1(t)P0(t)

is a martingale.

Binomial measure with respect to time discretization n: Q(n).

Expectation with respect to that measure: E (n).F (n)

i

i∈0,1,...,n filtration generated by the price process

P(n)

1 (i)

i∈0,1,...,n.

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Moment conditions:

(r − 1

2σ2)∆t = EQ

(ln(

P1(∆t)P1(0)

))= E (n)

(ln(

P(n)1 (1)

P(n)1 (0)

))

= ln(u) · q + ln(d) · (1 − q),

(r − 1

2σ2)2(∆t)2 + σ2∆t = EQ

(ln(

P1(∆t)P1(0)

)2)

= E (n)

(ln(

P(n)1 (1)

P(n)1 (0)

)2)

= ln(u)2 · q + ln(d)2 · (1 − q).

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Moment conditions:

(r − 1

2σ2)∆t = EQ

(ln(

P1(∆t)P1(0)

))= E (n)

(ln(

P(n)1 (1)

P(n)1 (0)

))

= ln(u) · q + ln(d) · (1 − q),

(r − 1

2σ2)2(∆t)2 + σ2∆t = EQ

(ln(

P1(∆t)P1(0)

)2)

= E (n)

(ln(

P(n)1 (1)

P(n)1 (0)

)2)

= ln(u)2 · q + ln(d)2 · (1 − q).

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Unknown parameters:

u, d "incremental factors",

q "probability of an upwards movement"

Moment conditions allow free choice of one of the parameters if

u, d > 0 and q ∈ (0, 1).

Popular choices:

u =1d

, d < 1 or q =12

.

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Special case: q = 12

Moment conditions:

ln(u · d) = 2(r − 1

2σ2)∆t

ln(u)2 + ln(d)2 = 2(r − 1

2σ2)(∆t)2 + 2σ2∆t

Symmetric in u and d

Ansatz:u = eB+C , d = eB−C .

⇒ B =(r − 1

2σ2)∆t , C = |σ| ·

√∆t .

⇒ u = e(

r−12σ2)∆t+|σ|·

√∆t , d = e

(r−1

2σ2)∆t−|σ|·

√∆t .

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Special case: q = 12

For r > 0 we have

0 < d < u and d < er∆t .

To obtainer∆t < u

we must have|σ| ·

√∆t − 1

2σ2∆t > 0.

Time discretization must be sufficiently fine:

n >T · σ2

4.

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The option price in the binomial model

The stock price model given by a binomial tree together with thepossibility of a bond investment at times i · T

n (with a bond priceP0(t) = ert ) constitute a complete market.

Price of an option = discounted expectation of the payoff B int = T with respect to the unique equivalent martingale measureQn.

Qn is given by the "upwards probability" q = qn.

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For given u and d with

0 < d < er∆t < u

we obtain q from the martingale requirement

0 = EQn

(P(n)

1 (i)

P0(i · T

n

) − P(n)1 (i − 1)

P0((i − 1) · T

n

)∣∣∣∣F

(n)i−1

)

=P(n)

1 (i − 1)

P0((i − 1) · T

n

) ·(

(q · u + (1 − q) · d) e−r Tn − 1

)

as

q =er T

n − du − d

.

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For given u and d with

u = e(

r−12 σ2)∆t+|σ|·

√∆t and d = e

(r−1

2σ2)∆t−|σ|·

√∆t

q differs from 1/2.

The value E (n)(e−rT Bn

)computed as an approximation for the

option priceEQ(e−rT B

)

in the continuous model will in general not coincide with the optionprice

EQn

(e−rT Bn

)

in the binomial model.

327 / 477


The use of binomial trees serves us only as a method ofnumerical approximation of the expectation EQ

(e−rT B

).

The fact that this expectation is an option price has no meaning forthe numerical method.

There is no reason why the approximating sequence for thisexpectation should be option prices in some discretized models.

328 / 477


Another method of approximation of EQ(e−rT B

)is to determine

q, u, d via requiring the equality of the first two moments of theincrements of the discrete- and the continuous-time price process.

Equality of the first moment of the increments with theindependence and identical distribution of all increments in thebinomial model imply the martingale condition.

Choose u, d such that the second moment of the increments ofboth price models coincide.

The option price in the binomial model (given by n, q, u, d ) iscomputed as an approximation for the option price in thecontinuous model.

329 / 477


The concept

"approximate the option price in the continuous model by thediscrete-time model option price"

cannot be justified by weak convergence arguments.

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Computation of the expected discounted payoff in thebinomial tree

The possibility of an efficient calculation of the expectationE (n)

(e−rT Bn

)depends on the type of the functional B

(resp. its discretized variant Bn).

Two examples for n = 2:

European call

Double-barrier knockout call

331 / 477


The European call

Choose q = 12 and

market parameters r = 0, σ = 0.5, T = 2, p = 1.

332 / 477

Computation of the expected discounted payoff in thebinomial treeBinomial tree for P(2)

1 (i):

t = 0 1 2 ”time”

2.117

1.455

1/2nnnnnnnnnnnnn

1/2 PPPPPPPPPPPPP

1

1/2nnnnnnnnnnnnnnn

1/2 PPPPPPPPPPPPPPP 0.779

0.535

1/2nnnnnnnnnnnnn

1/2 PPPPPPPPPPPPP

0.287 ”price”

333 / 477


The European call

Aim: Value a European option with a payoff of the form

B = f (P1(T )).

Approximate its value by the two-period binomial tree and thediscretized variant of the payoff

B2 = f(P(2)

1 (2)).

Discounted expected payoff (discretized model)

E (2)(B2) =12

(12

[f (2.117) + f (0.779)

])

+12

(12

[f (0.779) + f (0.287)

])

334 / 477


European call option with strike K = 0.5

Approximate value

E (2)(B2) =14· 1.617 +

12· 0.279 = 0.54375

Black-Scholes value 0.5416

335 / 477


Backwards induction principle

Payoff:B = f

(P1(T )

).

LetBn = f

(P(n)

1 (n))

and

V (n)(

i · Tn

, P(n)1 (i)

):= E (n)

(e−r(

T−i ·Tn)· Bn

∣∣∣P(n)1 (i)

)

be the expected payoff in t = T discounted back to t = i · Tn , if the

stock price in the binomial model at time t = i · Tn attains the value

P(n)1 (i).

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Expected discounted payoff of the option in the binomial model

V (n)(

T , P(n)1 (n)

)= f(P(n)

1 (n)),

V (n)(

i · Tn

, P(n)1 (i)

)

=12

[V (n)

((i + 1) · T

n, uP(n)

1 (i))

+ V (n)((i + 1) · T

n, dP(n)

1 (i))]

· e−r Tn , for i = n − 1, . . . , 0

E (n)(

e−rT Bn

)= V (n)(0, p).

337 / 477


Advantage of the induction scheme:

Calculate onlyn · (n − 1)

2arithmetic means (although the stock price can follow 2n differentpaths).

Binomial tree = recombining tree

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Backwards induction principle (modification)

How about path-dependent options?

In case of path-dependent options path dependency has to be takeninto account

⇒ Modification of backward induction recursion.

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Example: Double-barrier knockout call

Payoff:

BCallDB = (P1(T ) − 0.5)+ · 1

P1(t)∈[0.4, 1.4] for all t∈[0,T ]

Payoff (discrete):(

BCallDB

)2

=(P(2)

1 (2) − 0.5)+ · 1

P(2)1 (i)∈[0.4, 1.4], i=0,1,2

= 0.279 · 1P(2)

1 (2)=0.779, P(2)1 (1)=0.535

Double-barrier knockout call price (discrete approximation):

E (2)(B2) =14· 0.279 = 0.06975.

