Portfolio Optimization

Gerhard-Wilhelm Weber1           Erik Kropat2      Zafer-Korcan Görgülü3

                   1 Ins...
Outline I


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


2   The Continuous-Time Market Model...
Outline II




3   Option Pricing
      Introduction
      Examples
      The Replication Principle
      Arbitrage Opport...
Outline III




4   Pricing of Exotic Options and Numerical Algorithms
       Introduction
       Examples
       Examples...
Outline IV



5   Optimal Portfolios
      Introduction and Formulation of the Problem
      The martingale method
      O...
Outline


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




                                                       ...
Outline


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




                                    ...
Introduction




MVA     Based on H. M ARKOWITZ
OPM   • Decisions on investment strategies only at the beginning of the
  ...
The one-period model




Market with d traded securities
   d different securities with positive prices p1 , . . . , pd at...
Securities in a OPM


Returns of Securities
             Pi (T )
Ri (T ) :=     pi          (1 ≤ i ≤ d )

Estimated Means,...
Securities in a OPM




Each security perfectly divisable
    Hold ϕi ∈ R shares of security i        (1 ≤ i ≤ d )
    Neg...
Budget equation and portfolio return

The Budget Equation
Investor with initial wealth x > 0 holds ϕi ≥ 0 shares of securi...
Budget equation and portfolio return



Remark
   πi . . . fraction of total wealth invested in security i

              ...
Budget equation and portfolio return


Remark (continued)
   Portfolio Return
                                            ...
Selection of a portfolio–criterion




 (i) Maximize mean return (choose security of highest mean return)
    −→ risky, bi...
Selection of a portfolio–approach by Markowitz (MVA)

Balance Risk (Portfolio Variance) and Return (Portfolio Mean)
 (i) M...
Solution methods



 (i) Linear Optimization Problem with quadratic constraint
    −→ No standard algorithms, numerical in...
Relations between the formulations (i) and (ii)


Theorem
Assume:
    σ positive definite
                    min µi ≤ c2 ≤...
The diversification effect–example


Holding different Securities reduces Variance
Both security prices fluctuate
    random...
The diversification effect–example



Holding different Securities reduces Variance
                                       ...
Example

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

                         1 ...
Example




                             1             0.5
   Consider Portfolios              and          (does not sati...
Example

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

          min    0.1 · π1 + 0.1...
Example



                        1.0




                       π2

                        0.5

                       ...
Example


                                0.15




                                    0.1

                              ...
Stock price model




OPM
   No assumption on distribution of security returns
   Solving MV Problem just needed expectati...
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 · p...
Stock price model




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

   ...
Comments on MVA



   Only trading at initial time t = 0
   No reaction to current price changes possible
   ( −→ static m...
Outline


2   The Continuous-Time Market Model




                                       30 / 477
Outline

2   The Continuous-Time Market Model
      Modeling the Security Prices
      Excursion 1: Brownian Motion and Ma...
Modeling the security prices


Market with d+1 securities
    d risky stocks with
    prices p1 , p2 , . . . , pd at time ...
The bond price



Assume: Continuous compounding of interest at

    a constant rate r :
           Bond price: P0 (t) = p...
The stock price

Stock price = random fluctuations around an intrinsic bond part

                       2


              ...
The stock price



Randomness is assumed

   to have no tendency, i.e., E("randomness") = 0,
   to be time-dependent,
   t...
The stock price



Deviation at time t
                        Y (t) := ln(Pi (t)) − ln(pi ) − bi · t
with
               ...
The stock price




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

Distribution of the increments of the dev...
The stock price




Existence and properties of the stochastic process

                            {Y (t)}t∈[0,∞)

will b...
Outline

2   The Continuous-Time Market Model
      Modeling the Security Prices
      Excursion 1: Brownian Motion and Ma...
General assumptions




General assumptions
Let (Ω, F, P) be a complete probability space with sample space Ω,
σ-field F an...
Filtration



Definition
Let {Ft }t∈I be a family of sub-σ-fields of F, I be an ordered index set
with Fs ⊂ Ft for s < t, s,...
Stochastic process




Definition
A set {(Xt , Ft )}t∈I consisting of a filtration {Ft }t∈I and a family of
Rn -valued rando...
Remark




Remark
   I = [0, ∞) or I = [0, T ].
   Canoncial filtration (natural filtration) of {Xt }t∈I :

                ...
Sample path




Sample path
For fixed ω ∈ Ω the set

                  X .(ω) := {Xt (ω)}t∈I = {X (t, ω)}t∈I

is called a s...
Identification of stochastic processes

Can two stochastic processes be identified with each other?

Definition
Let {(Xt , Ft...
Identification of stochastic processes



Remark
    X , Y indistinguishable ⇒ Y modification of X .


Theorem
Let the stoch...
Brownian motion



Definition
The real-valued process {Wt }t≥0 with continuous sample paths and
  i) W0 = 0 P-a.s.
 ii) Wt ...
Brownian motion




Remark
By an n-dimensional Brownian motion we mean the Rn -valued process

                    W (t) =...
Brownian motion and filtration



Brownian motion can be associated with

    natural filtration

                   FtW := ...
Brownian motion and filtration

Requirement iii) of a Brownian motion with respect to a filtration
{Ft }t≥0 is often stated ...
Existence of the Brownian motion




How can we show the existence of a stochastic process satisfying the
requirements of ...
Brownian motion and filtration

Theorem
The Brownian filtration {Ft }t≥0 is right-continuous as well as
left-continuous, i.e...
Martingales

Definition
The real-valued process {(Xt , Ft )}t∈I with E |Xt | < ∞ for all t ∈ I
(where I is an ordered index...
Interpretation of the martingale concept



Example: Modeling games of chance

Xn : Wealth of a gambler after n-th partici...
Interpretation of the martingale concept




Example: Tossing a fair coin

"Head": Gambler receives one dollar

"Tail":   ...
Interpretation of the martingale concept


Theorem
A one-dimensional Brownian motion Wt is a martingale.

Remark
    Each ...
Interpretation of the martingale concept


Theorem
(1) Let {(Xt , Ft )}t∈I be a martingale and ϕ : R → R be a convex
    f...
Interpretation of the martingale concept



Remark
(1) Typical applications are given by

                         ϕ(x) = ...
Stopping time


Definition
A stopping time with respect to a filtration {Ft }t∈[0,∞)
(or {Fn }n∈N ) is an F-measurable rando...
The stopped process



The stopped process
Let {(Xt , Ft )}t∈I be a stochastic process, let I be either N or [0, ∞),
and τ...
The stopped filtration



The stopped filtration
Let τ be a stopping time with respect to a filtration {Ft }t∈[0,∞) .
    σ-fi...
The stopped filtration


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

Theorem: Optional sampling
Let {(Xt...
The stopped filtration




Corollary
Let τ be a stopping time and {(Xt , Ft )}t∈[0,∞) a right-continuous
sub-martingale (or...
The stopped filtration




Theorem
Let {(Xt , Ft )}t∈[0,∞) be a right-continuous process. Then Xt is a
martingale if and on...
The stopped filtration


Definition
Let {(Xt , Ft )}t∈[0,∞) be a stochastic process with X0 = 0. If there is a
non-decreasin...
The stopped filtration

Remark
(1) Each martingale is a local martingale.
(2) A local martingale with continuous paths is c...
The stopped filtration

Theorem: Doob’s inequality
Let {Mt }t≥0 be a martingale with right-continuous paths and
     2
E (M...
Outline

2   The Continuous-Time Market Model
      Modeling the Security Prices
      Excursion 1: Brownian Motion and Ma...
Continuation: The stock price




log-linear model for a stock price
               ln(Pi (t)) = ln(pi ) + bi · t + ”rando...
Continuation: The stock price

Market with one stock and one bond (d=1)
                  ln(P1 (t)) = ln(p1 ) + b1 · t + ...
Continuation: The stock price




Distribution of the logarithm of the stock prices
                                      ...
Continuation: The stock price


Lemma
                     m
                 1          2
Let bi := bi +   2         σij ...
Interpretation of the stock price model

The stock price model

                                              m
          ...
Interpretation of the stock price model



   Vector of mean rates of stock returns

                             b = (b1 ...
Summary: Stock prices




Bond price and stock prices
 P0 (t) = p0 · ert
                                                 ...
Extension

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

2   The Continuous-Time Market Model
      Modeling the Security Prices
      Excursion 1: Brownian Motion and Ma...
ˆ
The Ito integral




Is it possible to define the stochastic integral
                                 t

               ...
ˆ
The Ito integral



Theorem
P-almost all paths of the Brownian motion {Wt }t∈[0,∞) are nowhere
differentiable.


