Credit Risk : Firm Value Model

Credit Risk : Firm Value Model Prof. Dr. Svetlozar Rachev Institute for Statistics and Mathematical Economics University of Karlsruhe and Karlsruhe Institute of Technology (KIT) Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 1 / 39

Copyright These lecture-notes cannot be copied and/or distributed without permission. Prof. Svetlozar (Zari) T. Rachev Chair of Econometrics, Statistics and Mathematical Finance School of Economics and Business Engineering University of Karlsruhe and Karlsruhe Institute of Technology (KIT) Kollegium am Schloss, Bau II, 20.12, R210 Postfach 6980, D-76128, Karlsruhe, Germany Tel. +49-721-608-7535, +49-721-608-2042(s) Fax: +49-721-608-3811 http://www.statistik.uni-karslruhe.de Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 2 / 39

Risk Market Risk (30% of the total risk) economic factors : current state of the economy government band rate : default free rate FX-rate (Currency) unemployment rate Credit Risk (40% of the total risk) loans defaultable bonds (Corporate bonds) defaultable fixed income securities Operational Risk (30% of the total risk) business lines event types management internal / external fraud Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 3 / 39

Contents Firm value model The defaults are endogenous Option pricing method APT (Arbitrage Pricing Theory) : Stochastic calculus, risk neutral valuation and no-arbitrage markets Intensity based model The defaults are exogenous. The model is designed for large portfolios of corporate bonds. Rating based model Markov chains Rating agencies - nationally recognized statistical rating organizations (NRSROs) Credit derivatives Description Valuation Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 4 / 39

Merton s Firm Value Model The defaultable bond and the stock price are derivatives with underlying the value of the firm. The default time is endogeneous for the model. Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 5 / 39

Example Firm value: V t = D t }{{} debt + E }{{} t = B }{{} t + S }{{} t equity defaultable bond price stock value Merton: What is B t the value of the corporate bond at time t? What is S t the stock value at time t? Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 6 / 39

Simplest Model Suppose V t is the value of the firm at t. Geometric Brownian motion: dv t = µv t dt + σv t dw t (1) where (W t ) t 0 is the Brownian motion on the market measure P (natural world). Let r t be the risk free rate at t, and assume r t r. The bank account: b t = b 0 e rt, b 0 = 1. (2) Discount factor: β t = 1 b t = e rt *(1) and (2) : classical Black-Scholes model for option pricing. Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 7 / 39

B t under Black-Scholes model Let B t = B(t, T ). At maturity t = T, Hence B(t, T ) { D, VT > B(T, T ) = D, V T D V T = min(v T, D) = D max( D V T, 0). = the value of European contingent claim withb(t, T ) = min(v T, D) ] [e r(t t) B(T, T ) Ft = E P = discounted final payoff under risk-neutral measure P given F t = σ(w u, u t) Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 8 / 39

B t under Black-Scholes model Recall that under P (risk-neutral world), dv t = rv t dt + σv t d W t (3) where ( W t ) t 0 is the Brownian motion on P. * On the natural world, Wt = W t + θt where θ = (µ r)/σ is the market price of risk. The solution for (3) is By (4), given information F t, σ2 (r V t = V 0 e 2 )t+σ W t, t 0. (4) σ2 (r V T = V t e 2 )(T t)+σ( W T W t ) on P. Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 9 / 39

B t under Black-Scholes model Therefore, B(t, T ) = e r(t t) ] E P [ B(T, T ) Ft = e r(t t) [ ] E P D max( D VT, 0) F t = e r(t t) D e r(t t) [ ] E P max( D VT, 0) F t. By the Black-Scholes put option price formula, B(t, T ) = e r(t t) D e r(t t) DN( d2 ) + V t N( d 1 ) where = e r(t t) D(1 N( d2 )) + V t N( d 1 ) = e r(t t) DN(d2 ) + V t N( d 1 ) (5) d 1 = ln(v t/ D) + (r σ 2 /2)(T t) σ T t d 2 = d 1 σ T t and N(x) is the cumulative density function of the standard normal distribution. Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 10 / 39

S t under Black-Scholes model Answer 1: Having (5) and V t = B(t, T ) + S t, we have where S t = V t B(t, T ) = V t (1 N( d 1 )) e r(t t) DN(d2 ) = V t N(d 1 ) e r(t t) DN(d2 ) d 1 = ln(v t/ D) + (r σ 2 /2)(T t) σ T t d 2 = d 1 σ T t Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 11 / 39

