Inverse problems in derivative pricing: stochastic control formulation and solution via duality

Transcription

1 Inverse problems in derivative pricing: stochastic control formulation and solution via duality Rama CONT Columbia University, New York & Laboratoire de Probabilités et Modèles Aléatoires (CNRS, France) Joint work with: Andreea MINCA (Université Pierre et Marie Curie, France)

2 Based on: R Cont & A Minca (2008) Recovering portfolio default intensities implied by CDO quotes, Columbia Financial Engineering Report No R Cont & I Savescu : Forward equations for portfolio credit derivatives, Chapter 11, New Frontiers in Quantitative Finance: Credit risk and volatility modeling, Wiley (2008). Available on:

3 Outline Option pricing models and the calibration problem Calibration by relative entropy minimization Stochastic control formulation and dual solution Worked out example: portfolio default risk model Calibration of a CDO pricing model Interpretation of dual problem as intensity control problem Solution of HJB equations Algorithm and implementation Default probabilities implied by CDO spreads.

4 Option pricing models Consider a financial market whose evolution is described by a probability space (Ω, F, P) and the market history (F t ) t 0. P= joint law of evolution of market prices (S t,t 0) A financial contract with maturity/expiry T is described by a F T -measurable random variable H describing its (random) payoff. A pricing rule V associated a value process V t (H) toeach contract/payoff H. Probabilistic representation of arbitrage-free option pricing models: if a pricing rule is arbitrage-free then there exists a probability measure Q P such that V t (H) =B(t, T )E Q [H F t ] This is a non-constructive result which does not tell us how to model/choose/specify Q.

5 Calibration problem On derivatives markets we observe the prices (C i,i I) ofvarious contracts (payoffs) (H i,i I): these are the traded/liquid derivatives (at, say t =0). Examples: Equity derivatives: H i = call/put options Interest rate markets: H i = caps, floors, options on swaps Credit markets: H i = credit default swaps, index CDOs In all these cases the payoff is of the form H i = h i (X Ti )whereh i is a deterministic function and X t the price of some underlying asset / observable state variable.

6 A pricing rule is said to be market-consistent or calibrated to the market prices (C i,i I) if i I, C i = V 0 (H i )=E Q [H i ] Problem 1 (Calibration problem). Construct a probability measure Q P on (Ω, F) such that i I, C i = V 0 (H i )=E Q [H i ] This is a (generalized) moment-problem for the law Q of a stochastic process: it is a typical examples of ill-posed inverse problem.

7 Markovian state variables Equity indices: diffusion models dx t = μdt + σ(t, X t )dw t Modeling price discontinuities: Lévy process X t specified by a triplet (γ,σ, ν) whereγ,σ are real and ν is a positive measure on R {0} with ν(dx)(1 x 2 ) < Credit risk models: point process N t representing number of defaults in [0,t], described by the default intensity λ(t, n) = lim Δt 0 1 Δt Q[N t+δt = N t +1 N t = n] defined as the conditional probability per unit time of the next default event. parametrization of Q by a (functional) coefficient α E (diffusion coefficient, intensity function, jump measure,..)

8 Forward problem can be stated as a partial differential equation (diffusion case), integro-differential equation (Lévy case) or system of ODEs (Markovian default models).

9 Model selection by Relative entropy minimization under constraints Diffusion models: Avellaneda Friedman Holmes Samperi 1997, Samperi 2002, Denis & Martini 2004, Carmona & Xu 2000 Monte Carlo setting: Avellaneda et al 2001 Lévy processes: RC & Tankov 2004, 2006 Point processes/credit risk models: RC & Minca 2008 Idea: pick the model consistent with market prices which is the closest to some prior model Q 0 Problem 2 (Calibration via relative entropy minimization). Given a prior model Q 0, find model parameter α E which minimizes inf α E EQ 0 [ dqα ln dqα ] under E Q α [H i ]=C i (1) dq 0 dq 0 In many cases, the Radon-Nikodym derivative dqα dq 0 expressed as a function of the state variable. can be

10 The primal optimization problem Dual formulation inf α E EQ 0 [ dqα ln dqα ] under E Qα [H i ]=C i (2) dq 0 dq 0 admits a dual given by sup μ R I inf α E EQ 0 [ dqα ln dqα ] dq 0 dq 0 i I μ i (E Qα [H i ] C i ) (3) The inner optimization problem inf α E EQ 0 [ dqα ln dqα dq 0 dq 0 i I μ i (h i (X Ti ) C i ) ] (4) is then a stochastic control problem where the control variable is the model parameter α.