⇒ A defi...
ˆ
The Ito integral


Theorem
With the definition
                       2n
           Zn (ω) :=         W i (ω) − W i−1 (ω)...
General assumptions




General assumptions for this section
Let (Ω, F, P) be a complete probability space equipped with a...
Simple process


Definition
A stochastic process {Xt }t∈[0,T ] is called a simple process if there exist
real numbers 0 = t...
Simple process

Remark
   Xt is Fti−1 -measurable for all t ∈ (ti−1 , ti ].
   The paths X (., ω) of the simple process Xt...
Stochastic integral


Definition
For a simple process {Xt }t∈[0,T ] the stochastic integral I.(X ) for
t ∈ (tk , tk +1 ] is...
Stochastic integral


Theorem: Elementary properties of the stochastic integral
Let X := {Xt }t∈[0,T ] be a simple process...
Stochastic integral


Remark
(1) By (2) the stochastic integral is a square-integrable stochastic
    process.
(2) For the...
Stochastic integral

Remark
(1) Integrals with general boundaries:
                    T               T                  ...
Measurability

Definition
A stochastic process {(Xt , Gt )}t∈[0,∞) is called measurable if the
mapping

                   ...
Measurability




Definition
A stochastic process {(Xt , Gt )}t∈[0,∞) is called progressively
measurable if for all t ≥ 0 t...
Measurability



Remark
(1) If the real-valued process {(Xt , Gt )}t∈[0,∞) is progressively
    measurable and bounded, th...
Measurability


Theorem
If all paths of the stochastic process {(Xt , Gt )}t∈[0,∞) are
right-continuous (or left-continuou...
Extension of the stochastic integral to
L2 [0, T ]-processes

Definition

   L2 [0, T ] := L2 [0, T ], Ω, F, {Ft }t∈[0,T ] ...
Extension of the stochastic integral to
L2 [0, T ]-processes




     ·   2   L2 -norm on the probability space
         T...
Extension of the stochastic integral to
L2 [0, T ]-processes


  ˆ
Ito isometry
Let X be a simple process. The mapping X →...
Extension of the stochastic integral to
L2 [0, T ]-processes



   Use processes X ∈ L2 [0, T ] approximated by a sequence...
Extension of the stochastic integral to
L2 [0, T ]-processes



                                 J(.)
                    ...
Extension of the stochastic integral to
L2 [0, T ]-processes



Theorem
An arbitrary X ∈ L2 [0, T ] can be approximated by...
Extension of the stochastic integral to
L2 [0, T ]-processes




Lemma
Let {(Xt , Gt )}t∈[0,∞) be a martingale where the fi...
Extension of the stochastic integral to
L2 [0, T ]-processes


                      ˆ
Construction of the Ito integral fo...
Extension of the stochastic integral to
L2 [0, T ]-processes



Definition
For X ∈ L2 [0, T ] and J as before we define by
 ...
Extension of the stochastic integral to
L2 [0, T ]-processes




Theorem: Special case of Doob’s inequality
Let X ∈ L2 [0,...
Extension of the stochastic integral to
L2 [0, T ]-processes
Multi-dimensional generalization of the stochastic integral
{...
Further extension of the stochastic integral


Definition

    H 2 [0, T ] := H 2 [0, T ], Ω, F, {Ft }t∈[0,T ] , P

       ...
Further extension of the stochastic integral
Processes X ∈ H 2 [0, T ]
    do not necessarily have a finite T -norm
    → n...
Further extension of the stochastic integral



Stochastic integral:

                       It (X ) := It (X (n) ) for 0 ...
Further extension of the stochastic integral




Stopping times:
                             n→∞
                        ...
Outline

2   The Continuous-Time Market Model
      Modeling the Security Prices
      Excursion 1: Brownian Motion and Ma...
ˆ
The Ito formula




General assumptions for this section
Let (Ω, F, P) be a complete probability space equipped with a fi...
ˆ
The Ito formula


Definition
Let {(Wt , Ft )}t∈[0,∞) be an m-dimensional Brownian motion.
                               ...
ˆ
The Ito formula



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

         ...
ˆ
The Ito formula




Remark
   Hj ∈ H 2 [0, T ] for all T > 0.
                                 ˆ
   The representation o...
ˆ
The Ito formula

Definition
                                 ˆ
Let X and Y be two real-valued Ito processes with
        ...
ˆ
The Ito formula

Definition
Quadratic variation of X

                                           X   t   := X , X t .


N...
ˆ
The Ito formula
                           ˆ
Theorem: One-dimensional Ito formula
                                      ...
ˆ
The Ito formula


Remark
         ˆ
   The Ito formula differs from the fundamental theorem of calculus
   by the additi...
ˆ
The Ito formula


Lemma
Let X be a martingale with |Xs | ≤ C for all s ∈ [0, t] P-a.s.
Let π = {t0 , t1 , . . . , tm }, ...
′
                       ˆ
Some applications of Ito s formula

                          ′
                       ˆ
Some a...
′
                       ˆ
Some applications of Ito s formula II


                       ˆ′
Some applications of Ito s fo...
′
                       ˆ
Some applications of Ito s formula III

                                  ′
                   ...
ˆ
The Ito formula
                             ˆ
Theorem: Multi-dimensional Ito formula
                                  ...
Product rule or partial integration
Corollary: Product rule or partial integration
                                 ˆ
Let ...
The stock price equation
Simple continuous-time market model (1 bond, one stock).
Stock price influenced by a one-dimension...
The stock price equation


The stock price equation
                                   t                       t

        ...
The stock price equation




The stock price equation in differential form

                    dP(t) = P(t) b dt + σ dWt
...
The stock price equation


Theorem: Variation of constants
Let {(W (t), Ft )}t∈[0,∞) be an m-dimensional Brownian motion.
...
The stock price equation

Theorem: Variation of constants
Then the stochastic differential equation
                      ...
The stock price equation


Theorem: Variation of constants
Hereby is
                         t                           ...
The stock price equation


Remark
The process {(X (t), Ft )}t∈[0,∞) solves the stochastic differential
equation in the sen...
Outline

2   The Continuous-Time Market Model
      Modeling the Security Prices
      Excursion 1: Brownian Motion and Ma...
General assumptions

General assumptions for this section
(Ω, F, P) be a complete probability space,
{(W (t), Ft )}t∈[0,∞)...
General assumptions (continued)

General assumptions for this section (continued)
    r (t), b(t) = (b1 (t), . . . , bd (t...
Bond and stock prices

Bond and stock prices are unique solutions of the stochastic
differential equations

    dP0 (t) = ...
Possible actions of investors




(1) Investor can rebalance his holdings
    → sell some securities
    → invest in secur...
Requirements on a market model



(1) Investor should not be able to foresee events
    → no knowledge of future prices.
(...
Requirements on a market model




(6) Securities are perfectly divisible.
(7) Negative positions in securities are possib...
Negative bond positions and credit interest rates



Negative bond positions and credit interest rates


    Assume: Inter...
Mathematical realizations of some requirements

Market with 1 bond and d stocks

    Time t = 0:   – Initial capital of in...
Discrete-time example: self-financing strategy


Market with 1 riskless bond and 1 stock

Two-period model for time points ...
Discrete-time example: self-financing strategy




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


   ...
Discrete-time example: self-financing strategy

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

C...
Discrete-time example: self-financing strategy
t =2
Invest remaining capital at the market

Wealth:
                 X (2) ...
Discrete-time example: self-financing strategy


Self-financing trading strategy:
    wealth before rebalancing - consumptio...
Discrete-time example: self-financing strategy



Continuous-time setting

Wealth process corresponding to strategy ϕ(t):
 ...
Trading strategy and wealth processes

Definition
(1) A trading strategy ϕ with
                                           ...
Trading strategy and wealth processes

Definition
    The value
                                    d
                     ...
Trading strategy and wealth processes



Definition
(3) A non-negative progressively measurable process c(t) with
    respe...
Trading strategy and wealth processes


Definition
A pair (ϕ, c) consisting of a trading strategy ϕ and a consumption rate
...
Trading strategy and wealth processes


Remark
We have
       t                          t

           ϕ0 (s) dP0 (s) =   ...
Self-financing portfolio process



Definition
Let (ϕ, c) be a self-financing pair consisting of a trading strategy and a
con...
Portfolio processes


Remark
(1) The portfolio process denotes the fractions of total wealth invested
    in the different...
The wealth equation




The wealth equation

        dX (t) = [r (t) X (t) − c(t)] dt

               + X (t) π(t)T (b(t) ...
Alternative definition of a portfolio process



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



Definition
A self-financing pair (ϕ, c) or (π, c) consisting of a trading strategy ϕ or
a portfolio process...
An example

Portfolio process:
                       π(t) ≡ π ∈ Rd constant
Consumption rate:
                        c(t...
An example

   Wealth equation:

                    dX (t) = [r (t) − γ] X (t) dt
                           + X (t)π T (...
Outline

2   The Continuous-Time Market Model
      Modeling the Security Prices
      Excursion 1: Brownian Motion and Ma...
Properties of the continuous-time market model



Assumptions:
    Dimension of the underlying Brownian motion
    = numbe...
General assumption / notation