S t under Black-Scholes model Answer 2: If B(t, T ) is unknown, we view S t as European contingent claim on V t. By Black-Scholes theory, we have ] [e r(t t) S T F t. S t = E P At the terminal time (i.e. at the maturity T ), { VT S T = D, V T > D 0, V T D = max(v T D, 0). Therefore, S t = E P [e r(t t) max(v T D, 0) F t ] * Note : B(t, T ) = V t S t. = V t N(d 1 ) e r(t t) DN(d2 ) Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 12 / 39

Remark Under Merton s model, regardless how complex a defaultable instrument is, price of a structural instrument at time t is given by F t = price of the structural instrument = F( B(t, T ), t), B(t,T )= B(V t,t,t ) ======= F(V t, t, T ) which is an European contingent claim on V t. Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 13 / 39

Hedge Q:How do the bond holders hedge their risk? A: The bond holders are long in the bond. The only security they can use for hedging is the stock. The stock is the only security available for trade. They can buy or sell the stock. Let t ( Delta position at t) be the number of stock shares bought (or sold) at t. The bond holders form a riskless portfolio. Π t = 1 B(t, T ) + t S t (=riskless, like risk free bank account, complete immunization, perfect hedge) * Bond holders typically (in US) immunize 7% of their holding. Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 14 / 39

Hedge The hedge strategy (a t, b t ) = (1, t ) should be self-financing Then Π t = a t B(t, T ) + bt S t t = Π 0 + a s d B(s, T ) + b s ds s 0 0 }{{}}{{} total gain from keeping the bond total gain from trading the stock t dπ t = a t d B(t, T ) + b t ds t = d B(t, T ) + t ds t Because the bond holders want full immunization, i.e. Π t = C t e rt : like a bank account (no randomness). So, dπ t = [ ]dt + 0dW t }{{} no risk Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 15 / 39

Hedge Under the Merton s model B(t, T ) = B(V t, t, T ) = B(V t, t) S t = S(V t, t) = V t B(V t, t) Thus dπ t = 1 d B(V t, t) + t ds(v t, t) = [ ]dt + 0dW t ( Instantaneously risk free portfolio ). Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 16 / 39

Hedge By the Ito formula and we obtain and dv t = µv t dt + σv t dw t, (dv t ) 2 = σ 2 V 2 t dt, d B(V t, t) = B B dt + t V dv t + 1 2 B 2 V 2 (dv t) 2 ( B = + σ2 t 2 V t 2 2 B V 2 + µv B t V ds(v t, t) = S S dt + t V dv t + 1 2 S 2 V 2 (dv t) 2 ( S = + σ2 t 2 V t 2 2 S V 2 + µv S t V ) dt + ) dt + ( σv B ) t dw t V ( ) S σv t dw t. V Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 17 / 39

Hedge Therefore, dπ t [ B = t + σ2 2 V t 2 2 B V 2 + µv B ( S t V + t t + σv }{{} t B V + S t }{{ V dw t } >0 =0 + σ2 2 V t 2 2 S V 2 + µv S t V )] dt and hence we obtain Perfect Hedge!! t = B V S V. Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 18 / 39

Generalization of Merton s model In general V (t) follows Itô process (Continuous diffusion process): dv (t) = µ(t)v (t)dt + σ(t)v (t)dw (t) on P natural world b(t) = b 0 e t 0 r(u)du, r(t) : FRB or ECB rate, F(t)adapted, b 0 = 1. Suppose, there exists unique equivalent martingale measure P. Then every security (portfolio) price P(t) after discounting with b(t) is P-martingale. i.e. [ ] P(t) P(s) b(t) = E P b(s) F(t), 0 < t < s. (6) then it implies, on P, dp(t) = r(t)p(t)dt + [ ] }{{} historical diffusion coefficient d W (t) = known from historical data Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 19 / 39

Generalization of Merton s model In our case P(t) = V (t). Hence dv (t) = r(t)v (t)dt + σ(t)v (t)d W (t) where W (t) is a Brownian motion on P. (Note that the process (σ(t)) t 0 is estimated from historical data.) Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 20 / 39

Generalization of Merton s model The interest rate on P will have general form dr(t) = µ r (t)dt + σ r (t)d W r (t) where W r (t) is a Brownian motion on P. Example (CIR (Cox Ingersol Ross) model.) In real application, µ r (t) and σ r (t) have simple form. dr(t) = (a r b r r(t))dt + σ r r(t)d W (t) (7) :mean reverting Ornstein-Uhlenbeck process. We estimate a r > 0, b r > 0, and σ r > 0 by calibrating the default free term structure interest rate, that is we found the best a, b and σ so that [ t B(0, T i ) = E P e 0 r (u)du] ar,br,σr, i = 1, 2,, M are as closed as possible to the market prices B market (0, T i ) at time t = 0 (today). Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 21 / 39