11 Parameter calibration via stochastic control Solve the stochastic control problem V (μ) = inf α E EQ 0 [ dqα dq 0 ln dqα dq 0 i I μ i (h i (X Ti ) C i ) ] by dynamic programming. Denote by α (μ) E the optimal control and V (μ) the value function. Optimize over the Lagrange multiplier μ R I : sup V (μ) μ (5) μ R I This step involves an unconstrained concave maximization problem in dimension = number of observations. The solution of the inverse problem is given by α = α (μ ) and E Q [H i ]=C i

12 Parameter calibration via stochastic control Duality reduces infinite dimensional optimization into a finite dimensional one. The dual is in fact easier tosolveifwehave few observations (few constraints). BUT Duality may or may not hold (lack of convexity etc). Need efficient methods for solving the dynamic programming equations for each μ.

13 Example: calibration of a default risk model via intensity control

14 Portfolio credit risk models Idea: model risk neutral/ market implied dynamics of portfolio loss L t. Loss L t is a jump process with increasing, piecewise constant sample paths, whose jump times T j are the default events and whose jump sizes L j are default losses: L t = 1 N n N i (1 R i )1 τi t = i=1 N t j=1 L j (6) where N t = n i=1 1 τ i t is the number of defaults in portfolio before t and L j is loss at j-th default event.

15 Sample path of the loss process

16 Clustering of defaults

17 Default intensity Idea: model the occurrence of jumps via the aggregate default intensity λ t. N t is said to have (F t ) t [0,T ] intensity λ t under Q if N t t 0 λ u du is a Q-local martingale. Intuitively: probability per unit time of the next default conditional on current market information λ t = lim Δt 0 1 Δt Q[N t+δt = N t +1 F t ] Market convention: L j =(1 R)/N is constant. No specific assumption on filtration.

18 Wide variety of specifications for portfolio default intensity: Intensity λ t of (next) default event: λ t (ω) = lim Δt 0 Q(N(t +Δt) =N(t )+1 F t ) Δt Compound Poisson: λ t = f(t) (Brigo & Pallavicini 05) Cox process: default intensity driven by other market factors, not by default itself (Longstaff & Rajan) dλ t = μ(t, λ t )dt + σ(λ t )dw t Continuous-time Markov process: λ t = f(t, N t ) Example: Herbertsson model λ t =(n N t ) N t k=1 b k Dependence on history of defaults/ losses (Hawkes process, Giesecke & Goldberg): λ t = g(t j,l j,j =1..N t 1)

19 CDOs Idea: insurance against a portion (tranche) of default losses in a given portfolio Portfolio (index) with n names ( e.g. n = 125) Number of defaults in [0,t]: N t Discount factor B(t, T ) Portfolio loss (as percentage of total nominal): L t = 1 N n i=1 (1 R i)1 τi t Tranche attachment/detachment points 0 a<b 1. Tranche loss: L a,b (t) =(L(t) a) + (L(t) b) +

20 Cash flow structure of a CDO tranche Default leg: tranche loss due to defaults between t j 1 and t j Cash flow at t j N[L a,b (t j ) L a,b (t j 1 )] Value at t =0 J N B(0,t j )E Q [L a,b (t j ) L a,b (t j 1 )] (7) = N j=1 J B(0,t j )E Q [(L(t j ) a) + (L(t j ) b) + j=1 (L(t j 1 ) a) + +(L(t j 1 ) b) + ] Similar to pricing of a portfolio of calls on L(t). Requires knowledge of the risk neutral distribution of total portfolio loss L(t)

21 Premium leg: pays fixed spread S(a,b) at dates t j on remaining principal Cash flow at t j S(a, b)n(t j t j 1 )[(b L(t j )) + (a L(t j )) + ] J Value at t =0 S(a, b)n B(0,t j )(t j t j 1 ) j=1 E Q [(b L(t j )) + (a L(t j )) + ] Computation of E Q [(L(t j ) K) + ] requires knowledge of the (risk neutral) distribution of total loss L(t j ) which depends on dependence among defaults

