Pricing Foreign Exchange Risk

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Pricing Foreign Exchange Risk
- by Glen Dixon, Associate Lecturer, Griffith University

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  • People have been borrowing, lending and exchanging money for centuries. Indeed, as long as trade and investment continue in a world of sovereign governments controlling national currencies, they must continue to do so. The foreign exchange market exists to facilitate this conversion of one currency into another. Developments over two decades have radically altered the size and nature of foreign exchange markets. The progressive liberalisation of cross-border financial flows and reform within various financial sectors has been a common development throughout industrialised countries during this time. This, combined with developments in the field of information and communications technology and the establishment of an international banking and settlements system, set the stage for rapid expansion in the foreign exchange market. As a result, the foreign exchange market today is the largest and most truly global financial market in the world.   Estimates suggest that some 70% of world trade transactions and a significant proportion of international capital transactions are denominated in US dollars.   AUS/USD 0.7550 means AUD 1 = USD 0.7750
  • A study done by ASX over a 10 year period reveals 2-3% volatility for interest rates, 8-9% volatility for foreign exchange, 20-25% volatility for commodities, Over a shorter period of time for electricity (5 years) over 50% volatility.
  • The following listing provides some detail about the factors that have influenced the direction of the AUD since it was floated in 1983. Source: Securities Institute Education, Graduate Diploma in Applied Finance and Investment, Course: Foreign Exchange Markets and Trading. 1983 5 March Australian Labor Party elected 8 March AUD devalued by 10%, to USD 0.8549 28 October Government announces that forward foreign exchange market would be allowed to float on an intra day basis 8 December Reserve Bank of Australia (RBA) closes dealing in foreign exchange markets, after a high level of speculative capital inflow during the previous week. 9 December Treasurer Paul Keating announces the Australian dollar will be floated effective from Monday 12 December 1983, and that the majority of exchange controls will be abolished. 12 December First day of the floating for the AUD. Exchange rate opens at USD 0.9160 , rose to USD 0.9285 , but fell later to close at USD 0.9120 13 December The Australian Financial Review (AFR) noted: “The ANZ Banking Group and the Australian Bank Ltd executed the first transaction under the new arrangements, at USD 0.9160 ” Initial spreads stood at about 30 points, but these narrowed over the day to finish at about a 10 point average. First major transaction expected to be AUD 700 million in overseas funds for Elders IXL to finance the purchase of Carlton and United Breweries. 1984 5 March AUD reaches a post-float high of USD 0.9495 February till Strong USD sees AUD consistently weaken to USD 0.8150 early 1985 1985 6-8 February MX missile crisis. Cabinet overturns Prime Minister (PM) on US test of MX missile; PM Bob Hawke tells US Secretary of State that Australia would not be able to provide facilities for MX missile testing. AUD falls 5 cents, from USD 0.8150 to US 0.7650 , during the crisis, and then continues to fall to USD 0.7150 over the rest of the month. 1986 13 May Current account deficit comes in at AUD 1.5 billion, well above market expectations. AUD falls from USD 0.7480 prior to the release to USD 0.7340 at the end of trade. 14 May Treasurer Keating makes ‘Banana Republic’ comments in radio interview with John Lawes, on radio stations 2GB and 3AW. AUD falls from just over USD 0.7300 to USD 0.7000 at the close of trade. 2 July Government announces removal of withholding tax exemptions for securities and overseas borrowings by public authorities. Change create uncertainty in markets, and AUD falls to a low of USD 0.6450 4 July Negative sentiment pushes AUD further lower, reaching USD 0.6420 25 July Higher than expected Consumer Price Index (CPI) outcomes (+1.7% versus market expectation of 1-1.5%) undermines AUD, which falls to USD 0.6000 in New York trade 28 July Wildest day on record for AUD. Opens at USD 0.6000 , with Japanese selling taking the currency to its record ever low of USD 0.5710. RBA raises rediscount rate from 14.6% to 16% at 12:30pm, and Government announces reversal of withholding tax changes instituted on 2 July at 4pm. AUD spikes to USD 0.6300 , before closing at USD 0.6100. 19 August Treasurer Keating delivers 1986/87 Budget. Tough spending cuts gives support to AUD. 1987 20 October Share market crash in Australia. AUD holds during share crash, but then falls to 4 cents, from USD 0.7100 to USD 0.6700 November/ Commodity prices rise strongly, dragging AUD with them December 1988 April till Australian economic growth surges, and official short-term December 1989 interest rates are increased from 10% to 18%. AUD rises to USD 0.8900 1989 14 February Comments by Treasurer Keating warning AUD is overvalued at current levels cause AUD to fall 1.5 cents to USD 0.8750 Late February till AUD continues to fall to USD 0.8000 by end February, and April then again to USD 0.7500 in April May Moody’s announces review of Commonwealth foreign currency debt August Moody’s confirms downgrade in Commonwealth debt rating to AA2 1990 23 January RBA announces easing in monetary policy, with cash rates cut by 1% to around 17%. The Australian official short-term interest rates cut on 15 separate occasions, falling from 17% to 4.75%. August Iraq invades Kuwait, oil prices sky-rocket, and AUD is afforded safe-haven status 1991 3 June Treasurer Keating reveals special leadership pact between Hawke and Keating had been broken by Hawke. Revelation prompts Keating to challenge Hawke for leadership of the Australian Labor Party; he resign as Treasurer when challenge is unsuccessful 19 December Paul Keating again challenges PM Hawke for leadership, and is elected leader of ALP and Prime Minister 1992 January ‘One Nation’ statement is announced. Fears that new PM Keating will boost spending weaken bond market and AUD 26 February PM Keating delivers ‘One Nation’ Economic Statement February till March Fading hopes of further rate cuts boost AUD June Renewed rate cuts speculation and falling commodity prices weaken the AUD October Moody’s downgrades Victorian debt by two notches December Election uncertainty and rate cut speculation  