General assumption for this section
                                        d =m


Notation...
Properties of the continuous-time market model


   b, r uniformly bounded
   σσ T uniformly positive definite
   ⇒ θ(t) 2 ...
Completeness of the market



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


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

   H(t) can be regarded as the appropriate discounting process that
   determines the initial ...
Completeness of the market




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

                         ...
Outline

2   The Continuous-Time Market Model
      Modeling the Security Prices
      Excursion 1: Brownian Motion and Ma...
Excursion 4: The martingale representation theorem



General assumptions
(Ω, F, P) complete probability space.
{(Wt , Ft ...
The martingale representation theorem

Martingale representation theorem
Let {(Mt , Ft )}t∈[0,T ] be a square-integrable B...
The martingale representation theorem


Corollary
Let {(Mt , Ft )}t∈[0,T ] be a local martingale with respect to the Brown...
The martingale representation theorem




Remark
    Each local martingale with respect to the Brownian filtration can
    ...
Outline




3   Option Pricing




                     169 / 477
Outline



3   Option Pricing
      Introduction
      Examples
      The Replication Principle
      Arbitrage Opportunit...
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Portfolio Optimization
Portfolio Optimization
Portfolio Optimization
Portfolio Optimization
Portfolio Optimization
Portfolio Optimization
Portfolio Optimization
Portfolio Optimization
Portfolio Optimization
Portfolio Optimization
Portfolio Optimization
Portfolio Optimization
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Portfolio Optimization

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AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 1.
More info at http://summerschool.ssa.org.ua