22 Fair spread of a CDO tranche swap with attachment point a and detachment b initiated at t =0: S 0 (a, b) = J j=1 B(0,t j)e Q [L a,b (t j ) L a,b (t j 1 )] J j=1 B(0,t j)(t j t j 1 )E Q [(b L(t j )) + (a L(t j )) + ] Constraint S 0 (a, b) =C canbewrittenase Q [H] =0where H = S 0 (K i,k i+1,t k ) B(0,t j )(t j t j 1 )[(K i+1 L(t j )) + (K i L(t j )) + ] t j T k B(0,t j )[(K i+1 L(t j )) + (K i L(t j )) + (K i+1 L(t j 1 ) + +(K i L(t j 1 )) + )) ] (8) t j T k

23 Data: ITRAXX CDO tranches

24 Maturity Low High Bid\ Upfront Ask\ Upfront 0% 3% 11.75% 12.00% 3% 6% Y 6% 9% % 12% % 22% % 100% Y 0% 3% 26.88% 27.13% 3% 6% % 9% % 12% % 22% % 100%

25 Maturity Low High Bid\ Upfront Ask\ Upfront 10Y 0% 3% 41.88% 42.13% 3% 6% % 9% % 12% % 22% % 100% Table 1: ITRAXX tranche spreads, in bp. For the equity tranche the periodic spread is 500bp and figures represent upfront payments.

26 Information content of credit portfolio derivatives Market observations consist of fair spreads for (index) CDO tranches. These can be represented in terms of expected tranche notionals C(t j,k i )=C i = E Q [(K i L tj ) + ] (9) Common procedure is to strip CDO spreads to get expected tranche notionals C(t j,k i ) and then calibrate these using a model. Problem: we need C(t j,k i ) for all payment dates t j : many more than data observed! Ill-posed linear problem parametrization of C(.,.) / interpolation usually used Here we will avoid this step and use a nonparametric approach

27 Information content of credit portfolio derivatives Proposition 1. Consider any non-explosive jump process (L t ) t [0,T ] with a intensity process (λ t (ω)) t [0,T ] and IID jumps with distribution F. Define ( L t ) t [0,T ] as the Markovian jump process with jump size distribution F and intensity λ eff (t, l) =E Q [λ t L t = l, F 0 ] (10) Then, for any t [0,T ], L t and L t have the same distribution conditional on F 0. In particular, the flow of marginal distributions of (L t ) t [0,T ] only depends on the intensity (λ t ) t [0,T ] through its conditional expectation λ eff (.,.). Analogy with local volatility.

28 Corollary 1 (Information content of non-path dependent portfolio credit derivatives). The value E Q [f(l T ) F 0 ] at t =0of any derivative whose payoff depends on the aggregate loss L T of the portfolio at on a fixed grid of dates, only depends on the default intensity (λ t ) t [0,T ] through its risk-neutral conditional expectation with respect to the current loss level: λ eff (t, l) =E Q [λ t L t = l, F 0 ] (11) In particular, CDO tranche spreads and mark-to-market value of CDO tranches only depends on the transition rate (λ t ) t [0,T ] through the effective default intensity λ eff (.,.).

29 Forward equation for expected tranche loss (Cont & Savescu 2007) Proposition The expected tranche loss C(T,K)=E Qλ [(K L T ) + ] solves a (Dupire-type) forward equation C(T,K) C(T,K δ)λ k (T )+λ k 1 (T )C(T,K) T k 2 + [λ j+1 (T ) 2λ j (T )+λ j+1 (T )] C(T,jδ) = 0 (12) j=1 where λ k (T )=λ eff (T,kδ)andδ =(1 R)/N. This bidiagonal system of ODEs can be solved efficiently with high-order time stepping schemes (e.g. Runge Kutta).