340 / 477



The final payoff(

BCallDB

)2

at T = 2 in the state P(2)1 (2) = 0.779 can

attain two possible values 0 and 0.279→ typical for path-dependent options.

Path-dependent final payoff can lead to a situation that each pathof the stock price yields a different final payment.

Maximum number of different values of final payments is 2n

→ complexity of computations and storage.

341 / 477

Backwards induction principle (modification)A non-recombining tree:

t = 0 1 2 ”time”

2.117

1.455

1/2nnnnnnnnnnnnn

1/2 PPPPPPPPPPPPP

0.779

1

1/2

1/2

AAAA

AAAA

AAAA

AAAA

AAAA

0.779

0.535

1/2nnnnnnnnnnnnn

1/2 PPPPPPPPPPPPP

0.287 ”price”

342 / 477



Simplifications to keep backward induction algorithm effici ent:

The binomial tree which corresponds to the above computation of

E (2)(

BCallDB

)2

has the usual recombining form of a binomial tree.

The option prices in the node P(2)1 (2) = 0.779 are not unique.

The payoff depends on the paths reaching this node

343 / 477


Principle of backward induction stays valid.

For a general path-dependent option we have to calculate up to2i−1 arithmetic means at time i (i + 1 in the non-path-dependentcase).

In the double-barrier knockout case, we can simply proceedbackwards in the recombining tree, but setting the option price tozero in all nodes where the knockout-condition is satisfied).

Computational complexity is comparable to that of a Europeannon-path-dependent option.

344 / 477


In other cases such as

an average option with final payment of

BCallAv =

(1T

T∫

0

P1(t) dt − K)+

respectively its discrete variant

(BCall

Av

)n

=

(1

n + 1

n∑

i=0

P(n)1 (i) − K

)+

the full non-recombining tree has to be considered for theapproximative calculation of the option price.

345 / 477

Convergence of the model

If the moment conditions are satisfied the process P(s,n)1 (t),

obtained from P(n)1 (i) by linear interpolation, converges weakly to

the process P1(t).

If the family

Bs,n := B(

P(s,n)1 (t), t ∈ [0, T ]

)

is uniformly integrable then we obtain the convergence

E (n)(e−rtBs,n

) n→∞−−−→ EQ(e−rT B

).

Q(n) is defined on the paths of P(s,n)1 (t) by identifying them with

the corresponding paths of P(n)1 (i).

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We obtainE (n)

(e−rtBn

) n→∞−−−→ EQ(e−rT B

),

iflim

n→∞E (n)

(e−rt(Bs,n − Bn)

)= 0.

Remark: The latter convergence has to be proved for each type ofoption explicitly. It is satisfied if (Bs,n − Bn) converges to zero uniformly.

It is fulfilled for European lookbacks, barrier and double barrier options,and Asian options.

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Advantages/Disadvantages of the method

Advantages

Easy to implement (but their efficiency can depend strongly on theoption type).

Binomial methods converge faster than Monte-Carlo-simulation.

Hybrid method via combination of binomial tree and Monte-Carlomethod for very big binomial trees available.

Disadvantages

Slow and irregular convergence behaviour for double barrieroptions.

Accuracy of approximation does not necessarily increase withfineness n.

348 / 477

Trinomial trees and explicit finite-difference methods

Option prices can be obtained as solutions of correspondingCauchy problems under certain assumptions.

In particular for options with a final payment of the form

B = f (P1(T )).

If the Cauchy problem does not admit an explicit solution we mustsolve it numerically.

Numerical methods: Connection between stochastic methods andPDE methods similar to Feynman-Kac theorem.

349 / 477


Approximation of the Black-Scholes model by a recombiningtrinomial tree

Discrete-time stock price process P(n)1 (i), i = 0, 1, . . . , n,

with possible paths in a trinomial tree.

Assume1u

< er · Tn < u.

Probability for an upwards movement: q1

Probability for an downwards movement: q2

Probability for the stock price to rest at the same level:

q3 = 1 − (q1 + q2).

350 / 477

Trinomial trees and explicit finite-difference methodsTrinomial tree:

t = 0 1 · T/n 2 · T/n T ”time”

unp

u2p

rrrrrrrrrrrrrr un−1p

up

nnnnnnnnnnnnnnnnn

PPPPPPPPPPPPPPPPPP up up

p

ooooooooooooooooo

OOOOOOOOOOOOOOOOO p

nnnnnnnnnnnnnnnnnn

PPPPPPPPPPPPPPPPPPP p

1u p

nnnnnnnnnnnnnnnnnn

PPPPPPPPPPPPPPPP

1u2 p

rrrrrrrrrrrrrrrr

LLLLLLLLLLLLLL 1un−1 p

1un p ”price”

351 / 477



Assume 0 < q1, q2 < 1, q1 + q2 ≤ 1.

q1 + q2 = 1 ⇒ binomial situationAssume: q1 + q2 < 1.

Donsker’s theorem ⇒ weak convergence of P(n)1 (i) to the stock

price process (in risk-neutral market)

P1(t) = p · exp((

r − 12

σ2)t + σW (t)),

if the first two moments of the increments of ln(P1(t)) between(i − 1) · T

n and i · Tn

ln(

P1(i · T

n

)

P1((i − 1) · T

n

))

coincide with the corresponding increments of ln(P(n)1 (i)).

352 / 477



This leads to

(r − 1

2σ2)∆t = ln(u) · q1 + ln

(1u

)· q2,

(r − 1

2σ2)2

(∆t)2 + σ2∆t = ln(u)2 · q1 + ln(

1u

)2

· q2.

For given u > 0 we can determine q1, q2 (compare to the binomialmodel).

353 / 477



Method of Cox-Ross-Rubinstein:

u = eλσ√

∆t := e∆x

for some λ ∈ [1,∞) (neglect terms of higher order than ∆t above).

Error negligible for small ∆t .

λσ√

∆t(q1 − q2) =(r − 1

2σ2)∆t

λ2σ2∆t(q1 + q2) = σ2∆t .

354 / 477



Solutions:

q1 =12

((r − 1

2σ2) 1

λσ

√∆t +

1λ2

),

q2 =12

(1λ2 −

(r − 1

2σ2) 1

λσ

√∆t)

.

q1, q2, q3 ∈ (0, 1) for small ∆t (i.e., large n).

λ = 1 ⇒ binomial model.

355 / 477


Algorithm: Approximation by trinomial trees(1) For n ≫ 1 set up a suitable trinomial tree for the discrete-time

price process P(n)1 (i).

(2) Compute the expected discounted final payment E (n)(e−rT Bn

)

in the discrete-time model as an approximation for EQ(e−rT B

).

356 / 477

Computation of E (n)(e−rT Bn

)

Backward induction

LetX (n)

1 (i) := ln(P(n)

1 (i)), i = 0, 1, . . . , n,

and

V (n)(

i · ∆t , X (n)1 (i)

):= E (n)

(e−r(

T−i ·∆t)· Bn

∣∣∣P(n)1 (i)

)

357 / 477

Computation of E (n)(e−rT Bn

)

Backward induction

Compute recursively

V (n)(

T , X (n)1 (n)

)= f(

exp(X (n)

1 (n)))

,

V (n)(

i · ∆t , X (n)1 (i)

)

=

[q1V (n)

((i + 1) · ∆t , X (n)

1 (i) + ∆x)

+ q3V (n)((i + 1) · ∆t , X (n)

1 (i))

+ q2V (n)((i + 1) · ∆t , X (n)

1 (i) − ∆x)]

· e−r∆t ,

for i = n − 1, . . . , 0

E (n)(

e−rT Bn

)= V (n)(0, p).