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Portfolio Optimization

  1. 1. Portfolio Optimization Gerhard-Wilhelm Weber1 Erik Kropat2 Zafer-Korcan Görgülü3 1 Institute of Applied Mathematics Middle East Technical University Ankara, Turkey 2 Department of Mathematics University of Erlangen-Nuremberg Erlangen, Germany 3 University of the Federal Armed Forces Munich, Germany 2008 1 / 477
  2. 2. Outline I 1 The Mean-Variance Approach in a One-Period Model Introduction 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 2 / 477
  3. 3. Outline II 3 Option Pricing Introduction Examples The Replication Principle Arbitrage Opportunity Continuation Partial Differential Approach (PDA) Arbitrage & Option Pricing 3 / 477
  4. 4. Outline III 4 Pricing of Exotic Options and Numerical Algorithms Introduction Examples Examples Equivalent Martingale Measure Exotic Options with Explicit Pricing Formulae Weak Convergence of Stochastic Processes Monte-Carlo Simulation Approximation via Binomial Trees The Pathwise Binomial Approach of Rogers and Stapleton 4 / 477
  5. 5. Outline IV 5 Optimal Portfolios Introduction and Formulation of the Problem The martingale method Optimal Option Portfolios Excursion 8: Stochastic Control Maximize expected value in presence of quadratic control costs Introduction Portfolio Optimization via Stochastic Control Method 5 / 477
  6. 6. Outline 1 The Mean-Variance Approach in a One-Period Model 6 / 477
  7. 7. Outline 1 The Mean-Variance Approach in a One-Period Model Introduction 7 / 477
  8. 8. Introduction MVA Based on H. M ARKOWITZ OPM • Decisions on investment strategies only at the beginning of the period • Consequences of these decisions will be observed at the end of the period (−→ no action in between: static model) 8 / 477
  9. 9. The one-period model Market with d traded securities d different securities with positive prices p1 , . . . , pd at time t = 0 Security prices P1 (T ), . . . , Pd (T ) at final time t = T not foreseeable −→ modeled as non-negative random variables on probability space (Ω, F , P) 9 / 477
  10. 10. Securities in a OPM Returns of Securities Pi (T ) Ri (T ) := pi (1 ≤ i ≤ d ) Estimated Means, Variances and Covariances E (Ri (T )) = µi , Cov Ri (T ), Rj (T ) = σij (1 ≤ i ≤ d ) Remark The matrix σ := σij i,j∈{1,...,d} is positive semi-definite as it is a variance-covariance matrix. 10 / 477
  11. 11. 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 11 / 477
  12. 12. Budget equation and portfolio return The Budget Equation Investor with initial wealth x > 0 holds ϕi ≥ 0 shares of security i with ϕi · pi = x 1≤i≤d The Portfolio Vector π := (π1 , . . . , πd )T ϕi · pi πi := (1 ≤ i ≤ d ) x Portfolio Return R π := πi · Ri (T ) = π T R 1≤i≤d 12 / 477
  13. 13. Budget equation and portfolio return Remark πi . . . fraction of total wealth invested in security i ϕi · pi 1≤i≤d x πi = = =1 x x 1≤i≤d X π (T ) . . . final wealth corresponding to x and π X π (T ) = ϕi · Pi (T ) 1≤i≤d 13 / 477
  14. 14. Budget equation and portfolio return Remark (continued) Portfolio Return ϕi · pi Pi (T ) X π (T ) Rπ = πi · Ri (T ) = · = x pi x 1≤i≤d 1≤i≤d Portfolio Mean and Portfolio Variance E (R π ) = πi · µi , Var (R π ) = πi · σij · πj 1≤i≤d 1≤i,j≤d 14 / 477
  15. 15. 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 15 / 477
  16. 16. Selection of a portfolio–approach by Markowitz (MVA) Balance Risk (Portfolio Variance) and Return (Portfolio Mean) (i) Maximize E (R π ) under given upper bound c1 for Var (R π )   πi ≥ 0 (1 ≤ i ≤ d )    π πi = 1 max E (R ) subject to π∈Rd  1≤i≤d    Var (R π ) ≤ c1 (ii) Minimize Var (R π ) under given lower bound c2 for E (R π )   πi ≥ 0  (1 ≤ i ≤ d )  min Var (R π ) subject to πi = 1 π∈Rd   1≤i≤d  E (R π ) ≥ c2 16 / 477
  17. 17. Solution methods (i) Linear Optimization Problem with quadratic constraint −→ No standard algorithms, numerical inefficient (ii) Quadratic Optimization Problem with positive semidefinite objective matrix σ −→ efficient algorithms (i.e., G OLDFARB/I DNANI, G ILL/M URRAY) Feasible region non-empty if c2 ≤ max µi 1≤i≤d σ positive definite and feasible region non-empty −→ unique solution (even if one security riskless) 17 / 477
  18. 18. Relations between the formulations (i) and (ii) Theorem Assume: σ positive definite min µi ≤ c2 ≤ max µi c2 ∈ R+ 0 1≤i≤d 1≤i≤d min σ 2 (π) ≤ c1 ≤ max σ 2 (π) c1 ∈ R+ 0 πi ≥0, 1≤i≤d πi =1 πi ≥0, 1≤i≤d πi =1 Then ∗ (1) π ∗ solves (i) with Var R π = c1 =⇒ π ∗ solves (ii) with ∗ c2 := E R π (2) π solves (ii) with E R π = c2 =⇒ π solves (i) with c1 := Var R π 18 / 477
  19. 19. The diversification effect–example Holding different Securities reduces Variance Both security prices fluctuate randomly σ11 , σ22 > 0 independent σ12 = σ21 = 0 0.5 Then for the Portfolio π = we get 0.5 σ11 σ22 Var (R π ) = Var (0.5 · R1 + 0.5 · R2 ) = + 4 4 19 / 477
  20. 20. The diversification effect–example Holding different Securities reduces Variance 0.5 −→ If σ11 = σ22 then the Variance of Portfolio is half as big 0.5 1 0 as that of or 0 1 −→ Reduction of Variance . . . Diversification Effect depends on number of traded securities 20 / 477
  21. 21. Example Mean-Variance Criterion Investing into seemingly bad security can be optimal. Let be 1 0.1 −0.1 µ= , σ= 0.9 −0.1 0.15 Formulation (ii) becomes (II) min Var (R π ) = min 2 2 0.1 · π1 + 0.15 · π2 − 0.2 · π1 π2 π π   π1 , π2 ≥ 0 subject to π1 + π2 = 1  E (R π ) = π1 + 0.9 · π2 ≥ 0.96 21 / 477
  22. 22. Example 1 0.5 Consider Portfolios and (does not satify 0 0.5 expectation constraint) T T Var R (1,0) = 0.1 , E R (1,0) =1 T T Var R (0.5,0.5) = 0.125 , E R (0.5,0.5) = 0.95 22 / 477
  23. 23. Example Ignore expectation constraint and remember π1 , π2 ≥ 0 π1 + π2 = 1. Hence min 0.1 · π1 + 0.15 · (1 − π1 )2 − 0.2 · π1 · (1 − π1 ) 2 π 2 = min 0.45 · π1 − 0.5 · π1 + 0.15 π 0.5 −→ Minimizing Portfolio (No solution of (II) but better than ) 0.5 1 5 π= · 9 4 T −→ Portfolio Return Variance Var R ( 9 , 9 ) 5 4 ¯ = 0.001 T −→ Portfolio Return Mean E R ( 9 , 9 ) 5 4 ¯ = 0.95 23 / 477
  24. 24. Example 1.0 π2 0.5 0.4 0.0 0.0 0.5 0.6 1.0 π1 Pairs (π1 , π2 ) satisfying expectation constraint are above the dotted line Intersect with line π1 + π2 = 1 −→ Feasible region of MeanVariance Problem (bold line) 24 / 477
  25. 25. Example 0.15 0.1 Var 0.05 0 0 0.5 0.6 1.0 1.5 π1 Portfolio Return Variance (as function of π1 ) of all pairs satisfying π1 + π2 = 1 Minimum in feasible region π ∈ [0.6, 1] is attained at π = 0.6 Optimal Portfolio in (II) →∗ = − 0.6 ∗ ∗ π with Var R π = 0.012 , E Rπ = 0.96 0.4 25 / 477
  26. 26. Stock price model OPM No assumption on distribution of security returns Solving MV Problem just needed expectations and covariances 26 / 477
  27. 27. 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 d 1−q : probability of increasing by factor u (u > d ) Mean and Variance of Return P1 (T ) E (R1 (T )) =E = q · u + (1 − q) · d p1 P1 (T ) Var (R1 (T )) = Var = q · u 2 + (1 − q) · d 2 p1 − (q · u + (1 − q) · d )2 27 / 477
  28. 28. 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 · u Xn · d n−Xn with Xn ∼ B(n, p) number of up-movements of price in n periods 28 / 477
  29. 29. 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 Models Discrete-time multi-period models (many periods −→ no fast algorithms) 29 / 477
  30. 30. Outline 2 The Continuous-Time Market Model 30 / 477
  31. 31. Outline 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 31 / 477
  32. 32. Modeling the security prices Market with d+1 securities d risky stocks with prices p1 , p2 , . . . , pd at time t = 0 and random prices P1 (t), P2 (t), . . . , Pd (t) at times t > 0 1 bond with price p0 at time t = 0 and deterministic 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 ]. 32 / 477
  33. 33. 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): t r (s) ds Bond price: P0 (t) = p0 · e 0 for t ∈ [0, T ] 33 / 477
  34. 34. The stock price Stock price = random fluctuations around an intrinsic bond part 2 1.8 1.6 1.4 1.2 1 0.8 0 0.2 0.4 0.6 0.8 1 log-linear model for a stock price ln(Pi (t)) = ln(pi ) + bi · t + ”randomness” 34 / 477
  35. 35. 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 of ln(Pi (t)) from ln(pi ) + bi · t on [0, T ], ∼ N (0, σ 2 t) for some σ > 0. 35 / 477
  36. 36. The stock price Deviation at time t Y (t) := ln(Pi (t)) − ln(pi ) − bi · t with Y (t) ∼ N (0, σ 2 t) Properties: E (Y (t)) = 0, Y (t) is time-dependent. 36 / 477
  37. 37. 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, σ 2 (t − δ) 37 / 477
  38. 38. The stock price Existence and properties of the stochastic process {Y (t)}t∈[0,∞) will be studied in the excursion on the Brownian motion. 38 / 477