30 Proposition 2. The expected tranche notional C k (T )=C t0 (T,kδ) solves the following forward equation, where λ k (T )=λ eff (T ): C k (T ) = λ k (T )C k 1 (T ) λ k 1 (T )C k (T ) T k 2 C j (T )[λ j+1 (T ) 2λ j (T )+λ j 1 (T )] j=1 for T t 0, with the initial condition C k (t 0 )=(K L t0 ) + (13)

31 Problem 3 (Calibration problem). Given a set of observed CDO tranche spreads (S 0 (K i,k i+1,t k ),i=1..i 1,k =1..m) for a reference portfolio, construct a (risk neutral) default rate/ loss intensity λ =(λ t ) t [0,T ] such that the spreads computed under the model Q λ match the market observations S 0 (K i,k i+1,t k )= E Qλ t j T k B(0,t j )[L Ki,K i+1 (t j ) L Ki,K i+1 (t j 1 )] E Qλ t j T k B(0,t j )(t j t j 1 )[(K i+1 L(t j )) + (K i L(t j )) + ]

32 Calibration by Relative entropy minimization under constraints One period case: Buchen & Kelly, Avellaneda 1998 Diffusion models: Avellaneda Friedman Holmes Samperi 1997 Monte Carlo setting: Avellaneda et al 2001 Lévy processes: Cont & Tankov 2004, 2006) Given market prices C(K i ) of tranche payoffs and a prior guess λ 0 for the loss intensity process, the reconstruction of the default intensity process (λ t ) t [0,T ] can be formalized as inf E Q 0 [ dqλ ln dqλ ] (14) Q λ Λ dq 0 dq 0 under the constraint that the model Q λ prices correctly the observed CDO tranches, where Q λ is the law of the point process with intensity process λ and Q 0 is the law of the point process with intensity λ 0.

33 Problem 4 (Calibration via relative entropy minimization). Given a prior loss process with law Q 0, find a default intensity (λ t ) t [0,T ] which minimizes inf E Q 0 [ dqλ ln dqλ ] under E Qλ [H i,k ] = 0 (15) Q λ Λ dq 0 dq 0 H ik = S 0 (K i,k i+1,t k ) B(0,t j )(t j t j 1 )[(K i+1 L(t j )) + (K i L(t j )) + ] t j T k B(0,t j )[(K i+1 L(t j )) + (K i L(t j )) + (K i+1 L(t j 1 ) + +(K i L(t j 1 )) + )) ] (16) t j T k and Q λ denotes the law of the point process with intensity (λ t ) t [0,T ] and Q 0 is the law of the point process with intensity λ 0. Using the previous result we can restrict Λ to Markovian intensities λ(t, L t ).

34 Computation of entropy Equivalent change of measure for point processes (Jacod 1980, Bremaud 1981) Proposition 3. Let N t be a Poisson process with intensity γ 0 on (Ω, F t, Q 0 ) and λ =(λ t ) t [0,T ] be an F t -predictable process with t 0 λ s ds < Q 0 a.s. (17) Define the probability measure Q λ on F T by dq λ = Z T where Z t = { λ t τj exp dq 0 γ 0 τ j t 0 } (γ 0 λ s ) ds Then N t is a point process with F t intensity (λ t ) t [0,T ] under Q λ.

35 Proposition 4 (Computation of relative entropy). Denote by Q 0 the law on [0,T] of a (standard unit intensity) Poisson process and Q λ the law on [0,T] of the point process with intensity (λ t ) t [0,T ] verifying t 0 λ sds < Q 0 a.s. The relative entropy of Q λ with respect to Q 0 is given by: E Q 0 [ dqλ dq 0 ln dqλ dq 0 ]=E Qλ [ T 0 λ t ln λ t dt + T T 0 λ t dt] (18)

36 Duality Define the Lagrangian T L(λ, μ) =E Qλ [ λ s ln λ s ds + T 0 T 0 λ s ds I i=1 Using convex duality arguments, the primal problem: m μ i,k H ik ] k=1 inf E Q 0 [ dqλ ln dqλ ] under E Qλ [H ik ] = 0 (19) Q λ Λ dq 0 dq 0 is equivalent to the dual problem sup μ R m.i T inf λ Λ EQλ [ 0 T λ s ln λ s ds+t 0 I m λ s ds μ i,k H ik ] (20) i=1 k=1

37 Intensity control problem An intensity control problem is an optimization problem with a criterion of the type T E Qλ [ 0 ϕ(t, λ t,l t )dt + J Φ j (t j,l tj )], j=1 where ϕ(t, λ t,n t )isarunning cost and Φ j (t j,l tj )representsthe terminal cost. Here ϕ(t, λ, L) =λ ln λ +1 λ and Φ j (t j,l tj )= I M ij (K i L tj ) + i=1 where M ij are constants depending on contract features and the initial discount curve.

