A Simple Robust Link Between American Puts and Credit Protection

Transcription

1 A Simple Robust Link Between American Puts and Credit Protection Liuren Wu Baruch College Joint work with Peter Carr (Bloomberg) The Western Finance Association Meeting June 24, 2008, Hawaii Carr & Wu American Puts & Credit Protection 1 / 26

2 Background: Linkages between equity and debt markets Merton (74): Structural Models Equity is a call on the firm. American puts on stock are linked to credit [in conjunction with leverage and firm value dynamics] in complicated ways. (Black, Cox (76),... Hull, Nelken, White (04),... Cremers, Driessen, Maenhout, 08) Merton (76): Reduced-Form Models When a company defaults, its stock price drops to zero. American puts are linked to credit through the joint specification of stock price, return volatility, default arrival dynamics. (Carr, Wu (05), Carr, Linetsky (06)) Empirical evidence: credit spreads co-vary with stock price, realized return volatility, option implied volatilities, and implied volatility skew (Collin-Dufresne, Goldstein, Martin (01), Cremers, Driessen, Maenhout, Weinbaum (04), Zhang, Zhou, Zhu (06), Cao, Yu,Zhong (07), Berndt, Ostrovnaya (07)... Carr & Wu American Puts & Credit Protection 2 / 26

3 A new, simple, robust linkage between American puts on the stock and credit protection on the company. What s new? We use an American put (spread) to replicate a standardized default insurance contract that pays one dollar whenever default occurs. The linkage is based on cash flow matching, not stat. co-movements. The linkage is direct: It does not operate through firm value, leverage, or assumed co-movements between stock volatility and default arrival. How simple? The American put spread has the same payoff as a pure credit contract. No Fourier transforms, MC simulation, PIDEs, or trees. No model parameter estimation/calibration. How robust? The linkage remains valid, regardless of specifications on default arrival, interest rates, and pre- and post-default price dynamics (as long as the stock price stays out of a default corridor). Carr & Wu American Puts & Credit Protection 3 / 26

4 The Default Corridor We assume: There exists a flat lower barrier B (0, S 0 ) which the stock price S stays above before any default time. There exists another flat barrier A [0, B) which the stock price drops below at any default time and stays below afterwards. [A, B] defines a default corridor that the stock price can never be in. Mnemonic: B is below S before default; A is above S at and after default. Why might a default corridor exist? Strategic default: Debt holders have incentives to induce default before the equity value vanishes (B > 0). (Anderson, Sundaresan (96), Mella-Barral, Perraudin (97), Fan, Sundaresan (00), Broadie, Chernov, Sundaresan (07), Carey, Gordy (07), Hackbarth, Hennessy, Leland (07).) Sudden default generates sudden drops in equity value (A < B) due to legal fees, liquidation costs, and loss of continuation value on projects... Carr & Wu American Puts & Credit Protection 4 / 26

5 Linking equity American puts to credit protection Suppose that we can trade in two out-of-the-money (OTM) American puts on the same stock with the same expiry T and whose strikes lie inside the company s default corridor: A K 1 < K 2 B. The scaled spread between the two American put options, U p (t, T ) (P t (K 2, T ) P t (K 1, T ))/(K 2 K 1 ) is the cost of replicating a standardized default insurance contract that: pays one dollar at default if the company defaults prior to the option expiry T and zero otherwise. We call this contract a Unit Recovery Claim (URC). The URC s time-t value is: U(t, T ) = E Q t [e rτ 1(τ < T )]. The value of a URC is slightly smaller (due to discounting) than the risk-neutral default probability over the contract horizon, Q t {τ < T } = E Q t [1(τ < T )]. Further assuming a constant risk-neutral hazard rate λ, U(t, T ) = T t λe (r+λ)s ds = λ 1 e (r+λ)(t t) r+λ. Carr & Wu American Puts & Credit Protection 5 / 26