  39. 39. Outline 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 39 / 477
  40. 40. General assumptions General assumptions Let (Ω, F, P) be a complete probability space with sample space Ω, σ-field F and probability measure P. 40 / 477
  41. 41. Filtration Definition Let {Ft }t∈I be a family of sub-σ-fields of F, I be an ordered index set with Fs ⊂ Ft for s < t, s, t ∈ I. The family {Ft }t∈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 to determine its value from the information given at time t. 41 / 477
  42. 42. Stochastic process Definition A set {(Xt , Ft )}t∈I consisting of a filtration {Ft }t∈I and a family of Rn -valued random variables {Xt }t∈I with Xt being Ft -measurable is called a stochastic process with filtration {Ft }t∈I . 42 / 477
  43. 43. Remark Remark I = [0, ∞) or I = [0, T ]. Canoncial filtration (natural filtration) of {Xt }t∈I : Ft := FtX := σ{Xs | s ≤ t, s ∈ I}. Notation: {Xt }t∈I = {X (t)}t∈I = X . 43 / 477
  44. 44. Sample path Sample path For 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. 44 / 477
  45. 45. Identification of stochastic processes Can two stochastic processes be identified with each other? Definition Let {(Xt , Ft )}t∈[0,∞) and {(Yt , Gt )}t∈[0,∞) be two stochastic processes. Y is a modification of X , if P{ω | Xt (ω) = Yt (ω)} = 1 for all t ≥ 0. Definition Let {(Xt , Ft )}t∈[0,∞) and {(Yt , Gt )}t∈[0,∞) be two stochastic processes. X and Y are indistinguishable, if P{ω | Xt (ω) = Yt (ω) for all t ∈ [0, ∞)} = 1. 45 / 477
  46. 46. Identification of stochastic processes Remark X , Y indistinguishable ⇒ Y modification of X . Theorem Let the stochastic process Y be a modification of X . If both processes have continuous sample paths P-almost surely, then X and Y are indistinguishable. 46 / 477
  47. 47. Brownian motion Definition The real-valued process {Wt }t≥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. 47 / 477
  48. 48. Brownian motion Remark By an n-dimensional Brownian motion we mean the Rn -valued process W (t) = (W1 (t), . . . , Wn (t)), with components Wi being independent one-dimensional Brownian motions. 48 / 477
  49. 49. Brownian motion and filtration Brownian motion can be associated with natural filtration FtW := σ{Ws | 0 ≤ s ≤ t}, t ∈ [0, ∞) P-augmentation of the natural filtration (Brownian filtration) Ft := σ{FtW ∪ N | N ∈ F, P(N) = 0}, t ∈ [0, ∞) 49 / 477
  50. 50. Brownian motion and filtration Requirement iii) of a Brownian motion with respect to a filtration {Ft }t≥0 is often stated as iii)∗ Wt − Ws independent of Fs , 0 ≤ s < t. {Ft }t≥0 natural filtration (Brownian filtration) ⇒ iii) and iii)∗ are equivalent. {Ft }t≥0 arbitrary filtration ⇒ iii) and iii)∗ are usually not equivalent. Convention If we consider a Brownian motion {(Wt , Ft )}t≥0 with an arbitrary filtration we implicitly assume iii)∗ to be fulfilled. 50 / 477
  51. 51. Existence of the Brownian motion How can we show the existence of a stochastic process satisfying the requirements of a Brownian motion? Construction and existence proofs are long and technical. Construction based on weak convergence and approximation by random walks [Billingsley 1968]. Wiener measure, Wiener process. 51 / 477
  52. 52. Brownian motion and filtration Theorem The Brownian filtration {Ft }t≥0 is right-continuous as well as left-continuous, i.e., we have Ft = Ft+ := Ft+ε and Ft = Ft− := σ Fs . ε>0 s<t Definition A filtration {Gt }t≥0 satifies the usual conditions, if it is right-continuous and G0 contains all P-null sets of F. General assumption for this section Let {Ft }t≥0 be a filtration which satisfies the usual conditions. 52 / 477
  53. 53. Martingales Definition The 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. . 53 / 477
  54. 54. 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-martingale non-favorable game = super-martingale 54 / 477
  55. 55. Interpretation of the martingale concept Example: Tossing a fair coin "Head": Gambler receives one dollar "Tail": Gambler loses one dollar ⇒ Martingale 55 / 477
  56. 56. Interpretation of the martingale concept Theorem A one-dimensional Brownian motion Wt is a martingale. Remark Each stochastic process with independent, centered increments is a 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 a sub-martingale if µ ≥ 0. 56 / 477
  57. 57. Interpretation of the martingale concept 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. 57 / 477
  58. 58. Interpretation of the martingale concept Remark (1) Typical applications are given by ϕ(x) = x 2 , 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. 58 / 477
  59. 59. Stopping time Definition A stopping time with respect to a filtration {Ft }t∈[0,∞) (or {Fn }n∈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). Theorem If τ1 , τ2 are both stopping times then τ1 ∧ τ2 := min{τ1 , τ2 } is also a stopping time. 59 / 477
  60. 60. The stopped process The stopped process Let {(Xt , Ft )}t∈I be a stochastic process, let I be either N or [0, ∞), and τ a stopping time. The stopped process {Xt∧τ }t∈I is defined by Xt (ω) if t ≤ τ (ω), Xt∧τ (ω) := Xτ (ω) (ω) if t > τ (ω). Example: Wealth of a gambler who participates in a sequence of games until he is either bankrupt or has reached a given level of wealth. 60 / 477
  61. 61. The stopped filtration The stopped filtration Let τ be a stopping time with respect to a filtration {Ft }t∈[0,∞) . σ-field of events determined prior to the stopping time τ Fτ := {A ∈ F | A ∩ {τ ≤ t} ∈ Ft for all t ∈ [0, ∞)} Stopped filtration {Fτ ∧t }t∈[0,∞) . 61 / 477
  62. 62. The stopped filtration 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 (or martingale). Let τ1 , τ2 be stopping times with τ1 ≤ τ2 . Then for all t ∈ [0, ∞) we have E (Xt∧τ2 | Ft∧τ1 ) ≥ Xt∧τ1 P-a.s. (or E (Xt∧τ2 | Ft∧τ1 ) = Xt∧τ1 P-a.s.). 62 / 477
  63. 63. The stopped filtration Corollary Let τ be a stopping time and {(Xt , Ft )}t∈[0,∞) a right-continuous sub-martingale (or martingale). Then the stopped process {(Xt∧τ , Ft )}t∈[0,∞) is also a sub-martingale (or martingale). 63 / 477
  64. 64. The stopped filtration Theorem Let {(Xt , Ft )}t∈[0,∞) be a right-continuous process. Then Xt is a martingale if and only if for all bounded stopping times τ we have EXτ = EX0 . → Characterization of a martingale 64 / 477
  65. 65. The stopped filtration Definition Let {(Xt , Ft )}t∈[0,∞) be a stochastic process with X0 = 0. If there is a non-decreasing sequence {τn }n∈N of stopping times with P lim τn = ∞ = 1, n→∞ such that (n) Xt := (Xt∧τn , Ft ) t∈[0,∞) is a martingale for all n ∈ N, then X is a local martingale. The sequence {τn }n∈N is called a localizing sequence corresponding to X . 65 / 477
  66. 66. The stopped filtration Remark (1) Each martingale is a local martingale. (2) A local martingale with continuous paths is called continuous local martingale. (3) There exist local martingales which are not martingales. E (Xt ) need not exist for a local martingale. However, the expectation has to exist along the localizing sequence t ∧ τn . The local martingale is a martingale on the random time intervals [0, τn ]. Theorem A non-negative local martingale is a super-martingale. 66 / 477
  67. 67. The stopped filtration Theorem: Doob’s inequality Let {Mt }t≥0 be a martingale with right-continuous paths and 2 E (MT ) < ∞ or all T > 0. Then, we have 2 2 E sup |Mt | ≤ 4 · E (MT ). 0≤t≤T Theorem Let {(Xt , Ft )}t∈[0,∞) be a non-negative super-martingale with right-continuous paths. Then, for λ > 0 we obtain λ·P ω sup Xs (ω) ≥ λ ≤ E (X0 ). 0≤s≤t 67 / 477
  68. 68. Outline 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 68 / 477
  69. 69. 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 to model the "randomness" 69 / 477
  70. 70. Continuation: The stock price Market with one stock and one bond (d=1) ln(P1 (t)) = ln(p1 ) + b1 · t + σ11 Wt P1 (t) = p1 · exp b1 · t + σ11 Wt Market with d stocks and one bond (d>1) m ln(Pi (t)) = ln(pi ) + bi · t + σij Wj (t), i = 1, . . . , d j=1 m Pi (t) = pi · exp bi · t + σij Wj (t) , i = 1, . . . , d j=1 70 / 477
  71. 71. Continuation: The stock price Distribution of the logarithm of the stock prices m 2 ln(Pi (t)) ∼ N ln(pi ) + bi · t, σij · t j=1 ⇒ Pi (t) is log-normally distributed. 71 / 477
  72. 72. Continuation: The stock price Lemma m 1 2 Let bi := bi + 2 σij for i = 1, . . . , d . j=1 (1) E (Pi (t)) = pi · ebi t . m (2) Var (Pi (t)) = pi2 · exp(2bi t) · exp 2 σij t −1 . j=1 m 1 (3) Xt := a · exp cj Wj (t) − cj2 t with a, cj ∈ R, j = 1, . . . , m 2 j=1 is a martingale. 72 / 477
  73. 73. Interpretation of the stock price model The stock price model m 1 2 Pi (t) = pi · exp(bi t) · exp σij Wj (t) − σij t , 2 j=1 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 m 1 2 exp σij Wj (t) − σij t 2 j=1 which models the stock price around its mean value. 73 / 477
