1. PhD workshop in Mathematical Finance, Oslo 2006 1 Pricing Vulnerable Options with Good Deal Bounds Agatha Murgoci, SSE, Stockholm

2. PhD workshop in Mathematical Finance, Oslo 2006 2 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. PhD workshop in Mathematical Finance, Oslo 2006 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.

4. PhD workshop in Mathematical Finance, Oslo 2006 4 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

5. PhD workshop in Mathematical Finance, Oslo 2006 5 Vulnerable Options In Complete Markets oBased on the paper of Klein(1996) → structural model oCalculating the price of vulnerable European call using the change of numeraire (old result) oSince the computations are more tractable, one can extend the result by pricing other vanilla vulnerable options, e.g an exchange option

6. PhD workshop in Mathematical Finance, Oslo 2006 6 Vulnerable Options in Incomplete Markets pricing in incomplete markets → no unique EMM → no unique price Classical solutions: Arbitrage pricing pricing bounds are too large Utility based pricing too sensitive to a particular model choice GOOD DEAL BOUNDS

7. PhD workshop in Mathematical Finance, Oslo 2006 7 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

8. PhD workshop in Mathematical Finance, Oslo 2006 8 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

9. PhD workshop in Mathematical Finance, Oslo 2006 9 Structure of the problem in incomplete markets (2 ) Model definition under the objective probability measure EMM oidentified by the usual existence conditions onot unique othe dynamics of all defined processes under the possible EMM Q + constraint on the possible Girsanov kernels defining the set of admissible EMM

10. PhD workshop in Mathematical Finance, Oslo 2006 10 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

11. PhD workshop in Mathematical Finance, Oslo 2006 11 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: φ t is Markov

12. PhD workshop in Mathematical Finance, Oslo 2006 12 Structural model - Results Closed form formula for the price of the option

13. PhD workshop in Mathematical Finance, Oslo 2006 13 Intensity based models Modeling 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; HJB equation:

14. PhD workshop in Mathematical Finance, Oslo 2006 14 Intensity based models -Results Closed form solutions for the homogenous Poisson process case (see left) Numerical solutions for the other cases