Outline for this talk

1. SAMSI Credit Risk Working Group PresentationModel for Unified Valuation of Credit and Equity DerivativesApoorv Mathur(Ongoing work)Joint Work with Prof. Jean-Pierre FouqueNovember 16, 2005North Carolina State University

2. Outline for this talk • Objective • Background • Valuation of Credit and Equity Derivatives • Credit Risk: Structural and Intensity Based Approaches • Structural: Two factor Stochastic Volatility Model • Intensity Based: Cox Process (Doubly Stochastic Process) • Intuition • Model • Results

3. Objective A Model where Default may occur by • Hitting a Barrier • Jump to Default Valuation of Credit, Equity Derivatives • Implied Volatility • European Call Options • Credit Spreads • Defaultable Zero coupon Bonds with no recovery Explain/Fit Data, Economic Intuition, Tractability

4. Background • Structural Approach • Diffusion to Default • Merton Model, Black Cox Model • Intensity Based Approach • Jump to Default • Poisson Process • Hybrid Approach • Default may occur through diffusion or a jump

5. Background • Two factor Stochastic Volatility Model • “Stochastic Volatility Effects on Defaultable Bonds” by Jean-Pierre Fouque, Ronnie Sircar, Knut Solna (2004) • It is a Structural model, so Default Probability/ Credit Spreads for very short maturities go to zero • Cox Process (Doubly Stochastic Process) • “Credit Risk” SAMSI Course Notes by Ronnie Sircar (2005) • Need explanation for assuming intensity as stochastic process • Unified Valuation of Corporate Liabilities, Credit Derivatives (Short Maturities), and Equity Derivatives • “A Jump to Default Extended CEV Model” by Vadim Linetsky, Peter Carr (2005) • CEV assumes perfectly negative correlation between Stock price and the Volatility and therefore it is unrealistic

6. Intuition behind the Model • Stock Price and Value of the Firm are modeled using a single framework • Default can occur through jumps that kill the underlying stock price process (send it to a cemetery state) or by hitting a prescribed barrier • Default is an absorbing state • Two common underlying economic factors are driving the volatility and the jump intensity • Mathematically they are just slow and fast mean reverting stochastic processes • The stochastic volatility factors are negatively correlated with the stock price (leverage effect) • Choose Negative correlation between Stock Price & Volatility • Implied Volatility Skew • The default indicators (and jump intensities) are positively related to the historical volatilities • Choose the jump intensity to vary proportionately to Variance • Credit Spreads

7. Basic Notation

8. The Model Pre-default Stock Price (Post Default, St = 0)

9. The Model Payoff Price of Claim Computations

10. Special Cases

11. Monte Carlo Simulations • n = 1000 or 5000, dt = 0.001 or 0.0005 • So = 50, D = 0 or 35, Yo = 0, Zo = 0, m = 0; • r = 0.02, q = 0, σ1 = 0.2; • ρ01 = -0.1, ρ02 = -0.1, ρ12 = 0; • ε = 0.1, δ = 0.05; • ν1 = 0.5 or 1, ν2 = 0.5; • b = 0, 0.1 or 0.2, λ1 = 0 or 2; • K = [42,45,48,50,52,53,55,57]; • T = [1/8,1/4,1/2,3/4,1];

12. Results • Implied Volatility Skews for a Call Option • Term Structure (short maturities) for Credit Spreads of a Zero coupon Bond with no recovery

13. Implied Volatility 1 No Defaults (v1,v2 = 0.5) 2 Black Cox Model (v1,v2=0.5) 3 Intensity Model (b=0.2, λ1=0) 4 Intensity Model (b=0.2,λ1=2)

14. Yield Spreads 1 Structural Black Cox Model (v1,v2 = 0.5) 2 Structural Black Cox Model (v1=1,v2=0.5) 3 Intensity Model (b=0.2, λ1=0) 4 Intensity Model (b=0.2,λ1=2)

15. Yield Spreads 5 Hybrid Model (D=35, v1,v2 = 0.5, b=0.1,λ1=0) 6 Hybrid Model (D=35, v1=1,v2=0.5, b=0.1,λ1=0)

16. Thank You