sampled control for mean-variance hedging in a jump diffusion financial market

1704 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 7, JULY 2010

Sampled Control for Mean-Variance Hedgingin a Jump Diffusion Financial Market

O. L. V. Costa, A. C. Maiali, and Afonso de C. Pinto

Abstract—In this technical note we consider the mean-variance hedgingproblem of a jump diffusion continuous state space financial model with there-balancing strategies for the hedging portfolio taken at discrete times, asituation that more closely reflects real market conditions. A direct expres-sion based on some change of measures, not depending on any recursions,is derived for the optimal hedging strategy as well as for the “fair hedgingprice” considering any given payoff. For the case of a European call optionthese expressions can be evaluated in a closed form.

Index Terms—Discrete-time mean-variance hedging, optimal control,options pricing.

I. INTRODUCTION

Lately there has been a great deal of attention in the application ofstochastic control methods in the field of financial engineering. Oneexample is the classical mean-variance portfolio selection problem,recently extended in several directions using stochastic control tools(see for instance, [1]–[9], among others). The problem of hedgingderivatives has systematically been the focus of attention from bothresearchers and practitioners alike. When modeling the dynamics ofan asset price, its derivatives and the corresponding hedging process,the choices of state space and time parameter are determined so as tosimplify the model’s complexity. However, with respect to hedging,the situation that more closely follows what is observed in real marketconditions is the use of discrete times for representing portfoliore-balancing instants, and continuous state spaces for the possiblevalues assumed by the prices. Indeed, decisions regarding re-balancingthe hedged position naturally occur at discrete times, whereas thesmallest possible price variation (“market ticks”) can be more ade-quately modeled within a continuous state space framework. In thiswork we consider the mean-variance hedging problem of derivativesin a continuous state space financial model with the re-balancingstrategies for the hedging portfolio taken at discrete times. Moststudies of mean-variance hedging to date have considered the caseof re-balancing strategies taken at continuous-time. For discrete-timere-balancing, various intertemporal mean-variance criteria were an-alyzed in [10] for the case of a constant investment opportunity set.A solution for the general problem with one asset and non-stochasticinterest rate was presented in [11].

A recursive dynamic programming solution to the hedging problemwas presented in [12], [13], and [14], involving a backward evaluationin time of some conditional expectations related to the fair hedgingprice of the derivative to be hedged, and the optimal control to be ap-plied at the re-balancing instants. In [14] it is assumed a finite samplespace and the calculation of these conditional expectations is obtainedin a recursive way from the associated tree structure of the model (seeSection III of [14]). The numerical procedure proposed in [12] consists

Manuscript received January 28, 2009; revised January 18, 2010. First pub-lished April 01, 2010; current version published July 08, 2010. This work wassupported in part by the Brazilian National Research Council-CNPq, under theGrant 301067/2009-0. Recommended by Associate Editor Z. Wang.

O. L. V. Costa and A. C. Maiali are with the Departamento de Engenharia deTelecomunicações e Controle, Escola Politécnica da Universidade de São Paulo,São Paulo, SP, 05508-900 Brazil (e-mail: [email protected]).

A. de C. Pinto is with the Fundação Getulio Vargas, EESP, CEP: 01332-000,São Paulo, SP, Brazil (e-mail: [email protected]).

Digital Object Identifier 10.1109/TAC.2010.2046923

of a spatial grid-point discretization combined with a quadratic inter-polation (see Section IV of [12]). By doing this the parameters of theoptimal replication problem can be numerically derived in a recursiveway. The purpose of the present work is to, based on the results in [12]and [14] to the case where the dynamics of a risky asset price is repre-sented by a jump Itô diffusion equation with constant parameters, ob-tain direct expressions for both the fair hedging price of the derivativeto be hedged, and the optimal re-balancing control. These expressionsare derived based on some change of probability measures of the under-lying process and do not depend, as in previous papers, neither on anybackward recursion nor on any state-space discretization. In particularfor the case of European vanilla call options we derive closed-form so-lutions which eliminate the recursiveness and state-space discretizationof previous methods, and thus producing considerable computationalgains.

