Abstract
A combined financial and insurance model is introduced. We consider a Black-Scholes financial model with stochastic coefficients. We use a step process with a stochastic intensity and a random transition kernel to model claims. We investigate a stream of liabilities which consists of annuity, death and survival benefits. We define a set of admissible investment strategies for an insurer (an investor) who trades in the financial market and aims to replicate the stream of liabilities.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ankirchner, S., Imkeller, P.: Quadratic hedging of weather and catastrophe risk by using short term climate prediction. Preprint (2008)
Bening, V.E., Korolev, V.Y.: Generalised Poisson Models and Their Applications in Insurance and Finance. VSP, Utrecht (2002)
Bielecki, T., Jeanblanc, M., Rutkowski, R.: Hedging of defaultable claims. In: Carmona, R. (ed.) Indifference Pricing, Paris-Princeton Lectures on Mathematical Finance, pp. 1–132. Springer, Berlin (2004)
Bielecki, T., Jeanblanc, M., Rutkowski, R.: Pricing and trading credit default swaps in a hazard process model. Ann. Appl. Probab. 18, 2495–2529 (2008)
Blanchet-Scalliet, C., Eyraud-Loisel, A., Royer-Carenzi, M.: Hedging of defaultable claims using BSDE with uncertain time horizon. Preprint (2008)
Brémaud, P.: Point Processes and Queues. Springer, New York (1981)
Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman and Hall/CRC Press, London (2004)
Crépey, S.: About the pricing equation in finance. In: Carmona, R.A., Çinlar, E., Ekeland, I., Jouini, E., Scheinkman, J.A., Touzi, N. (eds.) Paris-Princeton Lectures in Mathematical Finance 2010, pp. 63–203. Springer, Berlin (2011)
Crépey, S., Matoussi, A.: Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison. Ann. Appl. Probab. 18, 2041–2069 (2008)
Dahl, M., Møller, T.: Valuation and hedging of life insurance liabilities with systematic mortality risk. Insur. Math. Econ. 39, 193–217 (2006)
Dahl, M., Melchior, M., Møller, T.: On systematic mortality risk and risk-minimization with survival swaps. Scand. Actuar. J. 2, 114–146 (2008)
Dassios, A., Jang, J.: Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity. Finance Stoch. 7, 73–95 (2003)
Dudley, R.M.: Winer functionals as Itô integrals. Ann. Probab. 5, 140–141 (1977)
Dupire, B.: Pricing and hedging with smiles. In: Dempster, M.A., Pliska, S.R. (eds.) Mathematics of Derivative Securities, pp. 103–111. Cambridge University Press, Cambridge (1997)
El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Finance 7, 1–71 (1997b)
European Commission: Fifth quantitative impact study: call for advice and technical specifications. http://ec.europa.eu/internal_market/insurance/solvency/index_en.htm (2010)
Filipovic, D.: Term-Structure Models. Springer, Berlin (2009)
Fouque, J.P., Papanicolaou, G., Sircar, K.R.: Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press, Cambridge (2000)
Gundlach, M., Lehrbass, F.: CreditRisk+ in the Banking Industry. Springer, Berlin (2004)
Jeanblanc, M., Le Cam, Y.: Progressive enlargement of filtrations with initial times. Stoch. Process. Appl. 8, 2523–2543 (2009)
Jeanblanc, M., Rutkowski, M.: Default risk and hazard process. In: Geman, H., Madan, D., Pliska, S.R., Vorst, T. (eds.) Mathematical Finance—Bachelier Congress 2002, pp. 281–313. Springer, Berlin (2000)
Jiao, Y., Kharroubi, I., Pham, H.: Optimal investment under multiple default risk: a BSDE decomposition approach. Ann. Appl. Probab. 23, 455–491 (2013)
Lim, A.: Mean-variance hedging when there are jumps. SIAM J. Control Optim. 44, 1893–1922 (2005)
Mikosch, T.: Non-life Insurance Mathematics. Springer, Berlin (2009)
Møller, T., Steffensen, M.: Market Valuation Methods in Life and Pension Insurance. Cambridge University Press, Cambridge (2007)
Protter, P.: Stochastic Integration and Differential Equations. Springer, Berlin (2004)
Rolski, T., Schmidili, H., Schmidt, V., Teugels, J.: Stochastic Processes for Insurance and Finance. Wiley, New York (1999)
Shreve, S.E.: Stochastic Calculus for Finance II: Continuous-Time Models. Springer, Berlin (2004)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag London
About this chapter
Cite this chapter
Delong, Ł. (2013). Combined Financial and Insurance Model. In: Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications. EAA Series. Springer, London. https://doi.org/10.1007/978-1-4471-5331-3_7
Download citation
DOI: https://doi.org/10.1007/978-1-4471-5331-3_7
Publisher Name: Springer, London
Print ISBN: 978-1-4471-5330-6
Online ISBN: 978-1-4471-5331-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)