Inverse problems in derivative pricing: stochastic control formulation and solution via duality Rama CONT Columbia University, New York & Laboratoire de Probabilit´ es et Mod` eles Al´ eatoires (CNRS, France) Joint work with: Andreea MINCA (Universit´ e Pierre et Marie Curie, France)
Based on: R Cont & A Minca (2008) Recovering portfolio default intensities implied by CDO quotes, Columbia Financial Engineering Report No. 2008-01. 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: www.cfe.columbia.edu
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.
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 ) to each 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 .
Calibration problem On derivatives markets we observe the prices ( C i , i ∈ I ) of various 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 T i ) where h i is a deterministic function and X t the price of some underlying asset / observable state variable.
A pricing rule is said to be market-consistent or calibrated to the market prices ( C i , i ∈ I ) if C i = V 0 ( H i ) = E Q [ H i ] ∀ i ∈ I, Problem 1 (Calibration problem) . Construct a probability measure Q ∼ P on (Ω , F ) such that C i = V 0 ( H i ) = E Q [ H i ] ∀ i ∈ 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.
Markovian state variables • Equity indices: diffusion models dX t = µdt + σ ( t, X t ) dW t • Modeling price discontinuities: L´ evy process X t specified by a triplet ( γ, σ, ν ) where γ, σ are real and ν is a positive measure ν ( dx )(1 ∧ x 2 ) < ∞ � on R − { 0 } with • Credit risk models: point process N t representing number of defaults in [0 , t ], described by the default intensity 1 λ ( t, n ) = lim ∆ t Q [ N t +∆ t = N t + 1 | N t = n ] ∆ t → 0 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,..)
Forward problem can be stated as a partial differential equation (diffusion case), integro-differential equation (L´ evy case) or system of ODEs (Markovian default models).
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 L´ evy 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 α ∈ E E Q 0 [ d Q α ln d Q α E Q α [ H i ] = C i inf ] under (1) d Q 0 d Q 0 In many cases, the Radon-Nikodym derivative d Q α d Q 0 can be expressed as a function of the state variable.
Dual formulation The primal optimization problem α ∈ E E Q 0 [ d Q α ln d Q α E Q α [ H i ] = C i inf ] under (2) d Q 0 d Q 0 admits a dual given by α ∈ E E Q 0 [ d Q α ln d Q α µ i ( E Q α [ H i ] − C i ) � µ ∈ R I inf sup ] − (3) d Q 0 d Q 0 i ∈ I The inner optimization problem α ∈ E E Q 0 [ d Q α ln d Q α � inf − µ i ( h i ( X T i ) − C i ) ] (4) d Q 0 d Q 0 i ∈ I is then a stochastic control problem where the control variable is the model parameter α .
Parameter calibration via stochastic control • Solve the stochastic control problem α ∈ E E Q 0 [ d Q α ln d Q α � V ( µ ) = inf − µ i ( h i ( X T i ) − C i ) ] d Q 0 d Q 0 i ∈ 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 : µ ∈ R I V ( µ ) sup → µ ∗ (5) 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 ∗ ) E Q ∗ [ H i ] = C i and
Parameter calibration via stochastic control • Duality reduces infinite dimensional optimization into a finite dimensional one. The dual is in fact easier to solve if we have 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 .
Example: calibration of a default risk model via intensity control
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: n N t L t = 1 � � N i (1 − R i )1 τ i ≤ t = L j (6) N i =1 j =1 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.
Sample path of the loss process
Clustering of defaults
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 � t N t − λ u du 0 is a Q -local martingale. Intuitively: probability per unit time of the next default conditional on current market information 1 λ t = lim ∆ t Q [ N t +∆ t = N t + 1 |F t ] ∆ t → 0 Market convention: L j = (1 − R ) /N is constant. No specific assumption on filtration.
Wide variety of specifications for portfolio default intensity: • Intensity λ t of (next) default event: Q ( N ( t + ∆ t ) = N ( t − ) + 1 |F t ) λ t ( ω ) = lim ∆ t ∆ t → 0 – 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)
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): � n L t = 1 i =1 (1 − R i )1 τ i ≤ t N • Tranche attachment/detachment points 0 ≤ a < b ≤ 1. • Tranche loss: L a,b ( t ) = ( L ( t ) − a ) + − ( L ( t ) − b ) +
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 )] J � B (0 , t j ) E Q [ L a,b ( t j ) − L a,b ( t j − 1 ) ] Value at t = 0 N (7) j =1 J B (0 , t j ) E Q [( L ( t j ) − a ) + − ( L ( t j ) − b ) + � = N 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 )
Premium leg: pays fixed spread S(a,b) at dates t j on remaining principal S ( a, b ) N ( t j − t j − 1 )[( b − L ( t j )) + − ( a − L ( t j )) + ] Cash flow at 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
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