## The Annals of Probability

### Approximation pricing and the variance-optimal martingale measure

Martin Schweizer

#### Abstract

Let X be a semimartiangale and let $\Theta$ be the space of all predictable X-integrable process $\vartheta$ such that $\int\vartheta dX$ is in the space $\varsigma^2$ of semimartingales. We consider the problem of approximating a given random variable $H\in L^2(P)$ by the sum of a constant c and a stochastic integral $\int_0^T\vartheta_s dX_s$, with respect to the $L^2(P)$-norm. This problem comes from financial mathematics, where the optimal constant $V_0$ can be interpreted as an approximation price for the contingent clam H. An elementary computation yields $V_0$ as the expectation of H under the variance-optimal signed $\Theta$-martingale measure $\tilda{P}$, and this leads us to study $\tilda{P}$ in more detail. In the case of finite discrete time, we explicitly construct $\tilda{P}$ by backward recursion, and we show that $\tilda{P}$ is typically not a probability, but only a signed measure. In a continuous-time framework, the situation becomes rather different: we prove that $\tilda{P}$ is nonegative is X has continuous paths and satisfies a very mild no-arbitrage condition. As an application, we show how to obtain the optimal integrand $\xi\in\Theta$ in feedback form with the help of a backward stochastic differential equation.

#### Article information

Source
Ann. Probab. Volume 24, Number 1 (1996), 206-236.

Dates
First available: 15 January 2003

http://projecteuclid.org/euclid.aop/1042644714

Mathematical Reviews number (MathSciNet)
MR1387633

Digital Object Identifier
doi:10.1214/aop/1042644714

Zentralblatt MATH identifier
0854.60045

Subjects
Primary: 60G48: Generalizations of martingales
Secondary: 90A09 60H05: Stochastic integrals

#### Citation

Schweizer, Martin. Approximation pricing and the variance-optimal martingale measure. The Annals of Probability 24 (1996), no. 1, 206--236. doi:10.1214/aop/1042644714. http://projecteuclid.org/euclid.aop/1042644714.

#### References

• ANSEL, J.-P. and STRICKER, C. 1992. Lois de martingale, densites et decomposition de ´ ´ Follmer Schweizer. Ann. Inst. H. Poincare 28 375 392. ¨ ´ Z.
• BARRON, E. N. and JENSEN, R. 1990. A stochastic control approach to the pricing of options. Math. Oper. Res. 15 49 79. Z.
• BLACK, F. and SCHOLES, M. 1973. The pricing of options and corporate liabilities. J. Political Economy 81 637 654. Z.
• BOULEAU, N. and LAMBERTON, D. 1989. Residual risks and hedging strategies in Markovian markets. Stochastic Process. Appl. 33 131 150. Z.
• CVITANIC, J. and KARATZAS, I. 1993. Hedging contingent claims with constrained portfolios. ´ Ann. Appl. Probab. 3 652 681. Z.
• DAVIS, M. H. A. 1994. A general option pricing formula. Preprint, Imperial College, London. Z.
• DELBAEN, F., MONAT, P., SCHACHERMAy ER, W., SCHWEIZER, M. and STRICKER, C. 1995. Weighted norm inequalities and closedness of a space of stochastic integrals. Preprint, Univ. Franche-Comte, Besanc ¸on. ´ Z.
• DELBAEN, F. and SCHACHERMAy ER, W. 1994. The variance-optimal martingale measure for continuous processes. Preprint, Univ. Vienna. Z.
• DUFFIE, D. and RICHARDSON, H. R. 1991. Mean-variance hedging in continuous time. Ann. Appl. Probab. 1 1 15. Z.
• EL KAROUI, N. and QUENEZ, M.-C. 1995. Dy namic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33 29 66. Z.
• FOLLMER, H. and SCHWEIZER, M. 1991. Hedging of contingent claims under incomplete informa¨ Z. tion. In Applied Stochastic Analy sis M. H. A. Davis and R. J. Elliott, eds. 389 414. Gordon and Breach, London. Z.
• FOLLMER, H. and SONDERMANN, D. 1986. Hedging of non-redundant contingent claims. In ¨ Z. Contributions to Mathematical Economics W. Hildenbrand and A. Mas-Colell, eds. 205 223. Z.
• HANSEN, L. P. and JAGANNATHAN, R. 1991. Implications of security market data for models of dy namic economies. J. Political Economy 99 225 262. Z.
• HARRISON, J. M. and KREPS, D. M. 1979. Martingales and arbitrage in multiperiod securities markets. J. Economic Theory 20 381 408. Z.
• HARRISON, J. M. and PLISKA, S. R. 1981. Martingales and stochastic integrals in the theory of continuous trading. Stochastic Process. Appl. 11 215 260.
• HARRISON, J. M. and PLISKA, S. R. 1983. A stochastic calculus model of continuous trading: complete markets. Stochastic Process. Appl. 15 313 316. Z.
• HIPP, C. 1993. Hedging general claims. Proceedings of the 3rd AFIR Colloquium, Rome, 2 603 613. Z.
• LEPINGLE, D. and MEMIN, J. 1978. Sur l'integrabilite uniforme des martingales exponentielles. ´ ´ ´ Z. Wahrsch. Verw. Gebiete 42 175 203. Z.
• MERTON, R. C. 1973. Theory of rational option pricing. Bell J. Economics Manage. Sci. 4 141 183. Z. Z. 2 Z. Z.
• MONAT, P. and STRICKER, C. 1994. Fermeture de G et de LL FF G. Seminaire de ´ T 0 T Probabilites XXVIII. Lecture Notes in Math. 1583 189 194. Springer, Berlin. ´ Z.
• MONAT, P. and STRICKER, C. 1995. Follmer Schweizer decomposition and mean-variance hedg¨ ing for general claims. Ann. Probab. 23 605 628. Z.
• MULLER, S. M. 1985. Arbitrage Pricing of Contingent Claims. Lecture Notes in Econom. and ¨ Math. Sy stems 254. Springer, Berlin. Z.
• PROTTER, P. 1990. Stochastic Integration and Differential Equations. A New Approach. Springer, Berlin. Z.
• SCHAL, M. 1994. On quadratic cost criteria for option hedging. Math. Oper. Res. 19 121 131. ¨ Z.
• SCHWEIZER, M. 1992. Mean-variance hedging for general claims. Ann. Appl. Probab. 2 171 179. Z.
• SCHWEIZER, M. 1994. Approximating random variables by stochastic integrals. Ann. Probab. 22 1536 1575. Z.
• SCHWEIZER, M. 1995a. Variance-optimal hedging in discrete time. Math. Oper. Res. 20 1 32. Z.
• SCHWEIZER, M. 1995b. On the minimal martingale measure and the Follmer Schweizer decom¨ position. Stochastic Anal. Appl. 13 573 599. Z.
• STRICKER, C. 1990. Arbitrage et lois de martingale. Ann. Inst. H. Poincare 26 451 460. ´
• FACHBEREICH MATHEMATIK, MA 7-4 STRASSE DES 17. JUNI 136 D-10623 BERLIN GERMANY E-mail: mschweiz@math.tu-berlin.de