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handout04_econ_default - Outline Structural models...

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Outline Structural models Intensity models Credit default swaps Econometric modeling of default risk Haipeng Xing Haipeng Xing AMS517, SUNY Stony Brook Econometric modeling of default risk Outline Structural models Intensity models Credit default swaps Outline 1 Structural models of default risk 2 Intensity modeling of corporate and sovereign bonds 3 Credit default swaps Haipeng Xing AMS517, SUNY Stony Brook Econometric modeling of default risk
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Outline Structural models Intensity models Credit default swaps Merton’s model of corporate debt and equity I Suppose that the asset of the corporate (or the total market value of the corporate’s future cash flows) at time t is A t , and the corporate asset (or capital) structure at time t consists of pure equity U t and debt V t in the form of a single zero-coupon bond maturing at time T , of face value D . In the event that the asset value A T of the corporate at maturity is less then the contractual payment D , default is triggered and the corporate gives its total value A T to debtholders. Hence the debtholders receive V T = min( D, A T ) at T and equityholders receive U T = A T - min( D, A T ) = ( A T - D ) + , where x + = max( x, 0) . For simplicity, we don’t assume other distributions (such as dividend) to debt or equity. To evaluate the values of debt and equity at time t < T , we assume the setting of the standard Black-Scholes-Merton model, which includes Haipeng Xing AMS517, SUNY Stony Brook Econometric modeling of default risk Outline Structural models Intensity models Credit default swaps Merton’s model of corporate debt and equity II (A1) The asset A t is a geometric Brownian motion with drift μ and volatility σ , dA t = μA t dt + σ A t dB t , (1) in which B t is a standard Brownian motion on a complete probability space ( Ω , F , ) . (A2) The borrowing and lending can be done at the same riskless interest rate r through a money-market account. (A3) Agents are price takers (i.e., trading in assets has no e ff ect on prices) and there are no transaction costs. (A4) Short selling is allowed, and the asset is perfectly divisible. Note that (A2) and a self-financing trading strategy suggests that there is at most one equivalent martingale measure. Let B t = B t + η t , where η = ( μ - r ) / σ . We have dA t = rA t dt + σ A t dB t . Haipeng Xing AMS517, SUNY Stony Brook Econometric modeling of default risk
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Outline Structural models Intensity models Credit default swaps Merton’s model of corporate debt and equity III By Girsanov’s theorem, B defines a standard Brownian motion under the equivalent probability measure defined by d d = exp ( - η B T - 1 2 η 2 T ) . By Ito’s formula, { e - rt A t ; t [0 , T ] } is a -martingale, and hence is the unique equivalent martingale measure. Therefore, the price U t of equity in the absence of arbitrage is given by U t = E t e - r ( T - t ) U T = E t e - r ( T - t ) ( A T - D ) + = C BS ( A t , D, σ , r, T - t ) , (2) Haipeng Xing AMS517, SUNY Stony Brook Econometric modeling of default risk Outline Structural models Intensity models Credit default swaps Merton’s model of corporate debt and equity IV in which C BS ( A t , D, σ , r, T - t )
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