An evolutionary computing approach to minimize dynamic hedging error

1
An evolutionary computing approach to minimize
dynamic hedging error
Saeid Nahavandi School of Eng and Information
Technology Deakin University, Australia

• Mohammad Khoshnevisan
• School of Accounting Finance
• Griffith University,
  Australia

2
Problem of endogenous capital guarantee

• Let a structured financial product be made up of
  an envelope of J different assets such that the
  investor has the right to claim the return on the
  best-performing asset out of that envelope after
  a stipulated lock-in period
• Then, if one of the J assets in the envelope is
  the risk-free asset then the investor gets
  assured of a minimum return equal to the
  risk-free rate i on his invested capital at the
  termination of the stipulated lock-in period
• This effectively means that his or her investment
  becomes endogenously capital-guaranteed as the
  terminal wealth, even at its worst, cannot be
  lower in value to the initial wealth plus the
  return earned on the risk-free asset minus a
  finite cost of portfolio insurance, which is paid
  as the premium to the option writer

3
Expected payoff from a multi-asset capital
guaranteed structured financial product

• The expected present value of terminal option
  payoff is obtained as follows
• Ê (r) tT Max w, Max j e-it E (rj) tT, j
  1, 2 J 1
• In the above equation, i is the rate of
  return on the risk-free asset and T is the length
  of the investment horizon in continuous time and
  w is the initial wealth invested i.e. ignoring
  insurance cost, if the risk-free asset
  outperforms all other assets then we get
• Ê (r)
  tT weiT/eiT w
• Now what is the probability of each of the (J
  1) risky assets performing worse than the
  risk-free asset? Even if we assume that there are
  some cross-correlations present among the (J 1)
  risky assets, given the statistical nature of the
  risk-return trade-off, the joint probability of
  all these assets performing worse than the
  risk-free asset will be very low over even
  moderately long investment horizons. And this
  probability will keep going down with every
  additional risky asset added to the envelope.
  Thus this probability can become quite negligible
  if we consider sufficiently large values of n

4
Formulation as a generalized stochastic
optimization model

• For an option writer who is looking to hedge his
  or her position, the expected utility
  maximization criterion will require the tracking
  error to be at a minimum at each point of
  rebalancing, where the tracking error is the
  difference between the expected payoff on the
  best-of option and the replicating portfolio
  value at that point
• At each point of re-balancing, the tracking error
  has to be minimized if the difference between the
  expected option payoff and the replicating
  portfolio value is to be minimized. The more
  significant this difference, the more will be the
  cost of re-balancing associated with correcting
  the tracking error and as these costs cumulate
  the less will be the ultimate utility of the
  hedge to the option writer at the end of the
  lock-in period. Then the cumulative tracking
  error over the lock-in period is given as
• ? ?t E (rt) vt
  
• Here E (r) t is the expected best-of option
  payoff at time-point t and vt is the replicating
  portfolio value at that point of time. Then the
  replicating portfolio value at time t is obtained
  as the following linear form
• vt (p0) t eit ? j (pj) t (Xj) t, j 1, 2
  J 1
• Here (Xj) t is the realized return on asset j at
  time-point t and p1, p2 pJ-1 are the respective
  allocation proportions of investment funds among
  the J 1 risky assets at time-point t and (p0) t
  is the allocation for the risk-free asset at
  time-point t. Of course
• (p0) t 1 ?j (pj) t
  
• It is the portfolio weights i.e. the p0 and pj
  values that are of critical importance in
  determining the size of the tracking error. The
  correct selection of these portfolio weights will
  ensure that the replicating portfolio accurately
  tracks the option

5
Casting the objective function as the total cost
of tracking error

• The problem of minimizing the randomness
  associated with the tracking error can be
  mathematically cast as a sequential, stochastic
  optimization problem with respect to the
  portfolio weight vector pt-1T corresponding to
  the last rebalancing time point. The latter
  rebalancing decisions are affected by the earlier
  decisions and also by the randomness or white
  noise component in the market information. The
  squared cost of tracking error for the tth
  rebalancing is obtainable as follows
• C (?t) 2 htdt Maxj E
  (rj) t pt-1T R t 2
• Here ht is a fixed rebalancing cost and dt is a
  binary variable such that dt 1 when Maxj E
  (rj) t ptT R t gt ht and dt 0 otherwise. That
  is, it will be feasible to rebalance at time
  point t only if the rebalancing cost at time
  point t is less than the cost of keeping the same
  portfolio weights as at time point t1. Rt is the
  vector of expected returns on the J assets
  constituting the portfolio

