Partial Hedging Using Malliavin Calculus

Under the constraint that the initial capital is not enough for a perfect hedge, the problem of deriving an optimal partial hedging portfolio so as to minimize the shortfall risk is worked out by solving two connected subproblems sequentially. One subproblem is to find the optimal terminal wealth that minimizes the shortfall risk. The shortfall risk is quantified by a general convex risk measure to accommodate different levels of risk tolerance. A convex duality approach is used to obtain an explicit formula for the optimal terminal wealth. The second subproblem is to derive the explicit expression for the admissible replicating portfolio that generates the optimal terminal wealth. We show by examples that to solve the second subproblem, the Malliavin calculus approach outperforms the traditional delta-hedging approach even for the simplest claim. Explicit worked-out examples include a European call option and a standard lookback put option.


Introduction
A replicating (self-financing) portfolio designed to eliminate the risk exposure of the target contingent claim completely is called a perfect hedge.Since the value of a perfect hedging portfolio achieves exact replication of the payoff of the target security at the expiration date T, one can offset the risk of the target claim by selling the replicating strategy.In a financial market that is complete and arbitrage free, a perfect hedging strategy exists for any contingent claim with a sufficiently integral terminal payoff .The cost of replication  

C
given by the expected value of the discounted payoff under the unique, risk neutral equivalent martingale measure 0 P , i.e.,   where is the price of the risk-free asset.that the in l hedging en   B T the One of drawbacks of a perfect hedge is itial cost of the exact replication (i.e.,   ) is high.In addition, avoiding risks complet ans losing out on the potential gain that accepting the risk may have allowed.To this end, we discuss the position of an agent who is unwilling to commit at time = 0 t the entire amount necessary for implementing a perfect hedge an us interested in a partial hedging strategy that offers the balance between the cost and the risk exposure.Our goal is to derive optimal partial hedging strategies for various target contingent claims.
Since the shortfall risk is intrinsic in a partia ely me d is th vironment, one natural way to find the optimal partial hedging strategy would be to minimize the shortfall risk under the constraint that the initial capital x is less than   0 C (i.e., the amount required for a p fect hedge).iterion used to quantify the shortfall risk is the expectation of the shortfall T is the value of the hedging portfolio.Compare he linear loss function criterion adopted by [1] and [2], convexity of the loss function g offers the flexibility to accommodate different types of market participants with different levels of risk tolerance.For example, pension funds and foundations are usually risk-averse whereas hedge funds are more likely to have risk-seeking behaveiors.Moreover, individual investors' attitudes towards risk are unique depending on their own personal and financial circumstances.
The problem of solvin d with t g for an optimal partial hedging portfolio so as to minimize the shortfall risk is decomposed into two subproblems.One subproblem is to find the attainable terminal wealth that minimizes the shortfall risk under the insufficient initial capital constraint.A convex duality approach is used to obtain an explicit formula for the optimal terminal wealth cating portfolio that generates the optimal terminal wealth   * X T .There are two different approaches to solving ond subproblem.One is the well-known delta-hedging approach and the other is the Malliavin calculus approach.[4] compared these two approaches in the Black and Scholes environment.The author commented that the difficulty of applying the delta-hedging approach is to verify the continuous differentiability condition of the price process of the target claim.In the case of perfect hedge, the difficulty noted above does not exist for a standard call option.The Malliavin calculus approach is only needed for certain path dependent options such as lookback options.However, this is no longer the case in the partial hedging environment.We find that to derive an optimal partial hedging portfolio even for the simplest claim such as an ordinary call option, it is not trivial to verify the continuous differentiability condition for the price process of the optimal terminal wealth.Nevertheless the machinery of the Malliavin calculus approach help circumvent this difficulty.Although the full range of cases remain to be investigated, we illustrate by examples that in the context of partial hedging, the Malliavin calculus approach is not only mathematically rigorous, but also straightforward and easy to implement.Explicit worked-out examples in previous partial hedging studies are only restricted to standard European options.In this paper, by applying the Malliavin calculus approach, we are able to obtain the explicit partial hedging formula for a lookback option.
It is worth noting that the Malliavin calculus approach the sec ha anized as follows.Section 2 se s gained considerable interest since it was first introduced to the portfolio theory literature by [5].For example, [6] applied the Clark-Ocone formula and the gradient operator in Malliavin calculus to derive an explicit representation for the optimal trading strategy in the case of partial information.The Malliavin calculus approach has also been used to derive perfect hedging strategies for lookback and barrier options (see [7,8]).[9] found the Malliavin calculus approach useful in deriving the hedging portfolios for an expected-utility-maximizing investor whose consumption rate and terminal wealth are subject to downside constraints.
The rest of the paper is org ts up the model for the financial market, presents the dynamics of the agent's wealth process   ,π x X  , and defines the class of admissible portfolios

 
x  ction 3 solves the problem of minimizing the expected shortfall loss using the convex duality approach.The main result is an explicit expression for the optimal terminal wealth   * .Se X T .The existence of an optimal hedging strategy wn as well.The Malliavin calculus approach is summarized in Section 4 and is used to derive the optimal partial hedging portfolios for two specific examples in Section 5.

