Currency Options as Central Bank Risk Management Tool

Transcription

1 Currency Options as Central Bank Risk Management Tool Preliminary version, April 2014 Helena Glebocki Keefe, Erick W. Rengifo Fordham University, Economics Department April 1, 2014 Abstract Many central banks in emerging markets and developing economies are concerned with excessive volatility in foreign exchange markets and wish to control the direction and speed with which the value of their currency changes. Historically, intervention has consisted of using foreign exchange reserves to purchase and sell foreign currency directly in the spot market. The research presented in this paper explores how currency options may be a viable central bank tool for intervention. Holding and issuing bundles of call and put options with multiple strike prices while dynamically delta hedging the portfolio position curbs excessive reserve accumulation, builds markets and domestic liquidity, establishes a more effective signaling process between policy makers and the market, and creates a more cohesive intervention plan than direct spot market intervention. We use the Garman-Kohlhagen options pricing model to analyze the case study of Colombia and to simulate the impact of using an alternative to a butterfly strategy as an intervention mechanism on the spot market position of the central bank, reserve accumulation and total costs accrued from intervention 1. 1 The research conducted and presented in this paper has been sponsored by the Global Association of Risk Professionals. 1

2 1 Introduction Holding 2 and issuing 3 bundles of call and put options while dynamically delta hedging 4 the net portfolio position allows central banks in developing economies to have a targeted approach to currency market intervention. Previous attempts to use options as an intervention mechanism by Mexico and Colombia have been abandoned. Deemed ineffective in curbing volatility by some, such as Mandeng (2003), past failure of options contracts may be due to their sporadic and unhedged issuance, leading to little sustained impact and no clear picture of how to operate in the spot market. This paper revisits the case of Colombia to analyze how using an alternative to the butterfly option strategy can provide central banks with an alternative policy tool to intervene in currency markets to control volatility, influence expectations, build markets and ensure domestic liquidity at a lower cost than pure spot market intervention. Central banks in emerging markets and developing economies are concerned with excessive fluctuations of their exchange rates. Such volatility can cause risks associated with banking crises, economic instability, slowed growth and diminished trade. According to a survey by the Bank of International Settlements of 19 central banks in developing economies, two-thirds conducted some type of currency market intervention and found it to be an effective tool for controlling exchange rate volatility (Mihaljek, 2004). Most developing countries are engaging in some type of intervention into currency markets to exert control over exchange rates. Many intervene to calm disorderly markets and relieve liquidity shortages, while others try to correct misalignment and stabilize volatile exchange rates. All policy makers surveyed stated that intervention which influences future expectations and signals a future stance of monetary policy is the most effective. Interventions are assumed to have primarily a short-term influence on currency markets. Since currency markets are very dynamic, even in emerging markets and developing economies, the most effective strategy for the central bank will be one that is consistent. Holding and issuing bundles of call and put options at various strike prices on a consistent basis while dynamically delta hedging the portfolio position in the spot market will allow the central bank to influence the expectations of traders, target volatility and signal their policy stance. It will also provide policy makers with a clear target for operating in the spot market while increasing liquidity domestically. Additionally, the costs associated with intervention will be lower, sterilization problems 2 Holding options contracts will refer to ownership of the contract. Specifically, when it is holding a contract, the central bank will be long the option contract and have the right, but not the obligation to exercise the contract. 3 Issuing options contracts will refer to writing and auctioning the option contract. Specifically, when it is issuing a contract, the central bank will be short the option contract and will be obligated to fulfill the contract if it is executed by the owner at maturity. 4 Dynamic delta hedging is the main hedging strategy considered in this research because it reflects a short-run strategy of intervention, which has been found to be most effective by policymakers (Mihaljek, 2004) and because drastic changes in day to day value of the currency are not anticipated. An alternative strategy, such as gamma hedging will be considered in future research that builds on the current findings. 2

3 will be alleviated, and signals between traders and the central bank will allow policy makers more time to react to speculative threats and deviations of the exchange rate from macroeconomic fundamentals. This paper explores how options may be a viable alternative policy tool for central banks in developing economies to use in currency market intervention. First, we analyze Colombia s past experimentation with options contracts as the baseline case. Next, we explore which option strategies are most appropriate to meet the goals of central bank currency market intervention. For the analytical approach, we use both a random process and an Ornstein-Uhlenbeck process with GARCH volatility to simulate the Colombian Peso-US Dollar (COPUSD) exchange rate. With the simulated series, we price options contracts at various strike prices and analyze the outcome of hedging a portfolio position that is tailored to counter persistent appreciation or depreciation in the exchange rate. We compare the spot market position and total costs to the central bank of such a strategy with the cost of daily interventions currently conducted by Colombia. The remaining sections of the chapter are structured as follows. Sections two presents an overview of options and basic strategies. Section three lays out the motivation for research and a review of the literature. Section four presents the data and historical analysis for the case of Colombia. Section five addresses the possible option strategies that are most effective to the goals of the central bank. Section six presents the analytical approach, models and methodology for simulation. Section seven reports the simulation results. Section eight discusses implications of the findings and future extensions of the research. 2 Overview of Options An options contract provides the owner the right, but not the obligation, to exercise their position at the given strike price. In other words, a call (put) option offers the owner the right to buy (sell) the underlying asset at the given strike price on or before maturity of the contract from the writer or issuer of the contract. A call (put) option will be exercised when the spot market price at the end of the contract is above (below) the strike price. To be long in an option contract is to have purchased the contract, and therefore hold the right to exercise the option upon maturity. To be short in an option is to have written or sold the option contract to a market participant. As can be seen in Figure (1), a long call option has limited downside risk when the value of the underlying asset changes (S) but unlimited payoffs with an increase in the value of S. As the value of the underlying asset increases, the owner of the call option will be able to exercise the option at the 3

4 agreed upon strike price (K). He is therefore able to buy the asset at a lower price than market value, in turn buying low, selling high. His downside risk is limited to the premium he must pay to own the contract if the contract is not exercised. In contrast, a short call option has unlimited downside risk and limited payoffs. The payoffs of a short call are limited to the premium received by the writer of the contract when the call is not exercised. As the value of the underlying asset increases, the writer of the call option is obligated to sell the underlying asset to the owner of the contract (the one in the long position) at the strike price, which is below the market value of the asset. He is therefore selling low, buying high, which puts him in a position of unlimited loss as the asset value rises. On the other side of the market, a long put option has limited downside risk as well as limited payoffs. The owner of the put option has the right to sell the underlying asset at the strike price K to the writer of the contract. The long put option will be exercised if the value of the underlying asset (S) is below the strike price (K). In this scenario, the owner of the long put option sells the asset at a price higher than the market value to the writer of the contract. He is therefore selling high, buying low. The downside risk of the long put option is limit to the premium the agent must pay for the right to own the contract, even if the contract is not exercised. The writer of the contract has a short put option position, and is also faced with limited payoffs and large but limited downside risks. He will be obligated to buy the underlying asset from the owner of the contract at the strike price K. Once again, his payoff is limited to the premium he receives if the contract is not exercised. Figure 1: Call and Put Option Strategies The above figure illustrates call and put option positions and payoffs. The value of the underlying asset in the spot market is represented by S and the strike price of the option is K. Long calls have unlimited gains, and limited losses, whereas long puts have limited gains and limited losses. Short calls have unlimited losses and limited gains, whereas short puts have limited gains and unlimited losses. In the context of the foreign exchange market, a currency call option on US dollars (USD) in Colombia (COP) gives the owner the right to buy USD from the writer of the contract at the strike price. The 4

