Hybrid Structural Default Modeling

1. Hybrid Structural Default Modeling Stephen D. Young Chief Risk Officer Affiliated Managers Division Wells Fargo Asset Management Marat V. Kramin DirectorFixed Income AnalyticsWells Fargo Securities March 25, 2013 UNCC

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3. Default Modeling • Statistical Approaches • Discriminant analysis • Logit models • Probit models • Reduced Form Approach • Jump process • Structural Default Approaches • Firm Value • First Passage Frequently used for default prediction of firms and individuals (i.e. “scoring models”). Frequently used for valuing credit derivatives by taking observable CDS or bond data and solving for market implied hazard rates or survival probabilities. Frequently used for ordinal ranking of firm credit quality. Also used for relative value trading (e.g. debt, CDS, and equity trading by hedge funds).

4. Structural Default Modeling: Introduction • The central distinguishing point of “structural models” is the view of debt, equity, and other claims issued by a firm as contingent claims on the firm’s asset value. • Merton (1974) is the seminal work on structural default modeling and is based on a firm defaulting if the value of its assets is below the value of debt at the expiration date of current debt contracts. • Key idea - M is the debt principal amount and VT is the value of the firm at date T. Equity is a call option on firm value and the debt holder’s position is analogous to long a risk-free bond and short a put option on the firm value.

5. Structural Default Modeling: Introduction • Thus we have the value from the perspective of equity and debt holders: Payoff to claimholders Equity M Debt M VT

6. Structural Default Modeling: Firm Value Approach • Merton (1974) assumes that the asset value is given by the following stochastic differential equation in the risk-neutral measure: • In Merton’s (1974) framework we have that the boundary condition for the firm’s equity holders at the terminal period is: • where ET represents the firm’s equity value at time T, VT is the value of the firm’s assets at time T, and M is the face amount of a single pure discount bond or an approximation to total liabilities. • Denoting a bond value as B(t,T) one can represent the payoff to the debt holder as:

7. Merton (1974) Model: Firm Value Approach • Following the Merton (1974) model, the value of the firm at time t is: • where N(.) is the standard normal cumulative density function and: • with r the risk-free interest rate, σv the instantaneous standard deviation of the return on the firm’s assets, t is the current time, and T is the maturity of the debt. And, in this model there is one risk-neutral probability of default given by: • And the following relationship between asset and equity volatility:

8. Merton (1974) Model: Firm Value Approach • The obvious appeal of the Merton (1974) model is the economic intuition whereby default occurs when the firm’s asset value (VT) is less than the debt due (M). • However, Merton’s model contains several restrictive assumptions including: • Simple debt structure (i.e. single debt due at time T where debt maturity is chosen and debt payments are mapped to single payment on debt maturity date in some manner) • Default possible only at maturity of debt (i.e. time T) • Constant interest rates • Default never a “surprise” (i.e. no jump to default)

9. Geske (1977) Model: Firm Value Approach • Geske (1977) generalizes the Merton (1974) model to cases where the firm is financed with coupon-paying debt or with debt maturity at different dates. At each payment date, shareholders decide either to meet their obligation or discontinue firm operations and leave firm assets to debt holders. • The value of the firm with two tranches of debt is given by: • where in the Geske (1977) model with two tranches of debt N2 is the bi-variate cumulative normal distribution function. Yi is related to M1 and M2 and “critical V” – (see paper) with M1short-term debt at time T1and M2long-term debt at time T2. N2 is the bivariate cumulative normal distribution and V is the critical firm value for bankruptcy at T1.

10. Geske Model: Firm Value Approach • Similar to Merton (1974) we have that: • With Geske (1977) and two tranches of debt we can compute the following total, short, and forward risk-neutral probabilities of default: • Delianedis and Geske (2003) find that the compound option formulation provides additional information about migration and default relative to the Merton framework. • However, one drawback to the compound option framework is that with each additional tranche of debt considered one must evaluate a more complex cumulative distribution function (e.g. three tranches tri-variate, …)

11. Structural Default Modeling: Extensions to Merton Framework • There have been numerous extensions to the original Merton (1974) framework including Geske (1977) and other works that account for: • Default before maturity • Stochastic interest rates • Jumps in the firm-value process • Many others • While firm value approaches are based on analyzing a firm’s capital structure and comparing asset value to debt, first passage models allow for default when the asset value drops for the first time below a pre-defined barrier, allowing for default at any time.

