Pricing Options On Foreign Currency With A Preset Exchange .

2y ago
41 Views
3 Downloads
1.14 MB
11 Pages
Last View : 2m ago
Last Download : 2m ago
Upload by : Annika Witter
Transcription

Journal of Mathematical Finance, 2012, 2, 214-224http://dx.doi.org/10.4236/jmf.2012.23024 Published Online August 2012 (http://www.SciRP.org/journal/jmf)Pricing Options on Foreign Currency with a PresetExchange RateAvner Wolf, Christopher HesselDepartment of Economics and Finance, Baruch College, New York, USAEmail: er.wolf@baruch.cuny.edu, Christopher.hessel@baruch.cuny.eduReceived May 4, 2012; revised June 8, 2012; accepted June 15, 2012ABSTRACTThis paper presents a new option that can be used by agents for managing foreign exchange risk. Unlike the GarmanKolhagen model [1], (GK), this paper presents a new model with a preset exchange rate (PE), that allows the agent totake advantage of the his/her view on both the direction and magnitude of rate movement and as such provides thisagent with more choices. The model has a provision for an automatic exchange of the payoff at a preset exchange rate,and upon expiration gives the agent the choice of keeping the payoff in the foreign currency or exchanging it back to thepricing currency. At the time of writing, the buyer selects the preset exchange rate. Depending on the value selected, thePE option’s price and payoff will be equal to, greater than or less than those of the GK model. A decision rule forchoosing between the PE and GK models is developed by determining the expiration spot rate that equates the twomodels’ returns. The range of spot rates that makes the PE option’s return greater than the GK’s return is the PE preferred range. If the agent expects the expiration spot rate will be within the preferred range, the PE option is purchased.The size of the preferred range is a decreasing function of time to expiration, a decreasing function of spot rate volatilityand an increasing function of the basis point spread between foreign and domestic interest rates.Keywords: Garman Kolhagen; Option Pricing; Currency; Foreign Exchange; Hedging; Speculating; Preset ExchangeRate1. IntroductionThe options markets have grown tremendously since theintroduction of Chicago Board Options Exchange(CBOE) in 1973. With the introduction of InternationalSecurities Exchange (ISE), the first electronic exchangein the USA, the market received a significant boost andthe options markets reached new heights. The equity options daily volume in 1997 was 1.079 million contracts,and at the end of the 1st quarter of 2011, the volume was17.609 million contracts for an average annual growthrate of 21.8%1. The foreign exchange (FX) options alsogrew rapidly. The FX option notional amount in 1995was 0.8 billion and by 2010 was 12.1 billion, for anaverage annual growth rate of 39.74%2.To a great degree growth of options trading was madepossible due to the applicability of options pricing models developed in the seminal papers of Black and Scholes(BS) [3,4], and Merton [5] which created a new era andrenewed interest in options pricing and has inspired re1Data for the number of contracts from 1997 to 2011 was provided bye-mail on March 20, 2011.2Data is from Bank of International Settlements website, “TriennialCentral Bank Survey of Foreign Exchange and Derivatives MarketActivity” reports for 1996-2010 [2].Copyright 2012 SciRes.searchers to expand the pricing model in many directionsincluding correcting for the unrealistic assumption ofconstant volatility. Hull and White [6,7], Ritchken andTrevor [8] and Wiggins [9] generalized the constantvolatility models to allow for stochastic volatility. Hestonand Nandi [10] develop a closed-form single laggedGARCH model. Later Christoffersen, Heston and Jacobs[11], and Mercuri [12] extended Heston and Nandi’smodel, where Christoffersen et al. utilizes inverse Gaussian innovations and Mercuri utilizes tempered stable innovations. Barone-Adesi, Rasmussen and Ravanelli [13]value the stock option using a GARCH diffusion model.While Badescu and Kulperger [14] capture skewness andleptokurtosis observed in stock price data. Gong, Thavaneswaran and Singh [15,16] model the expiration callprice as the expected price of a truncated normal distribution. The need to capture the radical changes in volatility over time lead to the development of jump andswitching models. Elliot, Sui and Chan [17] presentchange in volatility as a switching Markov process withthe transition accomplished through an Esscher transformation. Both Thavaneswaran and Singh (TS) [18,19]and Pillay and O’Hara’s [20] (PH) incorporate the lognormal distribution. TS has a jump diffusion model withJMF

