Portfolio Optimization with Stochastic Volatility and discrete-time observations

1
Portfolio Optimizationwith Stochastic
Volatility anddiscrete-time observations

• Natalia Batalova Frederi Viens
• Bank of America, London, UK
  Purdue University Dept. Statistics

2
Outline

• Problem Overview
• Model Description
• Algorithm
• Results
• Summary

3
Problem Overview

• The Problem
• selection of an optimal portfolio of
• stock and risk-free asset
• Model for the Stock Price
• Black-Scholes Model with constant
• volatility and rate of return
• Solution
• Hamilton-Jacobi-Bellman (HJB) PDE

4
The Black-Scholes Model

• Stock price modeled by continuous-time stochastic
  process (geometric Brownian Motion)
• Allows analytic solution for many problems such
  as
• derivatives pricing
• portfolio optimization
• Built on assumptions that are not satisfied in
  real markets
• mean rate of return and volatility are not
  constants
• information regarding the stock prices arrives at
  discrete times rather than continuous times
• Does not explain empirically observed phenomena
  such as
• volatility smile
• existence of the bull and bear markets

5
Model Extensions

• More flexibility in stock price modeling
• Model mean rate of return as a stochastic process
• Model volatility as a stochastic process
• Use marked point process, jump processes or Levy
  processes
• To deal with incomplete information
• Linear filtering with non-stochastic volatility
• ARCH/GARCH framework
• Nonlinear stochastic filtering
• stochastic volatility
• nonlinear filtering approach

Consider
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The SV Model
where
X risky asset price, B non-risky asset, µ
constant mean rate of return, W Brownian
motion
deterministic function of a stochastic
process
satisfies a diffusion equation driven by another
Brownian motion Z
Fast-mean-reverting process
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Filtering
information contained in a discrete sequence
of observed asset prices X0, X1, , Xi.
a fixed sequence of positive numbers
(observations)
the event
The stochastic volatility filtering problem
estimate the conditional probability
distribution
! Random (depends on X0, X1, , Xi)
.
Replace
by
Non-random
(depends only on )
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Portfolio
self-financing portfolio
wealth
Substitution
initial wealth (given)
The task
find a portfolio a that attains the supremum
the supremum is taken over all non-anticipating
(a,b) (depending only on )
U utility function (Typically
for some )
.
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Recursion Formula for the Filter pi(dy)
remains unchanged

• Change of variables
• under this change of variables, for all t i
• Decompose W(t)
• W?(t) is independent of Z(t),

?
,
?

• Write recursively as

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Recursion Formula for the Filter pi(dy)
EZ expectation with respect to Z only
For any realization of the increments
with respect to W? of the random variable
the density at
,
given
and
Since Y and Z are non-random with respect to W?
where
Hard to evaluate ?
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Assumptions

• Utility function U and its derivative U' are
  bounded
• where ?0 and ?1 are
  positive constants
• where is a
  constant depending only on an integer
• For a fixed integer for some
  positive integer k
• for all i
• for some A gt 0

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Notation

• Truncation. For each , for all
  , if
• let . If
  , let . If
  
• let . Each pair
  will be denoted either or
• depending on whether we
  truncate .
• Discretization. For each , let x be
  the largest element of (1/m')N that is smaller
  than x. For a pair we let
  .
• Euler Method. For any d-dimensional Markov
  process ? with infinitesimal generator ? given by
  
  ,
• with ? a square root of the matrix a, the
  m-step Euler scheme for ? on i,i1
• is the process defined by
  and
• where is a family of
  independent standard Brownian motions.
• We denote .

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The Monte-Carlo Approximation of Fi

• sequence of
  independent random variables that
• share the same distribution with an
  integrable random variable ?
• standard
  Monte-Carlo procedure for the
• approximation
  of EX (empirical mean)
• For fixed and for all bounded
  functions
• restriction to (x,w) of the
  basic Euler approximation
• of .
• The conditioning under M means that the common
  starting point of the n independent copies of
  is .
• Upon iteration, the above procedure yields
  Monte-Carlo approximation
• of utility function supremum V

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Algorithm

• 1. Initialization. Let ,
  and for all k 1,,n.
  For each let
  .
• 2. Calculation of the filter. For each
  , use the del Moral-Jacod-Protter method to
  calculate the particles for
  all i N, k n.
  