12. Structural Default Modeling: Firm Value and First Passage • Given a particular firm and a firm value (e.g. Merton ((1974)) and first passage model (e.g. Black and Cox (1976)) what is the probability that the firm will end up insolvent? V No default in either formulation No default for firm value M Default probability (the area) Default for first passage Default in both formulations T

13. Merton (1976) Model: Option Pricing under Discontinuous Asset Returns • Merton (1976) models the asset price S as a combination of a Brownian motion and a compound Poisson process: where is the Poisson process intensity, the log price jump size are all independent of each other. Jump Risk is diversifiable and earn no risk premium

14. Merton (1976) Model: Option Pricing under Discontinuous Asset Returns • Following Merton (1976), the value of the firm at time t is: • where EMertonis the standard Merton (1974) option value. We also have that: • and: • Thus the Merton (1976) model allows for jump diffusion. Similar to Merton (1974) and Geske (1977) we have that:

15. A Firm-Value Structural Default Lattice based Approach • A simple lattice based approach allows us to address shortcomings of other formulations such as Merton (1974) and Geske (1977). • The first lattice we present will allow for many tranches of debt, coupons, interest payments, etc. • This lattice will serve to “operationalize” the Geske framework so that the implementation to many tranches of debt is quite simple. • After we will see how one may use the lattice to create a hybrid firm value/first passage approach and allow for jumps in the underlying asset value-process.

16. A Firm-Value Structural Default Lattice based Approach • The lattice leverages both backward recursion and forward induction whereby given an underlying process for the firm’s asset value and a debt structure, using backward recursion one assumes that the shareholder’s are rational and will pay off debt obligations if the asset value is greater than the payment due at the time due. • Then, using simple forward induction one may compute not a single risk-neutral probability of default but rather an entire term structure. • For this lattice we will use the following parameterization: • which represent the up and down multipliers, the drift of log asset returns, the risk-neutral probability, and the time increment.

17. A Firm-Value Structural Default Lattice based Approach • Upon evolving the asset value V, we apply backward induction (recursion) to populate the lattice with firm values where default occurs should the asset value become insufficient to meet debt obligations (i.e. Vt ≤ Mt). We carry and indicator variable back through the lattice where one is to signify solvency and zero default. • We then apply forward induction to calculate state prices and calculate survival (default) probabilities from which we can calculate conditional measures.

18. Backward Induction • Set the value of the variable as one at the given time and backward induct it (with no discounting) to time zero resetting all the state (node) values where the n-order compound option on the firm’s assets is exercised (does not exist anymore) to zero along the way. • The calculated probability of option existence at the given time corresponds to survival probability up to this time. • Standard backward induction the iterative equation is as follows: • Ki,jis the set of numbers of lattice nodes at the next (i+1) time layer linked to the node (i,j). The sum is a probability weighted average of the values of the option at the corresponding nodes at the next time layer. Oi,j, di,jare an option value and a local discount factor at node (i,j) respectively. • The backward induction to compute the survival probability Qi,jat node (i,j): • Ii,jis a local indicator at the node (i,j), which is one when the option/company exists and zero if the option is optimal to exercise.

19. Terminal and Exercise Conditions • The terminal conditions in the case of the option valuation and the computation of unconditional survival probability are respectively as follows: • O, V, M are the values of the option (equity), the underlying (assets), and • the compound option strike (debt), and H is the Heaviside function. • At every time t where intermediate exercise may be optimal (i.e. a debt • tranche is due) the following reset calculations take place: • is the indicator variable, which is one when the • option/company exists and zero if the option is optimal to exercise • and are the continuation (i.e. backward-inducted) option and survival probability • and are corresponding values after the optimal option decision

20. Forward Induction • Standard Expectation: • Expectation for Forward Induction: • Standard Density: • Adjusted Density:

21. Forward Induction • The probability P of the option existence at a given time can be alternatively computed as a sum of state prices of security S that have the constant value of one at this time if the option exists (and zero otherwise): • The state prices of such securities can be built forward along the lattice taking into account the existence indicators Ii,jobtained as a by-product during the backward induction used to value the option: • where is the set of numbers of lattice nodes at the previous (i) time • layer linked to the node (i+1,j) and the sum is a probability weighted • average of the values of the state prices at the corresponding nodes at the • previous time layer. • The initial condition when it is not optimal to exercise the option at time zero:

22. Induction Process • We carry option values (i.e. equity) and indicator variables back through the lattice with the backward recursion. • Under the assumption that the firm’s managers are rational, option values are positive if the value of the firm’s assets exceeds the value of liabilities due at points where the firm has debt and perhaps coupons to pay. • In the event the firm’s asset value is less than the liabilities, or if this value is less than a pre-specified default boundary, default occurs and the equity value goes to zero. • With the backward recursion we carry an indicator variable back through the lattice where this variable is reset to zero in the case of default. • For the forward induction, we carry the state price adjusted with the indicator variable through the lattice in order to derive survival probabilities from which we can get unconditional and conditional default probabilities. • At any time slice, the survival probability is the sum of the adjusted state prices S. 22

23. Backward Induction Debt is due at times 2, 3, and 4 with the highlighted region those nodes where the asset value is less than the debt due

24. Forward Induction From forward induction we can calculate the probability of reaching a particular node as the product of one-half and the sum of the probabilities of the two prior nodes. The time dependent survival probabilities are given by the sum of the products of the state prices and indicator variables. From these we can calculate default probabilities and conditional measures

25. Merton (1974), Geske (1977), and Firm-Value Structural Default Lattice • Results for a hypothetical firm that has an asset value of $70B (Billion), volatility of assets of = 20%, seven tranches of liabilities due at each of 1, 2, 5, 7, 10, 20, and 30 years with amounts of $15B, $10B, $2.5B, $2.5B, $5B, $10B and $15B respectively with risk-free rate is assumed to be 3.5%. Liability structure is simplified to accommodate Merton (1974) and Geske (1977) approaches.

