This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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: Moody’s 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 : τ = 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 (Cont’d) 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 + σV t dW t . I Therefore, S t = C BS ( t,V t ,T,L,σ ) D t = V t C BS ( t,V t ,T,L,σ ) = LB ( t,T ) P BS ( t,V t ,T,L,σ ) where C BS ( t,V t ,T,L,σ ) and P BS ( t,V t ,T,L,σ ) 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,σ ) where C BS ( t,V,T,K,σ ) = V N ( d 1 ) KB ( t,T ) N ( d 2 ) d 1 = ln V K + ( r + σ 2 2 )( T t ) σ √ T t = ln V B ( t,T ) K + σ 2 2 ( T t ) σ √ T t d 2 = d 1 σ √ T t = ln V B ( t,T ) K σ 2 2 ( T t ) σ √ T t N ( u ) = 1 √ 2 π Z u∞ exp( z 2 2 ) dz By putcall parity, P BS ( t,V,T,K,σ ) = 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 σ 10 / 46 Credit spread I Define the continuously compounded yield...
View
Full
Document
 Spring '12
 HeXuedong

Click to edit the document details