Estimation of Value at Risk and ruin probability for diffusion processes with jumps

Transcription

1 Estimation of Value at Risk and ruin probability for diffusion processes with jumps Begoña Fernández Universidad Nacional Autónoma de México joint work with Laurent Denis and Ana Meda PASI, May 21 Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () May with21 jumps 1 / 18

2 We will consider a process X that satisfies t t X t = m + σ s db s + X t N t b s ds + = sup X u. u t i=1 We will obtain upper and lower bounds for P[X t > z] γ T Y i, t >, (1) i We will use these bounds for estimations to Ruin Pobabilities and VaR where For q = 1 α in ], 1[, the Value at Risk VaR associated with Xt will be given by VaR α (Xt ) = inf{z IR, P(Xt z) q} Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () May with21 jumps 2 / 18

3 t t X t = m + σ s db s + N t b s ds + i=1 γ T Y i, t >, i where B is a one-dimensional Brownian Motion, N is a Poisson Process independent of B, T 1, T 2,..., are the jump times for N, the random variables Y i, i 1 are i.i.d. and independent of the Poisson Process and the Brownian Motion. We assume the following hypotheses and denote them by (H): (1) b is an integrable process. (2) For all t >, E( t σ2 sds) < +. (3) The jumps of the compound Poisson process are non-negative, i.e., Y 1 P-a.e., and we assume that Y 1 is not identically equal to. (4) The process ( N t i=1 γ T i Y i, t > ) is well-defined and integrable. Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () May with21 jumps 3 / 18

4 Hypothesis UBï 1 2 (1) We assume that the law of the jumps admits a Laplace transform defined on ], c[ where c is a positive constant (or c = + ) and we put L(x) = E[e xy 1 ], x < c. (2) There exists γ > such that γ s γ, P-a.s. for all s [, t]. (3) There exist < δ < (c/γ ) and a constant K t (δ), such that, for all s [, t], δ s b u du + δ2 2 s σ 2 udu + λs(l(δγ ) 1) K t (δ) a.e. (2) Let us remark that Assumption (UB)(3) is fulfilled if we replace it by the stronger one: (3 ) There exist constants b (t), a (t) such that, t σ 2 udu a (t), s In this case one has, for all < δ < c/γ, b u du b (t) P-a.e. s [, t]. Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value 2 a atwork Risk (t) with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () May with21 jumps 4 / 18

5 The main result to get the upper estimate is: Lemma Assume (UB). Let δ ], c/γ [ be as in (3) of (UB), and let z m. Then, P(Xt z) exp{δ(m z) + K t (δ)}. Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () May with21 jumps 5 / 18

6 Proof. We have P(Xt z) = P (δxt δz) P ( sup M s exp{δ(z m) K t (δ)} s t where {M s, s } is the martingale defined by ), Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () May with21 jumps 6 / 18

7 ( [ ] s N t M s = exp δ σ u db u + γ Y i i=1 δ2 2 s σ 2 udu λs(l(γ δ) 1) ). {M s, s } is a martingale since it is the product of a continuous martingale and a purely discontinuous one. The maximal inequality for exponential martingales gives the result. Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () May with21 jumps 7 / 18

8 Proposition Let us assume (UB), and let A = {δ ], c/γ [ δ satisfies (UB)(3)}. Then { VaR α (Xt ) inf m + K t(δ) ln α }. δ A δ δ Proof. Thanks to Lemma 1.1, P(X t > z) < α is implied by which yields the result. z > m + K t(δ) δ ln α δ, Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () May with21 jumps 8 / 18

9 Hypothesis (LB): There exist constants b (t), a (t) >, and γ IR, such that t σ 2 udu a (t), s b u du b (t) and γ s γ P-a.e. s [, t]. The lower bound depends on the sign of γ, so we shall discuss each case separately. Lemma Let us assume (LB), and γ. Then, for all z IR and t, P(X t z) P( a (t) Z + γ Y i z m b (t)), i=1 Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () May with21 jumps 9 / 18 N t

10 Sketch of the proof Proof. Since by hypothesis the compound Poisson and the Brownian motion are independent, we use the product structure of the measure and adopt a natural notation, a change of time and the fact that we know the distribution of the sup of the Brownian Motion. P(X t z) = Ω 2 P 1 (X t (, w 2 ) z)dp 2 (w 2 ). For fixed w 2, set v = N t i=1 Y i(w 2 ). Then we have P 1 (Xt (, w 2 ) z) P 1 ( sup s [,t] As B is a P 1 -Brownian motion, R s = s s σ u (, w 2 )db u z m b (t) γ v). σ u (, w 2 )db u is a P 1 -continuous martingale. By the change of time property, we know that there exists a P 1 -Brownian motion B such that Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () Maywith 21 jumps 1 / 18

11 For γ we can give a lower bound which is always valid. For this, let us introduce the process s s Y s = m + b u du + σ u db u, s >. As γ, the process X dominates Y, so for all α, VaR α (Xt ) VaR α(yt ). Proposition If we assume hypotheses (LB), and γ, for all α, VaR α (X t ) VaR α (Y t ) m + b (t) + a (t) Φ 1 (α/2). Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () Maywith 21 jumps 11 / 18

12 Now, if we make the additional hypothesis (denoted by (LB )) that there exists a constant a (t) such that t σ 2 u du a (t), we can considerably improve the bounds. Lemma Under (LB), (LB ), and γ, for all z IR and t, P(Xt z) P( a (t)z 1 a (t) a (t) Z 2 + N t γ Y i z m b (t)), i=1 where Z 1 and Z 2 are N(, 1) independent random variables that do not depend on N and (Y i ). Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () Maywith 21 jumps 12 / 18