6 Linking CDS to URC A1: Bond recovery rate (R b ) is known. The value of the protection leg of the CDS is linked to the value of a URC by V prot (t, T ) = (1 R b )U(t, T ). A2: Deterministic interest rates. The value of the premium leg of the CDS can also be linked to the whole term structure of URCs, A(t, T ) = k(t, T )E Q t = k(t, T ) [ T t T t e s t e s t (r(u)+λ(u))du ds r(u)du U(t, s) + s t r(u)e s u r(v)dv U(t, u)du A1+A2: We can strip URC term structure from a term structure of CDS, without assuming how default occurs. A3: Constant bond recovery, interest rates, and default rates. we can infer the value of a URC from the CDS spread (k(t, T )): ] ds. λ c (t, T ) = k(t, T )/(1 R b ), U c (t, T ) = λ c (t, T ) 1 e (r(t,t )+λc (t,t ))(T t) r(t,t )+λ c (t,t ). Carr & Wu American Puts & Credit Protection 6 / 26

7 Empirical implications Known: Credit spreads co-move with implied volatility and volatility skew. This evidence is consistent with our model (and many other models). What are the unique empirical implications of our theoretical result? American put spreads struck within the default corridor replicate a pure credit contract (URC). We can also infer the value of the URC from other traded credit contracts, such as credit default swaps (CDS) and recovery swaps. The URC values calculated from the American puts and the CDS should be similar in magnitude, and move together. When they differ significantly, we should be able to design arbitrage trading strategies. Their deviation should not depend on the other usual suspects such as leverage, stock return volatility (RV, IV), which are also determinants of credit spreads in most other models. Carr & Wu American Puts & Credit Protection 7 / 26

8 Empirical Analysis: Sample Selection Collect data on American stock put options from OptionMetrics and CDS spreads from various sources, on a list of companies. Sample period: January 2005 to June 2007 Company selection criteria: OptionMetrics have non-zero bid quotes on one or more put options struck more than one standard deviation below the current spot price and with maturities over 180 days. Reliable CDS quotes are available at 1-, 2-, 3-year maturities. The average CDS spreads at 1-year maturity is over 30bps. Carr & Wu American Puts & Credit Protection 8 / 26

9 List of selected companies Equity Ticker Cusip Number Company Name AMR American Airline CTB Cooper Tire & Ribber DDS Dillard s Inc. EK Eastman Kodak Co F Ford Motor Co GM General Motors Corp GT Goodyear Tire & Rubber Co KBH 48666K10 KB Home Carr & Wu American Puts & Credit Protection 9 / 26

10 Construct URC from American stock put options In theory: U p Pt(K2,T ) Pt(K1,T ) (t, T ) = K 2 K 1, A K 1 < K 2 < B. In practice: How to identify the default corridor [A,B]? If we have put quotes for a continuum of strikes, we can identify the default corridor based on the option price behaviors across strikes. The put option prices are a convex function of the strike outside the corridor, but are a linear function of the strike within the corridor. With only discrete strikes available, we consider 3 choices: C1 Set K 1 = 0 and K 2 to the lowest strike with non-zero bid. C2 Set (K 1, K 2) to the two lowest strikes with non-zero bids. C3 Estimate the convexity of the put mid quotes across adjacent strikes, choose the strike at which the convexity is the lowest as K 2 < S t, and the adjacent lower strike as K 1. These different choices generate largely similar results. We report results using C1 based on its simplicity and the lower transaction cost of using one instead of two options. One can also try to optimize the choice via ex post analysis C1 represents a simple, conservative choice. Carr & Wu American Puts & Credit Protection 10 / 26

11 Infer the URC value from CDS Take CDS quotes available at fixed maturities (1, 2, 3 years). Linear interpolation to obtain CDS spread at the longest option maturity. From the interpolated CDS spread k(t, T ), infer the default arrival rate based on constant interest rate and default rate assumptions and an assumed 40% bond recovery: λ c (t, T ) = k(t, T )/(1 R b ). One can use default recovery swap to fix the recovery. Compute the unit recovery claim value according to, U c (t, T ) = λ c (t, T ) 1 e (r(t,t )+λc (t,t ))(T t) r(t, T ) + λ c. (t, T ) Carr & Wu American Puts & Credit Protection 11 / 26

12 Summary statistics of unit recovery claims Ticker U p U c Cross-market Mean Std Auto Mean Std Auto Correlation AMR CTB DDS EK F GM GT KBH Average Similar mean magnitudes for the unit recovery values from the two markets. High cross-market correlations. Carr & Wu American Puts & Credit Protection 12 / 26

