Pricing catastrophe options in incomplete market

Transcription

1 Pricing catastrophe options in incomplete market Arthur Charpentier Actuarial and Financial Mathematics Conference Interplay between finance and insurance, February 2008 based on some joint work with R. Élie, E. Quémat & J. Ternat. 1

2 A short introduction Agenda From insurance valuation to financial pricing Financial pricing in complete markets Pricing formula in incomplete markete The model : insurance losses and financial risky asset Classical techniques with Lévy processes Indifference utility technique The framework HJM : primal and dual problems HJM : the dimension problem Numerical issues 2

4 A general introduction : from insurance to finance Finn and Lane (1995) : there are no right price of insurance, there is simply the transacted market price which is high enough to bring forth sellers and low enough to induce buyers. traditional indemni industry parametr il reinsurance securitiza loss securitiza derivatives securitization traditional indemnity indust parametric il reinsurance securitization loss securitization derivatives securitiza 4

5 A general introduction Indemnity versus payoff Indemnity of an insurance claim, with deductible d : Payoff of a call option, with strike K : X = (S d) + oo X = (S T K) + oo Pure premium versus price Pure premium of insurance product : Price of a call option, in a complete market : π = E P [(S d) + ]oo V 0 = E Q [(S T K) + F 0 ]oo Alternative to the pure premium Expected Utility approach, i.e. solving U(ω π) = E P (U(ω X))oo Yaari s dual approach, i.e. solving ω π = E g P (ω X)oo Essher s transform, i.e. π = E Q (X) = E P(X e αx ) E P (e αx oo ) 5

6 Financial pricing (complete market) Price of a European call with strike K and maturity T is V 0 = E Q [(S T K) + F 0 ] where Q stands for the risk neutral probability measure equivalent to P. the market is not complete, and catastrophe (or mortality risk) cannot be replicated by financial assets, the guarantees are not actively traded, and thus, it is difficult to assume no-arbitrage, the hedging portfolio should be continuously rebalanced, and there should be large transaction costs, if the portfolio is not continuously rebalanced, we introduce an hedging error, underlying risks are not driven by a geometric Brownian motion process. = Market is incomplete and catastrophe might be only partially hedged. 6

7 Impact of WTC 9/11 on stock prices (Munich Re and SCOR) Munich Re stock price Fig. 1 Catastrophe event and stock prices (Munich Re and SCOR). 7

9 The model : the insurance loss process Under P, assume that (N t ) t 0 is an homogeneous Poisson process, with parameter λ. Let (M t ) t 0 be the compensated Poisson process of (N t ) t 0, i.e. M t = N t λt. The ith catastrophe has a loss modeled has a positive random variable F Ti -measurable denoted X i. Variables (X i ) i 0 are supposed to be integrable, independent and identically distributed. Define L t = N t i=1 X i as the loss process, corresponding to the total amount of catastrophes occurred up to time t. 9

10 The model : the financial asset process Financial market consists in a free risk asset, and a risky asset, with price (S t ) t 0. The value of the risk free asset is assumed to be constant (hence it is chosen as a numeraire). The price of the risky asset is driven by the following diffusion process, ds t = S t µdt }{{} trend + σdw t } {{ } volatiliy + ξdm } {{ } t with S 0 = 1 jump where (W t ) t 0 is a Brownian motion under P, independent of the catastrophe occurrence process (N t ) t 0. Note that ξ is here constant. Note that the stochastic differential equation has the following explicit solution ) ] S t = exp [(µ σ2 2 λξ t + σw t (1 + ξ) N t. 10

11 Underlying and loss processes Time Fig. 2 The loss index and the price of the underlying financial asset. 11

12 Financial pricing (incomplete market) Let (X t ) t 0 be a Lévy process. The price of a risky asset (S t ) t 0 is S t = S 0 exp (X t ). X t+h X t has characteristic function φ h with log φ(u) = iγu 1 2 σ2 u ( e iux 1 iux1 { x <1} ) ν(dx), where γ R, σ 2 0 and ν is the so-called Lévy measure on R/{0}. The Lévy process (X t ) t 0 is characterized by triplet (γ, σ 2, ν). 12

