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

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2003) and the specific case of exchange rate risk comes under currency risk management(Stephens, 2001; Xin, 2003).Risk management consists primarily of an accurate analysis of the risk factorsinvolved(FX rate, in this case) and take appropriate action to mitigate the risk. Any such actionis known as hedging. There are many alternatives available to the company for hedgingincluding conversion of FC to HC at an appropriate time or enter into different hedgingcontracts with some financial institution. Such contracts may involve FX instruments(also known as hedges) such as forwards, futures, swaps, and a variety of FX options(Stephens, 2001; Xin, 2003). Specifically, a FX forward is an agreement between thetwo parties to exchange currencies, namely, to buy or sell a particular currency at apredetermined future date and a predetermined exchange rate. It costs nothing to entera forward contract. The party agreeing to buy the currency in the future assumes along position, and the party agreeing to sell the currency in the future assumes a shortposition. The price agreed upon is called the delivery price, which is equal to the forwardrate at the time of initiation of the contract. An option sets a rate at which the companymay choose to exchange currencies. If the exchange rate at option maturity is morefavorable, then the company will not exercise this option. Forward FX contracts areconsidered to be the simplest and the most widely used instrument for currency hedging.Their popularity may be partially explained by their simplicity, their initial zero cost, andfeasibility of over-the-counter (OTC) trading that permits exact specifications regardingdates and amounts. In this paper, we primary focus on optimal hedging problem usingForward FX contracts as a hedging instrument although the overall methodology caneasily be extended to include other instruments. For a general introduction to various(equity-based) financial instruments and their utility in heading, see Wilmott (2007); Hull(2008).Ahedging strategy represents atradingstrategyinvolving differenttypesofhedgesthatcan reduce currency risk. The hedging strategy typically depends on information such asexchange rate fluctuations and possibly a predictive model for the exchange rate (the socalled market view). These hedging schemes involve forward-rate based transactions toessentially lock infutureexchange rate. Suppose a company isexpecting a certain amountofFCatafuturedate. Thecompanyhasachoiceofdoingnothing(no hedging),enter intoa hedging contract once (static hedging), or enter into multiple contracts at different times(dynamic hedging). The choice depends on various reasons including the companies viewon the market and service costs to enter hedging contracts. If the company can eliminateorachieve minimumpossibleriskusingnohedgingorstatichedgingthendynamichedgingoffers no advantage. However, in highly volatile periods, dynamic hedging may achievesignificantly less risk compared to static hedging. Currency risk hedging remains anactiveareaofresearch(Stephens, 2001;Xin,2003;Gagnonetal.,1998;LypnyandPowalla, 1998;Campbell etal.,2010;GlenandJorion,1993;Chang,2011). Mostoftheexisting literatureeither focuses on static hedging or specific risk measures such as variance. In this paper,we present a general framework for designing risk optimal discrete-time dynamic hedgingstrategy where the risk measure can be chosen arbitrarily.Below, we summarize our framework for obtaining the optimal hedging strategy:i) In this paper, we assume that the FX rate follows a geometric Brownian motion2
(Alexander, 2009; Wilmott, 2007). However, it should be noted that the currentframework can easily be extended to other models such as GARCH-based modelsor stochatic volatility models.ii) In this paper, we assume that hedging can be performed at a fixed set of times,that is, the position of the forward contracts in the hedging portfolio is updated atpre-determined fixed set of times using information available upto that time. Weassume that the hedging strategy is a function of FX rates at all past and present(but not future) hedging times. In order to perform numerical optimization, weassume that this function is in terms of a finite set of paraemeters represented on ascenario tree (see below).Notethat the number of hedging times andthe hedging times can also be optimized.However, this leads to an optimal stopping time problem, which is very difficult tosolve and we do not consider it here.iii) Here, we minimize a risk metric (Artzner et al., 1999; Alexander, 2009; Dowd, 2005)defined on the P&L over the set of discrete-time hedging strategies defined on a(given) scenario tree. Though the framework allows an arbitrary risk metric, in thispaper, we only consider conditional value-at-risk (CVaR), probability of loss, andconditional expected loss. For a given hedging strategy, we first simulate sufficientlylarge number of Monte Carlo paths and then the risk metric is computed using thecorresponding P&L values. The optimal hedging strategy is obtained using either apattern search algorithm for single metric minimization and a genetic algorithm formulti-objective minimization (Doherty et al., 2004). Both these algorithms requireonly the objective function value and not its gradient information.2. No HedgingIn this section, we consider the case of no hedging. Here, we assume that a company isexpecting a foreign currency of N units at a future time T (we denote the current time tobe T 0 (< T)). Furthermore, we assume that the company has an obligation (to be paid)of NB units of HC, where B is called the budget rate. Typically the budget rate B isdetermined by the company once a year based on many factors such as costs and profitexpectations. Let us denote by X t the exchange rate in number of HC units for one unitof FC at time t. If the company converts all the FC into HC at time T then the profit/loss(P & L) ψ T is given byψ T = N(X T −B). (1)A positive ψ T represents a profit and a negative ψ T represents a loss. Since X T is arandom variable it follows that ψ T is a random variable and hence we have to use probabilitybased risk metric/measure to predict the profit/loss. A variety of risk measures areavailable to capture the randomness of ψ T (Artzner et al., 1999; Alexander, 2009; Dowd,2005). Two measures that may be used are conditional expected loss (E(ψ T |ψ T < 0))and probability of loss (P(ψ T |ψ T < 0)). Other popular measures include Value-at-Risk(VaR), the maximum likely loss to occur in (1−α)% of cases (where α is typically 95%3
Page 1: A Monte Carlo Approach toCurrency R
Page 5 and 6: problemminζ 0 ∈Λ 0ρ(ψ T ), (3
Page 7 and 8: Figure 1: Dynamic Hedging Strategy
Page 9 and 10: 0.90.8Efficient FrontierProportiona

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