A Monte Carlo Approach to Currency Risk Minimization - IEOR @IIT ...

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or more and indicates a level of confidence), and the expected tail loss or ConditionalValue-at-Risk (CVaR). The mathematical expressions for VaR(ψ T ) and CVaR(ψ T ) aremore involved and hence we refer the reader to Alexander (2009); Dowd (2005); Jorion(2006) for a detailed study of VaR and CVaR.Notethateachofthesemeasureshavetheirownadvantagesanddisadvantages. Hence,one should look at more than one risk measure to assess the risk. If the risk measures arenot within allowable limits with no hedging then we have to consider (static or dynamic)hedging to minimize the risk. In the subsequent sections, we provide details of static anddynamic hedging.3. Static HedgingIn this paper, we consider only those hedging strategies that involve setting up a hedgingportfolio of liquid Forward FX contracts. If the company is expecting N units of FC andan obligation of NB units of HC at time T and the risk due to X T is beyond allowablelimits then the company has to employ a hedging strategy. In this section, we considerthe static hedging case where the company invests only once (at time T 0 ) in a forward FXcontract. The company has a choice of i) covering all the N units of FC, that is, enteringinto N forward contracts each of unit value, ii) covering none of the FC (though this isno hedging, it is technically a (trivial) static hedging strategy), or iii) or cover ζ 0 units ofFC. The determination of the decision variable ζ 0 (whether 0, N, or some other number)is the problem of static hedging. The objective is to minimize the risk where the P & Lwill take the formψ T = N(X T −B)+ζ 0 (F 0,T −X T ), (2)where the first term in (2) is the same as P & L expression (1) in the no hedging casewhile the second term is the P & L coming from the forward contracts. Here, F 0,T denotethe forward FX price (or forward rate) of a forward contract which is initiated at timeT 0 and matures at future date T. For simplicity it is assumed that there are no bid-askspreads in the forward contract quotes and there are no transaction costs or cost-to-carryin trading these contracts. Some of the regulatory requirements that enforce constraintson over-hedging and under-hedging position limits are also ignored.In a given static hedging strategy the company is buying ζ 0 units of the forwardcontract atF 0,T . Thisarrangement allowsthecompany tosell FCcurrency atanexchangerate of F 0,T at time T. By locking into a forward contract to sell FC, the company sets afuture exchange rate without any additional cost. Any remaining N −ζ 0 units of FC isthen exchanged at T (at spot rate).One way of determining ζ 0 is to proportionally cover FC based on an estimate ofprobability that X T is lower than the budget rate B. Since the case of X T to cover p 0 N units of FC, where p 0 is the probability suchthat X T < B. The strategy where ζ 0 = p 0 N is called static proportional strategy.Here, we propose a static hedging strategy based on an optimal hedging problem.Specifically, let ρ(ψ T ) denote a risk metric (perhaps, E(ψ T |ψ T < 0) or P(ψ T |ψ T < 0)or CVaR(Ψ T )) which in turn depends on ζ 0 . Now, consider the following optimization4
problemminζ 0 ∈Λ 0ρ(ψ T ), (3)where Λ 0 is the set of real numbers. Let ζ0 ∗ denote the solution to the optimizationproblem (3). Since the no hedging strategy and the static proportional strategy are bothstatic hedging strategy with ζ 0 = 0 and ζ 0 = p 0 and 0,p 0 ∈ Λ 0 it follows that the optimalstrategy ζ0 ∗ will be better than both no hedging and static proportional strategies in termsof the risk measure ρ(·).4. Dynamic HedgingIn the previous section, we have seen that (optimal) static hedging strategy reduces therisk compared to the no hedging strategy. In this section, we consider dynamic hedgingstrategies where the company chooses to enter into FX contracts at a fixed set of times.It is easy to see that the optimal risk reduces with increase in number of hedging times.In reality the best possible dynamic hedging strategy will be a continuous-time hedgingstrategy, where the company updates the hedging strategy at every instant of time. Anexample of a continuous-time strategy is the popular ∆-neutral hedging strategy. Needlessto say that it is practically impossible to implement a continuous-time strategy and inpractice, the presence of transaction costs precludes any type of high frequency hedging.The problem of determining the appropriate number of hedging times is a very difficultproblem and is not considered here. Even if the number of hedging times is fixed thenone may consider the problem of determining optimal hedging times. This leads to anoptimal stopping time problem. In this paper, we assume that the company choosesa set of fixed hedging times T 0 , T 1 ,...,T n−1 < T n = T based on several considerationsincluding transaction costs. At each time T k , the company enters into ζ k units of forwardFX contract whose forward rate is F k,T . In this case, the P & L expression becomes∑n−1ψ T = N(X T −B)+ ζ k (F k,T −X T ), (4)where, as in the static hedging strategy, the first term of (4) is the same as P & Lexpression (2) while the second term is the P & L coming from all the forward contracts.Recall that ζ 0 in the static hedging case is a real number using only informationavailable at T 0 . Since we need to determine ζ k at only T k it is more beneficial to useall the information available up to time T k , specifically, all the available exchange rateinformation, namely, X 0 , X 1 , ..., X k . Since X k , k > 0, is a random variable it followsthat ζ k is also a random variable. This poses a significant difficulty in the optimal hedgingproblem, which will be considered in detail in the next section.One way of determining ζ k is to proportionally cover FC based on an estimate ofprobability that F k,T is lower than the budget rate B. This is a generalization of thestatic proportional strategy and we refer to this as dynamic proportional strategy. In thiscase, ζ k = p k N where p k is the probability such that F k,T < B.k=05
Page 1 and 2: A Monte Carlo Approach toCurrency R
Page 3: (Alexander, 2009; Wilmott, 2007). H
Page 7 and 8: Figure 1: Dynamic Hedging Strategy
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