5 owner of the contract is long a call option in this case. Therefore, the call option will be exercised if the spot exchange rate of COPUSD is above the strike price, or in other words, COP has depreciated since the issuance of the option. On the other hand, a put option on USD in Colombia gives the owner the right to sell USD to the writer of the contract at the strike price. In other words, the owner is long a put option and the writer of the contract is short a put option. If the put option is exercised, then the spot price at maturity has fallen below the strike price and COP has appreciated in value. Currency option contracts are beneficial for traders and hedgers alike because they mitigate some of the risk of drastic movements in the future value of the exchange rate. Market participants interested in further offsetting risks associated with long or short options positions can also engage in dynamic delta hedging, which allows them to continuously rebalance their portfolio in the the underlying asset (DeRosa, 2011). As explained in Chen (1998), the standard technique for a trader to hedge their position in the options market against the risk of changing prices is referred to as dynamic delta hedging. Traders are able to reduce risks associated with the movement in the price of the underlying asset by taking an offsetting position in the spot market. For small movements in the exchange rate, the value of the hedge will change in an equal but opposite direction. The delta is the responsiveness of the price of the option to changes in the value of the underlying asset. The delta of an option will change as the contract nears expiration or when implied volatility or the exchange rate changes. As the delta changes, traders will adjust their offsetting position through buying or selling the underlying asset. For a long call option, the offsetting delta hedging position in the spot market will be to short the underlying asset. When an owner of the long call wants to hedge his position, he will sell a given amount of the underlying asset directly in the spot market. The amount sold is determined by the delta of the option at the time of hedging. Therefore, if the contract is for USD, the owner will sell USD in the spot market. For a short call option, an offsetting delta hedging position in the spot market will be to long the underlying asset. For the writer of the contract, he will purchase a given amount of the underlying asset in the spot market to hedge his option contract position. If the contract is in USD, he will buy USD in the spot market. For a long put option, the offsetting delta hedging position in the spot market will be to long the underlying asset. For a contract in USD, the owner of a put option will purchase a given amount of USD in the spot market. The size of the purchase will be determined by the delta of the contract at the time of hedging. For a short put option, the writer of the contract will offset his position in the options market by selling the underlying asset in the spot market. For a contract in USD, the writer of the contract will sell a given amount of USD in the spot market to hedge his short put option position. Once 5

6 again, the size of the offsetting spot market position is determined by the delta, or the responsiveness of the option price to changes in the value of the underlying asset at the time of hedging. By taking an offsetting position in the spot market, agents that are dynamically delta hedging their options contracts should theoretically cover their hedging costs by the premium or payoff they gain from the option contract. Even if the contract is not exercised, through consistent daily or weekly hedging, the agent will be able to purchase or sell off the underlying asset over the period to maturity and cover his costs with the premium or payoff derived from the option contract. Dynamically delta hedging allows the agent to hold a neutral portfolio position with lower costs than relying solely on the spot market or options market. 3 Literature Review The following section will first delve into details on the currency options market and past literature that has addressed how they can be used by central banks. bank intervention into foreign exchange markets in general. It will then detail literature on central Finally, it will address macroeconomic fundamentals that influence exchange rate movements as well as the linkages between inflation targeting goals and exchange rates in emerging markets. 3.1 Currency Options Currency options are used by various agents in the foreign exchange market, including currency traders, speculators, hedgers and portfolio managers. The options market is mainly an interbank over-the-counter market. The majority of currency options are European, meaning the option can only be exercised at expiration (DeRosa, 2011). The global daily average turnover on a net-net basis 5 in the foreign exchange market in April 2013 was US$ 5.3 trillion, of which spot transactions were 38 percent and options were less than 6 percent. Net-gross daily turnover 6 in emerging markets made up roughly 6 percent of the global foreign exchange market (BIS, 2013). 3 If the central bank is the main writer of options, it can crowd out all other writers who may engage in dynamic delta hedging that is potentially destabilizing (HKMA, 2000). Breuer (1999) argues that if market makers are net long positions, their dynamic delta hedging behavior can lower volatility. When option buyers purchase domestic currency in the spot market to hedge their positions when the currency 5 Net-net basis adjusts for local and cross-border inter-dealer double-counting 6 Net-gross basis adjusts for only local inter-dealer double-counting 3 This omits Singapore and Hong Kong turnover. Including these two would increase the share of turnover in emerging markets to 40.2 percent 6

7 is depreciating, and sell it when it is appreciating, this will help stabilize exchange rates. In other words, given the option is written for USD, in a long call a delta hedging position would sell USD when the domestic currency is depreciating and buy USD when it is appreciating, therefore canceling out volatile pressure. Archer (2005) argues that the transparency with which the central bank auctions options contracts to market participants introduces stability and additional hedging instruments into the market. Therefore, central bank use of currency options can be effective in stabilizing the foreign exchange market and controlling volatility when it influences market liquidity and expectations. The Hong Kong Monetary authority notes that options contracts can lower costs of hedging risk, enhance the liquidity of the underlying asset and work to stabilize the foreign exchange market when issued by the central bank (HKMA, 2000). Options contracts issued by the central bank can mitigate the destabilizing dynamic delta hedging behavior that would otherwise be conducted by private market participants in reaction to changing market conditions. Authors such as Garber and Spencer (1995) and Grossman and Zhou (1996) have found a positive link between dynamic delta hedging and spot market volatility. Therefore, one risk central banks must consider when writing options contracts while dynamically delta hedging their position is that such hedging activity may amplify the appreciation or depreciation pressure on the exchange rate, which is contrary to the objectives of the central bank. The authors take into consideration only one-sided positions, such as issuing only calls or only puts. The alternative butterfly strategy position, or issuing both calls and puts, establishes a net hedging position that is smaller than a one-sided position in the market, and therefore will be less destabilizing. As HKMA (2000) notes, this stabilization will occur even if the amount of options contracts sold remains constant because the price of the option changes in response to changes in the market value of the domestic currency. When the central bank is holding a long position in the domestic currency (or a short position in USD), as market pressures increase, the central bank s long position increases while the option buyers hold an offsetting short position. This would have a similar impact as a spot market intervention. The effectiveness of this strategy will depend on the extent to which market participants dynamically hedge their positions, and whether the size of the options contracts are large enough to have a significant impact. Wiseman (1999) argues that governments should commit themselves to frequent and regular auctions of short-dated physically-delivered currency options as a mechanism to stabilize exchange rates. Since almost all central bank authorities would like to reduce exchange rate volatility, without pushing it all the way to zero, official auctions would encourage private banks to buy options and exercise them when profitable. dynamic delta hedging on the part of the trader will substitute for actively pursuing the 7

8 same position in the market. 3.2 Central Bank Currency Market Intervention The main goals of the central bank when intervening in currency markets are to smooth exchange rate volatility, supply liquidity into foreign exchange markets and to control the amount of foreign exchange reserves (Moreno, 2005). The broad motives for intervention are driven by macroeconomic goals, such as inflation targeting, maintaining economic stability and competitiveness, preventing crises and boosting growth. Acosta-Ormaechea and Coble (2011) find that in emerging markets with high levels of dollarization and a strong exchange rate pass through, inflation targeting is more effective through policies that target exchange rates rather than interest rates. There are four main channels through which the central bank can intervene into currency markets (Archer, 2005). First, in the monetary channel, changes in the domestic interest rate relative to the foreign interest rate can alter the value of the domestic currency. This occurs through a change in the domestic monetary policy. Next, in the portfolio balance channel, relative scarcity of the domestic currency to the foreign currency can appreciate the value of the domestic currency. Here the central bank intervention into the spot market determines the relative scarcity or abundance of the domestic currency, in turn directly influencing the value of the nominal exchange rate. Third, through the signaling and expectations channel, the central bank can shape expectations on future monetary and exchange rate policy. Influencing expectations through the promise of future intervention can curb speculative behavior and coordinate the direction of the currency towards equilibrium. The credibility of the signal is also critical. Signals to control appreciation tend to be more credible than those to curb depreciation. Lastly, in the order flow channel, the central bank tracks order flows to predict subsequent price action. Central bankers can alter the order flow with their own orders that must be large relative to the total market turnover. Due to less liquidity in the market and better access to information on order flows, this channel may be more effective in emerging markets than advanced economies. Canales-Kriljenko (2003) finds that in emerging markets and developing economies, 82 percent of interventions take place in the spot market because this is the main or only currency market in the economy. If the intervention is unsterilized, it can directly influence the direction of nominal exchange rates through the monetary channel. If sterilized, the intervention will affect volatility through expectations and by attempting to curb speculative behavior. The success of the latter interventions in lowering volatility has been questionable (Breuer, 1999). Sterilized spot market interventions involve exchange rate intervention by the central bank without any change in the country s monetary base. The intervention occurs through the buying and selling 8