38 Single horizon case T λ inf λ Λ([0,T ]) EQ [ (λ t ln λ t +1 λ t )dt +Φ(T,L T )], 0 Solution by dynamic programming: introduce the value function V (t, k) =E Qλ [ T 0 (λ t ln λ t +1 λ t )dt +Φ(T,L T ) N t = k] The value function can be characterized in terms of a Hamilton Jacobi equation (Bismut 1975, Bremaud 1982).

39 Proposition 5. (Hamilton-Jacobi equations) Suppose there exists a bounded function V :[0,T ] N V (t, n) differentiable in t, such that V t (t, k)+ inf {λ[v (t, k +1) V (t, k)] + λ ln λ λ +1} = 0 (21) λ ]0,infty[ for t [0,T] and V (T,k)=Φ(T,kδ) (22) and suppose there exists for each n N + an F t -predictable mapping t u (t, N t ) such that for each n N +,t [0,T] λ (t, k) =argmin{λ[v (t, k +1) V (t, k)] + λ ln λ λ +1} (23) λ ]0, [ Then λ t = λ (t, N t ) is an optimal control. Moreover V (t 0,N t0 )=inf λ Λt E Q λ [ T t 0 ϕ(t, λ t,l t )ds +Φ T (λ) F t0 ].

40 In our problem, in the case of a single maturity, the dual problem is an intensity control problem with running cost (ln λ(t, N t ) 1)λ(t, N t )+1 and terminal cost is of the type Φ j (L) = M ij (K i L) +. The Hamilton Jacobi equations are given by V t (t, n)+ inf {λ[v (t, n+1) V (t, n)]+(ln λ(t, n) 1)λ(t, n)+1) = 0 λ Λ which is a system of n = 125 coupled nonlinear ODEs.

41 The maximum in the nonlinear term can be explicitly computed: λ (t, n) =e [V (t,n+1) V (t,n)] (24) V t (t, n)+1 e [V (t,n+1) V (t,n)] = 0 (25) V (T,k)=Φ(T,k) (26) Proposition 6 (Value function). Consider any terminal condition Φ such that Φ(x) =0for x>n 0 δ. Then the solution of (25)-26 is given by = ln[1 + n 0 k j=0 V (t, k) = ln E Q 0 [Φ(T,N T ) N t = k] (27) (T t) (T t)j e (e Φ(T,(k+j)δ) 1)] (28) j!

42 The key is to note that if we consider the exponential change of variable u(t, k) =e V (t,k) then u solves a linear equation u(t, k) + u(t, k +1) u(t, k) =0 with u(t,k)=exp( Φ(T,kδ)) t which is recognized as the backward Kolmogorov equation associated with the Poisson process (i.e. the prior process, with law Q 0 ). The solution is thus given by the Feynman-Kac formula u(t, k; μ) =E Q 0 [e Φ(T,δN T ) N t = k] =E Q 0 [e Φ(T,kδ+δN T t) ] using the Markov property and the independence of increments of the Poisson process. The expectation is easily computed using the Poisson distribution, where the sum over jumps can be truncated

43 using the fact that Φ(x) =0forx nδ: u(t, k; μ) = = n k j=0 n k j=0 which leads to (28). (T t) (T t)j e e Φ(T,(k+j)δ) + j! (T t) (T t)j e e Φ(T,(k+j)δ) +1 j! =1+ n k j=0 j>n k n k j=0 (T t) (T t)j e j! (T t) (T t)j e j! (T t) (T t)j e [e Φ(T,(k+j)δ) 1] (29) j!