  74. 74. Interpretation of the stock price model Vector of mean rates of stock returns b = (b1 , . . . , bd )T Volatility matrix   σ11 . . . σ1m  . .  σ= . . .. . .  . σd1 . . . σdm Pi (t) is a geometric Brownian motion with drift bi and volatility σi. = (σi1 , . . . , σim )T . 74 / 477
  75. 75. Summary: Stock prices Bond price and stock prices P0 (t) = p0 · ert Bond price P0 (0)= p0 m 1 2 Pi (t) = pi · exp(bi t) · exp σij Wj (t) − σ t 2 ij Stock prices j=1 Pi (0) = pi , i = 1, . . . , d . 75 / 477
  76. 76. Extension Extension: Model with non-constant, time-dependent, and integrable rates of return bi (t) and volatilities σ(t). Stock prices: t m 1 2 Pi (t) = pi · exp bi (s) − σij (s) ds 2 0 j=1 m t · exp σij (s) dWj (s) j=1 0 t Problem: σij (s) dWj (s) 0 ˆ ⇒ Stochastic integral (Ito integral) 76 / 477
  77. 77. Outline 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 77 / 477
  78. 78. ˆ The Ito integral Is it possible to define the stochastic integral t Xs (ω) dWs (ω) 0 ω-wise in a reasonable way? 78 / 477
  79. 79. ˆ The Ito integral Theorem P-almost all paths of the Brownian motion {Wt }t∈[0,∞) are nowhere differentiable. ⇒ A definition of the form t t dWs (ω) Xs (ω) dWs (ω) = Xs (ω) ds ds 0 0 is impossible. 79 / 477
  80. 80. ˆ The Ito integral Theorem With the definition 2n Zn (ω) := W i (ω) − W i−1 (ω) , n ∈ N, ω ∈ Ω 2n 2n i=1 we have n→∞ Zn (ω) − − ∞ −→ P-a.s. , i.e., the paths Wt (ω) of the Brownian motion admit infinite variation on the interval [0, 1] P-almost surely. The paths Wt (ω) of the Brownian motion have infinite variation on each non-empty finite interval [s1 , s2 ] ⊂ [0, ∞) P-almost surely. 80 / 477
  81. 81. General assumptions General assumptions for this section Let (Ω, F, P) be a complete probability space equipped with a filtration {Ft }t satisfying the usual conditions. Further assume that on this space a Brownian motion {(Wt , Ft )}t∈[0,∞) with respect to this filtration is defined. 81 / 477
  82. 82. Simple process Definition A stochastic process {Xt }t∈[0,T ] is called a simple process if there exist real numbers 0 = t0 < t1 < . . . < tp = T , p ∈ N, and bounded random variables 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 p Xt (ω) = X (t, ω) = Φ0 (ω) · 1{0} (t) + Φi (ω) · 1(ti−1 ,ti ] (t) i=1 for each ω ∈ Ω. 82 / 477
  83. 83. Simple process Remark Xt is Fti−1 -measurable for all t ∈ (ti−1 , ti ]. The paths X (., ω) of the simple process Xt are left-continuous step functions with height Φi (ω) · 1(ti−1 ,ti ] (t). 1 0.9 0.8 0.7 0.6 X(.,ω) 0.5 0.4 0.3 0.2 0.1 0 0 t t2 t3 T 1 t 83 / 477
  84. 84. Stochastic integral Definition For a simple process {Xt }t∈[0,T ] the stochastic integral I.(X ) for t ∈ (tk , tk +1 ] is defined according to t It (X ) := Xs dWs := Φi (Wti − Wti−1 ) + Φk +1 (Wt − Wtk ), 0 1≤i≤k or more generally for t ∈ [0, T ]: t It (X ) := Xs dWs := Φi (Wti ∧t − Wti−1 ∧t ). 0 1≤i≤p 84 / 477
  85. 85. Stochastic integral Theorem: Elementary properties of the stochastic integral Let X := {Xt }t∈[0,T ] be a simple process. Then we have (1) {It (X )}t∈[0,T ] is a continuous martingale with respect to {Ft }t∈[0,T ] . In particular, we have E (It (X )) = 0 for all t ∈ [0, T ]. t 2 t (2) E Xs dWs =E 2 Xs ds for t ∈ [0, T ]. 0 0 t 2 T (3) E sup Xs dWs ≤4·E 2 Xs ds . 0≤t≤T 0 0 85 / 477
  86. 86. 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 1 dWs = Wt 0 and t t 2 E dWs = E (Wt2 ) =t= ds. 0 0 86 / 477
  87. 87. Stochastic integral Remark (1) Integrals with general boundaries: T T t Xs dWs := Xs dWs − Xs dWs for t ≤ T . t 0 0 For t ≤ T , A ∈ Ft we have T T 1A (ω) · Xs (ω) · 1[t,T ] (s) dWs = 1A (ω) · Xs (ω) dWs . 0 t (2) Let X , Y be simple processes, a, b ∈ R. Then we have It (aX + bY ) = a · It (X ) + b · It (Y ) (linearity) 87 / 477
  88. 88. Measurability Definition A stochastic process {(Xt , Gt )}t∈[0,∞) is called measurable if the mapping [0, ∞) × Ω → Rn (s, ω) → Xs (ω) is B([0, ∞)) ⊗ F-B(Rn )-measurable. Remark Measurability of the process X implies that X (., ω) is B([0, ∞))-B(Rn )-measurable for a fixed ω ∈ Ω. Thus, for all t ∈ [0, ∞), t i = 1, . . . , n, the integral Xi2 (s) ds is defined. 0 88 / 477
  89. 89. Measurability Definition A stochastic process {(Xt , Gt )}t∈[0,∞) is called progressively measurable if for all t ≥ 0 the mapping [0, t] × Ω → Rn (s, ω) → Xs (ω) is B([0, t]) ⊗ Gt -B(Rn )-measurable. 89 / 477
  90. 90. Measurability Remark (1) If the real-valued process {(Xt , Gt )}t∈[0,∞) is progressively measurable and bounded, then for all t ∈ [0, ∞) the integral t Xs ds is Gt -measurable. 0 (2) Every progressively measurable process is measurable. (3) Each measurable process possesses a progressively measurable modification. 90 / 477
  91. 91. Measurability Theorem If all paths of the stochastic process {(Xt , Gt )}t∈[0,∞) are right-continuous (or left-continuous), then the process is progressively measurable. Theorem Let τ be a stopping time with respect to the filtration {Gt }t∈[0,∞) . If the stochastic process {(Xt , Gt )}t∈[0,∞) is progressively measurable, then so is the stopped process {(Xt∧τ , Gt )}t∈[0,∞) . In particular, Xt∧τ is Gt and Gt∧τ -measurable. 91 / 477
  92. 92. Extension of the stochastic integral to L2 [0, T ]-processes Definition L2 [0, T ] := L2 [0, T ], Ω, F, {Ft }t∈[0,T ] , P := {(Xt , Ft )}t∈[0,T ] real-valued stochastic process T {Xt }t progressively measurable, E Xt2 dt < ∞ 0 T Norm on L2 [0, T ]: X 2 T := E Xt2 dt . 0 92 / 477
  93. 93. Extension of the stochastic integral to L2 [0, T ]-processes · 2 L2 -norm on the probability space T [0, T ] × Ω, B([0, T ]) ⊗ F, λ ⊗ P . · 2 semi-norm ( X −Y 2 = 0 ⇒ X = Y ). T T X equivalent to Y :⇔ X = Y a.s. λ ⊗ P. 93 / 477
  94. 94. Extension of the stochastic integral to L2 [0, T ]-processes ˆ Ito isometry Let X be a simple process. The mapping X → I.(X ) induces by T T 2 2 2 2 I.(X ) LT := E Xs dWs =E Xs ds = X T 0 0 a norm on the space of stochastic integrals. ⇒ I.(X ) linear, norm-preserving (= isometry) ˆ ⇒ I.(X ) Ito isometry 94 / 477
  95. 95. Extension of the stochastic integral to L2 [0, T ]-processes Use processes X ∈ L2 [0, T ] approximated by a sequence X (n) of simple 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 . 95 / 477
  96. 96. Extension of the stochastic integral to L2 [0, T ]-processes J(.) C X ∈ L2 [0, T ] _ _ _ _ _ _ _/ J(X ) ∈ M2 O O · T · LT X (n) / I(X (n) ) I(.) simple process stochastic integral for simple processes 96 / 477
  97. 97. Extension of the stochastic integral to L2 [0, T ]-processes Theorem An arbitrary X ∈ L2 [0, T ] can be approximated by a sequence of simple processes X (n) . More precisely: There exists a sequence X (n) of simple processes with T (n) 2 lim E Xs − Xs ds = 0. n→∞ 0 97 / 477
  98. 98. Extension of the stochastic integral to L2 [0, T ]-processes Lemma Let {(Xt , Gt )}t∈[0,∞) be a martingale where the filtration {Gt }t∈[0,∞) satisfies the usual conditions. Then the process Xt possesses a right-continuous modification {(Yt , Gt )}t∈[0,∞) such that {(Yt , Gt )}t∈[0,∞) is a martingale. 98 / 477
  99. 99. Extension of the stochastic integral to L2 [0, T ]-processes ˆ 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 of continuous martingales on [0, T ] with respect to {Ft }t∈[0,T ] satisfying (1) X = {Xt }t∈[0,T ] simple process ⇒ P Jt (X ) = It (X ) for all t ∈ [0, T ] = 1 t (2) E Jt (X )2 = E 2 ˆ Xs ds Ito isometry 0 Uniqueness: If J, J ′ satisfy (1) and (2), then for all X ∈ L2 [0, T ] the processes J ′ (X ) and J(X ) are indistinguishable. 99 / 477
  100. 100. Extension of the stochastic integral to L2 [0, T ]-processes Definition For X ∈ L2 [0, T ] and J as before we define by t Xs dWs := Jt (X ) 0 ˆ the stochastic integral (or Ito integral) of X with respect to W . 100 / 477
  101. 101. Extension of the stochastic integral to L2 [0, T ]-processes Theorem: Special case of Doob’s inequality Let X ∈ L2 [0, T ]. Then we have t T 2 2 E sup Xs dWs ≤4·E Xs ds . 0≤t≤T 0 0 101 / 477
  102. 102. Extension of the stochastic integral to L2 [0, T ]-processes Multi-dimensional generalization of the stochastic integral {(W (t), Ft )}t : m-dimensional Brownian motion with W (t) = (W1 (t), . . . , Wm (t)) {(X (t), Ft )}t : Rn,m -valued progressively measurable process with Xij ∈ L2 [0, T ]. ˆ Ito integral of X with respect to W :  t  m  X1j (s) dWj (s)   t  j=1   0   . .  X (s) dW (s) :=  . , t ∈ [0, T ]   0  m t     Xnj (s) dWj (s) j=1 0 102 / 477
  103. 103. Further extension of the stochastic integral Definition H 2 [0, T ] := H 2 [0, T ], Ω, F, {Ft }t∈[0,T ] , P := {(Xt , Ft )}t∈[0,T ] real-valued stochastic process {Xt }t progressively measurable, T Xt2 dt < ∞ P-a.s. 0 103 / 477