A. WOLF, C. HESSELstochastic volatility, where the expiration price is a moment of a truncated log-normal distribution and PH hasthe stock price follow a mean reverting log-normal process with stochastic diffusion and jumps and the option’sprice determine by fast Fourier transformation methodology. Makate and Sattayatham [21] have stock pricefollow a jump diffusion process with square root stochastic volatility and mean reversion, and Sarisa Pinkham,Pairote & Sattayatham [22] model change in stock priceas a linear combination of the time-change Lévy processand the Vasicek stochastic interest rate process. Much ofthese innovations are discussed in Gong, Thavanewswaran and Liang [23] where they use partial differentialequations for various stochastic diffusion models to studyoption pricing with the pure jump process, jump diffusion process, stochastic volatility and jump diffusionwith stochastic volatility. Ivancevic [24] shifts from theBlack and Scholes option pricing equation to a Kolmogorov probability approach to develop an adaptive waveform nonlinear stochastic option pricing model.Leaving the realm of option pricing for stock and stockindices, GK [1] extended the BS model to price optionson foreign currencies. For over a quarter of a century, theGK option-pricing model has been the standard foreigncurrency option-pricing model in pricing European styleoptions and the base for modified pricing formulation forAmerican style options. In 2007 Ahn, Cho and Park [25]took issue with the constant volatility assumption in GKand modeled volatility as a jump diffusion process toaddress pricing currency options under shocks such aslarge changes in the monetary system introduced by central banks or catastrophic events such at 9/11, HurricaneKatrina and the tsunami that struck Japan. This papermoves in a different direction and presents a currencyoption that enhances the set of choices for the participating agents. Based on economic theory, moving thechoice set towards a complete one helps to improve thewell-being of the participating agents by allowing themto move to a higher utility function. In this paper, weintroduce the Preset Exchange rate option model. The PEoption model has three distinct characteristics: 1) Theoption’s user sets the automatic exchange rate, “E”,which converts the premium currency payoff to theforeign currency payoff; 2) The option’s buyer has thechoice of keeping the payoff in the foreign currency orchanging it to the premium currency at the spot rate; and3) If the anticipated spot rate change occurs, the returnfrom the PE option exceeds the return of the GK option.A specific value of E makes the PE value identical tothat of GK. We term that value of E, the breakeven pointand designate it EBE. Not only is the price the same, but ifthe expiration spot rate equals EBE, the payoffs are thesame. The agent, who chooses between the PE options orthe GK options, bases the decision on whether the agentCopyright 2012 SciRes.215expects the expiration spot rate will be greater or lessthan EBE. As with any option, the value paid is lost, if theoption expires out-of-the-money. With the PE optionssetting E EBE, causes the PE option value GK value,and setting E EBE, causes the PE option value GKvalue. The agent can alter the price (exposure to lossshould the option expire out-of-the-money) by theselection of E. This is a characteristic not available in theGK options. Both price and payoff are inversely relatedto the size of E. For E E BE , the criteria, for choosingbetween the two option models, is based on percentreturn on investment. By determining the expiration spotrate, S*BE , that equates the percent return on investmentsfor the two models, the criteria for choosing between thePE model and the GK model is identified. The choice isbased on whether the agent expects the expiration spotprice will be greater than or less than S*BE . Of importance to the agent is the size of the range of spot ratesthat makes the PE the favorable choice. Both the value ofS*BE and its proximity to the strike rate is a function ofthe option’s term, rate volatility and the basis pointspread between the foreign currency interest rate and thedomestic currency interest rate.The remainder of the paper is organized in five parts.We begin by presenting payoff values for the PE call andput options, and compare them to the payoffs values forthe GK calls and puts. Using numerical examples, wedemonstrate the impact of changing the value of E on thePE call, put prices, and present the EBE for both options.We start the analysis with the special case E EBE andpresent a decision rule for the agent to use to choosebetween ownership of or writing of PE and GK modeloptions. We then extend the analysis to E E BE thatallows the development of a more general decision rulefor choosing between ownership or writing of PE or GKoptions written for differing terms, written during differing levels of volatility and written during differingsize spreads between the domestic currency and foreigncurrency interest rates. Next, we present the procedurefor calculating price, payoff, and the necessary decisionrule for implementation of the model. The final section isthe conclusion.2. Payoff of the PE OptionsThis section presents the payoffs for both the GK and thePE models. The analysis of the payoff is a necessary stepin developing the pricing formula for the PE. As such,the PE payoffs are presented in two stages: first, with thepayoff in the foreign currency and second, after converting the foreign denominated payoff to the domestic usingthe expiration spot rate. We then compare the PE and GKpricing formula. Lastly, we compare the two models interms of return on investment.JMF