9 of domestic and foreign bonds by the central bank (Weber, 1986). The primary purpose of sterilized interventions has been to counter appreciation of the domestic currency in fixed or managed float exchange rate regimes without impacting real exchange rates to diminish inflationary pressure coming from changes in foreign currency inflows (Agenor, 2004). Weber (1986) finds that from a theoretical perspective, sterilized interventions can in fact influence exchange rates if bonds denominated in different currencies are not perfect substitutes, but empirical evidence from the US indicates that sterilized interventions do not impact exchange rates. Craig and Humpage (2001) agree that such interventions have been ineffective because they do not affect macroeconomic fundamentals and instead influence expectations and perceptions, whereas unsterilized interventions can conflict with price stabilization but are unnecessary because the same effect can be achieved through open market operations. In terms of the size, frequency and timing of intervention, Mihaljek (2004) cites that when the goal of the intervention is to influence the exchange rate, central banks find larger and less frequent interventions to be more effective. In contrast, when the goal is reserve accumulation, frequent but smaller interventions are more successful. Emerging market policy makers viewed small and less frequent interventions as more likely to be successful than large but less frequent interventions. Over the last decade, emerging markets have experienced a significant increase in international financial flows. Even though these flows are generally beneficial in terms of growth and welfare enhancement, emerging markets frequently experience surges or sudden stops in flows, creating economic instability. Such volatile flows contribute to large fluctuations in exchange rates, fueling of domestic asset bubbles, poor resource allocation, balance sheet risks and banking or financial crises. One way central banks have created a buffer against the downside of surges and sudden stops has been the build up of reserves and intervention directly in the spot market (IMF, 2010). 3.3 Macroeconomic Fundamentals, Inflation Targeting and Exchange Rates The value of one country s currency reflects the market s expectation about current and future macroeconomic conditions, and therefore reacts to changes in macroeconomic fundamentals, such as trade, monetary policy, balance of payments, aggregate demand and aggregate supply (Obstfeld and Rogoff, 1999). Many theoretical models have linked exchange rate movements to changes in macroeconomic conditions. These include the monetary model presented in Dornbusch (1976) where an increase in the money supply decreases domestic interest rates to adjust for the excess supply of real money balances. Through the uncovered interest rate parity, the decrease in the domestic interest rate requires a change in the nominal exchange rate. Due to short run sticky prices, the depreciation in the short 9

10 run is larger than in the long run equilibrium. In portfolio balance model presented in Dornbusch and Fischer (1980), the exchange rate determines the equilibrium between domestic money, domestic bonds and foreign bonds. Changes in money supply or supply of bonds will drive changes in the exchange rates to maintain equilibrium. An increase in the supply of domestic bonds, an increase in the foreign interest rate, or expectation of future depreciation will result in a depreciation of the domestic currency. An increase the supply of foreign bonds or an increase in the domestic interest rate result in an appreciation of the domestic currency. Inflation targeting has been adopted by a number of both emerging and advanced economies over the last two decades. Even though it has been considered advantageous as a framework for monetary policy, the macroeconomic effects of inflation targeting in empirical terms have been limited (Levin, Natalucci and Piger, 2004). In industrialized economies, inflation targeting has been most effective in controlling long run inflation expectations and lowering the persistence of inflation. Fraga, Goldajn and Minella (2003) argue that emerging markets face more acute trade-offs when choosing the design of their inflation targeting monetary policy, including higher output and inflation volatility. Due to a more volatile macroeconomic environment, the implementation and commitment to inflation targeting becomes more difficult in emerging markets than in advanced economies. The impact of exchange rates on inflation targets and on monetary policy goals has been a concern for many emerging economies due to the weaker financial system and their susceptibility to external shocks. Stone, Roger, Nordstrom, Shimizu, Kisinbay and Restrepo (2009) argue that the exchange rate is more important as a policy tool for inflation-targeting emerging markets than for their counterparts in advanced economies for a number of reasons. In emerging markets, a high exchange rate passthrough indicates lower policy credibility and translates to a closer link between price and exchange rate movements. Additionally, less developed financial systems in these countries correspond to more rigidity in currency markets, which amplifies the impact of exchange rate shocks on the domestic economy. Intervention into currency markets reflects the desire of central banks in emerging markets to mitigate the impact of short-term currency fluctuations on output. Finally, active management of the exchange rate is seen as a way to promote financial stability, which can also minimize the negative impact of sudden stops in foreign currency inflows. In contrast, Sek (2008) finds that the reaction of monetary policy 7 to exchange rate shocks in three inflation-targeting East Asian economies has declined after the East Asian crisis. 6 A high exchange rate pass-through in emerging markets makes it more difficult for central banks to target low inflation rates and maintain price stability (Minella, de Freitas, Goldfajn and Muinhos, 2003, Fraga et al., 2003). 7 The monetary policy measures used include money demand (M1), short-term interest rates, output gap, and inflation 6 The three economies are Thailand, Korea and Philippines. 10

11 Reyes (2013) argues that the lower pass-through effect is a natural reaction to the implementation of inflation-targeting policies in emerging economies, but the effects of nominal exchange rate fluctuations on inflation rates can still be felt. If the pass-through effect is on the decline, this may explain why Sek finds a lower reaction of monetary policy to exchange rate shocks post-crisis. An appreciation of the domestic currency can lead to lower output and inflation in future periods due to expenditure switching and because import prices will not rise as quickly with the appreciation (Taylor, 2001). The reaction of interest rates to an appreciation is indirect as interest rates react to changes in inflation and real GDP instead of directly to fluctuations in the exchange rate. Taylor concludes the reaction of policy makers to changes in the exchange rate by adjusting interest rates may not improve performance because this mechanism is already build into the policy rule indirectly and because the reaction may make swings in real output and inflation even worse. Additionally, changes in exchange rates under floating exchange rate regimes may indicate changing productivity and should not be negated. 4 Historical Analysis of Issuing Options: The Case of Colombia Colombia has experimented with many different intervention tools in its recent history. The Colombian central bank began systemic currency market intervention following the introduction of a floating exchange rate regime and adoption on inflation-targeting monetary policy in 1999 (Uribe and Toro, 2005). It first started with the introduction of currency options for the purposes of reserve accumulation and later to control for volatility. From 2000 to 2012, the average yearly purchase of US dollars by the Colombian Central Bank was US$ 2.2 billion 8, or an average of 1.7 percent of market transactions (Echavarria, Melo, Tellez and Villamizar, 2013). From 2005 to 2007 as well as from 2010 to 2012, the purchase of US dollars by the central bank was much larger, the latter reflecting a change in policy to daily discretionary purchases. Trading of Colombia s currency represents approximately 0.05 percent of all currencies traded on a net-gross basis, amounting to daily average trades of US$ 3.34 billion in The domestic interbank forex market makes up only 25 percent of the total market for COPUSD. Domestic foreign exchange markets in Chile and Peru represent similar characteristics, as can be seen in Table (1). As discussed above, the majority of domestic forex transactions are interbank transactions. In Chile, for example, interbank spot market transactions make up approximately 52 percent of all domestic spot transactions. 2 8 Sales were smaller at US$ 571 million 2 Based on data from Central Bank of Chile. Data from statistics on forex trading in the formal market. Represents sum of interbank transactions, total sales and total purchases in USD in Chile in

12 Table 1: Foreign Exchange Markets: Global vs. Domestic Colombia Global Amount (USD Mil) 802 1,860 2,794 3,343 Percent World Total 0.03% 0.04% 0.06% 0.05% Domestic Amount (USD Mil) Percent of Total Traded 49.38% 41.93% 37.26% 25.28% Chile Global Amount (USD Mil) 2,462 4,003 5,544 11,956 Percent World Total 0.09% 0.09% 0.11% 0.18% Domestic Amount (USD Mil) 1,295 1,698 1,518 2,488 Percent of Total Traded 52.59% 42.42% 27.38% 20.81% Peru Global Amount (USD Mil) ,425 2,171 Percent World Total 0.01% 0.02% 0.03 % 0.03% Domestic Amount (USD Mil) Percent of Total Traded 26.39% 17.42% 33.50% 38.72% Global amount traded reflects average daily net-gross transactions. Domestic amount traded reflects interbank trading volume as reported by the central banks. Data for global transactions from Bank of International Settlements. The Colombian peso has been experiencing steady appreciation since From 2002 to 2009, it experienced a number of periods with high volatility, where the bid-ask prices on the market exchange rate were notably different official exchange rate. Figure (2) illustrates the differences between the bid price, ask price, and official exchange rate in Colombia from 2002 to 2014, as well as the differences between the official rates and bid or ask prices. Since 2012, the spread between official rates and market prices has been much lower than in previous periods. Colombia is one of the few countries to date that have auctioned call and put options to mitigate exchange rate volatility and accumulate reserves. For the purposes of reserve accumulation and decumulation, the central bank auctioned options contracts on a monthly basis. The options were exercised when the exchange rate appreciated or depreciated over than 20-day moving average mean, and the amount to be auctioned in the subsequent month was determined at the end of each contract. The volatility options with 30-day maturity were auctioned whenever the exchange rate changed more that 4 percent of the 20-day moving average. The maximum exercise amount was US$ 180 million. From 1999 to 2009, there were a total of 38 options contracts auctioned by the Colombian central bank. The options intervention strategy was abandoned when the central bank switched intervention strategies to a daily discrete intervention plan, where the central bank purchased an average of US$ 20 million per day. From August 2012, the amount purchased varied from US$ 20 million to US$ 50 million daily. The average intervention was 3.72 percent of total USD traded in the Colombian FOREX market, with 12

