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1 SAMSI Credit Risk Working Group Presentation Model for Unified Valuation of Credit and Equity Derivatives Apoorv Mathur (Ongoing work) Joint Work with Prof. Jean-Pierre Fouque November 16, 2005 North Carolina State University

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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

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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

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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

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5 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 Background

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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

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7 Basic Notation

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8 The Model Pre-default Stock Price (Post Default, S t = 0)

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9 The Model PayoffPrice of Claim Computations

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10 Special Cases

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11 Monte Carlo Simulations n = 1000 or 5000, dt = or So = 50, D = 0 or 35, Yo = 0, Zo = 0, m = 0; r = 0.02, q = 0, σ 1 = 0.2; ρ 01 = -0.1, ρ 02 = -0.1, ρ 12 = 0; ε = 0.1, δ = 0.05; ν 1 = 0.5 or 1, ν 2 = 0.5; b = 0, 0.1 or 0.2, λ 1 = 0 or 2; K = [42,45,48,50,52,53,55,57]; T = [1/8,1/4,1/2,3/4,1];

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

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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)

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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)

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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)

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16 Thank You

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