Du#e D. and Richardson H.R., 1991, "Mean-Variance Hedging in Continuous Time", Annals of Applied Probability, 1, 1-15.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....matching the sensitivities of the hedging and target portfolio to changes in the underlying asset price. In an attempt to improve the performance of the hedges we can choose our positions in the hedging assets in order to minimize the variance of the partially hedged position as described in [10, 20, 19]. Each time the hedge is rebalanced, we solve an optimization problem and select our hedging positions so that the variance of the hedged position is minimized. In Table 3 we see that the minimum variance optimal (MVO) hedge using only the underlying asset as a hedging instrument still o#ers very ....

D. Du#e and H. R. Richardson. Mean-variance hedging in continuous time. The Annals of Applied Probability, 1:1--15, 1991. 24

....see [3] An illustration from Hubalek and Schachermayer [13] is when the two assets are di erent brands of crude oil, only one of which is liquidly traded. This problem is an example of the problem of pricing a claim in an incomplete market and is similar to those considered in [7] 6] 20] [4] and many others. In common with [4] and [2] we model our agents as maximisers of expected utility. Another common approach is to select a martingale measure (for example the minimal martingale measure) and to use that for pricing. For the problem detailed above Hubalek and Schachermayer [13] ....

....and Schachermayer [13] is when the two assets are di erent brands of crude oil, only one of which is liquidly traded. This problem is an example of the problem of pricing a claim in an incomplete market and is similar to those considered in [7] 6] 20] 4] and many others. In common with [4] and [2] we model our agents as maximisers of expected utility. Another common approach is to select a martingale measure (for example the minimal martingale measure) and to use that for pricing. For the problem detailed above Hubalek and Schachermayer [13] show that no arbitrage arguments give ....

DUFFIE, D. and RICHARDSON, H.R. : Mean-variance hedging in continuous time. Ann. App. Prob., 1, 1-15, 1991.

....to justify a particular choice. One choice of measure is the variance optimal measure. Under certain regularity conditions this is the mean variance option pricing measure introduced in the martingale case by F ollmer and Sondermann [6] and extended to the general case by Due and Richardson [5], Schweizer [23] and Gouri eroux et al. [9] amongst others. The use of the variance optimal measure in stochastic volatility models has been extensively investigated by Laurent and Pham [16] Biagini et al. [1] and Heath et al. [11] and these studies provide a starting point for our investigations. ....

Due, D. and Richardson, H.L.; Mean-variance hedging in continuous time. Annals of Applied Probability. 1, 1-15, 1991.

....the terminal wealth and to minimize the risk using the variance as a criterion, which stems from the investors goal of seeking highest return upon specifying their acceptable risk level. Owing to its practical value, the mean variance model has drawn continuing attention; see for example, [15, 18, 8, 5, 6, 7, 17, 4, 19] among others. Recently, using the stochastic LQ theory developed in [2] a stochastic linear quadratic (LQ) control framework for studying mean variance optimization hedging problems was introduced in [27] along with a closedform solution of the optimal portfolio policy and an explicit ....

D. Du#e and H. Richardson, Mean-variance hedging in continuous time, Ann Appl Probab. 1 (1991), 1-15.

....as follows: Consider an individual who is free to trade in an asset with price process P t . How much should the individual be prepared to pay now in order to receive a random payment Y at some pre determined time T in the future This problem is similar to those considered in [4] 3] 11] and [2]. In common with [2] we model our individuals as maximers of expected utility. In x2 we outline the philosophy which underlies our pricing mechanism. We de ne two prices, the bid price and the ask price which correspond to the price at which the individual is prepared to buy or sell the claim Y . ....

....an individual who is free to trade in an asset with price process P t . How much should the individual be prepared to pay now in order to receive a random payment Y at some pre determined time T in the future This problem is similar to those considered in [4] 3] 11] and [2] In common with [2] we model our individuals as maximers of expected utility. In x2 we outline the philosophy which underlies our pricing mechanism. We de ne two prices, the bid price and the ask price which correspond to the price at which the individual is prepared to buy or sell the claim Y . The bid price never ....