The Economy is sho
Since our ultimate in sions for the optimal p terest is to obtain explicit expresartial hedging portfolios, the model under consideration here is a typical Black and Scholes economy as in [10], wherein there are one riskless asset of price B and one risky asset of price S .We shall assume that the riskless asset B earns a constant instantaneo s rate of interest r , and that e price S of the risky asset follows a geometric Brownian motion.More specifically, the respective prices All our problems are treated on a finite time-[0 horizon ,T].In Equation ( 3), is a standard Brownian motion on a complete probability space   , ,P   endowed with an augmented filtration . We ass  (stock retur rate), n  (stock volatility), and s are positive constants.
Set "the market-price-of-risk" he auxiliary probability measure and t defined on According to the Girsanov theorem the process In the context of the above market model, consi agen who is endowed with der an t initial wealth > 0 x , can decide, at each time , which amount   π t to invest in the risky asset without affecting its p .We shall denote by rice   X t the wealth of this agent at time t .With   π t chosen, the investor places the amount in the bank account.The agent's wealth process satisfies the equation Formally, we say that a trading strategy   π  over the time interval   0,T is self-financing if its wealth process satisfies (7).We require that the wealth process   X  in (7) . The other is the investor's attitude towards the shortfall risk, which is ss functi captured by a lo on g .We assume that g is an increasing and strictly convex function defined on   0,  , with   0 = 0 g .We f her assume that urt g is in

al definition hortfall risk
We now Definition pe give the form 3.1 The s of the risk measure. is defined as the exctation of the shortfall by the loss func weighed tion g .Remark 3.1 A special case of the risk measure defined [ with above is the lower partial moment (e.g., 11]) Our aim is to find an admissible portfolio   π  which ptimization lem solves the o prob for any T that mini mizes the shortfall risk in (13) under the constraint that . We first make the follow ul observation.

Let
The proo view of Lem f of the lemm ma 3.
T is assumed to be nonnegative.Hence for any admissible portfoli we have  as the inverse of g .In the case of . We shall adopt use from convex duality: starting wit the fu The supr of the reader, we summarize ies of the function For the c below some g  .

Lemma 3.2
The function g  enjoys the following properties.
1) for all 2) The function g  is convex and continuous.Proof.a) It follows from the explicit expr sio es n of g  in ( 16) that g  is nondecreasing.From the facts that g  is nondecreasing and fo  follows directly from the explicit expression of g  i ( 16).
It follows that for any initial capital most surely.Thus, in conjunction with ( 6) and , we obtain

< (21)
Now assume that for every , we have 0. ( Remark 3.2 Note that (22) is not as Proof.First we show that the function , then by assumption (22), and the dominated m i convergence theore lies the continuity of follows similarly using the bound To derive the maximum of and the dom ated convergence theo From ( 16) it follows that is convex and continuous on   0,  , and continuously differentiable on In particular, , we have For every 0 Indeed, by assumption (23),

E H T g H T E H T I H T H T C
  We now establish the following auxiliary number that be an arbitrary positive such ost su ely, and by the convexity of Relation (25) and the dominated convergence theorem imply (28) The right hand side of ( 28) is exactly 24) and (20).We know from Lemma 3.3 that n in th lity approach.
conditions for st uality of (19) holds as equality for some The following is the crucial observatio e dua Remark 3.3 (Sufficient and necessary rong duality) The ineq and with i we have and Now we are ready to state the main r oposition 3.1 For every the value esult of this section.
Proof.From (20), (1), and th t that we see that 4 Notice that whenever the optimal terminal wealth in (33) has a rm of In particular, it is equal to that portfolio which replicates the claim  

 
π  s (see [12], p. 93).The process  and the quirement that it's bounded from belo re zero, so w by is then a consequence of Remark 3.3.