Generalization of Merton s model Recall dv (t) = r(t)v (t)dt + σ(t)v (t)d W (t) dr(t) = µ r (t)dt + σ r (t)d W r (t). Here W (t) and W r (t) are Brownian motions on P and they are correlated d W (t)d W r (t) = ρdt, where ρ is correlation coefficient with 1 ρ 1. More precisely, W, W r t = ρt Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 22 / 39

Generalization of Merton s model Typically ρ < 0: when the interest rate r(t) goes up, the firm cannot easily borrow money and default are more likely, and hence the firm value V (t) goes down. ρ must be calibrated from defaultable term structure interest rate. In some cases, in practice, ρ is estimated from historical data with the hope that the model is flexible enough to avoid arbitrages. Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 23 / 39

General Default Boundary Value of the default free zero with maturity T evaluated at t (0 t T ): B(t, T ) = E P [e ] T t r(u)du F t where r(t) is default free Term Structure Interest Rate. Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 24 / 39

Default Structure Case 1: Remark: At time τ < T (stopping time), V (τ) hits the boundary, i.e. V (τ) = S B(τ, T ). Then at τ the bond holders sell the company cost C > 0, and get B(τ, T ) : the value of corporate (defaultable) bond. B(τ, T ) = V (τ) C B(τ, T ) = V (τ) C = S B(τ, T ) C = (S C)B(τ, T ) where C is relative cost (e.g. 0.05) such that C B(τ, T ) = C. Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 25 / 39

Default Structure Case 2: Then at T, B(T, T ) is the value of corporate bond at maturity T. { D, VT > B(T, T ) = D, V T D V T = min( D, V T ) Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 26 / 39

Valuation of the Corporate Bond Since the market is complete with unique equivalent martingale measure P, for any payoff P s at s > 0, we have present value at time t < s as P t = E P [e ] s t r(u)du P s F t In our case, P t = B(t, T ) is the value of the corporate bond at t. Hence B(t, T ) [1 τ<t e τ t = E P r(u)du (V (τ) C) + 1 τ T e T t r(u)du min( D, V (T )) F t ] (8) Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 27 / 39

Monte-Carlo Valuation of (8) on P We have to do M-C simulation : dv (t) = r(t)v (t)dt + σ(t)v (t)dw V (t) dr(t) = µ r (t)dt + σ r (t)dw r (t) d W (t)d W r (t) = ρdt (9) We simulate r(t), the default free Term Structure Interest Rate, and obtain the value B(t, T ), 0 t T T. (T : Time horizon. e.g. 30 years ) Thus we know the boundary S B(t, T ) for 0 t T. Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 28 / 39

Monte-Carlo Valuation of (8) on P Simulate the joint process (V (t), r(t)) on P. Discrete version of (9): V (t + t) = V (t) + r(t)v (t) t + σ(t)v (t)(w V (t + t) W V (t)) r(t + t) = r(t) + µ r (t) t + σ r (t)(w r (t + t) W r (t)) corr(w V (t + t) W V (t), W r (t + t) W r (t)) = ρ t where t = 0, t, 2 t,, (N 1) t. Remark: (10) converges with probability 1 to (9) as t 0 if (9) has unique strong solution. The drift and diffusion coefficients must be linear., (10) Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 29 / 39

Monte-Carlo Valuation of (8) on P For t = 0, { V ( t) = V (0) + r(0)v (0) t + σ(0)v (0) tεv r( t) = r(0) + µ r (0) t + σ r (0) tε r, (11) Furthermore, corr( tε V, tε r ) = ρ t corr(ε V, ε r ) = ρ where ε V N(0, 1) and ε r N(0, 1). Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 30 / 39

Monte-Carlo Valuation of (8) on P To simulate the pair (ε V, ε r ), we simulate two independent standard normal random variables (N 1, N 2 ) (i.e. N 1 N(0, 1), N 2 N(0, 1), and corr(n 1, N 2 ) = 0), we set ε V := N 1 ε r := ρn 1 + 1 ρ 2 N 2. Then E[ε V ] = E[ε r ] = 0, Var[ε V ] = Var[ε r ] = 1, corr(ε V, ε r ) = ρ. Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 31 / 39