13 Get a martingale measure : the Esscher transform A classical premium in insurance is obtained using Esscher transform, π = E Q (X) = E P(X e αx ) E P (e αx ) Given a Lévy process (X t ) t 0 under P with characteristic function φ or triplet (γ, σ 2, ν), then under Esscher transform probability measure Q α, (X t ) t 0 is still a Lévy process with characteristic function φ α such that log φ α (u) = log φ(u iα) log φ( iα), and triplet (γ α, σ 2 α, ν α ) for X 1, where σ 2 α = σ 2, and γ α = γ + σ 2 α (e αx 1)ν(dx) and ν α (dx) = e αx ν(dx), see e.g. Schoutens (2003). Thus, V 0 = E Qα [(S T K) + F 0 ] 13

15 Indifference utility pricing As in Davis (1997) or Schweizer (1997), assume that an investor has a utility function U, and initial endowment ω. The investor is trading both the risky asset and the risk free asset, forming a dynamic portfolio δ = (δ t ) t 0 whose value at time t is Π t = Π 0 + t 0 δ uds u. = Π 0 + δ S) t where (δ S) denotes the stochastic integral of δ with respect to S. A( strategy δ is admissible if) there exists M > 0 such[ that ] T P t [0, T ], (δ S) t M = 1, and further if E P 0 δ2 t St 2 dt < +. x R +, U L (x) = log(x) : logarithmic utility x R +, U P (x) = xp where p ], 0[ ]0, 1[ : power utility p ) x R, U E (x) = exp ( xx0 : exponential utility. 15

16 Indifference utility pricing If X is a random payoff, the classical Expected Utility based premium is obtain by solving u(ω, X) = U(ω π) = E P (U(ω X)). Consider an investor selling an option with payoff X at time T ] either he keeps the option, u δ (ω+π, 0o) = sup δ A E P [U(ω + (δ S) T X), ] either he sells the option,o u δ (ω + π, X) = sup δ A E P [U(ω + (δ S) T X). The price obtained by indifference utility is the minimum price such that the two quantities are equal, i.e. π(ω, X) = inf {π R such that u δ (ω + π, X) u δ (ω, 0) 0}. This price is the minimal amount such that it becomes interesting for the seller to sell the option : under this threshold, the seller has a higher utility keeping the option, and not selling it. 16

17 Set V δ,ω,t = ω + (δ S) t. Indifference utility pricing A classical idea to obtain a fair price is to use some marginal rate of substitution argument, i.e. π is a fair price if diverting of his funds into it at time 0 will have no effect on the investor s achievable utility. Hence, the idea is to find π, solution of { ]} ε max E P [U (V δ,ω ε,t + εx/π) ε=0 = 0, under some differentiability conditions of the function, where δ is an optimal strategy. Then (see Davis (1997)), the price of a contingent claim X is π = E P(U (V δ,ω,t )X) U, (ω) where again, the expression of the optimal strategy δ is necessary. 17

18 The model : the dimension issue Set Y t = (t, Π t, S t, L t ), taking values in S = [0, T ] R R 2 +. The control δ takes values in U = R. Assume that random variables (X i ) s have a density (with respect to Lebesgue s measure) denoted ν, and denote m its expected value. The diffusion process for Y t is dy t = b (Y t, δ t ) dt } {{ } trend + Σ (Y t, δ t ) dw t } {{ } volatility + R γ (Y t, δ t, x) M(dt, dx) } {{ } jumps 18

19 The model : the dimension issue where functions b, Σ and γ are defined as follows b : R 4 U R 4 Σ : R 4 U R 4 t 1 t 0 π, δ s µδs π, δ µs s σδs σs l λm l 0 γ : R 4 U R R 4 t 0 π, δ, x s ξδs ξs l x and where M(dt, dx) = N(dt, dx) ν(dx)λdt, and N(dt, dx) denotes the point measure associated to the compound Poisson process (L t ) (see Øksendal and Sulem (2005))., 19

20 Solving the optimization problem The Hamilton-Jacobi-Bellman equation related to this optimal control problem of a European option with payoff X = φ(s T ) is then { sup δ A A (δ) u(y) = 0 (1) u(t, π, s, l) = U(π φ(l)) where A (δ) ϕ(y) = ϕ ( (y) + s(µ ξλ) δ(y) ϕ ) ϕ (y) + t π s (y) + 1 ( ) 2 s2 σ 2 δ(y) 2 2 ϕ π 2 (y) + 2δ(y) 2 ϕ π s (y) + 2 ϕ s 2 (y) [ ] + λ ϕ(t, π + sξδ(y), s(1 + ξ), l + x) ϕ(t, π, s, l) ν(dx). R 20