358 / 477

The option price in the trinomial model

In general the final payment of a European call cannot bereplicated by a trading strategy in stock and bond in the trinomialmodel.

For this model exists a whole family of equivalent martingalemeasures.

The alternative method

"compute the option price in an approximating model"

cannot be performed without further modifications.

Until now we have not developed a method to compute an optionprice in incomplete markets.

359 / 477

Relations between trinomial trees and explicitfinite-difference methods

The option price solves the Cauchy problem

Vt +12

σ2p2Vpp + rpVp − rV = 0, (t , p) ∈ [0, T ] × (0,∞)

V (T , p) = f (p), p > 0.

Substitution:x = ln(p).

Notation:V (t , x) := V (t , p).

Transformed problem:

Vt +12σ2Vxx +

(r − 1

2σ2)Vx − r V = 0, (t , p) ∈ [0, T ] × R

V (T , x) = f (ex), x ∈ R.

360 / 477

Explicit finite-difference method

Time discretization: 0, ∆t , 2∆t , . . . , T

Space discretization: ln(p1), ln(p1) ± ∆x , ln(p1) ± 2∆x , . . .

∆t V (n)(t , x) :=V (n)(t + ∆t , x) − V (n)(t , x)

∆t,

∆x V (n)(t , x) :=V (n)(t + ∆t , x + ∆x) − V (n)(t + ∆t , x − ∆x)

2∆x,

∆xx V (n)(t , x)

:=V (n)(t + ∆t , x + ∆x) − 2V (n)(t + ∆t , x) + V (n)(t + ∆t , x − ∆x)

(∆x)2 .

361 / 477


Notation:

ti := i · ∆t , i = 0, 1, . . . , n,

X (j) := ln(p1) + j · ∆x , j ∈ Z.

V (n)(ti , X (j))

=1

1 + r∆t

(12

σ2 ∆t(∆x)2 +

12

(r − σ2

2

)∆t∆x

)V (n)(ti + ∆t , X (j) + ∆x)

+

(1 − σ2 ∆t

(∆x)2

)V (n)(ti + ∆t , X (j))

+

(12σ2 ∆t

(∆x)2 +12

(r − σ2

2

)∆t∆x

)V (n)(ti + ∆t , X (j) − ∆x)

.

362 / 477


At time T we know all values of V (n)(T , x).We obtain

V (n)(T − ∆t , X (j))

from the explicit representation.Via backward induction with step size ∆t we reach the startingtime t = 0 in n steps and obtain

V (n)(0, x)

as an approximation for the option price V (0, x).The recursion in the trinomial tree can be seen as a finitedifference scheme.V (n)(0, x) converges to V (0, x) if the stability condition

0 <∆t

(∆x)2 ≤ 1σ2

is satisfied.363 / 477

Outline


364 / 477

The pathwise binomial approach of Rogers andStapleton

Description of the basic idea

Binomial method: Only the distribution of P1(t) is approximated bya simpler, discrete distribution.

Method of Rogers and Stapleton: Approximate each single path ofP1(t) by a step function.

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Description of the basic idea (continued)

The step function is only allowed to attain values in a givendiscrete set and should at most deviate by a given accuracy ε fromthe corresponding path of P1(t).

Idea: Interprete the set of all paths of a step function as an infinitebinomial tree.

Compute the (approximate) discounted expected payment of anoption in the infinite tree as an approximation for the option pricein the Black-Scholes model.

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Algorithm: Pathwise binomial approach of Rogers and Stapleton

(1) For a given accuracy ∆y and starting point y = ln(p1) set up aninfinite binomial tree.

(2) Compute the discounted payoff E (∆y)(e−rT B∆y

)of the option in

the infinite binomial tree as an approximation for EQ(e−rT B

).

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Construction of the infinite binomial tree

i) Approximation

Logarithm of the stock price:

Y (t) = ln(P1(t)) = ln(p1)︸︷︷︸=:y

+σ · W (t) +(

r − 12σ2)· t .

For a given accuracy ∆y > 0 and for each ω ∈ Ω, t ∈ [0, T ], wedefine an approximating step function Z (t) via

τ0(ω) := 0,

τn(ω) := inf

t ∈ [0, T ] | t > τn−1(ω),

|Y (t , ω) − Y (τn−1(ω), ω)| > ∆y, n = 1, 2, . . . ,

ξ0(ω) := y ,

ξn(ω) := Y (τn, ω),

Z (t , ω) :=∞∑

n=0

ξn(ω) · 1[τn,τn+1)(t).

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i) Approximation (continued)

This means: as soon as Y (t) deviates from the current value ofthe step function Z (t) by ∆y , the step function will be set atexactly this value of Y (t).

By construction of Z (t) we have

sup0≤t≤T

|Y (t) − Z (t)| ≤ ∆y .

For given y and ∆y the step function Z (t) only attains values inthe set

y ± i∆y | i ∈ N.

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i) Approximation (continued)

Z (t) can only jump to adjacent values Z (t) ± ∆y

y−2∆

y−∆

y

y+∆

y+2∆

τ1

τ2

τ3 τ

4τ

5τ

6τ

7

Y(t),Z(t)

t

370 / 477


i) Approximation (continued)There is no a priori upper bound for the number of values thatZ (t , ω) can attain on [0, T ].

⇒ Identify the sequence of values of the step function Z (t , ω)on [0, T ] with a finite path in the infinite binomial tree.

y + 3∆y

y + 2∆yjjjj

TTTTT

y + ∆yjjjj

TTTTTTTTy + ∆y

yooooo

OOOOO yjjjjjjjjj

TTTTTTTTT . . .

y − ∆y

jjjjjjjj

TTTTy − ∆y

y − 2∆y

jjjjj

TTTT

y − 3∆y371 / 477


ii) Computation of the transition probabilities

Example: Double-barrier knockout call for Y (t) = ln(P1(t)).

Final option payment:

B = (P1(T ) − K )+ · 1ln(P1(t))∈(b∗ ,b∗) for all t∈[0,T ],

where K ≥ 0 is the strike price.

The real numbers b∗ < y < b∗ define the interval in which Y (t)has to stay so that the call is still valid in t = T .

If Y (t) leaves the interval (b∗, b∗) before T then the option runsout worthless.

372 / 477


ii) Computation of the transition probabilities (continue d)

Price of the call:

xB = EQ

(e−r(T )(P1(T ) − K )+ · 1ln(P1(t))∈(b∗ ,b∗) for all t∈[0,T ]

).

Calculate the price of the call approximately with the infinitebinomial tree.

373 / 477


ii) Computation of the transition probabilities (continue d)

Decompose the infinite tree in finite subtrees.

For a fixed n ∈ N the possible paths of the step function Z (t)containing exactly n jumps on [0, T ] will be identified with ann-period binomial tree.

Calculate the expected discounted payment of the option in thisfinite tree if the transition probabilities from a node to itssuccessors in the tree are determined.

As we have coincidence of the values of Z (t) and Y (t) in both thetimes τn−1 and τn, the transition probabilities in the tree coincidewith those of Y (t) to Y (t) ± ∆y .

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TheoremFor a given ∆y > 0 the probability for an upwards movement of Z (t) inτn, n ∈ N, is given by

q =s(0) − s(−∆y)

s(∆y) − s(−∆y)

with

s(x) = −e−2cx and c =r − 1

2σ2

σ2 .

The probability for a downwards movement of Z (t) in τn, n ∈ N, is

(1 − q).