  • The following listing provides some detail about the factors that have influenced the direction of the AUD since it was floated in 1983. Source: Securities Institute Education, Graduate Diploma in Applied Finance and Investment, Course: Foreign Exchange Markets and Trading. 1993 January Poor Balance of Payments (BoP) outcomes further undermine AUD April till June Weak commodity prices and rate cut speculation weaken AUD 17 August Treasurer Dawkins delivers 1993/94 Budget. In subsequent months the Government is forced to negotiate with minor parties in the Senate in order to have Budget passed, causing AUD instability. October Australian commodity price indices reach record post-war lows, and AUD falls to low of USD 0.6410. Native land title claim (known as the ‘Mabo case’) creates investor uncertainty Late 1993 till Improved sentiment about global economic recovery boosts early 1994 commodity price expectations, Australian sharemarket and AUD 1994 February AUD stalls at USD 0.72-0.73 after Fed lifts US interest rates, share and bond markets weaken 7-8 April Favourable OECD report on Australia released. AUD soars US 2 cents in two days Early May The Government’s unemployment white paper and concerns about the 1994/95 Budget undermined AUD sentiment Late June Renewed bond markets weakness relecting lack of confidence in local policy makers’ resolve to fight inflation hit AUD sentiment. July till October AUD range trades with no new direction shown 1995 June-December AUD trading up to 77 cents 1996 January-February Election uncertainty March Change of Federal Government after 13 years. AUD strengthened. May AUD hits 80 cents as commodity prices jump November till AUD spikes 0.79 to 0.816 before returning to 79 cents after December RBA cuts official interest rates 1997 February AUD falls below 76 cents as US dollar surges against all major currencies. 1998 1999 2000 2001 2002 2003
  • Garman, Mark B. and Steven W. Kohlhagen
  • History and Size of the Market Trading in foreign currency options began in the 1970s and 1980s in the venue of the listed futures and options markets of Chicago, Philadelphia, and London. Trading was concentrated in options and futures options on only a handful of major exchange rates. A structural change occurred in the 1990s, when the bulk of trading in currency options migrated “upstairs” to bank dealing rooms, to the detriment of the organized exchanges. Once installed in the domain of the interbank foreign exchange market, option trading exploded in volume. What is more, currency options began to key off of the full gamut of exchange rates. The topic of this seminar is the specialized area of options on foreign exchange. Attention will be focused on plain-vanilla European puts and calls on foreign exchange as well as on some of the more popular exotic varieties of currency options. Commercial and investment banks run the currency option market. The same money-center dealers that constitute the core of the spot and forward foreign exchange market are the most powerful market makers of currency options. For this reason, this book generally uses the conventions and terminology of the interbank foreign exchange option market. Currency options are used by currency hedgers, traders, speculators, portfolio managers, and, on occasion, central banks. In the mid 1990s, trading in exotic currency options began to develop at a rapid pace. Today, dealers routinely supply two-way bid-ask prices for a wide spectrum of exotic currency options. How-ever, the largest appetites for exotic currency options are for barrier options. The market for basket options, average rate currency options, compound currency options, and quantos options is smaller, yet not insignificant. The currency option market can rightfully claim to be the world’s only truly global, 24-hour option market. The currency option market is among the largest of the option markets by trading volume. It is impossible to be precise about its overall size because the majority of trading in currency options is done in the private interbank market. But some rough estimates are reported in a survey done by the Bank for International Settlements (BIS). The most recent BIS survey estimated that the daily volume of currency option trading was $86.9 billion in face value in April 1998. Notable is that the number was only $41.2 billion when the same survey was done in April 1995. The BIS survey reported that the largest portions of currency option trading are done in U.S. dollar/Japanese yen and the German mark (against a variety of currencies). In January 1999, the launch of the first round of the European Monetary Union meant that the new common currency, the euro, began to replace the German mark in foreign exchange derivatives trading. The underlying asset for currency options is foreign exchange. The elementary foreign exchange transaction, called a spot deal, is a contract to deliver and receive sums of foreign currency for value in two bank business days. Foreign exchange also trades on forward basis for value beyond the spot value date. Forward deals are routinely transacted for settlement on future spot value dates in one week, one month, three months, six months, and one year. The market exchange rate for forward dealing is called the forward outright. Currency options are calls and puts on sums of foreign currency. They are the right but not the obligation to buy or sell a sum of currency at a fixed strike price on or before the option’s expiration date.
  • Consider the following option on dollar/yen: USD call/JPY put Face amount in dollars $10,000,000 Option put/call Yen put Option expiry 90 days Strike 120.00 Exercise European This option is a call option on the U.S. dollar (USD) or equivalently, a put option on the Japanese yen (JPY). It grants its owner the right but not the obligation to receive $10,000,000 in exchange for delivery of 1,200,000,000 yen ($10 million * 120.00) at option expiration. This is an example of a standard, or vanilla, currency option. European exercise means that an option can be exercised only on the last day of its life. Upon exercise, a currency option triggers a spot foreign exchange transaction done at the strike price and for settlement on the spot value date. Barring exceptional circumstances, this option should be exercised if the spot exchange rate is above 120.00 on the expiration date. Likewise, a USD put/JPY call struck at the 120.00 should be exercised at expiration if the spot exchange rate is below 120.00. Interbank currency options can be transacted at practically any strike, but the most popular exercise price is at the prevailing forward outright (this is called at-the-money-forward ). European exercise currency options are surprisingly well understood using a simple adaptation of the Black-Scholes option pricing model. In fact, Black-Scholes concepts and terminology permeate the currency option market. Option prices are quoted not in terms of dollars and cents but rather in units of Black-Scholes implied volatility. Currency options themselves are identified by their Black-Scholes delta more frequently than by their actual strike price. Moreover, Black-Scholes concepts, like delta, gamma, theta, and vega, have become the basic vocabulary of option risk management. Although the great mass of interbank currency options is European exercise, there are some American exercise currency options. By definition, an American exercise option can be exercised at any time in its life. American exercise currency options can be priced with the binomial option-pricing model and also with a variety of numerical approximation techniques.
