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Unformatted text preview: IEOR E4731: Credit Risk and Credit Derivatives Lecture 04: Structural models of default. Notes originally written by Prof. Rama Cont Instructor: Xuedong He Spring, 2012 1 / 46 Capital structure of a firm I The structural approach to default modeling was initiated by [Black and Scholes, 1973] and [Merton, 1973] . I The starting point of this approach is the capital structure of the firm. I A firm can raise capital either by issuing equity (stocks) or by issuing debt (bonds). I The assets of the firm are divided into debt and equity . I V t : total assets of the firm I S t : equity I D t : value of debt I Equity + Debt = S t + D t = V t . 2 / 46 Lehman Brothers stock price in 2008 3 / 46 Enron stock price 20002001 4 / 46 Winstar Communication before default in April 2001 [source: Moodys KMV] 5 / 46 Default defined in terms of capital structure I Default happens when assets are not sufficient to pay back debt. I Debt: modeled as zerocoupon bond with nominal L and maturity T I Default at t = T if firm is unable to pay back debt: V T < L . I Default only happens at maturity T : = T 1 V T <L + 1 V T L I Payoff to bondholders: D T = min( V T ,L ) = L max( L V T , 0) I Shareholders get what is left after debt is paid back: S T = max( V T L, 0) 6 / 46 Default defined in terms of capital structure (Contd) I The above setting is the capital structure in Merton (1974) model I Equity behaves like a call option on V t with maturity T and strike L I Bond behaves like a portfolio of a defaultfree bond of nominal L and a short position in put option on V t with maturity T and strike L . I Riskneutral pricing (assuming the short rates are independent of the asset value) S t = B ( t,T ) E Q [max( V T L, 0)  F t ] D t = B ( t,T ) E Q [ L max( L V T , 0)  F t ] = V t S t . 7 / 46 Merton (1974) model I [Merton, 1974] assumes I constant short rate (interest rate) r , so B ( t,T ) = exp[ r ( T t )] . I Black Scholes dynamics for asset value (under Q ): dV t = rV t dt + V t dW t . I Therefore, S t = C BS ( t,V t ,T,L, ) D t = V t C BS ( t,V t ,T,L, ) = LB ( t,T ) P BS ( t,V t ,T,L, ) where C BS ( t,V t ,T,L, ) and P BS ( t,V t ,T,L, ) are call price and put price in standard BlackScholes formulae. 8 / 46 Black Scholes formula If the interest rate is r , then a call option with strike K and maturity T is valued at C BS ( t,V t ,T,K, ) where C BS ( t,V,T,K, ) = V N ( d 1 ) KB ( t,T ) N ( d 2 ) d 1 = ln V K + ( r + 2 2 )( T t ) T t = ln V B ( t,T ) K + 2 2 ( T t ) T t d 2 = d 1 T t = ln V B ( t,T ) K 2 2 ( T t ) T t N ( u ) = 1 2 Z u exp( z 2 2 ) dz By putcall parity, P BS ( t,V,T,K, ) = KB ( t,T ) N ( d 2 ) V N ( d 1 ) 9 / 46 Sensitivity to parameters The value of the risky debt is I increasing in V I increasing in L I decreasing in r I decreasing in timetomaturity I decreasing in volatility 10 / 46 Credit spread I Define the continuously compounded yield...
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 Spring '12
 HeXuedong

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