  104. 104. Further extension of the stochastic integral Processes X ∈ H 2 [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 {Ft }t ): t 2 τn (ω) := T ∧ inf 0 ≤ t ≤ T Xs (ω) ds ≥ n , n ∈ N 0 Sequence of stopped processes: (n) Xt (ω) := Xt (ω) · 1{τn (ω)≥t} ⇒ X (n) ∈ L2 [0, T ] ⇒ Stochastic integral already defined. 104 / 477
  105. 105. Further extension of the stochastic integral 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 ∈ H 2 [0, T ] 105 / 477
  106. 106. Further extension of the stochastic integral Stopping times: n→∞ τn − − +∞ P-a.s. −→ ⇒ It (X ) local martingale with localizing sequence τn . ⇒ Stochastic integral is linear and possesses continuous paths. 106 / 477
  107. 107. Outline 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 107 / 477
  108. 108. ˆ The Ito formula General assumptions for this section Let (Ω, F, P) be a complete probability space equipped with a filtration {Ft }t satisfying the usual conditions. Further, assume that on this space a Brownian motion {(Wt , Ft )}t∈[0,∞) with respect to this filtration is defined. 108 / 477
  109. 109. ˆ The Ito formula Definition Let {(Wt , Ft )}t∈[0,∞) be an m-dimensional Brownian motion. ˆ (1) {(X (t), Ft )}t∈[0,∞) is a real-valued Ito process if for all t ≥ 0 it admits the representation t t X (t) = X (0) + K (s) ds + H(s) dW (s) 0 0 t m t = X (0) + K (s) ds + Hj (s) dWj (s) P-a.s. 0 j=1 0 109 / 477
  110. 110. ˆ The Ito formula X (0) F0 -measurable, {K (t)}t∈[0,∞) , {H(t)}t∈[0,∞) progressively measurable with t t |K (s)| ds < ∞, Hi2 (s) ds < ∞ P-a.s. 0 0 for all t ≥ 0, i = 1, . . . , m. (2) n-dimensional Ito process X = (X (1) , . . . , X (n) ) ˆ ˆ = vector with components being real-valued Ito processes. 110 / 477
  111. 111. ˆ The Ito formula Remark Hj ∈ H 2 [0, T ] for all T > 0. ˆ The representation of an Ito process is unique up to indistinguishability of the representing integrands Kt , Ht . Symbolic differential notation: dXt = Kt dt + Ht dWt 111 / 477
  112. 112. ˆ The Ito formula Definition ˆ Let X and Y be two real-valued Ito processes with t t X (t) = X (0) + K (s) ds + H(s) dW (s), 0 0 t t Y (t) = Y (0) + L(s) ds + M(s) dW (s). 0 0 Quadratic covariation of X and Y : m t X,Y t := Hi (s) · Mi (s) ds. i=1 0 112 / 477
  113. 113. ˆ The Ito formula Definition Quadratic variation of X X t := X , X t . Notation ˆ Let X be a real-valued Ito process, and Y a real-valued, progressively measurable process. We set t t t Y (s) dX (s) := Y (s) · K (s) ds + Y (s) · H(s) dW (s) 0 0 0 if all integrals on the right-hand side are defined. 113 / 477
  114. 114. ˆ The Ito formula ˆ Theorem: One-dimensional Ito formula ˆ Let Wt be a one-dimensional Brownian motion, and Xt a real-valued Ito process with t t Xt = X0 + Ks ds + Hs dWs . 0 0 Let f ∈ C 2 (R). Then, for all t ≥ 0 we have t t ′ 1 f (Xt ) = f (X0 ) + f (Xs ) dXs + f ′′ (Xs ) d X s 2 0 0 t t 1 = f (X0 ) + f (Xs ) · Ks + · f ′′ (Xs ) · Hs ds + ′ 2 f ′ (Xs )Hs dWs 2 0 0 114 / 477
  115. 115. ˆ The Ito formula Remark ˆ The Ito formula differs from the fundamental theorem of calculus by the additional term t 1 f ′′ (Xs ) d X s . 2 0 The quadratic variation X t ˆ is an Ito process. Differential notation: 1 ′′ df (Xt ) = f ′ (Xt ) dXt + · f (Xt ) d X t . 2 115 / 477
  116. 116. ˆ The Ito formula Lemma Let 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 := max |tk − tk −1 |. 1≤k ≤m Then we have m 2 2 (1) E Xtk − Xtk −1 ≤ 48 · C 4 k =1 m 4 π →0 (2) X continuous ⇒ E Xtk − Xtk −1 − − → 0. −− k =1 116 / 477
  117. 117. ′ ˆ Some applications of Ito s formula ′ ˆ Some applications of Ito s formula I (1) Xt = t : Representation: t t Xt = 0 + 1 ds + 0 dWs . 0 0 For f ∈ C 2 (R) we have t f (t) = f (0) + f ′ (s) ds. 0 ⇒ Fundamental theorem of calculus is a special case of Ito′ s formula. ˆ 117 / 477
  118. 118. ′ ˆ Some applications of Ito s formula II ˆ′ Some applications of Ito s formula (2) Xt = h(t) : For h ∈ C 1 (R) Ito′ s formula implies the chain rule ˆ t t ′ Xt = h(0) + h (s) ds + 0 dWs 0 0 t ⇒ (f ◦ h)(t) = (f ◦ h)(0) + f ′ (h(s)) · h′ (s) ds. 0 118 / 477
  119. 119. ′ ˆ Some applications of Ito s formula III ′ ˆ Some applications of Ito s formula (3) Xt = Wt , f (x) = x 2 : Due to t t Wt = 0 + 0 ds + 1 dWs 0 0 we obtain t t t 1 Wt2 = 2 · Ws dWs + · 2 ds = 2 · Ws dWs + t 2 0 0 0 ⇒ Additional term "t" (→ nonvanishing quadratic variation of Wt ). 119 / 477
  120. 120. ˆ The Ito formula ˆ Theorem: Multi-dimensional Ito formula ˆ X (t) = X1 (t), . . . , Xn (t) n-dimensional Ito process with t m t Xi (t) = Xi (0) + Ki (s) ds + Hij (s) dWj (s), i = 1, . . . , n 0 j=1 0 where W (t) = W1 (t), . . . , Wm (t) is an m-dimensional Brownian motion. Let f : [0, ∞) × Rn → R be a C 1,2 -function. Then, we have f (t, X1 (t), . . . , Xn (t)) = f (0, X1 (0), . . . , Xn (0)) t n t + ft (s, X1 (s), . . . , Xn (s)) ds + fxi (s, X1 (s), . . . , Xn (s)) dXi (s) 0 i=1 0 n t 1 + · fxi xj (s, X1 (s), . . . , Xn (s)) d Xi , Xj s . 2 i,j=1 0 120 / 477
  121. 121. Product rule or partial integration Corollary: Product rule or partial integration ˆ Let Xt , Yt be one-dimensional Ito processes with t t Xt = X0 + Ks ds + Hs dWs , 0 0 t t Yt = Y0 + µs ds + σs dWs . 0 0 Then we have t t t Xt · Yt = X0 · Y0 + Xs dYs + Ys dXs + d X,Y s 0 0 0 t t = X0 · Y0 + Xs µs + Ys Ks + Hs σs ds + Xs σs + Ys Hs dWs . 0 0 121 / 477
  122. 122. 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 − 1 σ 2 t + σWt 2 Choose t t 1 2 Xt = 0 + b− 2σ ds + σ dWs , f (x) = p · ex 0 0 ˆ Ito formula implies t t 1 2 1 2 f (Xt ) = p + f (Xs )(b − 2 σ ) + 2 f (Xs ) ·σ ds + f (Xs ) · σ dWs 0 0 122 / 477
  123. 123. The stock price equation The stock price equation t t P(t) = p + P(s) · b ds + P(s) · σ dWs 0 0 Remark The stock price equation is valid for time-dependent b and σ, if t t 1 2 Xt = b(s) − 2 σ (s) ds + σ(s) dWs . 0 0 123 / 477
  124. 124. The stock price equation The stock price equation in differential form dP(t) = P(t) b dt + σ dWt P(0) = p 124 / 477
  125. 125. The stock price equation Theorem: Variation of constants Let {(W (t), Ft )}t∈[0,∞) be an m-dimensional Brownian motion. Let x ∈ R and A, a, Sj , σj be progressively measurable, real-valued processes with t |A(s)| + |a(s)| ds < ∞ for all t ≥ 0 P-a.s. 0 t Sj2 (s) + σj2 (s) ds < ∞ for all t ≥ 0 P-a.s. . 0 125 / 477
  126. 126. The stock price equation Theorem: Variation of constants Then the stochastic differential equation m dX (t) = A(t) · X (t) + a(t) dt + Sj (t)X (t) + σj (t) dWj (t) j=1 X (0) = x possesses a unique solution with respect to λ ⊗ P : t m 1 X (t) = Z (t) · x + a(u) − Sj (u)σj (u) du Z (u) 0 j=1 m t σj (u) + dWj (u) Z (u) j=1 0 126 / 477
  127. 127. The stock price equation Theorem: Variation of constants Hereby is t t 1 2 Z (t) = exp A(u) − 2 · S(u) du + S(u) dW (u) 0 0 the unique solution of the homogeneous equation dZ (t) = Z (t) A(t) dt + S(t)T dW (t) Z (0) = 1. 127 / 477
  128. 128. The stock price equation Remark The process {(X (t), Ft )}t∈[0,∞) solves the stochastic differential equation in the sense that X (t) satisfies t X (t) = x + A(s) · X (s) + a(s) ds 0 m t + Sj (s) X (s) + σj (s) dWj (s) j=1 0 for all t ≥ 0 P-almost surely. 128 / 477
  129. 129. Outline 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 129 / 477
  130. 130. 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: t P0 (t) = p0 · exp r (s) ds bond 0 t m 1 2 Pi (t) = pi · exp bi (s) − σij (s) ds 2 0 j=1 m t + σij (s) dWj (s) stock j=1 0 for t ∈ [0, T ], T > 0, i = 1, . . . , d . 130 / 477
  131. 131. 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 {Ft }t , component-wise uniformly bounded in (t, ω). σ(t)σ(t)T uniformly positive definite, i.e., it exists K > 0 with x T σ(t)σ(t)T x ≥ Kx T x for all x ∈ Rd and all t ∈ [0, T ] P-a.s. Deterministic rate of return r (t) is not required r (t) can be a stochastic process ⇒ bond is no longer riskless. 131 / 477
  132. 132. Bond and stock prices Bond and stock prices are unique solutions of the stochastic differential equations dP0 (t) = P0 (t) · r (t) dt bond P0 (t) = p0 m dPi (t) = Pi (t) bi (t) dt + σij (t) dWj (t) , i = 1, . . . , d j=1 Pi (0) = pi stock ˆ ⇒ Representations of prices as Ito processes 132 / 477
  133. 133. 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
  134. 134. 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 in bonds. (5) Investors act in a self-financing way (no secret source or sink for money). 134 / 477
  135. 135. Requirements on a market model (6) Securities are perfectly divisible. (7) Negative positions in securities are possible bond → credit stock → we sold some stock short. (8) No transaction costs. 135 / 477