13 Relating cross-market deviations to URC levels U p t U c t = D i (U p t + U c t )/ ln K t + e t (0.0608) (0.0075) D i company dummy. R 2 = 85.24%. The put-implied URC values (U p ) are higher for low-urc firms. Our strike choice (for non-zero bid) may over-estimate the URC value. Choosing a higher strike increases this bias. The CDS-implied URC values are higher for high-urc firms. If equity recovery R(τ) > 0, P t (K, T )/K pays (K R(τ))/K at default, less than the $1 payoff from a URC. U p underestimates the URC value. The bond recovery assumption (R b = 40%) can bias U c. If the actual recovery is lower, our assumption would over-estimate the default probability, and hence the URC value. Market segmentation? Sell CDS, buy American puts. Carr & Wu American Puts & Credit Protection 13 / 26

14 Contemporaneous regressions on the two URC series Ut p = a + but c + U t, Ticker a b R 2 AMR ( 1.42 ) ( ) CTB ( 4.53 ) ( ) DDS ( 3.55 ) ( 0.24 ) EK ( 3.77 ) ( ) F ( 2.44 ) ( ) GM ( ) ( ) GT ( 1.88 ) ( ) KBH ( ) ( 2.85 ) Average ( 1.87 ) ( ) In parentheses are t-statistics against the null: a = 0 and b = 1. Positive slope (co-movements), high R-squared. U t captures the unexplained component of the American put Ut p = P t /K. Non-CDS driven variation in American puts. Carr & Wu American Puts & Credit Protection 14 / 26

15 American put bid-ask quotes and CDS Red CDS (a + bu c t ), Blue American put ask, Green American put bid. AMR CTB DDS EK Unit recovery claim Unit recovery claim Unit recovery claim Unit recovery claim F 0.35 GM 0.2 GT 0.18 KBH Unit recovery claim Unit recovery claim Unit recovery claim Unit recovery claim Carr & Wu American Puts & Credit Protection 15 / 26

16 Explain non-cds driven daily variation in American puts Stock price: U t = a + b S t + e t, Ticker a b R 2 AMR ( 0.33 ) ( ) CTB ( ) ( ) DDS ( 0.66 ) ( ) EK ( ) ( ) F ( ) ( ) GM ( ) ( ) GT ( 0.88 ) ( ) KBH ( ) ( ) There is a negative delta component in the American puts: Either OTM put prices are not purely driven by default risk (as captured in CDS spreads), or: S t contains credit risk information absent from the current CDS quotes. Carr & Wu American Puts & Credit Protection 16 / 26

17 Explain non-cds driven daily variation in American puts Realized volatility: U t = a + b RV t + e t, Ticker a b R 2 AMR ( 0.13 ) ( 0.57 ) CTB ( ) ( 0.68 ) DDS ( ) ( ) EK ( 0.17 ) ( 0.91 ) F ( 0.30 ) ( ) GM ( 0.04 ) ( 0.37 ) GT ( 0.15 ) ( ) KBH ( ) ( ) Nothing much here. Given the CDS quotes, OTM put prices do not respond to changes in RV, as our theory predicts. Carr & Wu American Puts & Credit Protection 17 / 26

18 Explain non-cds driven daily variation in American puts At-the-money implied volatility: U t = a + b ATMV t + e t, Ticker a b R 2 AMR ( 0.45 ) ( 3.23 ) CTB ( ) ( 2.81 ) DDS ( 0.84 ) ( 4.61 ) EK ( 0.33 ) ( 3.70 ) F ( ) ( 3.53 ) GM ( ) ( 2.78 ) GT ( 0.78 ) ( 5.25 ) KBH ( ) ( 2.09 ) Slopes are all positive and significant. There is a risk premium priced into options that is not priced into the contemporaneous CDS quotes. This risk premium could compensate for either: a market risk (such as stochastic volatility), or default risk - i.e., a default risk premium is priced into options but not CDS. Carr & Wu American Puts & Credit Protection 18 / 26