13 Figure 2: Colombian Peso Dynamics The top graph represents the value of the official COPUSD exchange rate, the bid price and the ask price in the market. The bottom graph illustrates the difference between the official exchange rate and the market exchange rates (bid price and ask price). The spread between official and market rates has diminished in recent years. Market rates from OANDA. Official exchange rate data from Banco Republica de Colombia. 13

14 a maximum intervention that totaled 33.6 percent of the market volume. The auction of options contracts in Colombia were fully transparent and the benefits of these auctions were derived from the hedging operations of market participants (Uribe and Toro, 2005). When issuing options contracts, the main objectives of the central bank were to avoid excessive volatility in the exchange rate in a way that would uphold inflation targets, to strengthen the international liquidity position domestically, and smooth any deviations of the exchange rate from its long run trend. From 2000 to 2005, call options were deemed successful in influencing both the value of foreign exchange rate and the volatility. The call options were able to mitigate the increasing depreciation trend in 2003 that threatened inflation targets. Mandeng (2003) finds that volatility call options issued until 2003 were only moderately successful. On the other hand, Uribe and Toro (2005) state that put options were successful in the accumulation of reserves from 1999 to They also find that Colombia s intervention policies have been largely consistent with its goals of inflation targeting, such that changes in monetary policy came first through interest rates, and then through intervention in currency markets. Starting in 2008, the Colombian central bank began purchasing US$ 20 million daily, first for two months in 2008, then in 2010 for five months, in 2011 for six months, and every month since Following the policies of Chile and Israel for daily discretionary intervention, US$ 20 million is the average of the daily purchases in those countries (Echavarria et al., 2013). Colombia abandoned the use of options-based intervention once it began the daily purchase of US dollars. The change in policy has been considered a good mechanism for accumulating reserves without promoting speculative behavior because it is a consistent and transparent intervention. Mandeng (2003) uses an event study to observe the impact of auctioning three call options on exchange rate volatility, comparing the volatility before and after the time of maturity. At the time of his paper, Colombia had only issued these three options as a means to control volatility, and Mandeng finds them to be only slightly successful in lowering volatility. Using a similar approach, Table (2) illustrates the same analysis for all the call and put options issued since Volatility is measured as the annualized standard deviation of the log difference in daily exchange rates over a 10 day rolling window. Comparing volatility 2, 5, and 10 days before and after the contract maturity yields similar results, where volatility is successfully lowered only in 30 to 40 percent of the cases. In Table (3) the volatility calculation spans a two day, five day and ten day window depending on which period is observed. Volatility is measured as the annualized standard deviation of the log difference in daily exchange rates with a rolling window of 2, 5, and 10 days. With this calculation volatility after contract expiration compared to volatility at the time of maturity is lower in 52 to 58 percent of the cases. Using the latter calculation with different rolling windows captures the volatility in exchange rates 14

15 related specifically to the period that is being observed. The previous calculation compares volatility that includes exchange rates before, during, and after maturity, yielding misleading results. Part of the reason that past volatility options contracts were only moderately successful in lowering volatility in Colombia may be due to their sporadic issuance that went unhedged. Figures (3) and (4) illustrate intervention with put options and call options respectively. The sporadic issuance of options yields inconsistent results in lowering volatility. The benefits of greater liquidity, building markets and increasing the flow of information between policy makers and traders occur when auctions of the contracts occur consistently, as discussed in (Breuer, 1999). Additionally, the Central Bank only issued either calls or puts, not bundles of calls and puts at different strike prices. Auctioning bundles of calls and puts at different strike prices, while holding an offsetting position increases the information flow between policy makers and traders of expectations, lowers the chances of speculative attacks 9, and mitigates some of the costs of hedging. The net hedging position is lower for a mixed portfolio, therefore the position of the central bank in the spot market is less disruptive than if it took only one side of the market. Though the Colombian central bank issued only one side of the market at a time, there is no evidence that it engaged in dynamic delta hedging to offset the risks associated with issuing volatility options. In Tables (4) and (5), a simple regression tests the impact of the volatility options on the change in the value of COPUSD from the day before maturity to the day the contract was exercised(t-1 to t), and from the day of exercise to one day after maturity (t to t+1). The issuance of volatility calls options had a significant, contemporaneous effect on exchange rate value at the time of maturity. At the day of exercise of the call options, the change in COPUSD from the previous day was lower. The issuance of volatility put options had a lagged effect on the exchange rate value. The exercise of put options impacted the difference in value of the COPUSD the following day. In Table (5), the dummy variable represents the size of the option relative to the total volume traded in one day. If the exercised volume at maturity was greater than 20 percent of total volume, the dummy variable took on the value of one, otherwise zero. Taking into consideration the relative size of the intervention did not significantly alter the results, which may reflect the influence of volatility options through the expectations channel rather than through the portfolio balance channel. 9 Speculative attacks in currency markets occur when there is a massive sell off of a particular currency, leading to a significant depreciation or devaluation of the currency, depending on whether the exchange rate regime is floating or fixed. 15

16 Table 2: Volatility Options Contracts Issued in Colombia - Part 1 Put Options Date Exercised Volatility Before Volatility After Success USD Mil 10 Days 5 Days 2 Days 2 Days 5 Days 10 Days Short Mid Long 17-Dec % 5.14% 5.0% 5.46% 4.75% 6.93% - Lower - 11-Jul % 13.81% 14.14% 13.50% 15.83% 11.27% Lower - Lower 31-Jul % 13.90% 13.93% 9.56% 9.99% 9.21% Lower Lower Lower 10-Aug % 9.99% 7.77% 9.00% 6.75% 4.86% - Lower Lower 30-Oct % 5.31% 4.22% 3.89% 4.32% 5.83% Lower Lower - 21-Dec % 3.01% 3.14% 4.14% 4.21% 3.22% - - Lower 30-Mar % 7.22% 6.21% 11.30% 12.37% 11.58% May % 7.06% 3.15% 5.51% 7.45% 9.76% May % 10.31% 9.76% 11.49% 8.18% 9.04% - Lower - 4-Jun % 12.56% 15.05% 12.77% 15.80% 9.40% Lower Sep % 14.04% 10.23% 19.59% 19.91% 14.09% Dec % 11.66% 9.09% 8.11% 6.73% 5.44% Lower Lower Lower 15-Jan % 4.25% 6.75% 13.16% 15.25% 18.77% Feb % 7.55% 6.91% 10.98% 10.41% 11.94% Mar % 16.72% 13.10% 15.26% 9.82% 11.15% - Lower Lower 4-Jun % 5.89% 5.20% 7.39% 7.81% 10.10% Dec % 9.37% 8.38% 14.47% 15.35% 6.61% Mar % 14.71% 12.31% 16.06% 17.80% 21.00% Apr % 16.53% 16.61% 19.96% 18.32% 18.46% Jun % 12.88% 14.93% 17.18% 16.20% 14.27% Jul % 17.85% 16.27% 13.69% 11.87% 24.92% Lower Lower - Call Options Date Exercised Volatility Before Volatility After Success USD Mil 10 Days 5 Days 2 Days 2 Days 5 Days 10 Days Short Mid Long 29-Jul % 9.90% 11.23% 10.55% 8.45% 6.57% Lower Lower Lower 01-Aug % 10.63% 10.55% 6.14% 6.57% 20.24% Lower Lower - 02-Oct % 10.70% 9.87% 9.72% 8.75% 6.25% Lower Lower Lower 10-Apr % 4.08% 8.62% 10.52% 10.16% 10.98% May % 6.49% 11.84% 17.90% 14.57% 13.37% May % 11.86% 16.60% 15.34% 15.92% 16.88% Lower May % 17.90% 14.57% 16.25% 16.88% 13.01% - Lower - 25-May % 15.34% 15.92% 17.57% 16.62% 12.78% Jun % 9.96% 7.72% 7.19% % 13.92% Lower Jun % 17.10% 16.28% % 11.00% 10.41% - Lower Lower 13-Aug % 8.62% 8.23% 9.01% 9.39% 5.38% - - Lower 22-Nov % 9.35% 8.81% 10.57% 6.78% 10.81% - Lower - 07-Oct % 32.35% 26.48% 34.99% 41.28% 49.22% Oct % 46.64% 41.88% 24.36% 21.19% 11.89% Lower Lower Lower 30-Jan % 12.32% 11.30% 15.13% 15.61% 14.12% Feb % 11.30% 15.07% 15.61% 17.55% 17.43% Feb % 17.55% 11.87% 18.07% 15.86% 15.81% - Lower - Volatility is measured as the annualized standard deviation of the log difference in daily exchange rates over a 10 day rolling window. Using this calculation to test whether volatility is lowered after the option is exercised yields only moderately successful results partly because it accounts of movements in the exchange rate before, during and after the contract maturity. 16