A. WOLF, C. HESSEL216Generally, the payoffs of the call and the put under thePE setting, prior to converting the payoff to the domesticcurrency are: Cfpe* max S* K E, 0 Ppef * max K S* E, 0 (2) max S* K E, 0 S* max S* K S* E , 0 (3)**2* Cpe max S KS E, 0 The put payoff in domestic currency is: Ppe* max KS* S*2 E, 0 (4)where:C*pe —The PE call payoff in the domestic currencyupon expirationand:Ppe* —The PE put payoff in the domestic currencyupon expirationIn comparison, the standard GK expiration date payofffor calls and puts are as follows: C*gk max S* K , 0 (5)And: Pgk* max 0, K S* (6)*Copyright 2012 SciRes. K S* K K 2(7)As K, the incremental strike price approaches zero,the approximation of the portfolio’s payoff approaches: S* K 22 S*2 2S*2 K K 2(8)We multiply and divide Equation (8) by E and rearranging it to get: E 2 S* K S* E S* K K E(9)Based on the above, the PE call option value canapproximately be replicated constructing the followingportfolio:A long position in 2/E equally weighted portfolio ofEuropean call options where the lowest strike price inthis portfolio is K and the strike price of each subsequentcall option is greater than its previous one by K, and along position in K/E European call option with a strikeprice of K.Accuracy of duplicating the payoff of the original calloption by constructing the above-mentioned portfoliodepends on one’s ability to substantively reduce K, ifpossible close to zero, and simultaneously to increasingnumber of European call options in the approximatingportfolio.With this in mind and using BS [5] pricing setting, theunderlying price process S(t) can be described by thefollowing equation: S t Sxexp rd rf 1 2 2 t w t (10)where:t—Time to expiration expressed as a fraction of a year.S—Current exchange rate.rd—Domestic interest rate.rf—Foreign interest rate. —Standard deviation of S rate of change.E—Preset expiration exchange rate.w(t)—Standard Weiner process.Define y as follows: y ln S* S N , t ,3. Pricing Formula of the PE OptionsHart and Ross [26] introduced the concept of optionswith continuous strike prices. In their paper, they constructed a portfolio of European call options with different strike prices, but in all other respects identical. InHR, the strike prices are set so that the most expensivecall option is the one constructed with the lowest strikeprice.Thereafter, each subsequent option in the portfolio isset with a different strike price that is higher than the S(1)where:S* is the prevailing domestic exchange rate upon expiration, for example in the US could be the US dollarper Euro exchange rate ( / ) upon the option’s expirationdate.Cfpe* is the PE call payoff in the foreign currency uponexpiration.Ppef * is the PE put payoff in the foreign currency uponexpiration.K—The strike price.E—The preset exchange rate.The payoffs in the domestic currency using the expiration rate S* are:The call payoff is: previous one by K.For all S* K 0 the portfolio’s payoff uponexpiration becomes:where: rd rf 1 2 2 tThe call option value can be calculated as the expecteddiscounted risk neutral of the option’s payoff:C pe e rd t12π Sln K S K SEe21 y 2 2 dy (11)JMF