  • Exotic Options An exotic currency option is an option that has some nonstandard feature that sets it apart from ordinary vanilla currency options. The foreign exchange market is a fertile ground for the invention of new exotic options. The most popular exotic currency option is the barrier option. One type of barrier option is the knock-out option. A knock-out option is similar to a vanilla option except for the existence of a barrier exchange rate, called the out-strike, which when breached would cause the option to extinguish at any time during the option’s life. For example, one could add an out-strike at 115.00 to the previously mentioned USD call/JPY put. This would mean that if dollar/yen were to trade at 115.00 before expiration, the option would cease to exist. Naturally, this option must cost less than its vanilla counterpart, which cannot be knocked out regardless of where dollar/yen trades. There are also double-barrier currency options, which can be extinguished by either of two out-strikes. A binary option pays a lump sum of cash if the option is in-the money at expiration. Binary options too can have one or two barrier out-strikes. Another exotic binary option is the one-touch. This option pays a lump sum of cash to the holder of the option if its barrier level is traded. A special form of binary exotic option, called the doublebarrier range binary, has caught the attention of the trading community. This option pays a lump sum of cash at expiration provided that no barrier event takes place in the option’s life. Double-barrier range binary options are used in volatility trading. Although barrier currency options are mainly popular with traders and speculators, other types of exotic currency options are favorites with foreign exchange risk managers. A basket option is a put or a call on the aggregate value of a portfolio of foreign currencies. Basket options are popular because they offer economical solutions for hedging foreign exchange risk. Compound options are options on options. When one acquires a compound call option, one has the right but not the obligation to buy a vanilla call or put for a fixed compound strike price. Average rate (Asian) currency options are cheap compared to vanilla options because their payoff at expiration is a function not of a single observation of the spot exchange rate but rather of an average of the spot exchange rate over a period of time. Still another exotic option for hedgers is the quantos option. The quantos option has a floating face amount that adjusts to the market value of the underlying portfolio that is the subject of the hedging program.
  • Definitions at, in and out of money
  • Garman, Mark B. and Steven W. Kohlhagen
  • The Black-Scholes (1973) option pricing formula prices European put or call options on a stock that does not pay a dividend or make other distributions. The formula assumes the underlying stock price follows a geometric Brownian motion with constant volatility . It is historically significant as the original option pricing formula published by Black and Scholes in their landmark ( 1973 ) paper with the Journal of Political Economy. Here, log denotes the natural logarithm, and: s = the price of the underlying stock  x = the strike price  r = the continuously compounded risk free interest rate  t = the time in years until the expiration of the option  σ = the implied volatility for the underlying stock Φ = the standard normal cumulative distribution function.
  • The Greeks are a set of factor sensitivities used extensively by traders to quantify the exposures of portfolios that contain options . Each measures how the portfolio's market value should respond to a change in some variable—an underlier , implied volatility , interest rate or time. There are five Greeks:           delta measures first order (linear) sensitivity to an underlier;           gamma measures second order (quadratic) sensitivity to an underlier;           vega measures first order (linear) sensitivity to the implied volatility of an underlier;           theta measures first order (linear) sensitivity to the passage of time;           rho measures first order (linear) sensitivity to an applicable interest rate. They are called the Greeks because four out of the five are named after letters of the Greek alphabet. Vega is the exception. For reasons unknown, it is named after the brightest star in the constellation Lyra. At times, vega has been called kappa, but the name vega is now well established. Four of the five are risk metrics . Theta is not because the passage of time in certain—it entails no risk. Theta is akin to the accrual of interest on a bond. The Greeks are defined as first—and in the case of gamma, second—partial derivatives. See articles on each (linked above) for more information.
  • The Greeks —delta, gamma, vega, theta and rho—for a call are: Here, log denotes the natural logarithm, and: s = the price of the underlying stock  x = the strike price  r = the continuously compounded risk free interest rate  t = the time in years until the expiration of the option  σ = the implied volatility for the underlying stock Φ = the standard normal cumulative distribution function.   delta = Φ( d 1 )  where denotes the standard normal probability density function. For a put, the Greeks are:   delta = Φ( d 1 ) – 1 Note that gamma formulas [6] and [11] are identical for puts and calls , as are vega formulas [7] and [12].
  • The Greeks —delta, gamma, vega, theta and rho—for a call are: Here, log denotes the natural logarithm, and: s = the price of the underlying stock  x = the strike price  r = the continuously compounded risk free interest rate  t = the time in years until the expiration of the option  σ = the implied volatility for the underlying stock Φ = the standard normal cumulative distribution function.   delta = Φ( d 1 )  where denotes the standard normal probability density function. For a put, the Greeks are:   delta = Φ( d 1 ) – 1 Note that gamma formulas [6] and [11] are identical for puts and calls , as are vega formulas [7] and [12].
  • The Merton (1973) option pricing formula generalization the Black- Scholes (1973) formula so it can price European options on stocks or stock indices paying a known dividend yield. The yield is expressed as an annual continuously compounded rate q . A shortcoming of the Merton formula is its assumption that dividends are paid out continuously. For a stock index, this is an imperfect but usually reasonable approximation. For individual stocks, which typically distribute dividends in two payments each year, it is more problematic. The stock's annual yield is immaterial. The quantity q needs to reflect the dividends that will be earned prior to the option's expiration. If the stock has no ex-dividend date prior to the option's expiration, set q = 0. Otherwise, calculate the stock's dividend yield through expiration and annualize. Here, log denotes the natural logarithm, and: (Merton 1973) s = the price of the underlying stock  x = the strike price  r = the continuously compounded risk free interest rate  q = the continuously compounded annual dividend yield  t = the time in years until the expiration of the option  σ = the implied volatility for the underlying stock Φ = the standard normal cumulative distribution function. Here, log denotes the natural logarithm, and: (Black Scholes 1973)  s = the price of the underlying stock  x = the strike price  r = the continuously compounded risk free interest rate  t = the time in years until the expiration of the option  σ = the implied volatility for the underlying stock Φ = the standard normal cumulative distribution function.
  • The Merton (1973) option pricing formula generalization the Black-Scholes (1973) formula so it can price European options on stocks or stock indices paying a known dividend yield. The yield is expressed as an annual continuously compounded rate q . Here, log denotes the natural logarithm, and: (Merton 1973) s = the price of the underlying stock  x = the strike price  r = the continuously compounded risk free interest rate  q = the continuously compounded annual dividend yield  t = the time in years until the expiration of the option  σ = the implied volatility for the underlying stock Φ = the standard normal cumulative distribution function.