13 We will assume in this section that the process X s, s [, t] is given by s X s = m + bs + N s σ u db u i=1 γ T Y i, m, (9) i where b is a constant and we assume that there exist constants a and γ > such that s [, t], σ 2 s a, γ s γ a.e. (1) If γ = 1, we have the classical risk process (see [Asm, Gra]). If the insurer invests in a risky asset we obtain this general model, see for example [GaGrSc]. We can apply Lemma 1.1 to estimate the ruin probability. Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () Maywith 21 jumps 13 / 18

14 Proposition Let θ = E(Y 1 ). Let X be as in (9), with coefficients bounded as in (1) above. Assume in addition that a > and c = ; or that lim x c/γ L(x) = + ; and that the following safety load condition holds: b λθγ >. (11) Denote by δ the greatest positive root in ], c/γ [ of Then, h(δ) = bδ + δ2 2 a + λ(l(γ δ) 1) =. P( sup X s ) e δ m. s t As a consequence, we have the following upper bound for the ruin probability: P(X s < for some s in ], [) e δ m. Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () Maywith 21 jumps 14 / 18

15 The jumps have exponential law with parameter ν >. Corollary Suppose (UB), (LB), and (LB ) hold. Assume that for all s [, t], γ γ s γ, a.e., with γ >. If γ > then γ ν lim inf α VaR α (Xt ) lim sup ln α α VaR α (Xt ) γ ln α ν. If γ then 2a (t) lim inf α VaR α (Xt ) and lim sup ln α α The geometric Brownian motion model. VaR α (Xt ) γ ln α ν. Now we consider a case widely used in finance: imagine X satisfies Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () Maywith 21 jumps 15 / 18

16 So for all t, Y t = ln m + t t σ s db s + where the processes b and γ are defined by and γ s = N t b s ds + b s = b s 1 2 σ2 s, s + i= i=1 ln(1 + γ s Y i ) Y i 1 {s ]Ti,T i+1 ]}, γ T Y i, i where T = and we adopt the convention that ln(1 + )/ = 1. If Y is integrable, we can apply all the results of the previous sections to get an estimate for VaR α (Yt ) and hence for VaR α(xt ), thanks to the following result, whose proof is now obvious: Proposition Let X be as in (12). Under (GBM), for all α ], 1[, Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () Maywith 21 jumps 16 / 18

17 The work with heavy tails is in process and it will appear soon!!!!!!!!!!! Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () Maywith 21 jumps 17 / 18

18 Thanks Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () Maywith 21 jumps 18 / 18

19 S. Asmussen; Ruin Probabilities, World Scientific, Singapore, 2. M. Borkovec, C. Klüppelberg; Extremal Behavior of Diffusion Models in Finance, Extremes 1, no. 1, pp 47-8, R. Cont, P. Tankov; Financial modelling with jump processes, Chapman& Hall-CRC press, 23. J. Danielsson, and C. de Vries. Tail index and quantile estimation with very high frequency data. Journal of Empirical Finance, 4, , R. Davis; Maximum and minimum of one-dimensional diffusions, Stoch. Proc. Appl. 13, pp 1-9, L. Denis, B. Fernández, A. Meda; Estimates of dynamic VaR and mean loss associated to diffusion processes, preprint. Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () Maywith 21 jumps 18 / 18

20 P. Embrechts, C. Klüppelberg, T. Mikosch; Modelling Extremal Events for Insurance and Finance, Springer Verlag, Berlin-Heidelberg-New York, P. Embrechts, S. Resnick, and G. Samorodintsky. Extreme value theory as a risk management tool North American Actuarial Journal, 3 (2), 3-41, P. Embrechts, S. Resnick, and G. Samorodintsky. Living on the edge. RISK Magazine, 11 (1), 96-1, J. Gaier, P. Grandits, W. Schachermayer; Asymptotic Ruin Probabilities and Optimal Investment, Annals of Applied Probability, Vol.13 no.3, pp , 23. J. Grandell; Aspects of Risk theory, Springer, Berlin, C. Hipp, H. Schmidli; Asymptotics of ruin probabilities for controlled risk processes in the small claims case., Scand. Actuar. J., no. 5, , 24. Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () Maywith 21 jumps 18 / 18

21 N. Ikeda, S. Watanabe; Stochastic Differential Equations and Diffusion Processes, North-Holland, Tokyo, F. Longin. Beyond the VaR. Discussion paper 97-11, CERESSEC, F. Longin From value at risk to stress testing, the extreme value approach. Discussion paper 97-4, CERESSEC, A. McNeil; Extreme Value Theory for Risk Managers, In Extremes and Integrated Risk Management, P. Embrechts (Ed.) Risk Books, Risk Waters Group, London, pp 3-18, 2. A. McNeil, R. Frey; Rüdiger and P. Embrechts, Quantitative risk management: Concepts, techniques and tools, Princeton Series in Finance, Princeton University Press. Princeton, NJ. 25. D. Revuz and M. Yor; Continuous Martingale and Brownian Motion, Springer Verlag, Berlin-Heidelberg-New York, Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () Maywith 21 jumps 18 / 18

22 D. Talay, Z. Zheng; Quantiles of the Euler Scheme for Diffusion Processes and Financial Applications, Conference on Applications of Malliavin Calculus in Finance (Rocquencourt, 21). Math. Finance 13 no. 1, pp , 23. Begoña Fernández Universidad Nacional Autónoma Estimation de Méxicojoint of Value atwork Risk with and ruin Laurent probability Denis and for diffusion Ana Meda processes PASI, () Maywith 21 jumps 18 / 18