A. WOLF, C. HESSELAfter integration, we get: C pe e rd t E 2 2 rd rf 1 2 t r r t2 N w t KSe d f N w S e (12)Rearranging terms on the right had side of Equation(12) yields:C pe S E (13) rd 2rf 2 t N w t Ke rf t N w Se where:N w is the cumulative standard normal distributionfunction of w ; and: 1 (14)w ln S K rd rf 2 t t2 The put’s price can is derived using the same procedure as the one used for the call:Ppe S E (15) rd 2rf 2 t N Z t rf t Ke N z Se where: 1 (16)z ln K S rd rf 2 t t2 In comparison, the GK model results in the followingpricing formula:Cgk Se f N d1 Ke rd t N d 2 rt (17) (18)d1 ln S* K rd rf 0.5 2 t t (19)d2(20)rtPgk Ke rd t xN d 2 Se f N d1 where: d t 1217owner of the PE option earns a higher return on investment than a comparable investment in the GK option,regardless of the specification of E. All of these differences between the PE and GK models are shown in thissection.4.1. Price ComparisonsWe compare call prices using Equations (13) for PE and(17) for GK and put prices using Equations (15) for PEand (18) for GK and then compare the payoffs for callsand puts using Equations (3) and (4) for PE options, and(5) and (6) for GK options.By selecting the preset exchange rate E, the optionbuyer sets the price and resulting exposure to lossesshould the option expire out-of-the-money. Figure 1focuses attention on the relationship of E on the value ofat-the-money PE call option’s value and how the PEoption’s value compares with the GK call option’s value.In Figure 2 the analysis is repeated forat-the-money PEand GK putoptions.For the illustrations s k 1.00, σ 10%, t 1.0 years, rd rf 7%.The intersection of the lines is the value of E thatequates price, or exposure, EBE. Figure 1 shows the existence of an inverse relationship between E and price.The EBE for our illustration is 1.14. For E greater than1.14, the PE call costs less than the GK call. While for Eless than 1.14, the PE call is more expensive. The lineshows slight convexity indicating as E is lowered, theprice increases at an increasing rate, albeit it the increaseis very small.Figure 2 shows a similar relationship between E andthe put price. As in the case of the call, an inverse relationship exists between the E value and put price. Thereis a value of E that equates the prices of the PE and GKputs. EBE occurs at the intersection of the lines, and forthis illustration EBE 0.88.4. Comparison of the PE and GK ModelsUsing Numerical IllustrationsThe PE model expands the dimension of possibilities forthe participants, including investors, hedgers, and speculators and makes the market more complete. The participant chooses between leaving the payoff in the foreigncurrency, or converting it to the pricing currency. Theparticipant selects the value of E that determines theprice, which is the potential loss, should the option expireout of the money. The participant speculates on morethan direction of change, but also magnitude of change.If the expiration spot rate is within a specified range, theCopyright 2012 SciRes.Figure 1. Comparison of call prices for differing E values.The horizontal axis shows the preset exchange rates for atthe-money PE and GK call options. While the vertical axisshows the option price. S K 1.00, σ 10%, t 1.0 years,rd rf 7%.JMF

