We study a problem related to asset-liability management in life insurance. As shown by Wüthrich, Bühlmann and Furrer in , an insurance company can guarantee solvency by purchasing a Margrabe option enabling it to exchange its asset portfolio for a valuation portfolio. The latter can be viewed as a replicating portfolio for the insurance liabilities in terms of financial instruments. Our objective in this paper is to investigate numerically a valuation technique for such an option in a situation when the insurance company is a “large” investor, implying that its trading decisions can affect asset prices. We view this situation through the framework employed in the Cvitanic and Ma’s 1996 paper  and use the method of finite differences to solve the resulting non-linear PDE. Our results show reliability of this numerical method. Also we find, similarly to other authors, that the option price for the large investor is higher than that for a Black-Scholes trader. This makes it particularly compelling for a large insurance company to purchase a Margrabe option at the Black-Scholes price.
Cite this paper
E. Bølviken, F. Proske and M. Rubtsov, "Pricing of Margrabe Options for Large Investors with Application to Asset-Liability Management in Life Insurance," Journal of Mathematical Finance
, Vol. 4 No. 2, 2014, pp. 113-122. doi: 10.4236/jmf.2014.42011
 M. V. Wüthrich, H. Bühlmann and H. Furrer, “Market-Consistent Actuarial Valuation,” Springer, Berlin, 2008
 J. Cvitanic and J. Ma, “Hedging Options for a Large Investor and Forward-Backward SDE’s,” The Annals of Applied Probability, Vol. 6, No. 2, 1996, pp. 370-398. http://dx.doi.org/10.1214/aoap/1034968136
 U. Cetin, R. Jarrow and P. Protter, “Liquidity Risk and Arbitrage Pricing Theory,” Finance and Stochastics, Vol. 8, No. 3, 2004, pp. 311-341. http://dx.doi.org/10.1007/s00780-004-0123-x
 U. Cetin, H. M. Soner and N. Touzi, “Option Hedging for Small Investors under Liquidity Costs,” Finance and Stochastics, 2007, Forthcoming.
 U. Cetin and L. C. G. Rogers, “Modelling Liquidity Effects in Discrete Time,” Mathematical Finance, Vol. 17, No. 1, 2007, pp. 15-29. http://dx.doi.org/10.1111/j.1467-9965.2007.00292.x
 L. C. G. Rogers and S. Singh, “The Cost of Illiquidity and Its Effects on Hedging,” Preprint, 2007.
 H. M. Soner and S. Gokay, “Cetin-Jarrow-Protter Model of Liquidity in a Binomial Market,” Preprint, 2009.
 E. Platen and M. Schweizer, “On Feedback Effects from Hedging Derivatives,” Mathematical Finance, Vol. 8, No. 1, 1998, pp. 67-84. http://dx.doi.org/10.1111/1467-9965.00045
 R. Frey and A. Stremme, “Market Volatility and Feedback Effects from Dynamic Hedging,” Mathematical Finance, Vol. 7, No. 4, 1997, pp. 351-374. http://dx.doi.org/10.1111/1467-9965.00036
 R. Jarrow, “Derivative Securities Markets, Market Manipulation and Option Pricing Theory,” Journal of Financial and Quantitative Analysis, Vol. 29, No. 2, 1994, pp. 241-261. http://dx.doi.org/10.2307/2331224
 R. Frey, “Perfect Option Hedging for a Large Trader,” Finance and Stochastics, Vol. 2, No. 2, 1998, pp. 115-141. http://dx.doi.org/10.1007/s007800050035
 G. Papanicolaou and R. Sircar, “General Black-Scholes Models Accounting for Increased Market Volatility from Hedging Strategies,” Applied Mathematical Finance, Vol. 5, No. 1, 1998, pp. 45-82. http://dx.doi.org/10.1080/135048698334727
 P. Bank and D. Baum, “Hedging and Portfolio Optimization in Financial Markets with a Large Trader,” Mathematical Finance, Vol. 14, No. 1, 2004, pp. 1-18. http://dx.doi.org/10.1111/j.0960-1627.2004.00179.x
 D. Cuoco and J. Cvitanic, “Optimal Consumption Choice for a Large Investor,” Journal of Economic Dynamics and Control, Vol. 22, No. 3, 1998, pp. 401-436. http://dx.doi.org/10.1016/S0165-1889(97)00065-1
 P. M. DeMarzo and B. Urosevic, “Ownership Dynamics and Asset Pricing with a Large Shareholder,” Journal of Political Economy, Vol. 114, No. 4, 2006, pp. 774-815. http://dx.doi.org/10.1086/506334
 D. J. Duffy, “Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach,” John Wiley and Sons, Hoboken, 2006. http://dx.doi.org/10.1002/9781118673447