375 / 477

Continuation: Double-barrier knockout call

For the double-barrier knockout call all paths in the tree whichexceed b∗ by below and b∗ by above have zero value.

If the process Z (t) differs from b∗ or b∗ by a value less than ∆ywe have to take extra care.

To prevent the option being knocked out by the Y (.)-processbefore it is knocked out by the Z (.)-process, we must modify thedefinition of the Z (.)-process.

376 / 477


Y (t) can reach the boundary value b∗ or b∗ (knocking out theoption), but it is possible that Y (t) never reaches a value thatwould cause Z (t) to jump again.

To avoid this, choose the corresponding node in the tree such thatit is exactly b∗ or b∗.

The step function Z (t) jumps exactly when b∗ or b∗ is reached andnot when Z (t) − ∆y or Z (t) + ∆y is reached.

377 / 477


Consequences of the modification of Z (t):

Z (t) only attains values in the modified binomial tree.

The final payment of the double-barrier knockout call in themodified binomial tree is given by

B∆y =(

eZ (t) − K)+

· 1Z (t)∈(b∗,b∗) for all t∈[0,T ].

Z (t) reaches one of the barriers b∗ or b∗ if and only if Y (t)reaches the same barrier, i.e.,

1Z (t)∈(b∗,b∗) for all t∈[0,T ] = 1Y (t)∈(b∗ ,b∗) for all t∈[0,T ]

The option is knocked out before T in the original model if it isknocked out before T in the modified binomial model.

378 / 477


For the pricing of the knockout call, it plays no role if we defineZ (t) to be constant after reaching b∗ or b∗ or extend it as it wasoriginally defined.

It is important for the pricing purpose that the transitionprobabilities in the modified tree change for

Z (t) ∈ (b∗ − ∆y , b∗) or Z (t) ∈ (b∗, b∗ + ∆y).

379 / 477


Theorem(1) If we have Y (τn) = y∗ with y∗ ∈ (b∗ − ∆y , b∗) then we obtain

q∗ = P(Z (τn+1) = y∗ − ∆y |Z (τn) = y∗) =

s(b∗) − s(y∗)s(b∗) − s(y∗ − ∆y)

.

Z (τn+1) reaches with probability 1 − q∗ the value b∗ and theoption runs out worthless.

(2) In case of Y (τn) = y∗ with y∗ ∈ (b∗, b∗ + ∆y), we obtain

q∗ = P(Z (τn+1) = y∗ + ∆y |Z (τn) = y∗

)=

s(y∗) − s(b∗)s(y∗ + ∆y) − s(b∗)

.

Z (τn+1) reaches with probability 1− q∗ the value b∗ and the optionruns out worthless.

380 / 477


PropositionLet Ψ(k , y) be the expected final payment of the option in the binomialmodel for an initial value of Z (0) = y ∈ (b∗, b∗) and a given numberk ∈ N ∪ 0 of upwards and downwards movements of Z (t) on [0, T ].Then Ψ(k , y) can be computed inductively according to the followingalgorithm

Ψ(0, y) = (ey − K )+

Ψ(n + 1, y) = q(y) · Ψ(n, y + ∆y) + q(y) · Ψ(n, y − ∆y), n = 0, 1, 2, . . .

381 / 477


Proposition (continued)Here the probabilities q(y) for y ∈ (b∗ + ∆y , b∗ − ∆y) are given by

q =s(0) − s(−∆y)

s(∆y) − s(−∆y)

with

s(x) = −e−2cx and c =r − 1

2σ2

σ2

and we haveq(y) = 1 − q(y).

382 / 477


Proposition (continued)For y ∈ (b∗, b∗ + ∆y) we have

q(y) = 0

and

q(y) =s(y) − s(b∗)

s(y + ∆y) − s(b∗).

For y ∈ (b∗ − ∆y , b∗) we have

q(y) = 0

and

q(y) =s(b∗) − s(y )

s(b∗) − s(y − ∆y).

383 / 477


Discounted expected final payment of the option in the modifiedbinomial tree:

E (∆y)(e−rT B∆y

)=

∞∑

n=0

P(v = n) · Ψ(n, y) · e−rT .

For the computation we need the probability distribution of thesum v of upwards and downwards movements of the stock pricein the binomial model.

Due toω | v(ω) ≥ n = ω | τn(ω) ≤ T

this distribution can be obtained from that of τn:

P(v = n) = P(v ≤ n) − P(v ≤ n − 1) = P(τn ≥ T ) − P(τn−1 ≥ T ).

384 / 477


Theorem(1) The random variables τn+1 − τnn∈N∪0 are independent and

identically distributed. Their Laplace transform ϕ(λ) is given by

ϕ(λ) = E(e−λτ1

)=

cosh(µσ−2∆y

)

cosh(γ∆y

)

with

µ = r − 12

σ2, γ =

√µ2 + 2λσ2

σ2 , λ > 0.

(2)E(τ1) =

∆yµ

· tanh(

µ

σ2 · ∆y)

for µ 6= 0,

E(τ21 ) = 2(E(τ1))

2 +σ2∆y

µ3 · tanh(

µ

σ2 ∆y)−(

∆yµ

)2

for µ 6= 0.

(3) τn+1 − τn is independent of ξn+1.

385 / 477


We have

τn =

n∑

i=1

(τi − τi−1).

Summands are independent and identically distributed.

Central limit theorem ⇒τn − n · E(τ1)√

n · Var(τ1)

n→∞−−−→ N(0, 1) in distribution.

Determine the distribution of τn approximately for large n.

For small n this approximation is not accurate enough.

386 / 477


TheoremWe have

P(

τn − n · E(τ1)√n · Var(τ1)

≤ x)

= Φ(x) +α3(1 − x2)e− x2

2√72πn

+ o(n−1

2)

with

α3 = E((

τ1 − E(τ1)√Var(τ1)

)3),

where Φ is the distribution function of the standard normal distribution.

387 / 477


Use Laplace transform ϕ(λ) to calculate α3:

α3 =∆y · (A + B − C)

(µσ2

)5σ6(s(∆y) − 1

)3 ,

A = 12 · µ

σ2 ∆y(s(2∆y) + s(∆y)

),

B = 8 ·( µ

σ2

)2(∆y)2(s(∆y) − s(2∆y)

),

C = 3 ·(1 + s(∆y) − s(2∆y) − s(3∆y)

).

388 / 477

Method of Rogers and Stapleton

Algorithm: Method of Rogers and Stapleton

(1) For a given initial value of y = ln(P1(0)) and a given accuracy of∆y , compute "all" values of

Ψ(k , y), k ∈ N ∪ 0.

(2) Compute P(v = n) = P(τn ≥ T ) − P(τn−1 ≥ T ) approximatelyfrom the distribution of τnn (by neglecting the o

(n−1/2

)-terms).

(3) Determine

E (∆y)(e−rT B∆y

)=

∞∑

n=0

P(v = n) · Ψ(n, y) · e−rT

as an approximation for EQ(e−rT B

).

389 / 477


As for fixed ∆y > 0 we have

sup0≤t≤T

|Y (t) − Z (t)| < ∆y

we get uniform convergence of Z (t) to Y (t) for ∆y → 0.

⇒ Estimates for the approximation error(depending on the type of option).