21 Solving the optimization problem Define C 0 as the convex cone of random variables dominated by a stochastic integral, i.e. C 0 = {X X (δ S) T for some admissible portfolio δ} and set C = C 0 L the subset of bounded random variables. Note that functional u introduced earlier can be written u( ) solution of u(x, φ) = sup X C 0 E P [U (x + X φ(l T ))]. (2) 21

22 Solving the optimization problem : the dual version Consider the pricing of an option with payoff X = φ(s T ). Define the conjugate of utility function U, V : R + R defined as V (y) = sup x DU [U(x) xy]. Denote by (L ) the dual of bounded random variables, and define { } D = Q (L ) Q = 1 and ( X C)( Q, X 0). so that the dual of Equation (2) is then v( ) solution of v(y, φ) = inf Q D {E P [ V (y dqr dp ) ]} yqφ(l T ). (3) x R +, U L (x) = log(x) x R +, U P (x) = xp p x R, U E (x) = exp ( x ) i.e. y R +, V L (y) = log(y) 1 y R +, V P (y) = yq q, q = p p 1 y R +, V E (y) = y(log(y) 1). 22

23 Merton s problem, without jumps Assume that ds t = S t (µdt + σdw t ) (i.e. ξ = 0) then for logarithm utility function ( )] α 2 u L (t, π) = U L [π exp (T t). 2σ2 for power utility function ( α u P (t, π) = U P [π 2 )] exp (T t) 2(1 p)σ2 for exponential utility function [ ] u E (t, π) = U E π + α2 (T t). 2σ2. 23

24 Merton s problem, with jumps Assume that ds t = S t (µdt + σdw t + ξdm t ) (i.e. ξ 0) then for logarithm utility function u L (t, π) = U L ( πe (T t)c ) where C satisfies { C = (α ξλ)d 1 2 σ2 D 2 + λ log(1 + ξd) 0 = (α ξλ σ 2 D)(1 + ξd) + λξ for power utility function u P (t, π) = U P ( πe (T t)c ) where C satisfies { C = (α ξλ)d σ2 (p 1)D 2 + λ p [(1 + ξd)p 1] 0 = (α ξλ) + σ 2 (p 1)D + λξ(1 + ξd) p 1 for exponential utility function u E (t, π) = U E (π + (T t)c) where C satisfies { C = α ξ + (α σ2 ξ ξλ)d 1 2 σ 2 D [ ] 2 0 = ξλ α + σ2 D ξλ exp ξd 24

25 Assume that the investor has an exponential utility, U(x) = exp( x/ ), Theorem 1. Let φ denote a C 2 bounded function. If utility is exponential, the value function associated to the primal problem, [ ( ) ] u(t, π, s, l) = max E P U Π T φ(l T ) F t δ A does not depend on s and can be expressed as u(t, π, l) = U is a function independent of π satisfying 0 = ξλ µ + σ2 sδ [ ξλ exp C t (t, l) = µ ξ C(T, l) = φ(l) where δ denotes the optimal control. + (µ σ2 ξ ξλ)sδ 1 2 σ 2 (sδ ) 2 Démonstration. Theorem 19 in Quéma et al. (2007). ( ) π C(t, l), where C ξsδ + C(t, l) ] ( ) E P e 1 x C(t,l+X) 0 25

26 Theorem 2. Further, given K > 0 and a distribution for X such that ( )] 4 E P [exp KX < if we consider the set A of admissible controls δ satisfying inequality E P ( T 0 δ2 t S 2 t dt) 2 <, the previous results holds for φ(x) = Kx and C(t, l) = Kl (T t)c, where C is a constant solution ] ( 0 = ξλ µ + σ2 sδ ξλ exp [ ) ξsδ E P e 1 x KX 0 C = µ ξ + (µ σ2 ξ ξλ)sδ 1 2 σ 2 (sδ ) 2. Démonstration. Theorem 19 in Quéma et al. (2007). 26