  • The Merton (1973) option pricing formula generalization the Black- Scholes (1973) formula so it can price European options on stocks or stock indices paying a known dividend yield. The yield is expressed as an annual continuously compounded rate q . Here, log denotes the natural logarithm, and: (Merton 1973) s = the price of the underlying stock  x = the strike price  r = the continuously compounded risk free interest rate  q = the continuously compounded annual dividend yield  t = the time in years until the expiration of the option  σ = the implied volatility for the underlying stock Φ = the standard normal cumulative distribution function.
  • In the interbank foreign exchange market, options are not quoted with prices. They are quoted indirectly with implied volatilities . The convention for converting volatilities to prices is the Garman and Kohlhagen (1983) option pricing formula . Mathematically, the formula is identical to Merton's (1973) formula for options on dividend-paying stocks. Only the term q , that did represent a stock's dividend yield, now represents the foreign currency's continuously compounded risk-free rate. Like the Merton formula, the Garman and Kohlhagen formula applies only to European options. Generally, OTC currency options are European. Here, log denotes the natural logarithm, and: s = the current exchange rate (units of domestic currency per unit of foreign currency)  x = the strike exchange rate  r = the continuously compounded domestic risk free interest rate  q = the continuously compounded foreign risk free interest rate  t = the time in years until the expiration of the option σ = the implied volatility for the underlying exchange rate Φ = the standard normal cumulative distribution function.
  • Because their prices are affected by two—one domestic and the other foreign—risk free rates, currency options have two rho sensitivities. All the Greeks—delta, gamma, vega, theta, domestic rho and foreign rho—for a call are: Here, log denotes the natural logarithm, and: s = the current exchange rate (units of domestic currency per unit of foreign currency)  x = the strike exchange rate  r = the continuously compounded domestic risk free interest rate  q = the continuously compounded foreign risk free interest rate  t = the time in years until the expiration of the option σ = the implied volatility for the underlying exchange rate Φ = the standard normal cumulative distribution function.
  • Here, log denotes the natural logarithm, and: s = the current exchange rate (units of domestic currency per unit of foreign currency)  x = the strike exchange rate  r = the continuously compounded domestic risk free interest rate  q = the continuously compounded foreign risk free interest rate  t = the time in years until the expiration of the option σ = the implied volatility for the underlying exchange rate Φ = the standard normal cumulative distribution function.
  • Comparisons of Currencies
  • Garman, Mark B. and Steven W. Kohlhagen
  • Sensitivity Analysis : Definition of Speculation and Hedging Speculative trading strategy is typically a large-return-for-large-risk strategy. It is typically very short-term, in terms of how long the positions are held as well as how far out the traded contracts tend to go. This type of strategy benefits from the great deal of liquidity for greater deal turnover and hence more opportunity. Hedging The process of entering into Hedge Contracts in order to minimise risk.
  • Scenario development Stress testing and VaR techniques, are designed to look at the risk to a foreign exchange trading portfolio from changes in market prices over a short period of time. These are valuable tools for the energy risk manager and provide important information to energy traders and energy trading managers about the risk in the trading portfolio. They cannot, however, account for those events that are potentially most catastrophic to a firm – namely major changes in the external macroeconomic environment that have an effect well beyond their immediate impact on the value of the trading portfolio. When used in combination with stress tests and VaR calculations, this methodology serves to round out a firm’s risk management practice. This seminar will alert the proactive foreign exchange risk manager to the role of scenario analysis, its subjective nature and its limitations. The first part provides a definition of scenario analysis, partly by a comparison and contrast with its look-alike technique, stress testing. This will lead into an explanation of how to prepare a scenario analysis, interpret it and, finally, recognize its shortcomings.
  • Defining scenario analysis   Scenario analysis is a strategic technique which enables a firm to evaluate the potential impact on its earnings stream of various different eventualities. It uses multidimensional projections, and helps the firm to assess its longer term strategic vulnerabilities. It is important to distinguish between the respective roles of scenario analysis and stress testing. Both are forward looking techniques which seek to quantify the potential loss which might arise as a consequence of unlikely events. Stress testing is designed to evaluate the short-term impact on a given portfolio of a series of predefined moves, in particular market variables. Scenario analysis on the other hand seeks to assess the broader impact on the firm of more complex and inter-related developments. Huge losses often occur due to a sequence of several adverse events. Scenario analysis can help to identify such potential problems in advance. The purpose of scenario analysis is to help the firm’s decision makers think about and understand the impact of unlikely, but catastrophic, events before they happen. A management team that learns its lessons from previous catastrophic situations is more likely to avoid losses in the future. Scenario analysis is an effective tool to assist management in that process.
  • Scenario analysis is an essential complement to the risk management tool-kit, which helps management understand the effects of major changes in the external environment that often have repercussions well beyond their immediate impact on the value of the trading portfolio. As I have tried to point out, the process involves defining a scenario, including all assumptions and our time horizon. The second step is the identification of scenario fields, eg, risk factors. Scenarios are then projected based on the likely movements of identified variables under the given scenario. The results are consolidated, validated, presented, and analysed. Most important is the follow-up, the action taken or not taken by the firm. Scenario analysis hinges on effective pre-analysis, interviews, and necessarily the ability to envision a wide range of possibilities. Because firms must leverage off inputs from various departments throughout the organization, the scenario analysis has a final feature that is of critical importance.
  • Political Risk A significant political upheaval occurs in a country where a business has a large investment. The firm needs to act very quickly to evaluate what kind of impact this would have on the investment. Operational Risk A new settlement system has been implemented at a energy company and it cannot be reconciled back to the original system to ensure data integrity. Thus, payments cannot be made and trades cannot be booked. Legal Risk A large energy corporation sues an investment bank because they did not disclose all necessary risks associated with a particular transaction in the prospectus. Credit Risk A business looks at its top 5 credit exposures, all with varied liquidity problems and the electricity market moves 10 per cent against them. Reputational Risk An employee of an energy company embezzles funds for an extended period of time before being caught.