390 / 477


Error estimate for the double-barrier knockout call(b∗ > ln(K ) > b∗)

∣∣∣BCallDB − B∆y

∣∣∣

=

∣∣∣∣∣

(eY (T ) − K

)+

−(

eZ (T ) − K)+∣∣∣∣∣ · 1Y (t)∈(b∗ ,b∗) for all t∈[0,T ]

≤∣∣∣∣∣

(eY (T ) − K

)+

−(

eZ (T ) − K)+∣∣∣∣∣ · 1Y (T )∈(b∗,b∗)

≤ max

maxY (T )∈[ln(K ),b∗)

∣∣∣∣∣

(eY (T ) − K

)+

−(

eZ (T ) − K)+∣∣∣∣∣ ,

maxY (T )∈(b∗,ln(K )]

∣∣∣∣∣

(eY (T ) − K

)+

−(

eZ (T ) − K)+∣∣∣∣∣

. . .

391 / 477


Error estimate for the double-barrier knockout call(b∗ > ln(K ) > b∗)

∣∣∣BCallDB − B∆y

∣∣∣ ≤ . . .

≤ max(

eb∗ − eb∗−∆y)

, K · e∆y − K

= max

eb∗

(1 − e−∆y), K (e∆y − 1)

392 / 477

Advantages/Disadvantages of the method

Advantages

Paths of the approximating process Z (t) converge uniformly to thepaths Y (t) of the logarithm of the price process.

Explicit estimate of the convergence error.

Flexibility which allows for a choice of the nodes in the binomialtree ensuring that in the double-barrier knockout call case theoption runs out worthless in the modified binomial tree if and onlyif it runs out worthless in the Black-Scholes model.

Disadvantages

Concept requires a deeper understanding as in the case of thebinomial model.

Bigger computational complexity.

393 / 477

Outline

5 Optimal Portfolios

394 / 477

Outline


395 / 477

Introduction

Until now

Trading strategies that generate a given payoff profile (replication)or a lower bound for a payment (hedging strategy).

Costs for replication strategy determined the price of the payoffprofile.

Now

Given a fixed initial capital and search for an admissibleself-financing pair of portfolio and consumption processes whichyields a payment stream as lucrative as possible.

396 / 477

Formulation of the portfolio problem

The continuous-time portfolio problem

Initial capital x > 0.

Determine an optimal consumption and investment strategy.

The investor has to determine

how many shares of which security he has to hold at which time instant

and

how much of his wealth he is allowed to consume

to maximize

his utility from consumption during the period [0, T ] and/orfrom the terminal wealth at the time horizon t = T .

397 / 477

Formulation of the portfolio problem (continued)

The continuous-time portfolio problem

The portfolio problem contains

a choice problem ("which" security)

a problem of volumes ("how many" shares, "how much",...)

a dynamic component with respect to time ("which time").

The investor

can decide on his actions at each time instant t ∈ [0, T ]

has only the information of past and present prices

should not have any knowledge of future security prices or insiderinformation.

398 / 477

General assumptions

General assumptions for this section(Ω,F , P) be a complete probability space,(W (t),Ft )t∈[0,∞) m-dimensional Brownian motion.

Dynamics of bond and stock prices:

P0(t) = p0 · exp( t∫

0

r(s) ds)

bond


0

(bi(s) − 1

2

m∑

j=1

σ2ij (s)

)ds

+

m∑

j=1

t∫

0

σij(s) dWj(s)

)stock

for t ∈ [0, T ], T > 0, i = 1, . . . , d .

399 / 477

General assumptions (continued)

General assumptions for this section (continued)

r(t), b(t) = (b1(t), . . . , bd (t))T , σ(t) = (σij(t))ij

progressively measurable processes with respect to Ftt ,component-wise uniformly bounded in (t , ω).

σ(t)σ(t)T uniformly positive definite,i.e., it exists K > 0 with

xT σ(t)σ(t)T x ≥ KxT x

for all x ∈ Rd and all t ∈ [0, T ] P-a.s.

(π, c) is a self-financing pair consisting of a portfolio process π anda consumption process c to be admissible with initial wealth x > 0:

(π, c) ∈ A(x).

400 / 477

Solution approaches (continuous-time market model)

Two approaches:

Stochastic control approach (Robert/Merton)

Martingale method (Cox/Huang, Karatzas/Lehoczky/Shreve,Pliska)

401 / 477

Formulation of the problem

Introduce a functional J which measures the utility of a paymentstream (large value = good payment stream).

For a given initial wealth x > 0 an investor looks for aself-financing pair (π, c) ∈ A(x) which maximizes the expectedutility from consumption and / or terminal wealth

J(x ;π, c) = E( T∫

0

U1(t , c(t)

)dt + U2

(X (T )

))

Notation:

X (t) wealth corresponding to x and (π, c).U1, U2 utility functions.

402 / 477

Utility functions

Definition(1) Let U : (0,∞) → R be a strictly concave and continuously

differentiable function with

U ′(0) := limx↓0

U ′(x) = +∞, U ′(∞) := limx→∞

U ′(x) = 0.

Then U is called a utility function.

(2) A continuous function U : [0, T ] × (0,∞) → R such that for allt ∈ [0, T ] the function U(t , ·) is a utility function in the sense of (1)is called a utility function.

403 / 477

Examples of utility functions

Examples :

(1) U(x) = ln(x)

(2) U(x) =√

x

(3) U(x) = xα for 0 < α < 1

(4) U(t , x) = e−ρt · U1(x), ρ > 0 with U1 as in (1) or (2).

404 / 477

Utility functions

RemarkA utility function is strictly increasing

⇒ each additional wealth leads to additional utility.

A utility function is strictly concave

⇒ U ′(x) is strictly decreasing

⇒ decreasing marginal utility

⇒ the gain of utility from one additional unit of moneydecreases with increasing x .

405 / 477

Utility functions

Remark (continued)The marginal utility in x = 0 is infinite"a tiny bit is much more than nothing"

Vanishing marginal utility in x = ∞ models a saturation effect.

A wider class of utility functions can be considered. In particularthe (popular but criticized) quadratic utility function

U(x) = −12(x − a)2.

406 / 477

Formulation of the problem

RemarkFor an arbitrary (π, c) ∈ A(x) the expectation in J(x ;π, c) is notnecessarily defined.

We can restrict the class of self-financing pairs (π, c) to all thosefor which the expectation in (π, c) ∈ A(x) is finite.

We require only a weak integrability condition for a feasible pair(π, c).

407 / 477

Continuous-time portfolio problem


max(π,c)∈A′(x)

J(x ;π, c)

with

A′(x) =

(π, c) ∈ A(x)

∣∣∣∣E( T∫

0

U1(t , c(t)

)− dt + U2(X (T )

)−)

< ∞

408 / 477


Remark(1) By restricting to the set A′(x), the integral in J is defined.

The expectation exists but can be equal to infinity.

(2) In the case of positive utility functions, U1(t , ·) > 0 andU2(t , ·) > 0, the equality A(x) = A′(x) is satisfied.

409 / 477

Outline


410 / 477

General assumption / notation→ completeness of the market


d = m

Notation

γ(t) := exp(−

t∫

0

r(s) ds)

θ(t) := σ−1(t)(b(t) − r(t) 1

)

Z (t) := exp(−

t∫

0

θ(s)T dW (s) − 12

t∫

0

‖θ(s)‖2 ds)

H(t) := γ(t) · Z (t)

411 / 477

The main idea

The martingale method is based on a separation of the dynamicalproblem into

a static optimization problem(determination of the optimal payoff profile)

a representation problem(compute the portfolio process correspondingto the optimal payoff profile).

412 / 477

Motivation

Portfolio problem without consumption (c ≡ 0, U1 ≡ 0)

The self-financing pair (π, 0) be admissiblefor an initial wealth of x > 0.

