Optimal Option Replication by Minimizing Initial Portfolio

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Optimal Option Replication by Minimizing Initial
Portfolio

• Joon-Hui Yoon
• Seong-Hee Han

2
Outline

• Introduction
• Model Improvement
• Case Study
• Conclusion

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Introduction - Definition

• Financial Option
• A stipulated privilege of buying or selling a
  stated (underlying) stock at a strike price
  within a specified time
• Black-Scholes formula prices the option based on
  the expected price of underlying stock in the
  future
• Option Replication
• A technique to create an option-like payoff
  pattern through a series of transactions in the
  underlying stock to hedge the price of option
  dynamically
• If market friction is considered, replication
  strategy substantially deviates from
  Black-Scholes option price

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Introduction Motivation

• What is the optimal replication strategy to hedge
  the option?
• Maximizing utility, given initial wealth
• Minimizing cost, given terminal wealth
• Objective
• To improve a model which minimizes the initial
  cost of portfolio to replicate an financial option

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Model Improvement Assumptions

• Environment
• No market friction
• No transaction cost or restriction
• Basically following Black-Scholes price
• Option
• European Call
• Privilege to buy a stock at specified price at a
  fixed day
• Stock Dynamics
• Geometric Brownian Motion
• Path-Independent
• Trinomial Lattice
• Hedging Portfolio
• Underlying Stock and Risk-Free Bond
• Self-Financing
• No infusion of cash after initial investment

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Model Improvement Stock Dynamics

• Geometric Brownian Motion
• Stock price is governed by drift (average
  increase) and volatility (uncertainty)

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Model Improvement Stock Dynamics

• Geometric Brownian Motion Example
• Paths of daily stock prices of 5 German companies
  for 3 years
• Excerpted from www.math.ucalgary.ca/aswish/finlab
  talk_28_10_04.ppt

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Model Improvement Stock Dynamics

• Trinomial Lattice
• Price of stock is monitored over successive short
  period of time, and assumed only three movements
  are possible
• Lattice model has innate error than continuous
  GBM model
• Known to have faster conversion than binomial
  lattice
• In geometric brownian motion, it has
  path-independent property

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Model Improvement Stock Dynamics

• Trinomial Lattice Example
• Path-Independent property

Scenarios
4 3 2 1 0 -1 -2 -3 -4
S0
S1
S2
S4
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Model Improvement Basic Model

• Minimizing cost of super-replication of option
  model
• Multistage Stochastic Programming Formulation
• By C. Edirsinghe, V. Naik, R. Uppal, Optimal
  Replication of Options with Transactions Costs
  and Trading Restrictions, 1993, Journal of
  Financial and Quantitative Analysis
  Vol.28(Mar.,1993), 117-138

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Model Improvement Basic Model

• Critics
• No path-independent constraints
• self-finance constraints explode by the
  exponential rate by the time period increases.
• C(T, i)s are not clearly defined
• the terminal constraints are going to be too many
• Difficult to evaluate which C(T, i) can cause
  infeasibility
• the option price and expected payoff of our
  terminal portfolio of all scenarios are not
  related
• the super-replication of initial portfolio is
  difficult to compare with the option price in
  formulation

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Model Improvement Revised Model

• CVaR constraint
• Introduced to replace terminal constraints
• Relates between option payoff and replicating
  portfolio to bind the initial portfolio around
  the real option price.
• very convenient to control, because it has only
  one value for all terminal scenarios.
• Nonanticipativity constraints
• To replace the exponential growth of
  self-financing constraints
• Reduce the constraints from exponential rate to
  first-order polynomial rate.
• Expected terminal payoff constraint
• Included for CVaR constraint not to lead the
  expected terminal payoff to be negative.
• Since the model allows the loss between payoff
  and portfolio, this is not a super-replication
  model
• Possible to think it as the imperfect-replication
  model

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Model Improvement SP Formulation
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Case Study Methodology

• Implemented revised model through Excel solver
• To verify the result readily
• Flexible to change the parameter and reoptimize
• Geometric brownian motion stock dynamics with
  trinomial lattice
• Path-Independent Stochastic Programming(25 days)
• Comparison with Black-Scholes Formula

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Case Study Excel Solution

• Excel Solver Crap!

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Case Study Excel Solution
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Case Study Analytics

• Comparison among Black-Scholes price, Trinomial
  price, Minimized Portfolio price
• In case study, supposing the parameters are

• Conjecture
• For trinomial tree not yet sufficiently
  converged
• For initial portfolio by expected value
  constraint

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Case Study Analytics

• How does minimization problem decreases CVaR?

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Case Study Analytics

• CVaR Bound
• How much is the lower bound of CVaR (Loss)?
• Consistency of the initial portfolio regardless
  of the variation of Lr.
• Initial portfolio is very close to the option
  price by the other methods
• Value of the initial portfolio is not so
  sensitive to Lr as long as it is feasible

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Case Study Analytics

• Investigates the affect of option values by
  change of strike price (K)
• the options deep out of the money (OTM) and deep
  in the money (ITM) have the tendency to have
  higher implied volatility than at the money (ATM)
• which means they are highly priced. called
  (Volatility Smile).

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Case Study Analytics

• What happens if investor doesnt hedge anything?
• The investor is exposed to the risk of
  uncertainty of future stock price
• Non-Hedging Case coincides with Value of
  Stochastic Solution
• Expected value solution can be found easily by
  setting volatility to be 0

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Conclusion

• Improving from basic model, revised model
  achieves approximation of option price as well as
  reduced number of constraints
• Trinomial Price, Min. Initial Portfolio, and
  Black-Scholes Price are mostly converged
• Min. Initial Portfolio reduces CVaR loss
  significantly
• VSS value shows that Non-Hedging strategy expose
  investor much more risk in terms of CVaR

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Further Study

• Replication with various options
• American Put
• Jump-Diffusion
• Considering Market Friction
• Transaction cost
• Fixed lot size
• Longer Term Test
• More that one year

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Reference

• C. Edirsinghe, V. Naik, R. Uppal, Optimal
  Replication of Options with Transactions Costs
  and Trading Restrictions, 1993, Journal of
  Financial and Quantitative Analysis
  Vol.28(Mar.,1993), 117-138
• J. A .Primbs, Y. Yamada, A moment computation
  algorithm for the error in discrete dynamic
  hedging, 2005, Journal of Banking and Finance
• J. Birge, F. Louveaux, Introduction to
  Stochastic Programming, 1997, Springer
• J. Cox, S. Ross, M. Rubinstein, Option Pricing
  A Simplified Approach, ,1979, Journal of
  Financial Economics 7 229-263
• M. H. Vellekoop, Contingent Claims On
  Defaultable Assets A Trinomial Tree Method,
  Working Paper, 2004, University Of Twente
• P. P. Boyle, T. Vorst, Option Replication in
  Discrete Time with Transaction Costs, 1992,
  Journal of Finance, Vol.47(Mar., 1992), 271-293
• S. Alexander, T. F. Coleman, and Yuying Li,
  Minimizing CVaR and VaR for a portfolio of a
  derivatives, 2004, International Conference on
  Modeling, Optimization, and Risk Management in
  Finance, Cornell University
• V. Ryabchenko, S. Sarykalin, S. Uryasev, Pricing
  European Options By Numerical Replication
  Quadratic Programming with Constraints, 2005,
  Research Report 2005-5, University of Florida
• T. Zariphopoulou, Comment on the valuation of
  contingent claims under portfolio constraints
  Reservation buying and selling prices', European
  Finance Review 3 (1999), 389392

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Questions? Comments?