19 Predicting implications: Hypotheses H1: U t U p t (a + bu c t ) is purely due to transient noise in the options market. A positive U t predicts a future decline in the American put value. H2: U t reflects credit risk information from the options market that has not yet shown up in the current CDS quotes. A positive U t predicts a future increase in the CDS spread. Error-correction regressions: U p t+ t = αp + β p U t + e t+ t, U c t+ t = α c + β c U t + e t+ t Under H1, β p < 0 Under H2, β c > 0. Carr & Wu American Puts & Credit Protection 19 / 26

20 Predicting Put Prices and CDS Spreads over 1-day horizon U p t+ t = αp + β p U t + e t+ t, Ut+ t c = αc + β c U t + e t+ t, t = 1 day. Ticker β p R 2 β c R 2 AMR ( ) ( 1.94 ) CTB ( ) ( ) DDS ( ) ( ) EK ( ) ( 2.13 ) F ( ) ( ) GM ( ) ( 1.70 ) GT ( ) ( 0.22 ) KBH ( ) ( 2.06 ) Average ( ) ( 0.57 ) H1: β p is significantly negative for 6 of the 8 companies. R 2 averages at 3%. H2: β c is significantly positive for 4 of 8 companies. R 2 averages at 0.8%. Carr & Wu American Puts & Credit Protection 20 / 26

21 Predicting Put Prices and CDS Spreads over 7-day horizon U p t+ t = αp + β p U t + e t+ t, Ut+ t c = αc + β c U t + e t+ t, t = 7 days. Ticker β p R 2 β c R 2 AMR ( ) ( 1.70 ) CTB ( ) ( ) DDS ( ) ( ) EK ( ) ( 0.59 ) F ( ) ( ) GM ( ) ( 1.62 ) GT ( ) ( ) KBH ( ) ( 2.76 ) Average ( ) ( 0.42 ) H1: β p is significantly negative for 6 of the 8 companies. R 2 averages at 6.9%. H2: β c is significantly positive for 3 of 8 companies. R 2 averages at 2.4%. Carr & Wu American Puts & Credit Protection 21 / 26

22 Predicting Put Prices and CDS Spreads over 30-day horizon U p t+ t = αp + β p U t + e t+ t, Ut+ t c = αc + β c U t + e t+ t, t = 30 days. Ticker β p R 2 β c R 2 AMR ( ) ( ) CTB ( ) ( ) DDS ( ) ( ) EK ( ) ( ) F ( ) ( ) GM ( ) ( 1.73 ) GT ( ) ( ) KBH ( 0.55 ) ( 3.77 ) Average ( ) ( ) H1: β p is significantly negative for 5 of the 8 companies. R 2 averages at 16.2%. H2: β c is significantly positive for 2 of 8 companies. R 2 averages at 5.1%. Carr & Wu American Puts & Credit Protection 22 / 26

23 Prediction Implications: Summary Error-correction regressions: U p t+ t = αp + β p U t + e t+ t, U c t+ t = α c + β c U t + e t+ t R-squares from the first regression is higher than that from the second regression. There are more significantly negative β p than significantly positive β c. Implications: The credit risk information mainly flows from the CDS market to the American put options market. For a few companies, the credit risk information also flows the other way around. Carr & Wu American Puts & Credit Protection 23 / 26

24 Concluding Remarks We propose a simple robust theory that links out-of-the-money American put prices to the stock issuer s credit risk: Simple: A simple spread between two American put options replicates a pure credit insurance contract. Robust: The replication is valid as long as there exists a default corridor, irrespective of pre-default and post-default stock price dynamics, interest rate movements, or credit risk fluctuations. The theoretical linkage has strong empirical support: The values of the credit contract inferred from American put options and CDS spreads have strong, positive correlations. Their deviations predict future movements on American put options. Carr & Wu American Puts & Credit Protection 24 / 26

25 An example: Linking American puts on GM stock... At K = 5 (high open interest), put mid value is $1.22 (Maturity Jan 2010) URC =1.22/5=24.2%. Default probability is slightly higher due to rates. Carr & Wu American Puts & Credit Protection 25 / 26

26 to CDS on GM bonds Default probability at 9/20/2009 is 19.87%. Default probability at 9/20/2010 is 40.16%. Default probability at 1/15/2010 (Linear interpolation) 26.4%. Carr & Wu American Puts & Credit Protection 26 / 26