This technical note is organized as follows. Section II presents thebasic model and the recursive algorithm for obtaining the hedging priceand the sampled control strategy. In Section III we present the main re-sult of the technical note, which consists of some change of probabilitymeasures which yields non-recursive expressions for the fair hedgingprice and the corresponding optimal control. Section IV applies themethodology described in Section III to the case of a European vanillacall option, deriving closed-form expressions for the option value andthe sampled optimal control. Numerical results comparing the hedgingstrategy proposed in this technical note and that by the Black and Sc-holes [15] approach are presented in Section V. Some brief conclusionsare presented in Section VI.

II. DISCRETE TIME MEAN-VARIANCE HEDGING STRATEGY

Along this technical note we shall be following essentially the samenotation as in [16]. On a complete probability space �� withfiltration �� we consider, following [17], a jump diffusion continuousstate space financial model as follows. We define �� as the condi-tional expectation (w.r.t. the probability measure� ) given the filtration�� . We shall write �� whenever we want to denote the expectationw.r.t another probability measure �. The value �� of the risk-freeasset follows the differential equation �� .The value of the risky asset �� follows the jump-diffusion stochasticdifferential equation:

��

where � �, �� , and � �� is a Brownian motion independentof the compound Poisson process ��, which is given by ��

�� . Here �� is a Poisson process with intensity

and �� is a sequence of independent and identically distributednormal random variables, each of them with mean �� and variance �� .It is assumed that the random variables �� are independent ofone another and also independent of the Poisson process ��. Setting�� it is easy to see that � �� andthat �� . As shown in [16]

��

� � ��

��

�

��

��

�� (1)

Let � be an �� random variable representing a derivative maturingat time � , whose underlying asset is ��. A position in � must behedged at discrete time instants �� , called re-bal-ancing instants, with �� , �� and ��

�, � � �� . A re-balancing strategy will be denoted by � �

0018-9286/$26.00 © 2010 IEEE

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 7, JULY 2010 1705

�� where each �� is �� -measur-able and represents the number of risky assets in the portfolio. Let � �

(for simplicity, just � ) be a self-financing portfolio composed of thetwo assets �� and ��. Setting �� and �� we have that the recursive equa-tion for the value of the portfolio � �� for � � �� is (see, forinstance, [14])

� �� (2)

with � ��, the initial wealth, being��-measurable. The value function�� is given by

��

� �� (3)

with � �� being ��-measurable, and �� -measurable,� � �� . It is convenient to also define the interme-diate value functions at times ��, �� given by

��

� ��

The following result, proved in Cerny [14] and Bertsimas [12], providesrecursive equations for the solution of problem (3).

Proposition 1: The optimal control strategy � �� for problem (2), (3) is given, for� � �� , by

��

��

� ��

� ��

� �� (4)

where for � � � � ��

��

�� (5)

��

��

�

�� (6)

��

��

�� (7)

��

��

� ��

�� (8)

Moreover, ��

� where

��

��

��

and �� is as in (4) replacing � �� by ��.The main goal of the next section is to obtain, by making some ap-

propriate change of measures, direct expressions for �� and ��.

III. MAIN RESULTS

In what follows we write �� . Westart by deriving an expression for �� . Set ��

�� , ��

��, and ��

��

��.We need the following lemma.Lemma 1: We have that

��

�� (9)

where �� , �� , and

�� (10)

��

��

��

��

�� (11)

�� (12)

Proof: First we notice that, after some algebraic manipula-tions, we get that ��

�� where

�� is as in (11). From (1) we have that

��

��

��

��

��

��

��

��

��

��

�� (13)

After multiplying and dividing (13) by �� we get (9).The next lemma provides an expression for �� .Lemma 2: �� can be expressed as �� !�

!��, where !� � � � !� and

!� ��

��

��

��

Proof: From (7) and (8) we have that

��

��

��

��

� (14)

By direct calculation we obtain from (13) that

� ��

�� (15)

� ��

��

��

�

"� (16)

" � ��

�� (17)

Replacing (9) and (17) into (14) we get after some long manipulationsthe desired expression for �� .