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DUFFIE, D. and RICHARDSON, H.R. : Mean-variance hedging in continuous time. Ann. App. Prob., 1, 1-15, 1991.

....problem being actively investigated is a mean variance hedging, in which EjX(T ) Gamma j 2 is to be minimized, where is a random claim which can not be replicated perfectly. For this problem, an explicit solution was obtained for a very general model (see e.g. Follmer and Sondermann (1986) Duffie and Richardson (1991), Pham et al. (1998) Laurent and Pham (1999) The corresponding optimal hedging strategies are mixture of Merton s strategies and Black Scholes strategies. For all these strategies, the volatility coefficient is assumed to be known, and the strategies depend on the future values of volatility. ....

Duffie, D., and Richardson, H. (1991): Mean-variance hedging in continuous time.

.... final time T , and U( Delta) is an utility function (see e.g. 5] 8] 9] 10] 15] 17] 19] Another problem being actively investigated is the problem involving a mean variance hedging, i.e. a problem in which EjX(T ) Gamma j 2 is to be minimized, where is a random claim (see e.g. [4], 7] 11] 12] 16] Explicit formulas for optimal strategies for all these optimal investment problems are now available for cases when appreciation rates are random, but can be observed. However, it is practically not possible to accurately estimate the appreciation rates from the real ....

D. Duffie and H. Richardson, Mean-variance hedging in continuous time. Ann. Appl. Probab., 1, (1991) 1-15.

.... natural to look for a best approximation of H by the terminal value c G T (#) of some pair (c, #) The use of a quadratic criterion to measure the quality of this approximation has been proposed by Bouleau Lamberton (1989) if X is both a martingale and a function of a Markov process, and by Du#e Richardson (1991) and Schweizer (1994) among others, in more general cases. To find such a mean variance optimal strategy, one therefore has to project H in L 2 (P ) on the space IR G T (#) of attainable claims. In particular, this raises the question whether the space G T (#) of stochastic integrals is ....

D. Du#e and H.R. Richardson (1991) "Mean-Variance Hedging in Continuous Time", Annals of Applied Probability 1, 1--15.

....a continuous process S = M ff Delta hM i such that there exist equivalent martingale measures Q (even with dQ dP uniformly bounded) but nevertheless the local martingale E( Gammaff Delta M ) is not uniformly integrable. Hence, despite of many appealing properties (see, e.g. FS 91] DR 91] AS 93] Schw 92a] Scha 94] one cannot rely on the existence of the minimal martingale measure, even if S is continuous and models a perfectly arbitrage free market. Another natural approach is to look at the element of M e (P) of smallest L 2 norm, in other words to look for the ....

....Q 0 martingale Z opt t = EQ 0 Theta Z opt 1 j F t : The next lemma shows that the process Z opt is independent of the choice of Q 0 and may be written as a constant c, given by kZ opt 1 k 2 L 2 (P) and a stochastic integral on S. This basic fact was already observed in [DR 91] Scha 94] Schw 94] in various degrees of generality. We refer to [Schw 94] for an account on these results. 2.2 Lemma. Let S be a locally bounded semi martingale such that M e (P) L 2 (P) 6= and fix Q 0 2 M e (P) L 2 (P) Letting c = kZ opt 1 k 2 L 2 (P) we may find a ....

D. Duffie, H.R. Richardson, Mean-Variance Hedging in Continuous Time, Annals of Applied Probability 1 (1991), 1-15.

....economies trading derivatives, on the so called factor mimicking portfolios. Mean Variance preferences are also employed to formulate hedging strategies in incomplete markets contexts, and lead to the minimal martingale law as the candidate for the pricing kernel (see Follmer and Schweizer (1990) Duffie and Richardson (1991). As shown in Colwell and Elliott (1993) and Runggaldier and Schweizer (1996) for continuous time models with jump discontinuities, and Elliott and Madan (1996) for discrete time models, the resulting minimal martingale law pricing kernel is no longer non negative. This again yields the ....