  • Step 1: Scenario definition The first step in the process is defining a plausible scenario. Although recognizing relevant scenarios is more an art than a science, there are two principles of scenario selection: knowing your portfolio and understanding the relevant events in the marketplace. Numerous factors could affect the portfolio – major events in the markets, political elections, world banking crises, major tax reforms, severe flooding, the Australian stock market at all time highs, or high unemployment to name a few. Scenarios are typically requested and broadly defined by senior management because they are most familiar with the firm’s business and the external factors that affect the firm’s earnings. These scenarios ask the question: “What is the impact of …” and are often accompanied by specific scenario conditions and basic assumptions. Sometimes the risk group itself initiates and develops scenarios in order to identify potential weaknesses in their risk management concept and to make sure that the firm will survive even unlikely but catastrophic complex events. For this purpose one might: Develop new scenarios, e.g. single events triggering unfavourable changes e.g. a war, a natural disaster, an unexpected economic event; long-term market scenarios eg, recession in a group of determined countries, failure of EMU; and ask managers for the worst thing they could imagine regarding their business and extend this scenario to the whole firm. It is advisable to specify enough detail and key assumptions in order to ensure all of your experts interpret the scenario in the same way. For example, it is very important to specify the time horizon of the scenario. If the objective is to find out how much the firm would lose if it had to liquidate its inventory in an emergency, then the time frame might be five days. A time horizon for a scenario of a portfolio’s worst mark-to-market would be one day. A longer time horizon of three months might be appropriate for a scenario holding a large inventory position during a steady market decline.
  • Step 2: Scenario-field analysis Having defined a scenario to analyse, the next step is to undertake an extensive interview process. This process incorporates the appropriate business areas and other experts within the organisation in order to further refine the scenario and compile the relevant data needed. This step is very closely linked to the next steps, which are usually integrated into the interview process as well.The interview process should be standardized to ensure consistent feedback. One approach would be to give all the interviewees a written description of the scenario, along with a list of questions, and ask them to respond by a specific date. The purpose of this scenario-field analysis is to identify all relevant risk dimensions and risk factors, also called “scenario fields” which are affected by the scenario. For example, strong movements in the electricity market might trigger defaults of some firms causing substantial credit losses and vice versa, weather and outage market movements which themselves might not be in other markets with severe consequences. Second order and third order effects also have to be taken into account, eg, effects on future earnings, upon staffing, etc.It can be seen that establishing meaningful scenario analysis is a complex process which requires the expertise of many people with diverse backgrounds in various departments. In an merchant/investment bank, These may include: market research analysts, traders/salespeople, technologists, accountants, lawyers and senior management. Although it is always possible business people may participate in developing scenarios in order to design an analysis that gives an attractive appearance to their own business, the risk can be largely diffused by ensuring that a broad enough range of people are involved in the process. Moreover, by entrusting the risk management department to conduct scenario analysis at the firm wide level, an appropriate degree of independence from the business areas can be built into the process.
  • Step 3: Scenario projections   This is the heart of the scenario analysis. For each scenario field identified, a prognostic of the potential development within the given time horizon and of the associated potential losses have to be determined. Again this step should be carried out by or in close contact with, the appropriate business areas and experts. In this context it should be mentioned that it is not the goal to find the most likely development, but to identify adverse or extreme scenarios. Therefore this process will sometimes result in more than one estimate of the potential returns. Furthermore, it should be emphasized that the process is not an exact science, but rather requires best estimates.
  • Step 4: Scenario consolidation Here the projections developed for each scenario field are consolidated into one consistent scenario. However, the scenario has not only to be checked for consistency errors but also for double counting or contradictory assumptions. Results must also be checked for reasonableness: start by looking for any outliers and make sure the feedback is consistent. If there are one or two outliers, go back to the person questioned to make sure they understood what you were asking them. Inconsistent information is frequently caused by misunderstood questions. In order to prevent receiving such information, it is a good idea to conduct some background research prior to the review. When checking electricity market move projections over a certain time period, it may be useful to ask them (the energy company) what a typical move in the electricity market would be in order to find out if the scenario results makes sense. If you have received inconsistent information, it may be time to redefine the scenario. Whenever possible, leverage off historical data to compare the results from the interview process. Analyse the profit and loss history for the various business units for clues that your results make sense. Examine the market value of the portfolio for any significant changes. Take a look at the volatility of the P&L to see whether it is in line with what you would have expected, given the results. It may be helpful to consider questions such as: were the right people interviewed? did they have the appropriate level of experience? how long have they been in the current position?   Independent checks on the results may indicate the questions were not clear or that the wrong questions were asked. Here you may need to redefine the scenario in order to obtain more meaningful results. Sometimes it may also be appropriate to ask more than one person to review the results in order to check them for reasonableness.
  • Step 5: Scenario presentation and follow-up The final analysis and presentation of results is a critical step in the process. Evaluating results and drawing conclusions can be quite challenging, as it is a highly subjective process. It therefore requires an interpreter with both experience and judgement. Scenario results are frequently surprising and sometimes doubted. Often people are shocked at the magnitude of potential loss. While the probability of a scenario event is typically very low, the results should not be discounted. Since the analysis is subjective, it is easy to poke holes in any of the assumptions being made. Given the circumstances surrounding most scenario discussions, the dialogue is iterative. Therefore, when presenting the results, it is very important to state clearly what the assumptions and objectives of the exercise were before the results of the analysis in order for the reader to correctly interpret the results and understand their value. The primary goal of a scenario analysis is to get the audience to recognize that damaging losses can happen. But more importantly, to allow management to take appropriate action to prepare for the unlikely outlier events. The presentation should be a learning experience for everyone involved. As a result of the successful presentation some plan of action must be deployed. Follow-up can be as simple as: Put on a hedge; Put on a specific trade; Unwind a portfolio modify the analysis; Develop a repeatable process; Do nothing.
  • Stress Testing If one could encapsulate the most fundamental concern of any part of a firm involved with taking risk, from the individual trader right up to the board of directors, it could probably be summarized in the one question, “How much could I lose?”. The challenge for risk management is to be able to find a way of answering that question. The process of stress testing involves first identifying These potential movements, including which market variables to stress, how much to stress them by, and what timeframe to run the stress analysis over. Stress testing is another form of risk management which tests exposure to: Low probability extreme market events; Hidden assumptions in models; Structural breakdowns in the market environment; Robustness of risk management systems.  
  • Stress testing reveals exposures which sensitivities and VaR can’t, because this technique provides risk identification as well as measurement. An example is to use the jump model to generate jump scenarios. Foreign Exchange forward curves can be shifted by large multiples of main factors.