17 Table 3: Volatility Options Contracts Issued in Colombia - Part 2 Put Options Date Volatility at Maturity Volatility After Success σ(2) σ(5) σ(10) 2 days 5 days 10 days 2 days 5 days 10 days 17-Dec % 9.97% 10.19% 2.45% 10.91% 22.23% Jul % 13.68% 11.11% 14.70% 9.51% 11.27% Lower Lower - 31-Jul % 10.43% 14.17% 5.22% 4.09% 9.21% - Lower Lower 10-Aug % 8.61% 9.21% 4.83% 2.91% 4.86% Lower Lower Lower 30-Oct % 5.47% 4.29% 4.39% 4.09% 4.17% - Lower Lower 21-Dec % 5.24% 4.34% 2.76% 1.90% 3.22% Lower Lower Lower 30-Mar % 5.39% 6.76% 21.91% 17.03% 11.58% May % 7.53% 5.82% 0.05% 7.66% 9.76% Lower May % 10.86% 11.94% 0.21% 4.14% 9.04% Lower Lower Lower 04-Jun % 17.37% 14.65% 0.32% 11.99% 9.40% - Lower Lower 20-Sep % 27.97% 20.43% 2.48% 8.03% 14.09% Lower Lower Lower 11-Dec % 5.29% 8.25% 1.40% 7.62% 5.44% Lower - Lower 15-Jan % 15.47% 12.06% 8.36% 12.47% 18.77% Lower Lower - 20-Feb % 9.01% 7.31% 24.88% 12.72% 11.94% Mar % 0.00% 12.69% 24.45% 14.70% 11.15% - - Lower 04-Jun % 7.63% 6.15% 13.33% 8.88% 10.10% Dec % 18.34% 14.09% 12.20% 5.13% 6.61% - Lower Lower 17-Mar % 18.51% 16.46% 4.44% 17.57% 21.00% Lower Lower - 27-Apr % 8.38% 15.26% 20.20% 24.92% 18.46% Jun % 19.55% 17.48% 1.25% 3.42% 14.27% - Lower Lower 22-Jul % 10.28% 14.96% 7.38% 14.13% 24.92% Lower - - Call Options Date Volatility at Maturity Volatility After Success σ(2) σ(5) σ(10) 2 days 5 days 10 days 2 days 5 days 10 days 29-Jul % 10.43% 10.59% 11.46% 7.07% 6.57% - Lower Lower 01-Aug % 7.89% 10.25% 2.60% 6.28% 6.25% Lower Lower Lower 02-Oct % 10.59% 11.44% 4.31% 11.33% 9.76% Lower - Lower 10-Apr % 10.46% 8.80% 15.29% 7.75% 10.98% - Lower - 16-May % 16.50% 16.60% 21.13% 11.51% 13.37% Lower Lower Lower 18-May % 19.50% 17.90% 15.17% 8.71% 17.57% Lower Lower Lower 23-May % 13.76% 15.92% 17.38% 24.77% 16.62% Lower May % 15.72% 16.25% 22.08% 19.10% 15.32% - - Lower 27-Jun % 9.39% 7.07% 0.84% 20.48% 13.92% Jun % 13.89% 14.45% 8.19% 14.58% 10.26% Lower - Lower 13-Aug % 18.11% 12.70% 19.09% 32.76% 27.18% Nov % 8.03% 10.13% 8.85% 5.52% 10.81% Lower Lower - 07-Oct % 40.19% 33.43% 31.23% 47.03% 49.22% Lower Oct % 28.82% 44.15% 14.50% 10.01% 14.69% Lower Lower Lower 30-Jan % 16.65% 14.44% 18.93% 15.11% 14.12% - Lower Lower 02-Feb % 16.76% 15.29% 20.85% 17.45% 17.43% Feb % 18.14% 17.43% 3.81% 6.17% 15.81% - Lower Lower Volatility is measured as the annualized standard deviation of the log difference in daily exchange rates with a rolling window of 2, 5, and 10 days. Comparing volatility at the time of maturity to 2, 5, and 10 days after maturity yields more successful results. After the option is exercised, volatility decreases in 52 to 58 percent of all cases. 17

18 Figure 3: Volatility Put Options Interventions Data from Banco Republica de Colombia. Figure 4: Volatility Call Options Interventions Data from Banco Republica de Colombia. 18

19 Table 4: Impact of Options Issued on COPUSD Dependent Variable: Change in COPUSD (t-1 to t) Calls Puts C 12.90** 0.02 (2.07) (0.23) Amount Issued (Calls) ** (2.07) Amount Issued (Puts) (-0.35) R N.obs Dependent Variable: Change in COPUSD (t to t+1) Calls Puts C *** (0.81) (2.64) Amount Issued (Calls) 0.02 (0.82) Amount Issued (Puts) *** (2.62) R N.obs The dependent variable is the change in exchange rate value of COPUSD during time of option maturity. It is calculated as the [S t/s t 1 ] 1. The maturity of put options has a lagged impact on the exchange rate, whereas the maturity of call options has a contemporaneous impact on the exchange rate. Table 5: Impact of Options Issued on COPUSD with Dummy Dependent Variable: Change in COPUSD (t-1 to t) Calls Puts C 12.42* 0.03 (1.94) (0.30) Amount Issued (Calls) * (-1.94) Amount Issued (Puts) (-0.35) Dummy (0.63) (0.37) R N.obs Dependent Variable: Change in COPUSD (t to t+1) Calls Puts C ** (0.73) (2.43) Amount Issued (Calls) (0.73) Amount Issued (Puts) ** (2.39) Dummy (0.62) (0.60) R N.obs The dependent variable is the change in exchange rate value of COPUSD during time of option maturity. It is calculated as the [S t/s t 1 ] 1. The maturity of put options has a lagged impact on the exchange rate, whereas the maturity of call options has a contemporaneous impact on the exchange rate. The dummy variable represents the size of the option relative to the total volume traded in one day. If the exercised volume at maturity was greater than 20 percent of total volume, the dummy variable took on the value of one, otherwise zero. The values in parenthesis are t-statistics, and *, **, *** represent significance of 10, 5, and 1 percent. 19