6
Mathematical formulation of the generalized
optimization model

• Let the probability density function for getting
  a specific vector p tT be given by ?t. Then over
  a period of t 1, 2 ... T time points, this is
  obtainable as the joint conditional likelihood
  function obtainable as follows
• ?t ? (ptT pt-1T) ? (pTT pT-1T) ? (pT-1T
  pT-2T) ... ? (p2T p1T) ? (p1T)
• Therefore the expected squared cost of
  tracking error is obtainable as follows
• E C (?t) 2 ? ... ? ?t ? (ptT pt-1T)
  htdt Maxj E (rj) t pt-1T R t 2 dp1T...
  dpTT
• The target is to minimize this expected squared
  cost of tracking error over the entire investment
  horizon i.e. for all t 1, 2 ... T. To allow the
  replicating portfolio to be self-financing, the
  elements of ptT must be unrestricted in sign
• The fundamental stochastic recurrence relations
  to be computed in the minimization procedure are
  obtainable using the standard derivation as
  follows
• Ft (st1) Mint S ?t Qt st, ptT, E C
  (?t) 2, 1 t T, where
• Qt st1, ptT, E C (?t) 2 Rect
  st1, ptT, E C (?t) 2 Ft-1, 2 t T and
• Q1 s2, p1T, E C (?1) 2 Rec1 s2, p1T, E C
  (?1) 2
• In the above minimization procedure
  introduction of random variables causes no
  increase in the number of state variables. Since
  Qt is a function of only one random variable E C
  (?t) 2, only one parameter at a time is
  introduced into the minimization procedure. This
  helps to reduce the considerable difficulties
  associated with multi-variate sequential
  optimization

7
Developing a Genetic Algorithm model as a
computational alternative

• Given a necessarily biological basis of the
  evolution of utility forms (Robson, 1968 Becker,
  1976), a haploid genetic algorithm model, which,
  as a matter of fact, can be shown to be
  statistically equivalent to multiple multi-armed
  bandit processes (Berry and Fristedt, 1985),
  should show satisfactory convergence with the
  Black-Scholes type expected utility solution to
  the problem of minimizing the target cost
  function. This would allow for estimation of the
  optimal weight vector ptT without explicitly
  solving the stochastic optimization problem
  outlined above
• A computational haploid genetic algorithm model
  has been programmed for this purpose in Borland
  C Release 5.02 and performs the three basic
  genetic functions of reproduction, crossover and
  mutation with the premise that in each subsequent
  generation x number of chromosomes from the
  previous generation will be reproduced based on
  the principal of natural selection (De Jong,
  1976). The model is presently restricted to n 3
  underlying assets within the structured financial
  product
• Following the reproduction function, 2(x 1)
  number of additional chromosomes will be produced
  through the crossover function, whereby every g
  th chromosome included in the mating pool will be
  crossed with the (g 1) th chromosome at a
  pre-assigned crossover locus. There is also a
  provision in the computer program to introduce a
  maximum number of mutations in each current
  chromosome population in order to enable rapid
  adaptation

8
A proposed haploid Genetic Algorithm model

• According to our proposed haploid genetic
  algorithm reproduction and crossover functions,
  the size of the nth generation i.e. the number of
  chromosomes in the population at the end of the
  nth generation is given by the following
  first-order, linear difference equation
• Gn Gn 1 2 (Gn 1 1) 3 Gn 1 2
  
• If x initial number of chromosomes are introduced
  at n 0, we have G0 x. Then, obviously, G1 x
  2(x 1) 3x 2 31 (x 1) 1. Extending
  the recursive logic to G2 and G3 we therefore get
  G2 9x 8 32 (x 1) 1 and G3 27x 26
  33 (x 1) 1. Thus, extending to Gt we can
  write the following relation
• Gt 3t (x 1) 1
• Therefore, Gt1 3t1 (x 1) 1. But Gt1
  3Gt 2. Substituting for Gt we thereby get, Gt1
  33t (x 1) 1 2 3t1 (x 1) 3 2
  3t1 (x 1) 1. Therefore the case is proved
  for Gt1. But we have already proved it for G1,
  G2 and G3. Therefore, by the principle of
  mathematical induction, the general formula is
  derived as follows
• Gn 3n (x 1) 1
  
• This simple genetic algorithm performs
  satisfactorily in terms of computational
  efficiency as well as the target minimization
  objective for n 3. However, to reduce
  computational complications at the onset we have
  ignored a part of the objective function i.e.
  htdt. However our proposed computational genetic
  algorithm model may be appropriately extended to
  cover the complete version of the stochastic
  optimization problem with the minimization of the
  expected squared cost of tracking error as the
  target