For the next results we fix � � �� and consider # �� . For $ � �� #� we define the set %�

�� as formed by a possible way of choosing $ differentelements out of the set �� , where the index & representsa possible choice. Of course we have �

�different ways of doing this,

and thus & varies from 1 to ��

. We set for each &, %��

�� %��, %

�� %�

��, andfor % �� , �� if � � % , 0 otherwise. Wedefine, for � � ' � � , the function (��'� as follows: (��'� � �if ' � � � � �� for some ) � %�

��, (��'� � � otherwise, and


the functions ��, ��, �

��, ��

�� and �� as

follows:��

�� (18)

�� (19)

�� (20)

Define also

��

� ��

� (21)

��

��

��

�

��

��

��

�� (22)

��

��

�� (23)

Notice that we can decompose the compound Poisson process�� as(for notational simplicity we omit the dependence on , �, �) �� where

��

��

��

�� (24)

��

��

��

��

�� (25)

According to (24) and (25) we have that �� is associated to thejumps that occur within the time interval �� ,while �� is associated to the jumps that occur within the time in-terval ��

�� (notice that ��

��

and �� ). Bearing that in mind and recalling from (19)

that �� for � � ��, 0 elsewhere, and �� for � � ��, 0 elsewhere, we have that �� and �� are com-pound non-homogeneous Poisson processes, with jump rates ��and �� respectively. More precisely, defining for � � �� , thenon-homogeneous Poisson process�� with jump rate �� and�� the random variable associated to the size of the �� jump that oc-curs within the time interval ��, we have that (24) and (25) can bewritten as

��

� ��

��

�� (26)

Notice that for � � �� , �� are independent and identicallydistributed normal random variables, each of them with mean �� andvariance �� , and that

��

�

�� (27)

From this we can re-write �� in a more convenient way as

��

� ��

��

��

�� (28)

The next result writes �� in terms of ��, �� and �� .

Lemma 3: For �� as in (6) we have that

��

�

��

��

� �

��

�� (29)

Proof: From (6) and Lemma 2 we have that

��

��

��

��

�

�

��

��

� �

��

�� (30)

But from (18)–(23) we have that ��

�� completing the proof.We need the following lemma, proved in the appendix.Lemma 4: The process ��

�� is a martingale and, in particular,��

�� for all � � .For � � �� and � � ��

�define the probability

measures �� through its Radon-Nikodym derivatives as follows:

��

�� and we write �� to denote the expectedvalue with respect to the measure ��

��. We have the following the-orem, proved in the appendix.

Theorem 1: Under the probability measure ��, the process

��

�

�� is a Brownian motion, �� and

�� as in (26) are compound (non homogeneous) Poisson processeswith rates given respectively by �� and �� as defined in(20), and the jump sizes ��, �� are independent sequenceswith �� identically normally distributed having density function �and �� identically normally distributed having density function .The processes ��

��,�� and �� are independent.We can now characterize �� and the control law as defined in

(5) and (4) respectively directly in terms of the expected values withrespect to the probability measures ��

��.Theorem 2: Let �� and �� be as defined in (5) and (4), re-

spectively. Then

��

��

��

� �

��

��

� �� (31)

and

�� !

��

"��#(32)

where

� �

�

��

��

� �

��

�$��

� ��$��

� ��

� ��

� �� (33)

and # is as defined in (17).Proof: From (5) and (29) we get that

��

�

��

��

� �

��

��

� ��

��

��

� �

��

��

� ��

��

��

� �

��

��

� ��

showing (31). From (4) and the fact that ! �� is -measurable wecan write

��

�

�� %��

&��

�

! �� %��

"�� %��

� �


From (16) we get that �� ,

��

� � ��. For �� we recall

from Lemma 2 and (30) that

��

��

��

��

From (9) and noticing that �� is �� -measurable, we get (34),shown at the bottom of the page. We have that

��

��

��

��

�

�

��

��

� �

��

��

� ��

� ��

��

�

��

��

� �

��

��

� ��

� ��

� ��

��

� �� (35)

From (34) and (35) we get (32).