By the completeness of the market we have for eachcorresponding wealth process Xπ(T )

E(H(T ) Xπ(T )

)≤ x for T ≥ 0.

The final payment B ≥ 0 be FT -measurable with E(H(T ) B) = x .Example:

B :=x

E(H(T )).

There exists a portfolio process

(π, 0) ∈ A(x) with B = Xπ(T ) P-a.s.

413 / 477

Motivation


Define

B(x) := B ≥ 0 |B FT -measurable,

E(H(T ) B) ≤ x , E(U2(B)−) < ∞.

B(x) represents the set of all final wealths with

E(U2(B)−) < ∞

that can be generated by trading in the securities starting withsome initial wealth y ∈ (0, x ] and satisfying E(U2(B)−) < ∞.

414 / 477

Motivation


For determining the optimal final wealth Xπ(T ) in the

portfolio problem

max(π,0)∈A′(x)

E(U2(Xπ(T )

))

it is enough to maximize over all random variables B ∈ B(x),i.e., it is sufficient to solve the

static optimization problem

maxB∈B(x)

E(U2(B)

)

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Motivation


If B∗ is an optimal final wealth in the static optimization problem,then to solve the portfolio problem we have to solve the

representation problem

Find a (π, 0) ∈ A′(x)

with Xπ(T ) = B∗ P-a.s.

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Lagrangian method

f : Rn → R strictly convex

g: Rn → R

k convexf , g ∈ C1.

x solves the optimization problem

maxx∈Rn

f (x)

subject to g(x) = 0

⇔ there exists a λ such that(x , λ

)∈ R

n+k satisfies

∂

∂xif (x) −

k∑

j=1

λj∂

∂xig(x) = 0, i = 1, . . . , n

gi(x) = 0, i = 1, . . . , k .

417 / 477

Lagrangian method

In words:(x , λ

)∈ R

n+k is a zero of the derivative of the

Lagrangian function

L(x , λ) = f (x) − λT g(x)

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The Lagrangian method for the portfolio problem

Consider the static optimization problem

maxB∈B(x)

E(U2(B)

).

Lagrangian function

L(B, y) := E(U2(B) − y ·

(H(T ) B − x

)), y > 0.

Formally differentiating L with respect to B and yand interchanging the expectation withthe differentiating process yields

0 = LB(B, y) = E(U ′

2(B) − y H(T )),

0 = Ly (B, y) = x − E(H(T ) B

).

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A random variable B satisfying

U ′2(B) − y H(T ) = 0 P-a.s.

solves the first equation.

The range of U ′2(.) equals R

+ and U ′2(.) is strictly decreasing.

Therefore, it can be inverted on R+ and we obtain

B =(U ′

2

)−1(y · H(T )).

Putting this in the second equation yields

0 = x − E(H(T ) ·

(U ′

2

)−1(y · H(T ))

)︸︷︷︸

=:χ(y)

.

If we can solve this equation uniquely for y then we have found acandidate for the optimal final wealth.

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DefineY (u) := χ−1(u), I2 :=

(U ′

2

)−1.

Candidate for the optimal final wealth

B∗ = I2(Y (x) · H(T )

)> 0.

Prove that B∗ is the optimal final wealth.

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Notation

Notation

I2(y) :=(U ′

2

)−1(y) for y ∈ (0,∞)

I1(t , y) :=(U ′

1

)−1(t , y) for y ∈ (0,∞)

χ(y) := E( T∫

0

H(t) I1(t , y ·H(t)

)dt + H(T ) I2

(y · H(T )

))

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Properties of χ(y)

LemmaAssume

χ(y) < ∞ for all y > 0.

Then, χ is continuous on (0,∞), strictly decreasing

and satisfies

χ(0) := limy↓0

χ(y) = ∞

χ(∞) := limy→∞

χ(y) = 0.

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Properties of χ(y)

RemarkThe previous lemma implies the existence of

Y (x) := χ−1(x)

on (0,∞) with

Y (0) := limx↓0

Y (x) = ∞

Y (∞) := limx→∞

Y (x) = 0.

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Utility functions

Lemma

Let U be a utility function with I :=(U ′)−1. Then we have

U(I(y)

)≥ U(x) + y(I(y) − x)

for 0 < y, x < ∞.

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Optimal consumption / optimal terminal wealth

Theorem: Optimal consumption and optimal terminal wealthConsider the portfolio problem. Let x > 0 and χ(y) < ∞ for all y > 0.Set

Y (x) := χ−1(x).

Then, for

B∗ := I2(Y (x) · H(T )

)optimal terminal wealth

c∗(t) := I1(t , Y (x) · H(T )

)optimal consumption

there exists a self-financing portfolio process π∗(t), t ∈ [0, T ], such that(π∗, c∗) ∈ A′(x), X x,π∗,c∗

(T ) = B∗ P-a.s.

and such that(π∗, c∗) solves the portfolio problem. Here, X x,π∗,c∗

(t) isthe wealth process corresponding to the pair

(π∗, c∗) and the initial

wealth x .

426 / 477

Example: Logarithmic utility


U1(t , x) = U2(x) = ln(x)

⇒ I1(t , y) = I2(y) =1y

⇒ χ(y) = E( T∫

0

H(t) · 1y · H(t)

dt + H(T ) · 1y · H(T )

)=

1y

(T + 1)

⇒ Y (x) = χ−1(x) =1x

(T + 1).

427 / 477


Optimal consumption

c∗(t) = I1(t , Y (x) · H(t)

)=

xT + 1

· 1H(t)

Optimal final wealth

B∗ = I2(Y (x) · H(t)

)=

xT + 1

· 1H(T )

428 / 477


Calculation of the portfolio process

We have

H(t) · X x,π∗,c∗

(t) = E( T∫

t

H(s) c∗(s) ds + H(T ) B∗ | Ft

)

= x · T − t + 1T + 1

.

This implies

x = x · T − t + 1T + 1

+ x · tT + 1

= H(t) · X x,π∗,c∗

(t) +

t∫

0

H(s) c∗(s) ds.

429 / 477



Application of Ito’s formula to the product H(t) · X x,π∗,c∗(t) yields

x = x +

t∫

0

H(s) · X x,π∗,c∗

(s)(π∗(s)T σ(s) − θ(s)T )

︸︷︷︸=: f (s)

dW (s).

Hence we must have

f (s) = 0 P-a.s. for all s ∈ [0, T ].

As H(s) · X x,π∗,c∗(s) is positive we thus obtain

π∗(t) =(σ(t)T )−1

θ(t) for all t ∈ [0, T ].

In the special case of d = 1 and constant coefficients r , b, σ we get

π∗(t) =b − rσ2 local risk for stock investment

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Representation of the consumption rate

c∗(t) =1

T − t + 1X x,π∗,c∗

(t).

The consumption rate is

proportional to the current wealth of the investor and

inversely proportional to the remaining time T − t .

431 / 477

Solution of the representation problem

Theorem: Solution of the representation problemConsider the portfolio problem.Let x > 0 and assume χ(y) < ∞ for all y > 0. Let

B∗ := I2(Y (x) · H(T )

), c∗(t) := I1

(t , Y (x) · H(T )

).

If there exists a function f ∈ C1,2([0, T ] × R

d)

with f (0, 0, . . . , 0) = xand

1H(T )

· E( T∫

t

H(s) c∗(s) ds + H(T ) B∗ | Ft

)= f(t , W1(t), . . . , Wd (t)

),

then for all t ∈ [0, t] we have

π∗(t) =1

X x,π∗,c∗(t)σ−1(t)∇x f

(t , W1(t), . . . , Wd (t)

).