Duffie, D. and H.R. Richardson (1991), "Mean-Variance Hedging in Continuous Time," Annals of Applied Probability, 1, 1-15.

.... depends on the distribution of income (demand) and the availability of low coupon bonds (supply) The latter can abruptly change. The tax induced premium on bond prices need not be a smooth function of the coupon, and it certainly depends on both the coupon and the duration. See [Sch82] and [DR91]. iv) Yield to maturity is the concept that rules today s bond markets. This is very nice and very simple, and an ingenious approximation but sometimes it is wrong. This could mean we observe a market inefficiency yet to be corrected. The tax effect is tricky and we have no complete solution ....

....incomplete models, were the principle of arbitrage no longer yields a single price, but arbitrage bounds. Other pricing principles like equilibrium pricing or individual optimization problems are needed to single out one of the prices in the arbitrage free range. The theory presented here (and in [DR91]) restricts itself to buy and hold strategies. Dynamic trading strategies would provide more realistic arbitrage bounds. They also require assumptions on the stochastic properties of bond price processes, which is beyond the scope of this paper. Dermody and Rockafellar (1991, 1993) go farther than ....

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Darrell Duffie and Henry R. Richardson. Mean-variance hedging in continuous time. The Annals of Applied Probability, 1(1):1--15, 1991.

.... further details the reader is referred to Stricker (1990) Back and Pliska (1991) Delbaen (1992) Lakner (1993) Schachermayer (1994) 2 Considerable advances are also currently being made in bridging this gap by when equivalent martingale measures are not unique, F llmer and Schweizer (1991) Duffie and Richardson (1991) and Colwell and Elliott (1993) have recently proposed and studied the use of the minimal martingale law as a choice for an equivalent martingale measure. This change of measure has the minimal property that only the martingales driving the asset prices are affected by the change of measure in ....

....j,k 1 N s (32) c (S , S ) V (S , S ) h S . k 1k Nk k 1k Nk t j,k 1 jk j=0 The minimal martingale law is consistent with investment strategies designed to choose h with a view to minimizing the F conditional expectation of the j,k 1 t 1 square of c (F llmer and Schweizer (1991) Duffie and Richardson (1991), Colwell and k Elliott (1993) As demonstrated in section 2, the minimal martingale law generally fails to deliver a probability law in discrete time. For the hedging strategy associated with the Girsanov principle we define the costs of hedging using risk adjusted asset prices in evaluating ....

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Duffie, D. and H.R. Richardson (1991), "Mean-Variance Hedging in Continuous Time," Annals of Applied Probability, 1, 1-15.

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Du#e D. and Richardson H.R., 1991, "Mean-Variance Hedging in Continuous Time", Annals of Applied Probability, 1, 1-15.

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DUFFIE, D. and RICHARDSON, H.R. : Mean-variance hedging in continuous time. Ann. App. Prob., 1, 1-15, 1991.

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DUFFIE, D. and RICHARDSON, H.R. : Mean-variance hedging in continuous time. Ann. App. Prob., 1, 1-15, 1991.

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Due, D. and Richardson, H.L.; Mean-variance hedging in continuous time. Annals of Applied Probability. 1, 1-15, 1991.

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D. Du#e and H. Richardson. Mean-variance hedging in continuous time. Annals of Applied Probability, 1:1--15, 1991.

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D. Duffie and H.R. Richardson, Mean-variance hedging in continuous time, Ann. Appl. Probab. 1 (1991) 1-15

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Due, D. and Richardson, H. (1991), 'Mean-Variance Hedging in Continuous Time`, Annals of Applied Probability 1,1, 1-15.

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Duffie, D., Richardson, H.: Mean-variance hedging in continuous time. Ann. Appl. Probab. 1, 1-15 (1991)

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