20 5 Trading Strategies for the Central Bank As seen Figure (1) above, holding only one side of the market, either call or put, would expose the central bank to risks associated with a drastic movement in the exchange rate that can be caused by speculative attacks or macroeconomic fundamentals. Because the central bank is such a large player in the market, hedging one side of the market may introduce adverse signals to traders. The central bank has the means to move the market in its favor. Therefore, market participants may be weary to enter into contracts with the central bank if there are any incentives for or suspicions of market manipulation. Entering into long contracts in either the put or call position will expose the currency market to excessive volatility through the hedging behavior of the traders on the opposite side of the contract. This is exactly the opposite to the goals of the central bank. It would also create opportunities for large market makers to hold short positions and diminish the control of the central bank over expectations of market participants. A short strangle trading strategy combines the short put option and short call option strategy. The short strangle strategy can be a good one for the central bank if there are limited risks of drastic exchange rate movements. The strategy allows the central bank to be the main market maker, exert control over the currency options market, and ensure liquidity in both the spot and options market. Despite the benefits of this strategy, there are a number of sizable drawbacks of pursuing such the short strangle strategy for the central bank specifically. Issuing call and put options exposes the central bank to unlimited downside risk with limited gains, as can be seen in Figure (5). Such a strategy would need to be dynamically delta hedged to protect the central bank from losses associated with drastic movement in the exchange rate. Hedging such a position through an offsetting spot market position would force the central bank to contribute to the persistent appreciation or depreciation of the exchange rate. The position of the central bank in the spot market to offset the risks of the short options counters the goals of the central bank to ensure stable exchange rate values and limit the volatility in the market. Authors such as Breuer (1999) have noted that dynamically delta hedging a long option contract position can introduce stabilizing forces into currency markets. By dynamically delta hedging a long call on USD, the trader will hold an offsetting short position in USD, for example. If the currency (COPUSD) depreciates, the trader will sell USD in domestic spot market. By doing so, the supply of USD in the spot market rises and therefore introduces appreciationary pressure that can counter the depreciation of the COP through the portfolio balance channel. Similarly, by holding a long put position, the offsetting spot market position would be long in USD. As the currency depreciates, the long put will expire out of the money and not be exercised. The trader will sell USD (or buy COP) in spot market to offset his position at maturity of the option contract. The offsetting spot market position contributes 20

21 Figure 5: Short Strangle Option Strategy Short strangle presents an options trading position with unlimited risks and limited gains. The strike price for the short call and short put position equal at K. to a counteracting appreciation pressure on the depreciation of the COP. Due to the limited risks associated with a long options position, few traders would have the incentives to engage in dynamically delta hedging the position. Therefore, despite the fact that the hedging of a long position may introduce a stabilizing force into the currency market, the central bank cannot rely on market participants to act in such a way. The optimal strategy for the central bank will be one that includes both long and short positions in call and put options. Taking on both long and short positions in the option contract is considered a butterfly spread trading strategy. It is a neutral strategy with limited gains and limited downside risks. Call butterfly spreads consist of the trader holding two long and two short positions in the call option. Put butterfly spreads consist of the trader holding two long and two short positions in the put option. In each, the options with a high and low strike price are purchased, whereas two options in the middle strike price are issued. The strike prices for the long position are K1 and K3 in Table (6). The strike price (K2) for the short position is the midpoint price between the long strike prices. The traditional butterfly strategy has some benefits and drawbacks to reaching the goals of the central bank. Dynamic delta hedging of such a strategy is typically unnecessary, since holding a long and short position in each contract already hedges the risks to the trader of any movement in value of the underlying asset. For the central bank to have a strategic position in the spot market that is determined by the dynamic delta hedge, which increases information flow, domestic liquidity and which lowers costs of intervention, the optimal portfolio position for the central bank will be an alternative to the butterfly strategy. Specifically, the optimal strategy will be for the central bank to write or short one call or one 21

22 Figure 6: Call and Put Butterfly Spreads Call and put butterfly spreads present a neutral position with limited risks and limited gains. The middle strike price for the short position is determined as K1+K3. 2 put option contract at one strike price, and buy or long two call options or two put options at strike prices that are slightly out of the money. The alternative strategy that is used to simulate the position of the central bank in the spot market is a derivative of the butterfly strategy. The gains and losses will be limited, and the net position of the central bank in the spot market will one similar to hedging a long position. The central bank will be able to issue contracts in the domestic market, purchase long positions in the global market, and hedge its portfolio in the domestic spot market. The long positions of the alternative butterfly strategy will hedge the risks associated with the short position. By dynamically delta hedging the net portfolio position with an offsetting position in the spot market, the central bank will have a position in the spot market that will stabilize the movement of the domestic currency, smooth volatility, and influence the expectations of traders in a favorable way so as to contribute to the stabilization goals of policymakers. By holding and issuing bundles of call and put options at varying strike prices, the central bank signals to the market that it is taking two positions. It is betting on the exchange rate to appreciate or depreciate, and therefore is protecting itself by nullifying the net impact when the exchange rate moves in either direction. The signal to the central bank from the market will come from how many of each option will be purchased by market participants. If traders anticipate the currency to depreciate, more call options will be purchased to hedge against the movement in the exchange rate. Between the date of issue and maturity of the contract, the central bank will hedge its net portfolio position in the spot market, which will provide a stabilizing force in the market. 22

23 6 Analytical Approach In the following section, we will address the analytical approach used to determine how options contacts may be used by central banks for intervention into currency markets. First, we will present the Garman- Kohlhagen option pricing model, the alternative butterfly strategy and dynamic delta hedging. Next, we will address the simulation of exchange rate movements used for analysis, as well as the derivation of option prices and hedging positions based on the simulated exchange rates. The main objective of the research is to test whether an alternative butterfly strategy with dynamic delta hedging can be a viable strategy for central bank currency market intervention. The goal is to understand whether dynamic delta hedging under this scenario is stabilizing and whether this strategy can provide the central bank with a low cost, targeted intervention plan. 6.1 Alternative Butterfly Strategy with Dynamic Delta Hedging The Garman-Kohlhagen option pricing model is a derivative of the Black-Scholes option valuation model. The valuation of call options can be defined by the following: ( ) St ln v call = e r τ K S t Φ + r r + σ2 2 τ σ (1) 2 The valuation of a put option is defined as: ( ) ln St v put = e r τ K S t Φ + r r + σ2 2 τ σ (2) 2 where τ = T t, or time to maturity, Φ is the standard normal distribution function, σ is the volatility of the underlying asset, r and r are the domestic and foreign risk free interest rates, S t is the spot rate, and K is the strike price. In the analysis presented in this segment of research, the purchaser of the call (put) option, or the agent that is long in the option, has the right but not the obligation to buy (sell) one unit of foreign currency or USD. The issuer, which will be the central bank, has the obligation to sell (buy) one USD to the call (put) holder upon maturity if the option is exercised. All options in this research are European options, and therefore cannot be exercised until maturity. Dynamic delta hedging allows the issuer of the option to take an offsetting position in the spot market to cover their risk. The delta of the option is the responsiveness of the option value to changes in the value of the underlying asset and is the basis for risk management using dynamic delta hedging. The call and put deltas the derivative of the option value with respect to the spot exchange rate, and 23

24 can be presented as follows: where x = ln S t K + ( r r + σ2 2 σ 2 ) τ δ vcall = e r τ Φ(x + σ 2) (3) ( δ vput = e r τ Φ(x + σ ) 2) 1 and 0 δ vcall 1 for call deltas and 1 δ vput 0 for put deltas. For traders in the short option position, the trader would take a long position in the spot market for the underlying asset by delta units. For small changes in the underlying, the value of the hedge will change by an equal amount in the opposite direction. The trader incrementally adjusts his position (4) throughout the time to maturity. In reality, traders hedge their entire portfolios, not single options contracts. Therefore, the trader takes into consideration their net position when determining the dynamic hedge (Chen, 1998, DeRosa, 2011). As discussed by a number of authors mentioned above, dynamic delta hedging of a short option position by the central bank may create additional destabilization in currency markets. These approaches consider only the scenario under which the central bank issues only calls or only put options, where in fact the hedging position would exacerbate the movement in the exchange rate. By positioning itself in the alternative butterfly strategy, the central bank can strategically hedge its net position in a way that would counteract the persistent appreciation or depreciation of the currency. If we consider an example where there has been persistent appreciation over the last 10, 20 or 30 days, the central bank can hold and issue a bundle of calls and puts. The put options will be exercised, while the call options will expire out of the money. By hedging the net position, the central bank would be buying USD in the spot market to hedge both the call and put option position. Through the portfolio balance channel, it would therefore introduce depreciationary pressure to counteract the persistent appreciation. In contrast, if there is a persistent depreciation over the last 10, 20 or 30 days, the call options will be exercised, while the put options will expire out of the money. By hedging the net portfolio position, the central bank would be selling USD in the spot market for both the call and put option position. Through the portfolio balance channel, it would therefore introduce appreciationary pressure to counteract the persistent depreciation. The stabilization effect of holding and issuing a tailored bundle of call and put options is twofold. First, as discussed in Machnes (2006) and Sarwar (2003), the amount of calls and puts that are traded will effect the movement of the exchange rate and future volatility. Specifically, Machnes finds that trading of calls (puts) corresponds to greater depreciation (appreciation) pressure from one day to the next. Secondly, the dynamic delta hedging of a tailored net position will contribute to a position in the 24

