Pricing Vulnerable Options with Good Deal Bounds

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Pricing Vulnerable Options with Good Deal Bounds

• Agatha Murgoci, SSE, Stockholm

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The product

• Vulnerable options options where the writer of
  the option may default, mainly trading on OTC
  markets
• BIS, the OTC equity-linked option gross market
  value in the first half of 2005 USD 627 bln.

3
Previous literature

• treatment in complete markets (Hull-White(1995),
  Jarrow-Turnbull(1995), Klein(1996))
• Hung-Liu (2005) market incompleteness and good
  deal bound pricing for vulnerable options. Only
  Wiener process setup.

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Contributions of the paper

• Streamlining the existing literature on
  vulnerable options in complete markets
• Applying the Bjork-Slinko (2005) method of
  computing good deal bounds to obtain higher
  tractability
• Allowing for a jump-diffusion set up, versus the
  previous Wiener
• Applying both structural and intensity based
  methods for default

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Vulnerable Options In Complete Markets

• Based on the paper of Klein(1996) ? structural
  model
• Calculating the price of vulnerable European call
  using the change of numeraire (old result)
• Since the computations are more tractable, one
  can extend the result by pricing other vanilla
  vulnerable options, e.g an exchange option

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Vulnerable Options in Incomplete Markets

• pricing in incomplete markets ? no unique EMM ?
  no unique price
• Classical solutions

Utility based pricing too sensitive to a
particular model choice
Arbitrage pricing pricing bounds are too large
GOOD DEAL BOUNDS
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Theory of Good Deal Bounds

• Cochrane and Saa Raquejo (2000)
• put a bound on the Sharpe ratio ? a bound on
  the stochastic discount factor ? eliminate
  deals too good to be true ? tighter pricing
  bounds (good deal bounds)
• Bjork and Slinko (2005), variation of Cochrane
  and Saa Raquejo (2000)
• good deal bounds ? bound on the Girsanov
  kernel associated to the change of measure ? can
  use martingale theory in solving the pricing
  problem ? more tractability

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Structure of the problem in incomplete markets (1
)

• Main idea
• set a bound on the possible Sharpe Ratio of
  any portfolio that can be formed on the market ?
  set a bound on the possible Girsanov kernels for
  potential EMM ? set a bound on the possible
  prices for the claim

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Structure of the problem in incomplete markets (2
)

• Model definition under the objective probability
  measure
• EMM
• identified by the usual existence conditions
• not unique
• the dynamics of all defined processes under the
  possible EMM Q
• constraint on the possible Girsanov kernels
  defining the set of admissible EMM

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Structure of the problem in incomplete markets (3
)

• optimization problem for the highest/lowest price
  given the set of admissible EMM
• Additional assumption the Girsanov kernel is
  Markov
• Hamilton Jacobi Bellman equation solved in 2
  stages
• static constrained optimization
• partial differential equation

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Structural Model

• Set-up
• Default depends on the assets of the
  counterparty, not traded in incomplete market
  set-up
• We look for the Girsanov kernel that maximizes/
  minimizes the price of the claim while keeping
  the SR under a certain bound

Additional assumption ft is Markov
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Structural model - Results

• Closed form formula for the price of the option

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Intensity based modelsModeling specifications

• Default modeled as a point process
• homogenous and inhomogeneous Poisson process,
• Cox process,
• continuous Markov chain (credit rating based
  setup )

• Payoff function
• zero and constant recovery
• recovery of treasury
• recovery to market value

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Intensity based models -Results

• Closed form solutions for the homogenous Poisson
  process case (see left)
• Numerical solutions for the other cases