  • Step 1: Picking what to stress   Choice of market variables : Sometimes it is not clear which market variables to stress and whether this should be done individually or in groups. It may be helpful to consider which collections of market variables move together and which move independently. Range of stress: Small stress tests around the current position should not be a substitute for VaR. Very large ranges of stress tests, however, may be discounted as improbable or beyond the realistic assumptions of the underlying model. Usefulness of stress information vs data overload: An effective risk manager needs to keep in mind that stress tests are a supplemental risk technique. If the time spent running, analyzing, and distributing the results begins to approach anywhere near the majority of time spent on risk monitoring, data overload may be a problem and the number of stress tests being run may need to be cut back.
  • Step 2: Identifying assumptions   Will correlations hold or break?: Should we assume correlations that hold under normal circumstances also hold when markets crash? Deciding on how much faith to put in correlations is largely a matter of judgement concerning the nature of the particular marketing question. Assumptions about correlations between different market variables should be cross checked against the choice of which variables to stress. In the case of a 3 month electricity contract, for example, it may be advisable to stress all short-term foreign exchange contracts together, in much the same way as an entire yield curve rather than the individual bond rates often appearing in stress reports. For correlations that break, what are the new assumptions?: Can we look to other countries, markets, or industries for proxies and histories of similar events? Stress tests typically assume zero correlation among the variables being stressed and other market variables affecting the portfolio. Correlations that are still assumed to hold are treated, by stressing the two correlated variables together – effectively assuming a correlation of 1. For correlations that break, however, more sophisticated techniques may be employed when we can replace the zero. Correlation assumption with better estimates. Scenario analysis, for example, works top-down to explicitly consider which variables will be affected by the scenario and then considers their combined effect on P&L. Does the underlying financial model still hold? Stress testing is most reliable if it is based on an objective assessment of historical market movements and actual market data are used. Nevertheless as we have seen, stress testing is often used to “push beyond the envelope” into hypothetical situations, in part because these can be the worst-case scenarios. The extent to which a market variable is stressed far beyond its historical range is based on assuming the underlying financial model remains robust and still holds in this range. In extreme hypothetical situations, scenario analysis may again be the better technique as it deals head on with the analytical problems of new states of the world, the top-down assessment of the model and the assumptions that drive the results.
  • Step 3: Revaluing the portfolio   Back of the envelope vs sophisticated modeling: For most stress tests, sophisticated theoretical modeling is probably unfounded. There are more complex techniques such as scenario analysis or Monte-Carlo simulations for when assumptions get complicated.
  • Step 4: Deciding on action steps Reporting: In many large energy companies, stress reports may be circulated every day to energy traders. It is important to avoid data overload. As regards what is reported, risk managers and management are often most concerned about the impact of various stresses on the P&L. Energy traders on the other hand, tend to focus on the “Greeks” – i.e. mostly delta along with gamma and vega. Foreign Exchange Traders want to know, for example, if delta shifts how difficult will it be to buy or sell the underlying security and whether they could be caught on the wrong side of an illiquid market. Energy traders are more often concerned with smaller stresses while risk managers tend to look at larger moves and typically want practical, boiled-down versions of stress reports. The energy risk manager’s temptation, on the other hand, may be to analyse and report a large variety of possible stresses. If a bad event does happen, the energy risk manager can say they did their job and reported the potential loss. There is no simple answer to resolving these different objectives. It does, however, point out the need for careful up-front planning involving both energy traders and the risk management department to agree on mutually useful reporting content and procedures. Cross-checks on model and pricing validity: Stress tests often reveal sizeable vulnerabilities in P&L, especially for extreme movements in key market variables. In many cases, this can indicate a flaw in the model or pricing algorithm rather than an actual market sensitivity. Stress tests, in fact, are one of the most important cross-checks on model accuracy. Where stress testing reveals unusual results of this sort, it is often a good policy to check the underlying model. Risk management departments in large energy companies (both retailers and generators), in fact, often employ quantitative experts whose job includes an independent check on model validity. Follow-up with traders: When models and pricing have been confirmed to be accurate, there is often an issue of how much follow-up with traders is needed or even appropriate. In many instances, delivering the stress report to the Foreign Exchange trader, no matter how startling the results are, is the final step in the process. The Foreign Exchange trader is assumed to take any necessary actions concerning his or her trading strategy. A closer working relationship between Foreign Exchange risk manager and the energy trading desk, however, will often lead to a dialogue and follow-up. This can include such issues as possible diversification or hedging strategies to reduce the areas of greatest downside risk as revealed in the stress analysis. Action plan for dealing with actual catastrophe situation: Clearly, it is a matter of experience and judgement as to whether normal VaR monitoring has any usefulness during these periods. Stress tests, however, can help determine an action plan before catastrophe strikes. For example, stress testing may be used to identify “discontinuities” or “breakpoints” in likely P&L impact. In a Foreign Exchange market crash, for instance, losses may tend to accumulate at a greater rate after the Foreign Exchange market drops. By identifying possible inflection points in returns, stress tests can help establish credible benchmarks for dealing with potentially difficult situations.
  • First of all, let examine the distributions of historical temperature The first histogram in Figure 5.7 for whole season is obviously not normally distributed, since the density function has two humps. However, the situations are changed when we separate the whole season to summer and winter season. Figure 5.7, 5.8 and 5.9 show three histograms for whole season, summer season and winter season in Sydney. The first histogram in Figure 1 for whole season is obviously not normally distributed, since the density function has two humps. However, the situations are changed when we separate the whole season to winter and summer season. (No Kurtosis in all cases). The second histogram in Figure 5.8 for winter season looks normally distributed with slight left skewness. The third histogram in Figure 5.9 for summer season looks normally distributed. As a result, a standard time-series and stochastic differential equation with a constant volatility may not fit well, in particularly, the skewness. What caused the skewness in the implied distribution? The reason could be that the random process in the model might not be normally distributed or the volatility may not be a constant at all. According to the empirical research from financial market s , the volatility will most likely be a random process if we still want the normal distribution in the original data. Tried New York Data from CME (1979-2002) very similar to Sydney. Also one year in Spain looks similar.