25 spot market that counters the persistent movement of the exchange rate. 6.2 Simulation of Options Strategy To understand the potential for central banks to use options as a currency market intervention tool, we approach the analysis in three steps. Based on data from Colombia for COPUSD spot rates, we first simulate exchange rate movements using first a random process and then an Ornstein-Uhlenbeck process with GARCH volatility. The second simulation allows for price volatility to remain non-constant and ensure positive exchange rate values. Next, we use data from Colombia for the domestic interest rate, an estimation for volatility, and US three month t-bill rates for the foreign interest rate to calculate call and put prices and the corresponding deltas with the simulated exchange rate values. Lastly, using the simulated time series, we calculate the dynamic delta hedging position, or spot market position during the period when the option is issued to date of maturity, for the central bank when it issues only calls, only puts, or a bundle of both calls and puts, or a alternative butterfly strategy Simulating Exchange Rates Since Colombia has been intervening on a daily basis in its currency markets since 2010, to appropriately analyze the potential for using an options-based intervention strategy without bias in the data, we employ a simulated time series to calculate options prices and the dynamic delta hedging position for the central bank. We simulate the exchange rate using first a random process with 90 repetitions and then the Ornstein-Uhlenbeck process with GARCH volatility for 200 repetitions each with a time period of 30 days. In the random process simulation of exchange rates, we create two scenarios that represent persistent appreciation and persistent depreciation of the COPUSD. We use a uniformly distributed psuedorandom generation of exchange rates based on the value of COPUSD on October 20, This process controls the simulation environment to test the option pricing and dynamic delta hedging position when the exchange rate is moving strongly in one direction. It provides a clear picture of how the strategic hedging behavior of the central bank may introduce stabilizing pressure into currency markets. Simulating the exchange rate movement using an Ornstein-Uhlenbeck process with GARCH volatility is an optimal approach because it allows for non-constant volatility while ensuring positive exchange rate values. The Ornstein-Uhlenbeck process is a stochastic process that is stationary, Gaussian and Markovian. Therefore, time shifts leave joint probabilities unchanged, the vector of values if multivariate normally distributed, and the future is determined only by the present and not past values (Finch, 2004). The process is also mean-reverting and has been used to model interest rates, exchange rates and 25

26 commodity prices in financial mathematics. The Ornstein-Uhlenbeck process must first satisfy the following linear stochastic difference equation: dx t = ρ(x t µ)dt + σdw t (5) where W t is a Brownian motion so that t 0. In the asymptotically stationary case, µ, ρ, σ are constants which yield the following moments: E(X t X 0 ) = µ + (c µ)e ρt (6) ( Cov(X s, X t ) = σ2 e ρ s t e ρ(s+t)) (7) 2ρ To follow a Brownian motion, µ = c = 0, σ = 1 and ρ tends to zero. Here, the variance, σ is positive and constant. To simulate exchange rate movements, it is preferable for the variance or volatility to be non-constant. Therefore, to alter the process, σ is determined with a GARCH(1,1) process. Modeling volatility based on GARCH(p,q) model is typical in the financial mathematics literature, and is used extensively by professionals and academics alike. Modeling stochastic volatility using the GARCH process assumes that the randomness of the variance process varies with the variance of the model, allowing volatility to be non-constant. The standard GARCH(p,q) model is defined as: σ 2 = α 0 + α 1 ɛ 2 t α q ɛ 2 t q + β 1 σt β p σt p 2 q p σ 2 = α 0 + α i ɛ 2 t i + β i σt i 2 (8) i=1 i=1 The structure of the volatility model can be defined as x t = µ t (θ) + ɛ t (9) ɛ t = σ t (θ)z t (10) where σ 2 (θ) = E (x t µ t (θ)) 2 F t 1 (11) where F t 1 is the information set available at time t, µ t (θ) is the dynamics of the conditional mean set by an ARMA(p,q) process, ɛ t is the residual term, and θ is the vector of unknown parameters (Jondeau, Poon and Rockinger, 2007). Volatility in this model is an exact function of a set of given variables. Specifically, for the process presented below the known variables used to calculate volatility are past 26

27 values of the COPUSD exchange rate. After simulating 200 processes, we calculate the average difference in exchange rates over thirty days for each process. Based on the average change in exchange rates over thirty days, we segment the processes into two groups for additional analysis. The first group experienced on average a negative change in exchange rates, or a period of appreciation of the COPUSD. The second group experienced on average a positive change in exchange rates, or a period of depreciation of the COPUSD. The groups will be referred to below as appreciation-periods and depreciation-periods to distinguish between the prior movements of the exchange rate. The reason for segmentation into two groups is to account for the appropriate distribution of calls and puts issued in the alternative butterfly strategy bundle. During periods of appreciation, a putcall ratio below one would ensures the central bank s dynamic delta hedging strategy will introduce depreciationary pressure into the market through net purchase of USD over the period to maturity. In addition, with more calls exercised at maturity, this will also contribute to counteracting the appreciation (Machnes, 2006). During depreciationary periods, a put-call ratio greater than one will yield pressure from the central bank that may counter the rise in COPUSD and introduce more stability into the market. Segmenting the simulated processes into two groups makes this analysis much easier to conduct and interpret Calculating Options Bundle and Dynamic Delta Hedging Position To calculate the dynamic delta hedging position and option prices, we first start with deriving the volatility that will be used for analysis. In this model, we use the standard deviation of the log difference of the exchange rate over ten days. 2 The next step is to determine the strike prices that will be used to calculate the option prices. The set of strike prices in the results presented below are defined as follows, where S t represents the spot market exchange rate at time t: K t,1 = S t K t,3 = S t K t,2 = Kt,1+Kt,3 2 After the volatility and strike prices are determined, we determine the call and put prices, as well as the deltas of each using the Colombia risk free interest rate, the US risk free interest rate, and the simulated exchange rate series discussed above. The time to maturity for all contracts is 30 days. To determine the dynamic delta hedging position of the central bank, we calculate for each option the total shares purchased on a daily basis, the daily interest and cumulative costs, and the end of 2 Volatility measured over a 5, 10 and 20 day period were estimated with little difference in analysis. 27

28 period cumulative costs, payoffs and profits. The daily total shares purchased are the daily option delta multiplied by the option contract size, which is assumed to be USD$ 10,000. The cumulative costs at t = 1 are the total shares of USD purchased in the domestic currency or Shares USD,t. For t > 1, cumulative costs are calculated as: The interest costs are determined as: Cost Cum,t = Cost Cum,t 1 + Cost Int,t 1 + Shares USD,t Cost Int,t = ) (e r (t+1) 365 t Cost Cum,t The end of period cumulative cost for the long position is calculated as: Cost Cum,T = Cost Cum,t=T + δ T S T Υ The end of period cumulative cost for the short position is calculated as: Cost Cum,T = Cost Cum,t=T δ T S T Υ where Υ is the number of shares per contract, or US$ 10,000. The end of period payoff for call options is: and for put options: P ayoff calls = max(s T K, 0)Υ P ayoff puts = max(k S T, 0)Υ For the long position, the net gains are equal to the payoff at the end of the period, and the net losses are equal to the cumulative delta hedging costs over the 30 days plus the premium paid for the contract. For the short position, the net gains are equal to the premium, and the net losses are equal to the cumulative delta hedging costs over the 30 days plus the payoffs. Table (6) presents a summary of calculations. Theoretically, the profit or loss for the issuer when dynamically delta hedging should be zero. 7 Currency Options as a Central Bank Intervention Tool The goals of research are to analyze whether the net dynamic delta hedging position for an alternative butterfly strategy for a bundle of call and put options is smaller than holding a one-sided position, to understand if such a hedging strategy may introduce pressure into the market to counteract the current movement of the exchange rate and to determine whether the costs to issue such a portfolio position while hedging are lower than the costs of daily currency market interventions. It is assumed that one contract is for $ 10,000 USD, and that the central bank issues a total of 10,000 contracts at one time. 28