IV. APPLICATION TO EUROPEAN CALL OPTIONS

Here we apply the results obtained in the previous sectionto the case in which the derivative to be hedged is a Europeanvanilla call option. We derive closed-form solutions for both thefair hedging price �� and the optimal control �� to beapplied at re-balancing instant � . These closed-form solutionseliminate the recursiveness and state-space discretization of previ-ously proposed models, thus producing considerable computationalgains. Similar procedures would lead to closed-form solutionsfor the case of European vanilla put options. First we define

�� , �� ,

and ��

�.

The main result of this section is presented next.Theorem 3: Consider an European vanilla call option whose payoff

is given by �� . Then under these conditions

��

� �� does not depend on �

��

��

�

��

�

� ��

and

��

� ��

��

��

� ��

�

� ��

��

��

� ��

��

��

��

� ��

�

� ��

��

��

��

��(36)

with ��

�

�� , and

�� !��" ��" ��

!��" ��"

�� !��" ��" ��

!��" ��

where

!��" ��" ��

�� #� ��

��

� �� "

!��" ��" �� !��" ��" �� "

!��" ��

Moreover, �� is as in (32) and $� in (33) can be written as

$� �

�

��

��

��

�

��

��

��

��

��

� �� (37)

Proof: First we recall from (1) and (24)–(26), that

�� % �� % �� #� ��

� �

��

� ��

��

��

��

�� (38)

where, from Theorem 1, under the probability measure &�� , the

process %�� is a Brownian motion, &�� and &�� as in (26)

are compound (non homogeneous) Poisson processes with rates givenrespectively by �� and �� as defined in (20), and thejump sizes �'��, �'�� are independent sequences with �'��identically normally distributed having density function �� and �'��identically normally distributed having density function � . Moreoverthe processes%�

� ��,&�� and &�� are independent. We have that

��

� ��

� ��

� ��

� ��

� �� (39)

��

��

��

��

��

��

��

��

��

��

��

��

��

�� (34)


Let us evaluate first

��

� ��

�� (40)

Set � � ��

��

��

�� . Since

this is the sum of independent normal random variables, we have that �is also normal, with mean � � ��

�� and variance

� � �� . From (38) we have that ��

can be expressed as �� , where �� and �� is as defined previously. But under the probabilitymeasure ��

�, �� is a standard normal variable, and thus (40) equals

�� . Let us evaluate ��

� �� now. From(38)

��

�� (41)

� � ��

�� (42)

� � ��

� ��

��

��

� (43)

��

� ��

��

� ��

�� (44)

Set the probability measure �� through its Radon-Nikodym deriva-

tive as follows: ��

�� , and we write ��

to denote the expected value with respect to the measure ��.

As in Theorem 1, under the probability measure ��, the process

��

�� is a Brownian motion and �� and�� as in (26) are compound (non homogeneous) Poisson pro-cesses with rates given respectively by ��

�� and

�� , and the jump sizes ��, �� areindependent sequences with �� identically normally distributedhaving density function �� and �� identically normally distributedhaving density function � . Moreover the processes ��

��, ��and �� are independent. From (41)–(44)

��

� ��

��

� ��

� ��

� ��

��

� ��

Let us evaluate

��

� ��

(45)Set ��

��

��

�� . Since

this is the sum of independent normal random variables, we have that�� is also normal, with mean ��

��

and variance �� . From (38) we

have that �� can be expressed as �� , where�� and �� is as defined previously. But underthe probability measure��

�, �� is a standard normal variable, and thus

(45) equals �� , and (36) follows. Since ��

� �� does not depend on �, we can evaluate the sum in (33) to obtain

� �

��

�!�

� �� !

� ��

TABLE IHEDGING ERRORS FOR B&S, DP, AND RELATIVE ERROR ( � �� AND 6)

�"� �

#

!

�"� �

"� #

�!�

and (37) follows.