28 Practical and numerical issues The main difficulty of Theorem 1 is to derive C(t, l) and δ (t, l) characterized by integro-differential system 0 = ξλ µ + σ2 sδ [ ξλ exp ξsδ + C(t, l) ] e 1 x C(t,l+x) 0 f(x)dx R + C(t, l) = µ t ξ C(T, l) = φ(l) + (µ σ2 ξ ξλ)sδ 1 2 σ 2 (sδ ) 2 If payoff φ has a threshold (i.e. there exists B 0 such that φ is constant on interval [B, + )) ; it is possible to use a finite difference scheme. Hence, given two discretization parameters N t and M l, C n i C(t n, l i ) and D n i sδ (t n, l i ) where { t n = n t where N t t = T and n [0, N t ] N l i = i l where N l l = B and i [0, N l ] N 28

29 Practical and numerical issues, with threshold payoff Calculation of the integral can be done simply (and efficiently) using the trapezoide method. Note that can restrict integration on the interval [0, B], since I(t n, l i ) = Ii n = e 1 x C(t n,l i +x) 0 f(x)dx R + = B li 0 N l i 1 k=0 e 1 C(t n,l i +x) f(x)dx exp [ C n Nl { [ lk+i exp ] F ( ) l Nl i ] f(l k+i ) + exp B l i e 1 C(t n,l i +x) f(x)dx [ lk+i+1 ] } f(l k+i+1 ) 29

30 Practical and numerical issues with threshold payoff For the control, the goal is to calculate Di n sδ (t n, l i ), where sδ (t n, l i ) is the solution of 0 = ξλ µ + σ2 sδ (t n, l i ) [ ξλ exp ξsδ (r n, l i ) + C(t n, l i ) ] I(t n, l i ). Thus, D n i is the solution (obtained using Newton s method) of G(x) = 0 where [ G(x) = ξλ µ + σ2 x ξλ exp ξx + ] Cn i Ii n 30

31 Practical and numerical issues, with affine payoff In the case of an affine payoff, we have seen already that the system is degenerated, and that its solution is simply C(t, l) = K + Kl (T t)c, where C is a constant solution of C = µ + (µ σ2 ξ ξ ξλ)sδ 1 σ 2 (sδ ) 2 [ 2 0 = ξλ µ + σ2 sδ ξλ exp 1x0 ξsδ ]E P (e ) 1 x KX 0. If the Laplace transform of X is unknown, it is possible to approximate it using Monte Carlo techniques. And the second equation can be solved using Newton s method, as in the previous section : we can derive sδ and then get immediately C. 31

32 4 Loi Exponentielle(1) 3.8 Loi Pareto(1,2) Fig. 3 Price as a function of the risk aversion coefficient with T = 1, µ = 0, σ = 0.12, λ = 4, ξ = 0.05 and B = 4 32

33 Loi Exponentielle(1) 0.5 Loi Pareto(1,2) Fig. 4 Nominal amount at time t = 0 of the optimal edging, as a function of with T = 1, µ = 0, σ = 0.12, λ = 4, ξ = 0.05 and B = 4 33

34 Properties of optimal strategies Two nice results have been derived in Quéma et al. (2007), Lemma 3. C(t, ) is increasing if and only if φ is increasing. Lemma 4. If φ is increasing and µ > 0, then the optimal amount of risky asset to be hold when hedging is bounded from below by a striclty positive constant. 34

35 Davis, M.H.A (1997). Option Pricing in Incomplete Markets, in Mathematics of Derivative Securities, ed. by M. A. H.. Dempster, and S. R. Pliska, Finn, J. and Lane, M. (1995). The perfume of the premium... or pricing insurance derivatives. in Securitization of Insurance Risk : The 1995 Bowles Symposium, Øksendal, B. and Sulem, A. (2005). Applied Stochastic Control of Jump Diffusions. Springer Verlag. Quéma, E., Ternat, J., Charpentier, A. and prices of catastrophe options. submitted. Élie, R. (2007). Indifference Schoutens, W. (2003). Lévy Processes in Finance, pricing financial derivatives. Wiley Interscience. Schweizer, M. (1997). From actuarial to financial valuation principles. Proceedings of the 7th AFIR Colloquium and the 28th ASTIN Colloquium,