9
A 2-asset numerical illustration

• The following table shows the hypothesized
  figures relating to the two correlated risky
  assets and the risk-free asset underlying the
  best-of option as have been used in computing the
  expected option pay-offs. These figures have been
  chosen so as to maximize the chances for a
  pay-off pattern whereby each of the risky assets
  may be seen to outperform the other for some
  length of time within the 12-month lock-in period
• A Monte Carlo simulation algorithm was used to
  generate the potential payoffs for the option on
  best of three assets at the end of each month for
  t 0, 2 11. The word potential is crucial in
  the sense that our option is essentially European
  and path-independent i.e. basically to say only
  the terminal payoff counts. However the
  replicating portfolio has to track the option all
  through its life in order to ensure an optimal
  hedge and therefore we have evaluated potential
  payoffs at each t

10
Numerical results

• For an initial input of 1, apportioned at t 0
  as 45 between S1 and S2 and 10 for S0, we have
  constructed five replicating portfolios according
  to a simple rule-based logic k of funds are
  allocated to the observed best performing risky
  asset and the balance (90 k) to the other
  risky asset (keeping the portfolio self-financing
  after the initial investment) at every monthly
  re-balancing point. We have reduced k by 10 for
  each portfolio starting from 90 and going down
  to 50. This simple hedging scheme performed
  quite well over the lock-in period when k 90
  but the performance falls away steadily as k is
  reduced every time. The cumulative tracking
  errors corresponding to choices of k are given in
  the following table

11
Numerical results continued

• What we have done next is to introduce the
  above choices of k into our haploid genetic
  algorithm model encoded in the form of bit
  strings as the initial chromosomes. Subsequently
  we have noted the dominance of chromosomes in the
  range 80 lt k ? 90 after three generations. The
  output of our genetic algorithm model is
  graphically depicted below

12
Adaptation of Evolutionary Optimization
Parameters Based on a Fuzzy Logic Controller

• Our computational result as obtained above does
  provide some experimental support to the premise
  that embedding a Black Scholes type of expected
  payoff (utility) maximization function indeed
  results in evolutionary optimality (Robson, 2001)
  
• Evolutionary optimization algorithms like a
  genetic algorithm principally work by trying to
  optimally prioritize between exploitation
  (existing knowledge) and exploration (new
  knowledge). One of the primary goals in an
  evolutionary optimization set-up is to avoid
  getting stuck in local optima (premature
  convergence) in the effort to optimally
  prioritize between exploitation and exploration.
  As the knowledge base (existing as well as new)
  often consists of vague and imprecise
  information, the genetic algorithm performance
  can be better controlled using fuzzy logic
  controllers (FLCs). The FLC control process
  allows for an ideal man-machine interface for the
  optimal prioritization between exploitation and
  exploration
• The principal goal is to use an FLC with an input
  that is any combination of the genetic algorithm
  performance measures e.g. in our 3-asset dynamic
  hedging problem, it could very well be the mean
  square tracking error of the replicating
  portfolio. In a feedback control mechanism, the
  current performance measure of the genetic
  algorithm are routed through the FLC which
  computes new control parameter values that will
  be subsequently fed into the genetic algorithm.
  Standard genotype diversity measures like Hamming
  Distance, Euclidian Distance and entropy measures
  are also possible input candidates for the FLC
  along with common phenotype diversity measures
  like the span measure span 1/(N-1)S (fi
  f)2½ 1/nSfi-1 for any pre-defined fitness
  criteria fi and the average fitness criterion f

13
Extension of the proposed evolutionary
optimization scheme for the 3-asset dynamic
hedging problem using a simple FLC

• The genetic algorithm we have used here to
  solve the stochastic optimization problem of
  minimizing the option tracking error becomes
  particularly susceptible to getting stuck in
  local optima especially when the return on the
  risky assets underlying the best-of option have
  temporally unstable correlation and volatility.
  Especially for many of the longer term options
  implied volatility measures are often based on
  subjective and imprecise probability assessments
  which can significantly slow down the search
  speed (Xu and Vukovich, 1993). Genetic algorithms
  controlled by FLCs can resolve the imprecise
  information dynamically during run-time thereby
  allowing faster adaptation (Herrera et. al.,
  1994). In the context of our 3-asset dynamic
  hedging problem, a fuzzy rule pseudocode could be
  implemented as follows
• IF
• 
  r1 MAX (r0, r1, r2) 2 is ltsmallgt
• THEN
• 
  raise k ltslightlygt
• ELSE
• 
  lower k ltslightlygt
• The actual computational implementation of a
  FLC based genetic algorithm then becomes the
  immediate step to advance in the direction we
  have shown