V. NUMERICAL RESULTS

In this section the results obtained in Section IV are applied to Eu-ropean call options maturing in 60 business days. We considered thatthe risk-free rate was $ � � per annum, the current value of the un-derlying asset was � � ��, the rate of return was � � � perannum and that the volatility was � �� per annum. The jumpparameters considered were � � ��, � � � and � � �.Results for three different strikes were compared: � � ��, � ��, and � � ��, corresponding to in-the-money, at-the-money andout-of-the-money options, respectively. We supposed that the portfoliowas re-balanced every ��% business days, yielding to % re-balancinginstants, with 2 values for %; % � � (portfolio re-balanced each 5days) and % � � (portfolio re-balanced each 10 days). Paths of the un-derlying asset were simulated according to (1). For each path there is apayoff, �� , which is compared with the value of the hedging port-folio at maturity, & �� . We run 20000 Monte Carlo simulations andthe hedging error, expressed as the square root of the mean-squared dif-ference between the option’s payoff and hedging portfolio at maturity,was calculated. The procedure was repeated for two hedging methods:(i) the dynamic programming approach (DP) proposed in Section IV;and (ii) the Black & Scholes (B&S) approach (delta-hedging) con-sidering an augmented volatility �� to take into account the jumpsof the process. This volatility is obtained from ��

�� , and represents the sum of the variancesof the diffusion and jump components. The hedging errors incurred byboth methods (columns “error B&S” and “error DP”) as well as theDP error relative to that of the B&S approach (column “relative error”)are presented for each combination of the striking parameters in, at, andout-of-the-money call options, and %, in Table I. The simulation resultsindicate that the error increases as the option goes more and more out ofthe money for both methods, no matter what the number of re-balancingis. As expected in all cases the results from the B&S model approachwere worse than those obtained by the DP model (due to the optimalityof the DP approach) and, as the number of re-balancing decreases, thetotal error increases as well as relative difference of performance be-tween DP and B&S model. This is so due to the fact that the portfolioswill be kept unchanged for longer periods of time, making the trackingof the option value harder, specially for the B&S model which assumesthe continuous trading hypothesis. Finally it should be noticed that evenaround 1% of difference between the methods could yield a great differ-ence of values between the portfolios, bearing in mind the large amountof money usually involved in this kind of contract.

VI. CONCLUSION

In this work we have analyzed the mean-variance hedging problemof a continuous state space financial model with the rebalancing strate-gies for the hedging portfolio taken at discrete-times. By making someappropriate changes of measures, not depending on any recursions, we


have derived direct expressions for the optimal self-financing mean-variance hedging strategy problem as well as for the ”fair hedgingprice,” considering any given payoff in an incomplete market envi-ronment. As an application of the proposed method, we have obtainedclosed-form solutions for the value European vanilla call options andfor the amount of the corresponding underlying asset to be bought orsold for hedging purposes (sampled optimal control law).

APPENDIX

We present in this appendix the proofs of Lemma 4 and Theorem 1,following the same ideas as in [16].

Proof of Lemma 4: First we show that�� as defined in (28)

is a martingale. Set

� ��

� ��

��

��

��

� ��

��

��

��

Since � �� and ��

�� we have that ��

��

��

�� is a martingale. Moreover since

�� it follows that�� . Recalling from

(28) that ��

� �� it follows fromItô’s rule for jump processes that (see also [16, p.499])

��

��

�

��

��

�

�

��

and Theorem 11.4.5 of [16] implies that�� is a martingale. From

the stochastic calculus for continuous processes we know that��

as defined in (21) is a martingale and, since �� has no Itô in-

tegral part, it follows from Itô’s product rule for jump processes that��

��

��

��.