The Malliavin Calculus Approach
To solve the second subproblem of deriving the partial hedging po o s that generate the optimal term wealth in (33), we consider different a .One is the ll-known  -hedging approach and the other is the Malliavin cal p ach.For easy reference and to make the paper self-contained, we first he concept of culu  -hedging.Then we introa random duce the definition of the Mallivin derivative of variable and the Clark-Ocone formula.Thereafter, we present a Mallivin calculus approach for deriving the replicating portfolio as first used in [5].
In the standard Black and Scholes framework, the  -hedging approach works in the following way.In a complete financial market, the optimal wealth process  are used interchangeably.) is given by the discounted conditional expectation of the optimal terminal wealth   * X T under the risk neutral probability measure 0 rtfolio is number of units to be held at sk-free asset , and denotes the ts to be held i stoc at tim the relationship between The replicating po denoted by

X t h t rS t h t S t
0) is the famous -hedging formula.As n [4], the major difficulty of using the Fro the uniqueness of the ô integ follows that we can use (37) and (38) to identify the re icating portfolio and identif bability with 2 0 0 = : 0, : is absolutely continuous with respect to the Lebesgue measure on   We note that the Malliavin derivative is well defined almost everywhere d d Now, as the Malliavin derivative is a closa le operator (see [13]), we define by the Banach space which is th  .[8] shows that the Clark-Ocone formula is valid for any L  and therefore the Malliavin calculus approach to deriving the replicating portfolio of a contingent claim as in [5] can be extended in a similar way.However, since all of the examples discussed later in the paper only use the Clark-Ocone form to the stochastic variables in 1,2  , we only state th inal Clark-Ocone formula in [13] an ula e orig de d the results in [5] as a theorem.We refer the interested rea rs to [8] for the extensions. .Then we have the representation formula Following the above Clark-Ocone formula and e results in [5], any optimal portfolio X T  can be replicated by the self-financing portfolio verify the required differentiability condition.The firs example in this section shows that this is no longer case in the partial hedging environment.Since the optimal wealth process for partial hedging a standard call option does not possess a closed-form formula, the verification of the continuous differentia ility condition is no longer a trivial issue.In the second example, we derive the explicit representation for the partial hedging portfolio of a standard lookback put option.For explicit computational purpose, the loss function we shall use in both . Note that our approach can be straightforwardly adapted to solve the problem with a more general convex loss function.

Example 5.1 Consider partial hedging a standard European call option with payoff function
. It follows from Remark 3.4 and the facts From (3), ( 4), (5), and (10), we have the time T stock price and the time T state price density To apply the  -hedging approach, one needs an expression for From the formulae (3) and ( 10 . The expression (48) can be rewritten as -measurabl the Brownian motion, we e.By the basi ties of can rewrite (49) as c proper

r T t y e S t e K S t y bT y y s
here  is the normal density function mean with with the -hedging approach, one has to assume that (50) is a function of owever, in general, (

T t y D h t e e t y bT y y
where the set .Recall the relationship       1 = π h t t S t .After further simplifications, we have We now derive the partial hedging portfolio using the Malliavin calculus approach.The following prope which are proved in the Appendix, hold.
Corollary 5.1  .The optimal terminal wealth is To pursue the  -hedging approach, one needs an expression for the optimal wealth process In the Ap e show that (56) can be rewritten as and  is given in (68).First of all, it is very hard, if not impossible to obtain a closed-form expression for (57).Furthermore, the integrand in (57) is not differentiable with respect to dition is not easy.Even if we ignore this technical difficulty, formal differentiation (disregarding t where it is non-differentiable) would be poss algebraically very messy in addition to being non-rigorous.For these reasons we abandon the delta-hedging h and pursue the Malliavin s he points ible but gets approac calculu approach instead.
Corollary 5.2 Let the optimal terminal wealth be defined by .
Now the last conditional expectation on the right ati hand side of Equ on (58) can be written as 58) and (60) yield the optimal partial hedging strategy chastic control problem of minimizing the expected shortfall risk (quantified by a general convex risk measure) under the constraint that the initial capital is insufficient for a perfect hedge.We showed by examples that the Malliavin calculus approach is for finding the replicating portfolios in the partia ing environment.Further research consists of investigating other target contingent claims and extending the results to incomplete markets.,π ,π ,π = 0 x x then ent of the lemma.Suppose   ,π 3 2 and = 0.   continuous with respect to the Lebesgue measure on 3   we get that

0 W
in its representation as a stoch spect to    for a suitable portfolio proces the Cameron-Martin space  such that

Theorem 4 . 1
Let the stochastic variable F belong to 1,2

2
Now let us consider deriving the partial hedging portfolio for a standard lookback put option with terminal payoff where which coincid (53 that the Malliavin calculus approach not only avoids the technical difficulty encountered by the  -hedging approach, but also uses a sequence in  and use Definition 4.1 together with the closability of the Malliavin derivative.Notice also that both and (63) im-De(57).We are goin rive an expression fo the conditional expectation

2 X
T  ,and as the joint law of