432 / 477


Corollary(1) The optimal terminal wealth B∗ of the problem

max(π,0)∈A′(x)

E(U2(X x,π(T )

))

is given by

B∗ := I2(Y (x) · H(T )

)

where in the definition of χ(y) we have to set I1(t , y) ≡ 0.

433 / 477


Corollary (continued)

(2) The optimal consumption process c∗(t) of the problem

max(π,c)∈A′(x)

E( T∫

0

U1(t , c(t)

)dt)

is given by

c∗(t) := I1(t , Y (x) · H(T )

)

where in the definition of χ(y) we have to set I2(y) ≡ 0.

434 / 477

Outline


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General assumption / notation


d = m

Notation

γ(t) := exp(−

t∫

0

r(s) ds)

θ(t) := σ−1(t)(b(t) − r(t) 1

)

Z (t) := exp(−

t∫

0

θ(s)T dW (s) − 12

t∫

0

‖θ(s)‖2 ds)

H(t) := γ(t) · Z (t)

436 / 477

Description of the market model

RestrictionThe market coefficients r , b, σ are constant.

437 / 477


Consider

a market, where a bond, d stocks, and d options on these stocksare traded

a Portfolio consisting of the bond and the options.

Assume

The options have a price process of the form

f (i)(t , P1(t), . . . , Pd (t)), i = 1, . . . , d , f ∈ C1,2

Option prices satisfy certain requirements(particularly satisfied for European puts and calls in theBlack-Scholes model).

438 / 477


Admissible trading strategy in bond and options:

ϕ(t) =(ϕ0(t), ϕ1(t), . . . , ϕd (t)

)

The integralst∫

0

ϕ0(s) dP0(s)

t∫

0

ϕi(s) df (i)(s, P1(s), . . . , Pd (s))

are assumed to be defined.ϕ(t) is assumed to be Ft -progressively measurable.Wealth progress

X (t) = ϕ0(t) P0(t) +

d∑

i=1

ϕi(t) f (i)(t , P1(t), . . . , Pd (t)).

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The problem

Consider the problem

maxϕ

E(U(X (T ))

)

where U is a utility function.

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Motivation of the solution of the problem

To motivate the solution of the problem, we take a look at the followingdiagrams representing

the solution of the option pricing problemvia the replication approach and

the solution of the portfolio problemvia the martingale approach.

The third diagram will show the solution of the problem.

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Option pricing

option priceEQ(e−rT B

) terminal payoffB

?oo

=

cost of replication

X ∗(0)

=

OO

replication strategyXπ∗

(T )

π∗(t)oo

Starting out from the final payment B of an option we replicate the finalpayment via following a suitable portfolio strategy leading to a finalwealth that coincides with final option payment. Then, the costs ofsetting up the cheapest replication strategy yield the option price.

442 / 477

Portfolio optimization with stocks

initial wealthx

? // optimal terminal wealthX x,π∗

(T )

optimal final paymentB∗

// replication strategyπ∗(t)

OO

The given inital wealth x will be invested according to a portfolioprocess π∗(t) with the aim to obtain a terminal wealth which promisesthe highest possible utility (we ignore the possibility of consumption).To do so, we first determine an optimal final payoff B∗ and then look fora replication strategy for B∗.

443 / 477

Portfolio optimization with options

initial wealthx

? //optimal terminal

wealthX ∗(T )

optimal finalpayment

B∗

((QQQQQQQQQQQ

inversion of thereplication strategy

ϕ = Ψ−1ξ

ϕ=(ϕ0(t),ϕ(t))

OO

replication strategy inbond and stocks

ξ(t) = (ξ0(t), . . . , ξd (t))

66mmmmmmmmmm

We look for an final wealth starting with an initial capital x . To do so, wefirst determine an optimal final payoff B∗ and then a replication strategyξ(t) = (ξ0(t), . . . , ξd (t)) in bond and stocks for the payoff B∗. As stocksshould not appear in our portfolio, we have to replicate the stockposition by bond and options. This bond and option strategy yields theoptimal terminal wealth X ∗(T ).

444 / 477


TheoremLet the Delta matrix

Ψ(t) =(Ψij(t)

)ij , i , j = 1, . . . , d

withΨij := f (i)

pj

(t , P1(t), . . . , Pd (t)

)

be regular for all t ∈ [0, T ]. Then, the option portfolio problempossesses the following explicit solution:

(1) The optimal terminal wealth B∗ coincides with the optimal terminalwealth of the corresponding stock portfolio problem.

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Theorem (continued)

(2) Let ξ(t) =(ξ0(t), . . . , ξd (t)

)be the optimal trading strategy in the

corresponding stock portfolio problem. Then, the optimal tradingstrategy ϕ(t) =

(ϕ0(t), ϕ1(t), . . . , ϕd (t)

)in the option portfolio

problem is given by

ϕ(t) =(Ψ(t)T )−1 · ξ(t),

ϕ0(t) =

(X (t) −

d∑

i=1

ϕi(t) f (i)(t , P1(t), . . . , Pd (t)))

P0(t),

withϕ(t) :=

(ϕ1(t), . . . , ϕd (t)

)

andξ(t) :=

(ξ1(t), . . . , ξd(t)

).

446 / 477


Remark(1) Under the given assumptions, the optimal final wealth only

depends on the utility functions but not on the choice of thetradable securities.

(2) The optimal trading strategy depends on the traded options viathe delta matrix (more precisely: via the replication strategy for theoptions).

447 / 477


Utility functionU(x) = ln(x).

Consider Black-Scholes model with d = 1.

Stock position in the optimal trading strategy

ξ1(t) =π∗(t) · X (t)

P1(t)=

b − rσ2 · X (t)

P1(t).

Optimal trading strategy in bond and options

ϕ1(t) =b − rσ2 · X (t)

Ψ1(t) · P1(t).

448 / 477


Optimal portfolio process in the stock portfolio problem

πstock (t)

Option portfolio process

πopt(t) : =ϕ1(t) · f (1)

(t , P1(t)

)

X (t)

=b − rσ2 · X (t) · f (1)

(t , P1(t)

)

X (t) · Ψ1(t) · P1(t)

=b − rσ2 · f (1)

(t , P1(t)

)

f (1)p1

(t , P1(t)

)· P1(t)

= πstock (t)· f (1)(t , P1(t)

)

f (1)p1

(t , P1(t)

)· P1(t)

449 / 477


TheoremWith the choice of

U(x) = ln(x),

in the Black-Scholes model with d = 1 we have

(1) πopt(t) = πstock (t) for all t ∈ [0, T ]

⇔ f (1)(t , P1(t)

)= k · P1(t) for a constant k ∈ R \ 0.

(2) In the case of a European call option we have

πopt(t) < πstock (t)

for all t ∈ [0, T ].

450 / 477


Remark(1) states that in the Black-Scholes model, πopt(t) is constant ifand only if the payoff of the contingent claims is a multiple of theunderlying stock price.

(2) says that with the choice of a European call option, in theoption portfolio problem the optimal capital which is invested in therisky asset is always smaller than the corresponding risky positionin the stock portfolio problem.

451 / 477

Example: European call

Market coefficients

r = 0, b = 0.05, σ = 0.25, T = 1, K = 100, P0(0) = 1.

50 150 250 350 4500

0.2

0.4

0.6

0.8

p1

P1(0)

pstock

(0)

popt

(0)

The deeper the option is in the money (i.e., P1(0) > K ), the closerπopt(0) gets to the optimal stock portfolio component πstock (0).

The more the option is out of the money (i.e., P1(0) < K ), thesmaller πopt(0) gets.