From Theorem 11.4.5 of [16] we have that �� is a martingale and,

from the fact that �� , it follows that ��

�� .Proof of Theorem 1: From the proof of Lemma 4 we have that

�� is a martingale. Set �� , ��

and �� , where �� is the expected value with respectto the density function ��, and

��

� ��

��

� ��

��

It is easy to see that � � �� .From this it follows that:

��

�

��

��

�

�

��

is a martingale. Set

� ��

�

��

From Itô’s rule for jump processes it follows:

��

��

� ��

�

��

� ��

�

�

��

(see also [16, pp. 500–502]) and since the processes ��, ��

�

�� and��

�

�� are martingales and the

integrands are left-continuous, it follows from Theorem 11.4.5 of [16]that � ��

�� is a martingale. From this we have that

��

��

so that, from independence of ��, ��, �� and Girsanov’s

theorem, we get that

��

��

��

��

��

��

� � ��

� ��

� � ��

showing the desired result.

ACKNOWLEDGMENT

The authors would like to thank Mr. Cláudio De Nardi Queiroz forthe computational support in the implementation of the numerical sim-ulations of Section V. The authors would also like to thank the editorand the referees for their very helpful suggestions which greatly im-proved the quality of the technical note.

REFERENCES

[1] N. Bäuerle and U. Rieder, “Portfolio optimization with Markov-modu-lated stock prices and interest rates,” IEEE Trans. Autom. Control, vol.49, no. 3, pp. 442–447, Mar. 2004.

[2] U.Çakmak and S.Özekici, “Portfolio optimization in stochastic mar-kets,” Math. Methods Oper. Res., vol. 63, pp. 151–168, 2006.

[3] O. Costa and M. Araujo, “A generalized multiperiod mean-varianceportfolio optimization with Markov switching parameters,” Auto-matica, vol. 44, pp. 2487–2497, 2008.

[4] O. Costa and R. B. Nabholz, “Multiperiod mean-variance optimizationwith intertemporal restrictions,” J. Optim. Theory Appl., vol. 134, pp.257–274, 2007.

[5] B. Rustem, R. G. Becker, and W. Marty, “Robust min-max portfoliostrategies for rival forecast and risk scenarios,” J. Econ. Dyn. Control,vol. 24, pp. 1591–1621, 2000.

[6] G. Yin and X. Zhou, “Markowitz’s mean-variance portfolio selectionwith regime switching: From discrete-time models to their continuous-time limits,” IEEE Trans. Autom. Control, vol. 49, no. 3, pp. 349–360,Mar. 2004.

[7] Q. Zhang, “Stock trading: An optimal selling rule,” SIAM J. ControlOptim., vol. 40, pp. 64–87, 2001.

[8] X. Zhou and D. Li, “Continuous-time mean-variance portfolio selec-tion: A stochastic LQ framework,” Appl. Math. Optim., vol. 42, pp.19–33, 2000.

[9] X. Zhou and G. Yin, “Markowitz’s mean-variance portfolio selectionwith regime switching: A continuous-time model,” SIAM J. ControlOptim., vol. 42, pp. 1466–1482, 2003.

[10] M. Schäl, “On quadratic cost criteria for option hedging,” Math. Oper.Res., vol. 1, no. 19, pp. 121–131, 1994.

[11] M. Schweizer, “Variance-optimal hedging in discrete time,” Math.Oper. Res., vol. 1, no. 20, pp. 1–32, 1995.

[12] D. Bertsimas, L. Kogan, and A. W. Lo, “Hedging derivative securitiesin incomplete market: An -arbitrage approach,” Oper. Res., vol. 3,no. 49, pp. 372–397, 2001.

[13] S. Gugushvili, “Dynamic programming and mean-variance hedging indiscrete-time,” Georgian Math. J., vol. 10, pp. 237–246, 2003.

[14] A. Cerny, “Dynamic programming and mean-variance hedging in dis-crete time,” Appl. Math. Finance, vol. 1, no. 11, pp. 1–25, 2004.

[15] F. Black and M. Scholes, “The pricing of options and corporate liabil-ities,” J. Political Econ., no. 81, pp. 637–654, 1973.

[16] S. E. Shreve, Stochastic Calculus for Finance II. New York: Springer,2004.

[17] R. Merton, “Option pricing when underlying stock returns are discon-tinuous,” J. Financial Econ., vol. 3, pp. 125–144, Jan./Mar. 1976.

sampled control for mean-variance hedging in a jump diffusion financial market

Documents