29 Table 6: Summary of Dynamic Delta Hedging Cost Calculations End of Period (T = 30) Long Call Position Short Call Position Long Put Position Short Put Position Cumulative Hedging Costs Cost Cum,t=T +δ T S T Υ Cost Cum,t=T δ T S T Υ Cost Cum,t=T +δ T S T Υ Cost Cum,t=T δ T S T Υ Payoff max(s T K, 0) Υ max(s T K, 0) Υ max(k S T, 0) Υ max(k S T, 0) Υ Premium CallP rice Υ CallP rice Υ P utp rice Υ P utp rice Υ Net Gains Payoff Premium Payoff Premium Net Losses Hedge Costs + Premium Hedge Costs + Payoff Hedge Costs + Premium Hedge Costs + Payoff Cost calculations to the central bank to dynamically delta hedge its net portfolio position over an option contract period of 30 days. S T represents the spot exchange rate at maturity, or t = T. Υ represents the contract size which is US$ 10,000. Cost Cum,t=T represents the cumulative dynamic delta hedging cost on the last day of the contract or t = T. δ T is the delta of the option contract at t = T where T = 30. To understand the potential for using currency options as a central bank tool for intervention into foreign exchange markets, we start first with the simulated exchange rates for COPUSD over 90 repetitions under scenarios with persistent appreciation and persistent depreciation. Figure (7) illustrates the simulation. Using these simulations introduces a controlled environment under which we can first test the alternative butterfly strategy with dynamic delta hedging. Figure 7: Series of Simulated Exchange Rates Exchange rates are simulated as a random distribution to represent periods of appreciation and depreciation. The start value of simulation is based on 10 day average COPUSD exchange rates ending in October 20, Figures (8) and (9) calculate the dynamic delta hedging position and daily cumulative costs of hedging during a period of depreciation. The daily shares purchased represent the actual amount of USD the central bank is buying or selling in the spot market based on the net portfolio position and is determined 29

30 by the deltas of each option contract in the portfolio. The cumulative daily hedging costs include the interest costs and costs of purchasing shares of the foreign currency in the spot market. There are three scenarios depicted in each figure. The first is a call butterfly strategy, where the central bank is holding (long) and issuing (short) only call options. The initial spot market position is substantial, requiring the central bank to sell approximately $ 60 million USD in the spot market in one day. Throughout the period to maturity, the net portfolio position will require the central bank to continue selling USD in the spot market to dynamically delta hedge its options portfolio. By doing so, it will in turn introduce a counteracting pressure to the persistent depreciation. Under a put butterfly position, the central bank initially conducts large purchase of USD in the spot market and subsequently sells off its initial spot market position throughout the period to maturity. The net dynamic delta hedging position of both strategies allows the central bank to have a smaller initial position in the spot market, but it continues to sell USD over the period to maturity. Through dynamically delta hedging its net portfolio position, the central bank has a clear strategic approach to its spot market position, one that also introduces a stabilizing force into currency markets through the portfolio balance channel. Altering the put-call ratio to represent more call options, as seen in Figure (9), strengthens the position of the central bank on the initial day of the contract, but does not change the spot market position throughout the period to maturity. Figures (10) and (11) calculate the dynamic delta hedging position and daily cumulative costs of hedging during a period of appreciation. Once again, three scenarios are presented in each figure. The first call butterfly strategy, shows initial spot market position that requires the central bank to sell approximately $ 60 million USD in one day. This can be potentially destabilizing to the market. Throughout the period to maturity, the net portfolio position will require the central bank to buy USD in the spot market to dynamically delta hedge its position. The daily purchases of USD may in turn introduce a counteracting pressure to the persistent appreciation through the portfolio balance channel by driving up the value of USD relative to the domestic currency. In the put butterfly position, the central bank initially conducts large purchase of USD in the spot market and continues purchasing USD throughout the period to maturity. The net position of both strategies allows the central bank to continue to sell USD over the period to maturity. Altering the put-call ratio to represent more call options, as seen in Figure (11), forces the central bank to sell USD in the spot market on the initial day of the contract, but to continue purchasing USD throughout the period to maturity. The total cumulative costs represent the end of contract period costs of dynamic delta hedging that accumulated from daily hedging. The outstanding position of the central bank from its hedging 30

31 Figure 8: Depreciation Dynamic Delta Hedging Outcomes - Put/Call: 1 The dynamic delta hedging position is calculated for 90 series of 31 days. The exchange rate is steadily depreciating over the period to maturity. The put/call ratio is 1. As can be seen, holding a position in both a call and put strategy is the least disruptive to the spot market. 31

32 Figure 9: Depreciation Dynamic Delta Hedging Outcomes - Put/Call: 0.25 The dynamic delta hedging position is calculated for 90 series of 31 days. The exchange rate is steadily depreciating over the period to maturity. The put/call ratio is As can be seen, holding a position in both a call and put strategy is the least disruptive to the spot market. 32

33 Figure 10: Appreciation Dynamic Delta Hedging Outcomes - Put/Call: 1 The dynamic delta hedging position is calculated for 90 series of 31 days. The exchange rate is steadily appreciating over the period to maturity. The put/call ratio is 1. As can be seen, holding a position in both a call and put strategy is the least disruptive to the spot market. 33

34 Figure 11: Appreciation Dynamic Delta Hedging Outcomes - Put/Call: 4 The dynamic delta hedging position is calculated for 90 series of 31 days. The exchange rate is steadily appreciating over the period to maturity. The put/call ratio is 4. As can be seen, holding a position in both a call and put strategy is the least disruptive to the spot market. 34

35 operations and its options contracts will be determined by the hedging costs, payoffs and premiums, presented in Table (7). If the central bank holds and issues the equivalence of US$ 100 million in option contracts through the alternative butterfly strategy, its costs to dynamically delta hedge the position will accumulate to less than US$ 1 million. Compared to the costs in terms of reserve accumulation to buy and sell USD on the spot market daily, as Colombia has done, these costs are very small. The net costs of the overall portfolio position are even smaller. Therefore, in terms of costs of intervention, clearly the options-based strategy is better than daily discretionary interventions. Table 7: End of Period Dynamic Delta Costs, Payoffs and Premiums Average Across Series (USD) Appreciation Depreciation Delta Hedging Costs 642, ,290 Call Payoff 0 2,335,900 Put Payoff 1,858,300 0 Premium 337, ,850 Net Costs (Calls) -48,215-19,059 Net Costs (Puts) -56,404-27,068 Net Costs (All) -52,310-23,064 Total cumulative costs represent the end of contract period cumulative costs of dynamic delta hedging that accumulated from daily hedging. Total dynamic delta hedging costs represent the end of period cumulative costs and payoffs if the contract was exercised. Each contract size in US$ 10,000, and there were 10,000 contracts issued. Next we use the Ornstein-Uhlenbeck process with GARCH volatility to simulate the COPUSD exchange rate over thirty days. In addition to all 200 scenarios, we also use the first three series to analyze a short sample of the data. The simulated values of exchange rates for each of the three series is also captured in Figure (12). As seen in the figure, there are some drastic outliers that will drive extreme hedging positions for the central bank. We want to consider all scenarios that the central bank may face. In Figure (13), the dynamic delta hedging position of the central bank is depicted in terms of the total daily shares of USD purchased in the spot market and the daily hedging costs. Here, the dynamic delta hedging position is shown for all 200 simulations of the COPUSD. Once again, the initial position in the spot market at the start of the contract is smaller when the central bank is holding a net position in both calls and puts. The cumulative daily hedging costs are limited as well, with an upper bound of $ 40 million USD per day, and lower bound of $ 45 million USD per day, which is comparable to the daily discretionary purchases of the Colombian central bank in In Figure (14), the simulated series are segmented into two groups, one where the COPUSD is on average appreciating and one where the COPUSD is on average depreciating over the period to maturity. The differences in the daily shares purchased if the central bank holds only calls, only puts, or both is limited to the initial spot market position at the beginning of the contract. Otherwise, the currency 35

36 Figure 12: OH-GARCH Simulated Exchange Rates Exchange rates are simulated with a GARCH volatility process with an Ornstein-Uhlenbeck process to keep price volatility non-constant but exchange rates positive. The start value of simulation is based on 10 day average COPUSD exchange rates ending in October 20, Figure 13: OH-GARCH Series Dynamic Delta Hedging Outcomes The dynamic delta hedging position is calculated for 200 series of 31 days. The series include periods of both appreciation and depreciation. The put/call ratio is 1. 36