26. Firm-Value Structural Default Lattice based Model • Benchmarking was done with simplifications to debt structure for Merton (1974) and Geske (1977). Below is full term structure of risk-neutral survival and default probabilities from the Firm-Value Structural Default Lattice and full debt structure using all liabilities with respective times.

27. A Hybrid Structural Default Model • Firm Value • Default can only occur when debt is due. • Empirically, fails to explain short-term credit spreads largely as a result of default only likely when debt is due and debt may not be due making the probability of default zero and thus credit spreads should be zero. • First Passage • Artificial default boundary is typically some function of asset value or debt. • Less economic intuition than with Firm Value where asset value versus debt are used to define default.

28. Amin (1993) Model • Amin (1993) develops a discrete time model to value options when the underlying follows a jump diffusion process. • Multivariate jumps are superimposed on to a standard binomial lattice. • Consistent with Merton (1976) and his assumption of diversifiable jump risk, in a risk neutral world the governing process for the underlying is given by: • where is the intensity of the jump process, which is the expectation of the distribution function of which are independent and identically distributed random variables corresponding to the Poisson jump magnitudes, is a standard Brownian motion under a risk neutral measure, and is the total number of jumps up to time .

29. Amin (1993) Model • With a partitioning of the trading interval into subintervals of length Amin (1993) derives the following magnitudes and risk-neutral probabilities for the discrete time approximating process: • where the above represent a move up, a jump, and down along with their respective probability measures with the drift of the logarithm of the asset value.

30. Amin (1993) Model • is an integer which spans all nodes at a particular time slice excluding a single move up or down. corresponds to the distribution of the probability mass associated with jumps over all states and for and all other integer is given by: • For we have a move along the center of the lattice. • With local and non-local moves, while built upon a standard binomial lattice with nodes at each time slice, Amin (1993) results in nodes at each time period which is consistent with many trinomial models and necessary to accommodate the case where . The number of total jumps emanating from all nodes at a time slice is equal to one for and .

31. Amin (1993) Model • To be consistent with Merton (1976) we specify a normal distribution for . To complete the specification we need only the risk-neutral measure with probability (see Amin (1993), Equation (25), p. 1847) given by : • It should be noted that the lattice specification can easily be adjusted to accommodate a constant dividend yield with minor modifications to the magnitudes and risk-neutral probability of the lattice equations. • Practical implementation of the lattice entails a truncation of the distribution of non-local moves as one is effectively integrating over a specified density and therefore has to choose the upper and lower limits accordingly.

32. A Hybrid Structural Default Model • The hybrid structural default model lattice construction is based on Amin (1993). • The hybrid lattice approach allows for local and non-local (i.e. jumps) moves in the asset value thereby allowing for default to come as a surprise. • The hybrid lattice allows for a full term structure of debt, coupons, interest, etc and leverages the same backward and forward induction from Jabbour, Kramin, and Young (2010) Journal of Derivatives, Summer 2012. • In addition, we allow for tranches of debt as well as a default boundary. • The Merton (1974), Geske (1977), and Merton (1976) models are particular cases of the hybrid lattice. Many extant models are particular cases of the hybrid lattice.

33. Backward Induction

34. Forward Induction

35. Merton (1974), Geske (1977), Merton (1976) and Hybrid Structural Default Lattice • Benchmark results based on same inputs as prior set (see slide 23). Liability structure is simplified to accommodate Merton (1974), Geske (1977), and Merton (1976) models.

36. Hybrid Structural Default Lattice

37. Conclusions • In this presentation we present a lattice based approach to structural default modeling. • The lattice is flexible and may accommodate complex capital structures and serves to “operationalize” the Geske (1997) model which is used in practice but typically limited to two tranches of debt as a result of the necessary integration. • As shown, the lattice may be extended to include jumps in the asset value process. We implement a hybrid based structural default model by leveraging Amin (1993). With jumps, and an assumption around the distribution when a jump occurs (we assumed that the log of jump magnitude is normally distributed – could be double exponential, etc.), and a more complete liability structure one may better model default and capture short-term spread behavior. With the introduction of a default boundary one may create a hybrid based structural default model that is flexible, computationally efficient, and could produce a wealth of term structures of default probabilities. • Structural default models have been proven to be useful. The proposed hybrid based approach should serve to further enhance their usefulness for ordinal ranking of credit worthiness, relative value trading, and, perhaps relative pricing in the future.