  136. 136. 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 the same rate as we would get for investing in bonds. Interest depends on the market situation ((t, ω) ∈ [0, T ] × Ω), but not on positive or negative bond position. 136 / 477
  137. 137. 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 T ϕ(0) = ϕ0 (0), ϕ1 (0), . . . , ϕd (0) Time t > 0: – Trading strategy: ϕ(t) (1) ⇒ trading strategy is progressively measurable with respect to {Ft }t Decisions on buying and selling are made on basis of information available at time t (→ modelled by {Ft }t ) (5) ⇒ only self-financing trading strategies should be used. 137 / 477
  138. 138. 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 138 / 477
  139. 139. Discrete-time example: self-financing strategy t =0 Investor uses initial capital to buy shares of bond and stock X (0) = x = ϕ0 (0) · P0 (0) + ϕ1 (0) · P1 (0). 139 / 477
  140. 140. Discrete-time example: self-financing strategy t =1 Security 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). 140 / 477
  141. 141. Discrete-time example: self-financing strategy t =2 Invest remaining capital at the market Wealth: X (2) = ϕ0 (2) · P0 (2) + ϕ1 (2) · P1 (2). Wealth = total wealth of shares held Total: 2 X (2) = x + ϕ0 (i − 1) · (P0 (i) − P0 (i − 1)) i=1 +ϕ1 (i − 1) · (P1 (i) − P1 (i − 1)) 2 − C(i). i=1 141 / 477
  142. 142. Discrete-time example: self-financing strategy 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) 142 / 477
  143. 143. Discrete-time example: self-financing strategy Continuous-time setting Wealth process corresponding to strategy ϕ(t): t t t X (t) = x + ϕ0 (s) dP0 (s) + ϕ1 (s) dP1 (s) − c(s) ds 0 0 0 ˆ ⇒ Price processes are Ito processes. 143 / 477
  144. 144. Trading strategy and wealth processes Definition (1) A trading strategy ϕ with T ϕ(t) := ϕ0 (t), ϕ1 (t), . . . , ϕd (t) is an Rd+1 -valued progressively measurable process with respect to {Ft }t∈[0,T ] satisfying T |ϕ0 (t)| dt < ∞ P-a.s. 0 d T 2 ϕi (t) · Pi (t) dt < ∞ P-a.s. for i = 1, . . . , d . j=1 0 144 / 477
  145. 145. Trading strategy and wealth processes Definition The value d x := ϕi (0) · pi i=0 is called initial value of ϕ. (2) Let ϕ be a trading strategy with initial value x > 0. The process d X (t) := ϕi (t)Pi (t) i=0 is called wealth process corresponding to ϕ with initial wealth x. 145 / 477
  146. 146. Trading strategy and wealth processes Definition (3) A non-negative progressively measurable process c(t) with respect to {Ft }t∈[0,T ] with T c(t) dt < ∞ P-a.s. 0 is called consumption (rate) process. 146 / 477
  147. 147. Trading strategy and wealth processes Definition A pair (ϕ, c) consisting of a trading strategy ϕ and a consumption rate process c is called self-financing if the corresponding wealth process X (t) satisfies d t t X (t) = x + ϕi (s) dPi (s) − c(s) ds P-a.s. i=0 0 0 current wealth = initial wealth + gains/losses - consumption 147 / 477
  148. 148. Trading strategy and wealth processes Remark We have t t ϕ0 (s) dP0 (s) = ϕ0 (s) P0 (s) r (s) ds 0 0 t t ϕi (s) dPi (s) = ϕi (s) Pi (s) bi (s) ds 0 0 m t + ϕi (s) Pi (s) σij (s) dWj (s), i = 1, . . . , d . j=1 0 148 / 477
  149. 149. Self-financing portfolio process Definition Let (ϕ, c) be a self-financing pair consisting of a trading strategy and a consumption process with corresponding wealth process X (t) > 0 P-a.s. for all t ∈ [0, T ]. Then the Rd -valued process T ϕi (t) · Pi (t) π(t) = π1 (t), . . . , πd (t) with πi (t) = X (t) is called a self-financing portfolio process corresponding to the pair (ϕ, c). 149 / 477
  150. 150. 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 ϕ0 (t) · P0 (t) 1 − π(t)T 1 = , where 1 := (1, . . . , 1)T ∈ Rd . X (t) (3) Given knowledge of wealth X (t) and prices Pi (t), it is possible for an investor to describe his activities via a self-financing pair (π, c). → Portfolio process and trading strategy are equivalent descriptions of the same action. 150 / 477
  151. 151. 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 151 / 477
  152. 152. 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 consumption process c(t) if the corresponding wealth equation possesses a unique solution X (t) = X π,c (t) with T 2 X (t) · πi (t) dt < ∞ P-a.s. for i = 1, . . . , d . 0 152 / 477
  153. 153. Admissibility Definition A self-financing pair (ϕ, c) or (π, c) consisting of a trading strategy ϕ or a portfolio process π and a consumption process c will be called admissible for the initial wealth x > 0, if the corresponding wealth process satisfies X (t) ≥ 0 P-a.s. for all t ∈ [0, T ]. The set of admissible pairs will be denoted by A(x). 153 / 477
  154. 154. An example Portfolio process: π(t) ≡ π ∈ Rd 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 of wealth invested in the different stocks and in the bond remain constant over time. Consumption rate is proportional to the current wealth of the investor. 154 / 477
  155. 155. An example Wealth equation: dX (t) = [r (t) − γ] X (t) dt + X (t)π T (b(t) − r (t) 1) dt + σ(t) dW (t) X (0) = 0 Wealth process: t 1 T X (t) = x · exp r (s) − γ + π T b(s) − r (s) · 1 − π σ(s) 2 ds 2 0 t + π T σ(s) dW (s) 0 155 / 477
  156. 156. Outline 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 156 / 477
  157. 157. 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 the investors ⇒ Choose Brownian filtration {Ft }t∈[0,T ] Aim: Final wealths X (T ) when starting with initial capital of x. 157 / 477
  158. 158. General assumption / notation General assumption for this section d =m Notation t γ(t) := exp − r (s) ds 0 θ(t) := σ −1 (t) b(t) − r (t) 1 t t T 1 2 Z (t) := exp − θ(s) dW (s) − θ(s) ds 2 0 0 H(t) := γ(t) · Z (t) 158 / 477
  159. 159. Properties of the continuous-time market model b, r uniformly bounded σσ T 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 with respect to {Ft }t∈[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. 159 / 477
  160. 160. 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 of x ≥ 0, i.e., (π, c) ∈ A(x). Then the corresponding wealth process X (t) satisfies t E H(t) X (t) + H(s)c(s) ds ≤ x for all t ∈ [0, T ]. 0 160 / 477
  161. 161. Completeness of the market Theorem: Completeness of the market (2) Let B ≥ 0 be an FT -measurable random variable and c(t) a consumption process satisfying T x := E H(T ) B + H(s)c(s) ds < ∞. 0 Then there exists a portfolio process π(t) with (π, c) ∈ A(x) and the corresponding wealth process X (t) satisfies X (T ) = B P-a.s. 161 / 477
  162. 162. Completeness of the market H(t) can be regarded as the appropriate discounting process that determines the initial wealth at time t = 0 T E H(s) · c(s) ds + E (H(T ) · B) 0 which is necessary to attain future aims. (1) puts bounds on the desires of an investor given his initial capital 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 attained exactly via trading according to an appropriate self-financing pair (π, c) if one possesses sufficient initial capital (completeness/complete model). 162 / 477
  163. 163. Completeness of the market Remark 1/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 ). 163 / 477
  164. 164. Outline 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 164 / 477
  165. 165. Excursion 4: The martingale representation theorem General assumptions (Ω, F, P) complete probability space. {(Wt , Ft )}t∈[0,∞) m-dimensional Brownian motion. {Ft }t Brownian filtration. Definition A real-valued martingale {(Mt , Ft )}t∈[0,T ] with respect to the Brownian filtration {Ft }t is called a Brownian martingale. 165 / 477
  166. 166. The martingale representation theorem Martingale representation theorem Let {(Mt , Ft )}t∈[0,T ] be a square-integrable Brownian martingale, i.e., EMt2 < ∞ for all t ∈ [0, T ]. Then there exists a progressively measurable Rm -valued process Ψ(t) with T 2 E Ψ(t) dt <∞ 0 and t Mt = M0 + Ψ(s)T dW (s) P-a.s. . 0 166 / 477
  167. 167. The martingale representation theorem Corollary Let {(Mt , Ft )}t∈[0,T ] be a local martingale with respect to the Brownian filtration {Ft }t . Then there exists a progressively measurable Rm -valued process Ψ(t) with T 2 Ψ(t) dt < ∞ 0 and t Mt = M0 + Ψ(s)T dW (s) P-a.s. 0 167 / 477
  168. 168. The martingale representation theorem Remark Each local martingale with respect to the Brownian filtration can ˆ be represented as an Ito process. ˆ Each Brownian martingale can be represented as an Ito process. ⇒ Quadratic variation and quadratic covariation are defined. 168 / 477
  169. 169. Outline 3 Option Pricing 169 / 477
  170. 170. Outline 3 Option Pricing Introduction Examples The Replication Principle Arbitrage Opportunity Continuation Partial Differential Approach (PDA) Arbitrage & Option Pricing 170 / 477

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