  • Power Spectrum (Is used to detect cycles, i.e. seasonality) Before we investigate if volatility is a constant, we need to determine its mean value over time. As we have noticed, seasonality is very strong in temperature. The periodogram is normally used to examine its seasonality. Figure 4 show a periodogram for summer season separately. The upper graphs on the both figures cannot be seen clearly. We will reproduce the same graphs with a short period so that they are clear enough to see the peaks. Table 1 recorded the frequencies for yearly effects. From Figure 4, we can see that the peaks match the frequency values of a year effect, half year effect and one-third year effect in Table 1. These periodical effects must be included in the mean value. In summary, we could have a mathematical formulation of mean function given as Mean = Trend + One Year Effect + Half Year Effect + One Third Year Effect where these yearly cycles can be represented by sine or cosine functions. More preciously, it can be formed with trend as (See above) So 0.00653 = 1/153, ……0.0196 = 1/51
  • Power Spectrum (Is used to detect cycles, i.e. seasonality) Before we investigate if volatility is a constant, we need to determine its mean value over time. As we have noticed, seasonality is very strong in temperature. The periodogram is normally used to examine its seasonality. Figure 5 shows a periodogram for winter season separately. The upper graphs on the both figures cannot be seen clearly. We will reproduce the same graphs with a short period so that they are clear enough to see the peaks. Table 2 recorded the frequencies for yearly effects. From Figure 5, we can see that the peaks match the frequency values of a year effect, half year effect and one-third year effect in Table 2. These periodical effects must be included in the mean value. In summary, we could have a mathematical formulation of mean function given as Mean = Trend + One Year Effect + Half Year Effect + One Third Year Effect where these yearly cycles can be represented by sine or cosine functions. More preciously, it can be formed with trend as (See above) So 0.00457 = 1/212, ……0.0141 = 1/70.667
  • 1.1        Examining Variance (Leads to Stochastic Volatility – varying volatility). For the purpose of examining only, we assume that the temperature X(t) follows a simple model given as X(t) =Theta (T) + Epsilon (t) where Theta (T) is given as above and Epsilon (t) is normally distributed. This simple formulation will give us Expected [X(t)] = Theta (T) and Var [X(t)] =Sigma^2(T) We may calculate the variance as Var [X]= Expected [X(t) - Theta (T)^2] The steps of calculating variance is listed as 1. Using ordinary least square to evaluate Theta (T) . 2. Calculating the differences between X(t) and Theta (T) Figure 6 shows the fitted results of mean and variance. For simplicity, we only introduced a yearly effect in our model. We could see that the variance is varying over times. A stochastic volatility model is desirable.
  • First Now we describe the methodology to estimate the parameters. Due to analytically intractable likelihood function, a traditional maximum likelihood estimator cannot be easily applied to estimate the parameters and its volatility. Here we will introduce a numerical method of M arkov C hain M onte C arlo (MCMC) . MCMC is a conditional simulation methodology that generates random samples from a given target distribution. Second (From Above) In our SV model, we have to estimate the parameter set: Theta = (kappa,vega,alpha,m,beta,rho) and its time varying volatility: (nu) based on the observed temperature X, that is, a complete joint distribution: (joint distribution (theta, mu conditional on X)). By Bayes Theorem, we could possibly decompose the joint distribution:(joint distribution (theta, mu conditional on X)) to (joint distribution (theta, mu conditional on X)) proportional to (joint distribution (X conditional on theta, mu)) * (joint distribution (mu conditional on theta)) * (joint distribution (theta)). Joint distribution is decomposed into 3 marginal distributions. This theorem implies that knowing the marginal distributions of (joint distribution (X conditional on theta, mu)) and (joint distribution (mu conditional on theta)) would completely characterise the joint distribution:(joint distribution (theta, mu conditional on X)) Furthermore, the likelihood functions can be obtained as (joint distribution (X conditional on theta, mu)) = (joint distribution (X(o) to X(T) conditional on theta, mu)) =(t=0 Multiplication T)* (joint distribution (X(t+Change t) conditional on mu, theta)) and (joint distribution (mu conditional on theta)) = (t=0 Multiplication T)* (joint distribution mu(t+Change t) conditional on mu, theta)).
  • The iterative estimating procedure is defined by the following algorithm 1. Given initial values (theta (0), mu (0)) 2. Simulate (theta (1)) based on the given distribution (joint distribution (theta conditional on mu(0), X)) proportional to (joint distribution (X conditional on theta, mu(0))) *(joint distribution (theta)). where we can choose the prior distributions (joint distribution (theta)) for different parameters such as normal distribution or inverse gamma distribution. 3. Simulate (mu(1)) based on the given distribution (joint distribution (mu conditional on theta (0), X)) (S teps 2 and 3 will be repeated until it converges ) Bayes theorem => P(A/ B,C) = P (A/C)*P(B/A,C) P(B/C)
  • Used Euler Method within Monte Carlo. (In this model we only used the one year effect for simplicity.) phi (t, x) and u(t,x) Expectation of the payoff function. (where small x(o) initial value (constant - given condition). Double tilda = Approximately. Euler (Weak Convergence = 1, Strong Convergence = 1/2) Euler (Order 1) ((Used SPLUS 2000 (Professional) + MATLAB))
  • Baird, Allen J. (1993). Option Market Making should be the second book you read on options trading. Boyle, Phelim and Feidhlim Boyle (2001). Derivatives contains intriguing details about the historical origins of the Black-Scholes formula. Chriss, Neil A. (1997). Black-Scholes and Beyond is the definitive non-technical introduction to option pricing theory and financial engineering. Haug, Espen G. (1997). Option Pricing Formulas is an encyclopedia of published option pricing formulas. Hull, John C. (2002). Options, Futures and Other Derivatives is the standard introduction to financial engineering. Merton, Robert C. (1992). Continuous Time Finance is an edited collection of Merton's most important papers. It includes Merton (1973). Natenberg, Sheldon (1994). Option Volatility and Pricing . Most introductions to options trading are brief. This one isn't. I get my books free from all the publishers. So I am not biased with one textbook.