Monte-Carlo Valuation of (8) on P We continue using independent pairs (N 1, N 2 ) for every step, then we obtain one scenario for (V (t + t), r(t + t)) t=0, t,2 t,,(n 1) t, using N independent pairs of (N 1, N 2 ). We generate S-scenarios (V (j) (t + t), r (j) (t + t)) t=0, t,2 t,,(n 1) t, j = 1, 2,, J. (e.g. J = 10000) Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 32 / 39

Monte-Carlo Valuation of (8) on P V (s) (t) D=1 V(0) V (s') (t) S (S=0.4 or 0.5) S B(0,T) Default boundary τ T Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 33 / 39

Monte-Carlo Valuation of (8) on P The value of the corporate bond under scenario s is B (j) (t, T, ρ) = 1 τ (j) <T e τ t r (j) (u)du (V (j) (τ) C) + 1 τ (j) T e T t r (j) (u)du min( D, V (j) (T )) Given value ρ [ 1, 1], we get the M-C value of the corporate bond B(t, T, ρ) = 1 J J B (j) (t, T, ρ) (12) j=1 * ρ has to be calibrated. Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 34 / 39

Calibration of ρ For every ρ (m) = 1, 1 + ρ,, 1 + M ρ = 1 (e.g. M = 200), we calculate B(0, T, ρ (m) ) using (12). For given credit rating, say BBB, as the credit rating of our firm, we can have data for the market prices B market (0, T i ), i = 1, 2,, I. We find that ρ on the lattice for ρ, such that minimize the following error ( I Bmarket (0, T i ) B(0, ) 2 T i, ρ (m) ) i=1 B market (0, T i ) Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 35 / 39

Valuation of credit derivatives under Merton s model Ex1: The option of a corporate bond. Final payoff: F(T 1, T 2 ) = max(s B(T 1, T 2 ) B(T 1, T 2 ), 0), where 0 < t < T 1 < T 2 (European contingent claim) [ F(t, T 1, T 2 ) = E P e ] T 1 t r(u)du F(T 1, T 2 ) F t Having the M-C engine, generate (V (j) (t), r (j) ), j = 1,, J, then we compute F (j) (t, T 1, T 2 ) = e T 1 t r (j) (u)du max(s B(T 1, T 2 ) B (j) (T 1, T 2 ), 0). Finally the M-C value is F(t, T 1, T 2 ) = 1 J J F (j) (t, T 1, T 2 ). j=1 Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 36 / 39

Valuation of credit derivatives under Merton s model Credit Spread: (Default free) Yield Curve: Y (t, T ) = 1 B(t, T ) = E[e T t T t log B(t, T ) where r(u)du F t ] is obtained from the default free TSIR. Defaultable Yield Curve: Ȳ (t, T ) = 1 T t log B(t, T ) where B(t, T ) is the defaultable bond price given by (8). Because B(t, T ) B(t, T ), Credit Spread S(t, T ) := Ȳ (t, T ) Y (t, T ) 0. Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 37 / 39

Valuation of credit derivatives under Merton s model Ex2: Caplet. Insurance against potential future Credit Spread. It is designed for someone who want to have a protection on nominal principal L (say 10Mio). The terminal value at T 1 < T : F(T 1, T ) = Lδ(T 1, T ) max ( S(T 1, T ) S, 0 ) where S is fixed, and δ(t 1, T ) is the year fraction between T 1 and T. Since we have M-C scenario for B (j) (t, T ) and B (j) (t, T ), we have also S (j) (t, T ) = Ȳ (j) (t, T ) Y (j) (t, T ) = 1 T t (log B (j) (t, T ) log B (j) (t, T )) we have F (j) (T 1, T ) = Lδ(T 1, T ) max ( S (j) (T 1, T ) S, 0 ), and hence F(t, T ) = 1 J J j=1 e T 1 t r (j) (u)du F (j) (T 1, T ) Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 38 / 39

References D. LANDO (2004). Credit Risk Modeling Princeton Series in Finance P. J. SCHÖNBUCHER (2000). The Pricing of Credit Risk and Credit Derivatives http://www.schonbucher.de/papers/bookfo.pdf S. TRÜCK AND S. T. RACHEV (2005). Credit Portfolio Risk and PD Confidence Sets through the Business Cycle https://www.statistik.uni-karlsruhe.de/download/tr_credit_portfolio_risk.pdf Prof. Dr. Svetlozar Rachev (KIT) Firm Value Model 39 / 39