452 / 477

Outline


453 / 477

General Assumptions

General assumptions for this section

Let X (t) be an n-dimensional Ito process.

Controlled SDEControlled stochastic differential equation:

dX (t) = µ(t , X (t), u(t)) dt + σ(t , X (t), u(t)) dW (t)

where W (t) is an m-dimensional Brownian motion andu(t) a d -dimensional stochastic process (the control).

454 / 477

Outline


455 / 477

Example

Be X (t) = x + W (t) +t∫

0u(s) ds controlled process with 1-dimensional

Brownian motion W (t), as control action choose intensity u(t) of driftprocess at each time instant t ∈ [0, T ].Consider the problem to minimize (a, b > 0):

E

T∫

0

au2(t) dt − bX (T )

Under suitable requirements on u(t) we get

E(X (T )) = x + E

T∫

0

u(s) ds

and hence

E

T∫

0

(au2(t) − bu(t)) dt − bx

456 / 477

Example

Minimizing function u(t) under the integral leadsto the optimal choice of

u∗(t) =b2a

.

457 / 477

Outline


458 / 477

General assumptions

General assumptions for this section

For n, d ∈ N, 0 ≤ t0 < t1 < ∞, U ⊂ Rd closed, a constant C > 0 be

Q0 := [t0, t1) × Rn , Q0 := [t0, t1] × R

n

µ : Q0 × U → Rn , σ : Q0 × U → R

n,m

µ(., ., u), σ(., ., u) ∈ C1(Q0) for u ∈ U

|µt | + |µx | ≤ C , |σt | + |σx | ≤ C

|µ(t , x , u)| + |σ(t , x , u)| ≤ C (1 + |x | + |u|)

459 / 477

Notations

If the process X (t) solves the controlled SDE with an initial value of xat time t , then we indicate this by denoting its expectation at time t by

E t,x(X (s)).

460 / 477

Notations

Define for an open set O ⊂ Rn

Q := [t0, t1) × O,

Q := [t0, t1] × O,

τ := inft ≥ t0 | (t , X (t)) /∈ Q.

Define the cost function

J(t , x , u) = E t,x

τ∫

t

L(s, X (s), u(s)) ds + Ψ(τ, X (τ))

461 / 477

Notations

L,Ψ are continuous functions with

|L(t , x , u)| ≤ C(

1 + |x |k + |u|k)

,

|Ψ(t , x)| ≤ C(

1 + |x |k)

,

for some k ∈ N on Q × U or on Q.

L(s, X (s), u(s)) is called running costs and Ψ(τ, X (τ)) is calledterminal costs.

462 / 477

Notations

V (t , x) := infu∈A(t,x)

J(t , x ; u) (t , x) ∈ Q

is called value function, where A(t , x) denotes the set of all admissiblecontrols u(·).

463 / 477

Notations

(i) For G ∈ C1,2(Q), (t , x) ∈ Q, a := σσT , u ∈ Ulet

AuG(t , x) := Gt(t , x) + 0.5∑

1≤i ,j≤ n

aij(t , x , u)Gxi xj +

+∑

1≤i≤n

µi(t , x , u)Gxi (t , x)

(ii) ∂∗Q := ([t0, t1) × ∂O) ∪ (t1 × O).

464 / 477

Contolled SDE

Definition–Part IBe (Ω,F , Ftt∈[t0,t1]

, P) probability space with filtration. A U-valuedprogressively measurable process u(t), t ∈ [t0, t1] is called admissiblecontrol, if for all values x ∈ R

n the SDE with initial condition X (t0) = xpossess a unique solution X (t)t∈[t0 ,t1] and if we have

465 / 477

Contolled SDE

Definition–Part II

E

t1∫

t0

|u(s)|k ds

< ∞,

E t,x(||X (·)| |k

):= E t,x

(sup

s∈[t,t1]|X (s)|k

)< ∞

for all k ∈ N.

466 / 477

Controlled SDE

Theorem–Part I

Be G ∈ C1,2(Q) ∩ C(Q) with |G(t , x)| ≤ K(

1 + |x |k)

for some suitable

constants K > 0, k ∈ N, a solution to the Hamilton-Jacobi-Bellmanequation:

infu∈U

(AuG(t , x) + L(t , x , u)) = 0 (t , x) ∈ Q,

G(t , x) = Ψ(t , x) (t , x) ∈ ∂∗Q.

Then we have

467 / 477

Controlled SDE

Theorem–Part II(i) G(t , x) ≤ J(t , x , u) ∀ (t , x) ∈ Q, u(·) ∈ A(t , x).

(ii) If for all (t , x) ∈ Q there exists a u∗(·) ∈ A(t , x) with

u∗(s) = arg minu∈U

(AuG(s, X ∗(s)) + L(s, X ∗(s), u))

for all s ∈ [t , τ ], where X ∗ is controlled process corresponding tou∗, then

G(t , x) = V (t , x) = J(t , x ; u∗).

468 / 477

Outline


469 / 477

Main idea

General assumptions for this sectionMarket with constant market coefficients r , b, σ. Assume m ≥ d andσ ∈ R

d ,m has full rank.

Main idea–Part IIdentify wealth equation of investor with strategy (π, c) as controlledSDE of form

dX u(t) = µ(t , X u(t), u(t)) dt +

+σ(t , X u(t), u(t)) dW (t),

470 / 477

Main idea

Main idea–Part IIwhere µ, σ, u have special form

u = (u1, u2) := (π, c),

µ(t , x , u) = (r + uT1 (b − r 1))x − u2,

σ(t , x , u) = xuT1 σ.

471 / 477

Optimal consumption & wealth with finite horizon

Maximize utility functional

J(t , x , u) := E t,x

T∫

t

U1(t , u2(t)) dt + U2(Xu(T ))

Be V (t , x) = supu∈A(t,x) J(t , x ; u) be value function of the portfolioproblem. The corresponding Hamilton-Jacobi-Bellman (HJB) equationhas the form

maxu1∈[α1,α2]d ,u2∈[0,∞)

0.5uT

1 σσT u1x2Vxx (t , x)+

+((r + uT

1 (b − r 1))x − u2

)Vx (t , x)+

+U1(t , u2) + Vt(t , x) = 0

472 / 477


V (T , x) = U2(x)

for given constants α1, α2 ∈ R, α1 ≤ α2.

Solution of the corresponding HJB equationSolve the HJB equation for the special choice of

U1(t , c) =1γ

e−βtcγ , U2(x) =1γ

xγ

with β > 0 , γ ∈ (0, 1).

473 / 477


Proposition–Part IThe portfolio problem

max(π,c)∈A(x)

E

T∫

0

e−β t 1γ

c(t)γ dt +1γ

X (T )γ

is solved by strategy (π∗, c∗) according to

π∗(t) =1

1 − γ(σσT )−1(b − r 1),

c∗(t) =(

eβt f (t)) 1

γ−1X (t),

474 / 477


Proposition–Part IIwhere f (t) is given by

a1 := −0.5(b − r 1)T (σσT )−1(b − r 1)1

γ − 1+ r ,

a2 :=1 − γ

γe

βtγ−1 ,

g(t) = ea1

1−γ(T−t) +

+1 − γ

γ(a1 − β)

(e

a1−β

1−γT − e

a1−β

1−γt)

ea1

1−γ(T−t),

g(t) = f (t)γ

1−γ .

475 / 477

For Further Reading

R. Korn and E. KornOption Pricing and Portfolio Optimization -Modern methods of financial mathematics, AMS, 2001.

Gerhard-Wilhelm WeberFurther documents distributed in the course.

476 / 477

Education

Portfolio Optimization