A. WOLF, C. HESSEL218Figure 2. Comparison of put prices for differing E values.The horizontal axis shows the preset exchange rates for atthe-money PE and GK put options. While the vertical axisshows the option price. S 1.00, σ 10%, t 1.0 years, rd rf 7%.payout to the agent owner then if the GK call had beenwritten.When E is set for a value less than EBE, both the PE’sprice and payoff will exceed those of the GK call whilefor value of E EBE, both the PE’s price and payoff willbe less than those of the GK call. To determine the valueof S* in these situations, a different analysis is required.One involving return on price and is presented in Section4.3 immediately after the comparison of put payoffs.Figure 4 shows a comparison of PE and GK put payoffs for differing values of E. On the horizontal axis, weshow five expiration spot rates ranging from 1.05, abovethe spot rate (1.00), down to 0.85.Three PE puts are shown differing in E value; E EBE 0.884, E 0.95, between the strike rate and EBE and EFor E less than EBE, (0.88), the PE put price exceedsthat of the GK put and for E greater than EBE the PE putprice is less than the GK put price. As in the case of thecall, the E-Price line is slightly convex indicating as E islowered the price increases at an increasing rate.4.2. Payoff ComparisonsWe continue to use the same values of S, σ, t rf and rdwhen comparing payoffs. The PE call payoffs are shownfor three values of E: E 0.95, which is below the origination spot rate 1.00, E 1.05, which is above the origination spot rate and EBE 1.135.Examination of Figure 3 reveals the payoff lines to theright of 1.00 are slightly curved as compared with thestraight 45 angle of the conventional call. For the PE call,payoff increases at an increasing rate albeit at a smallrate of increase. S* is the spot rate at expiration and K isthe strike rate. The call payoff for the GK call isS* K and for the PE call S* K S* E . The PEpayoff is equal to the GK payoff when S* E. For allvalues of S* E the PE payoff is larger while for S* Ethe PE payoff is smaller.This relationship is most easily seen at EBE 1.135,which equates the two models’ values. For expirationspot rate, S* 1.135, the PE call was purchased for thesame price as the GK call, but has a larger payoff thanthe GK call. For S* 1.135, the PE call has the smallerpayoff. Agents have a basis for choosing between the PEand GK call. If the agent is a hedger that hedges cashflows and has the view that S* 1.135, he will buy thePE option, while if the view is S* 1.135 the agent willbuy the GK call. An agent willing to write options toincrease portfolio income and has the expectation S* 1.135 will choose to write PE calls over GK calls. This isbecause the writer receives the same price from bothcalls, but if S*

form nonlinear stochastic option pricing model. Leaving the realm of option pricing for stock and stock indices, GK [1] extended the BS model to price options on foreign currencies. For over a quarter of a century, the GK option-pricing model has been the standard foreign currency option-pricing model in pricing European style options and the .

Related Documents:

from an underlying asset which in foreign currency options is the exchange rate. There are two types of foreign currency options, a call currency option and a put currency option. A call option on a particular currency gives the holder the right but not an obligation to buy that currency at a predetermined exchange rate at a particular date and .

foreign currency notes, a foreign currency draft or cheque or foreign currency travellers cheques, ANZ may, in its absolute discretion, convert the deposit into Australian Dollars at the buying rate applicable on the day of the transaction and then reconvert the deposit back into the currency in which the FCA is denominated

a sterling/Euro currency account held with us.) Each payment up to the currency equivalent of 100 Free Each payment over the currency equivalent of 100 6 Sterling and foreign currency payments payable to an account held with us in a different currency from the payment Each payment up to 100 or the currency equivalent of 100 Free

Transaction currency The currency in which a transaction originates. Accounting currency The primary currency in which a legal entity maintains its financial records. The accounting currency is stored in the Ledger table. Reporting currency The reporting currency of the ledger. The reporting currency is stored in the Ledger table. It is optional.

Issue 1 – The pricing of foreign currency conversion services (a) what types of foreign currency conversion services your business offers to consumers and small businesses - XE/HiFX/Currency Online provide mo

A. Foreign Currency Accounts 7.1 Non-Resident Foreign Currency Account (NRFC) 34 7.2 Non-Resident Non-National Foreign Currency Account (NRNNFA) 35 7.3 Diplomatic Foreign Currency Account (DFA) 36 B. Rupee Accounts 7.4 Diplomatic Rupee Account (DRA) 37 8. Chapter 8 : Do I need Exchange Control approval to

the currency of the bond's denomination as the local currency and the chosen currency of the portfolio or index as the base currency. The return of this security in the base currency on day t can be computed using the following inputs. and 1 are the market values in local currency at the close of day t and t-1 respectively.

5541 (SCM 2034) for all animal species (EFSA-Q-2019-00319) A.02.02 Safety and efficacy of 31 flavouring compounds belonging to different chemically defined groups for all animal species (EFSA-Q-2020-00175) A.02.03 Benzoic acid for pigs and poultry as a flavouring compound. FAD-2016-0078 - Supplementary information