44 Case of several maturities Recursive algorithm via dynamic programming principle 1. Start from the last payment date j = J and set F J (k) =Φ J (t J,δk). 2. Solve the Hamilton Jacobi equations (25) on ]t j 1,t j ] backwards starting from the terminal condition V (t j,k)=f j (k) (30) which can be explicitly solved to yield V (t, k; μ) ont ]t j 1,t j ] using (28). 3. Set F j 1 (k) =V (t j 1,k)+Φ j 1 (t j 1,kδ) 4. Go to step 2 and repeat. Discontinuities may appear in value function at junction dates.

45 Calibration algorithm 1. Solve the dynamic programming equations (25) (26) μ R I to compute V (0, 0,μ). 2. Optimize V (0, 0,μ)overμ R I J using a gradient based method: inf V (0, 0,μ)=V (0, 0,μ )=V (0, 0) μ R I 3. Compute the calibrated default intensity (optimal control) as follows: λ (t, k) =e V (t,k) V (t,k+1) (31) 4. Compute the term structure of loss probabilities by solving the Fokker-Planck equations. 5. The calibrated default intensity λ (.,.) can then be used to compute CDO spreads for different tranches, forward tranches

46 etc. in a straightforward manner: first we compute the expected tranche loss C(T,K) by solving the forward equation: C(T,K) C(T,K δ)λ k (T )+λ k 1 (T )C(T,K) T k 2 + [λ j+1 (T ) 2λ j (T )+λ j+1 (T )] C(T,jδ)=0 (32) j=1 where λ k (T )=λ eff (T,kδ). In particular the calibrated default intensity can be used to fill the gaps in the base correlation surface in an arbitrage-free manner, by first computing the expected tranche loss for all strikes and then computing the spread/ base correlation for that strike.

47 Empirical results: ITRAXX

48 Maturity Low High Bid\ Upfront Ask\ Upfront 0% 3% 11.75% 12.00% 3% 6% Y 6% 9% % 12% % 22% % 100% Y 0% 3% 26.88% 27.13% 3% 6% % 9% % 12% % 22% % 100%

49 Maturity Low High Bid\ Upfront Ask\ Upfront 10Y 0% 3% 41.88% 42.13% 3% 6% % 9% % 12% % 22% % 100% Table 2: ITRAXX tranche spreads, in bp. For the equity tranche the periodic spread is 500bp and figures represent upfront payments.

50 Calibrated default intensity function λ(t,n) t 0 0 N Figure 1: Calibrated intensity function λ(t, L): ITRAXX Europe September 26, 2005.

51 60 Calibrated default intensity at t= N Figure 2: Dependence of default intensity on number of defaults for t = 1 year: ITRAXX September 26, 2005.

52 Calibrated (circle) and market (line) spreads Y 7Y 10Y bps (% for 0 3) Index Index/Tranche Figure 3: Observed vs calibrated CDO spreads. ITRAXX Europe, Sept

53 0.8 Implied loss distributions Loss level Figure 4: Implied loss distributions at various maturities: ITRAXX Europe Series 6, March

54 Calibrated default intensity function λ(t,n) t N 100 Figure 5: Calibrated intensity function λ(t, n): ITRAXX September 26, 2008

55 Calibrated default intensity function λ(t,n) Calibrated default intensity function λ(t,n) t 0 0 N t N 100 Figure 6: Before and after the credit crisis: 2005 vs 2008

56 Calibrated default intensity at t= N Figure 7: Dependence of default intensity on number of defaults for t =1year: ITRAXX March 25, 2008.

57 Calibrated (circle) and market (line) spreads 600 5Y 7Y 10Y 500 bps (% for 0 3) Index Index/Tranche Figure 8: Observed vs calibrated CDO spreads. ITRAXX Europe March 25, 2008.

58 Default intensity non-monotonic in observed number of defaults. Low initial default rate but sharp increase as soon as a few default occurs: contagion. Insurance against first losses was much cheaper before the crisis and was priced at a much lower default rate than insurance against large losses. Similar results obtained with parametric models for λ(n) (Herbertsson model).

59 Conclusion Stochastic control method for solving a model calibration problem. Rigorous methodology for calibrating a pricing model to market data. Convexity guarantees convergence. Nonparametric: no arbitrary functional form for the default intensity. No need to interpolate/smooth input data. Unconstrained convex minimization in dimension = number of observations sec on laptop. Results point to default contagion effects in the riskneutral loss process.