  
• Repeat step 3 for i N down to 0
• 3. Calculation of . We assume that we
  know for all
• and as well
  as the corresponding optimal strategy
  . From step 2, we know for all
  , k n. For each , ,
  do the following
• (i) For each k n
• (a) Simulate by the Euler
  Scheme with step 1/m for the pair (X,Y)
• starting from
  on
• (b) Calculate
• (ii) Calculate Monte-Carlo average
• (iii) The i-th step optimal strategy and
  maximum expected yield are

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Calculation of the filter

• Step 1. Simulate n i.i.d. variables
  according to the law ?, and for each i a
  variable according to the law
• Notation. At the end of step k we will know the
  variables ,
• so we can set for every function f on Rd
• Then we introduce the following random
  probability measure on Rd
• Step k?? 2. Simulate n i.i.d. variables
  according to the law .
• Then for each j 1,,n we simulate the random
  variable according to the law
  .
• We stop at the end of Step N. Our approximation
  of will be

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Implementation
utility function
Approximate by
where
?
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Implementation

• Maximum is attained when

?
- optimal strategy
ci proportion of money held in stock
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Algorithm

• (1) Initialization Let ,
  for all
• Repeat steps 2-3 for all i 0,1,, N -1
• (2) Calculation of optimal strategy ci
• (i) For each
• (a) Simulate by the Euler scheme with
  time step
• for the pair (X,Y) starting from
• (b) Calculate
• (ii) Construct Monte Carlo average
• where and find the i-th step optimal
• strategy by solving ,
  i.e. by finding
• such that

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Algorithm (cont.)
?

• (iii) The i-th step optimal strategy and
  maximum expected yield are
• (3) Calculation of the filter
• Use del Moral-Jacod-Protter method to
  calculate
• particles for the next time step
  for all

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Theoretical solution of HJB

• Assume
• - constant volatility ?
• - no consumption
• - power utility function

Wealth dynamics
Corresponding HJB
Solution
- optimal strategy
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Convergence Check

• In the case of constant volatility we can compare
  results with theoretical solution
• Example For the values of parameters
  , , ,
  the optimal strategy is .
• The optimal strategy c does converge to the
  theoretical value 0.5 with increasing number of
  steps in Euler scheme m as

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First Results

• Model
  , , ,
• Parameters values , ,
  , , .

• number of observations
• number of steps in Euler scheme

• 9 sample paths generated with the same values
  of parameters

xt stock price at time t?0,N
exp(yt) volatility at time t?0,N
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First Results (cont.)
wt wealth at time t?0,N
ct ?100 optimal strategy at time t?0,N
(proportion of money held in stock)

• Short-selling (c lt 0) and borrowing (c gt 1) is
  allowed
• Relatively low volatility (such as path 3
  (magenta) or 7 (red dash-dot)) leads to
  intensive borrowing (upto 1500)
• High volatility (such as path 0 (red solid))
  leads to very small c (less than 10 for tgt150)

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First Results Utility Function (sum over 9
paths)
Red optimal strategy Cyan 100
randomly chosen strategies (0ltclt4) Brown
all-money-in-stock strategy Green 100
randomly chosen strategies (-2ltclt6) Blue
all-money-in-bank strategy Black 1
random strategy (-1ltclt5) Magenta 100 randomly
chosen strategies 0ltclt1 (no short-selling or
borrowing allowed).

• Divided by 9 ( paths) this is a very rough
  approximation of our optimization goal, i.e.
  expected utility

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Results Utility Function (sum over 240 paths)
Solid Red optimal strategy Dashed Red
theor. optimal strategy (constant
volatility) Brown all-money-in-stock strategy
Cyan 100 randomly chosen strategies
(0ltclt4) Blue all-money-in-bank strategy
Green 100 randomly chosen strategies
(-2ltclt6) Magenta 100 randomly chosen
strategies (0ltclt1) Black 1 random strategy
(-1ltclt5)

• Number of MC trials 240 m 230

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Things To Do

• Reduce computational time to receive better
  precision in reasonable time
• Check algorithm performance on real data
• Include fixed transaction costs and bid-ask
  spreads
• Allow the portfolio to include several stocks

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Summary

• Optimal portfolio can be found using an
  algorithm that
• Allows for stochastic volatility estimated
  optimally in the setting of incomplete
  information
• Delivers optimal portfolio for the case of
  discrete access to the continuous-time stock
  prices
• Good for high frequency trading since the
  computations only need to be performed once
• Easily incorporates transaction costs