  • Pricing Foreign Exchange Risk

    1. 1. Currency Hedging in Turbulent times,Currency Hedging in Turbulent times, Executive Briefing Seminar, Grace Hotel, SydneyExecutive Briefing Seminar, Grace Hotel, Sydney 1010thth November 2003November 2003 ““Pricing Foreign Exchange Risk”Pricing Foreign Exchange Risk” By Glen DixonBy Glen Dixon Acting LecturerActing Lecturer School of Accounting,School of Accounting, Banking & Finance,Banking & Finance, Faculty of Commerce andFaculty of Commerce and ManagementManagement
    2. 2. OVERVIEW :OVERVIEW : An introduction to common derivative productsAn introduction to common derivative products Understanding the key components of the BlackUnderstanding the key components of the Black Scholes pricing methodologyScholes pricing methodology Constructing and using a forward price curveConstructing and using a forward price curve
    3. 3. An introduction to common derivative productsAn introduction to common derivative products Understanding the key components of the Black Scholes pricing methodologyUnderstanding the key components of the Black Scholes pricing methodology Constructing and using a forward price curveConstructing and using a forward price curve
    4. 4. ““Introduction plus History of FX”Introduction plus History of FX”
    5. 5. Introduction (Overview of FX) People have been borrowing, lending and exchanging money for centuries. The foreign exchange market exists to facilitate this conversion of one currency into another. As a result, the foreign exchange market today is the largest and most truly global financial market in the world.
    6. 6. Volatility in FX Markets Interest Rates Foreign Exchange Commodities Electricity
    7. 7. Shape of FX Forward Price CurveShape of FX Forward Price Curve Time (Minutes, Days, Weeks & Months) Price A$/US$ Time (Weeks) Time (Minutes) Time (Months) Time (Days)
    8. 8. • 1983 March1983 March - (5th , 8th ), OctoberOctober - (28th ), DecemberDecember - (8th , 9th ,12th and 13th ) • 1984 March1984 March - (5th ) February till early 1985February till early 1985 • 1985 February1985 February - (6th till 8th ) • 1986 May1986 May - (13th , 14th ), JulyJuly - (2nd , 4th, 25th and 28th ), AugustAugust - (19th ) • 1987 October1987 October - (20th ), November till DecemberNovember till December • 1988 April1988 April - till December 1989till December 1989 • 1989 February1989 February - Late February till AprilLate February till April - MayMay - AugustAugust • 1990 January1990 January – (23rd ), AugustAugust • 1991 June1991 June - (3rd ) – December (December (1919thth )) • 1992 January1992 January – February (26February (26thth ) – February till March, June, October,) – February till March, June, October, DecemberDecember History of FX in Australia Source: Securities Institute Education
    9. 9. • 1993 January1993 January –– April till June, August (17– April till June, August (17thth ), October, Late 1993 till), October, Late 1993 till early 1994early 1994 • 1994 February, April1994 February, April- (7th - 8th ), Early Mary, Late June, July tillEarly Mary, Late June, July till OctoberOctober • 1995 June till December1995 June till December • 1996 January till February1996 January till February, March, May, November till DecemberMarch, May, November till December • 1997 February1997 February • 2003 October2003 October History of FX in Australia (Cont.) Source: Securities Institute Education
    10. 10. ““Currency Futures and Options Market“Currency Futures and Options Market“
    11. 11. The Currency Futures and Options Markets • Foreign Currency Options – History and Size of Market – Options - General – Currency Options – Quotations • Foreign Currency Speculations
    12. 12. The Currency Futures and Options Markets (2) Foreign Exchange Contracts FX Portfolio FX Contracts AUS/US AUS/DEM AUS/SF FX Profiles
    13. 13. Foreign Currency Options History and Size of Market Attention will be focused on plain-vanilla European puts and calls on foreign exchange as well as on some of the more popular exotic varieties of currency options. The currency option market can rightfully claim to be the world’s only truly global, 24-hour option market. The underlying asset for currency options is foreign exchange.
    14. 14. Foreign Currency Options (2) Option: A contract that gives the option buyer (holder) the right (not obligation) to buy or sell a given amount of the underlying asset at a fixed price (exercise price) over a specified period of time (or at a specified date). • Underlying asset: e.g stock, commodities, stock indices, foreign currency etc. • Rule for exercise: – American - exercisable anytime until expiration – European - exercisable only at expiration • Types of option: – Call option: option to buy the underlying asset (e.g. foreign currency) – Put option: option to sell the underlying asset
    15. 15. Foreign Currency Options (3) Consider the following option on dollar/yen: USD call/JPY put Face amount in dollars $10,000,000 Option put/call Yen put Option expiry 90 days Strike 120.00 Exercise European
    16. 16. Foreign Currency Options (4a) An exotic currency option is an option that has some nonstandard feature that sets it apart from ordinary vanilla currency options. The most popular exotic currency options are the: 1) Barrier Option 2) Binary Option 3) Basket Option 4) Asian Option
    17. 17. Foreign Currency Options (4b) A Stock Simulation for the Barrier Option Source: Griffith University & Kerr 2000
    18. 18. Foreign Currency Options (5) Example: a $60 call (expiration in 3 months) on an ABC stock; option premium $1 Holder exercises if the spot price > $60 Payoff Profile S X=60 Premium Payoff $50 (60) (1) -1 out of the money 55 (60) (1) -1 60 (60) (1) -1 61 (60) (1) 0 at the money 62 (60) (1) 1 67 (60) (1) 6 in the money Payoff -1 Payoff -1 X=60 61 S
    19. 19. Foreign Currency Options (6) • Payoff Profile - Call option on DM – 1 option is for purchase of DM62,500 – exercise price $0.5850/DM – Option Premium $0.0050/DM or $312.50 • option in the money for spot > 0.5850 • option at the money for spot = 0.5850 • out of the money for spot < 0.5850 • Breakeven price = $0.5900/DM • Payoff Profile - Put Option on DM – exercise price $0.5850/DM – option premium $0.0050/DM • option in the money for spot < 0.5850 • at the money for spot = 0.5850 • out of the money for spot > 0.5850
    20. 20. Foreign Currency Options (7)
    21. 21. Foreign Currency Options (8)
    22. 22. Foreign Currency Options (9) An option hedge • A currency option is like one-half of a forward contract • An option to buy pound sterling at the current exchange rate – the option holder gains if pound sterling rises – the option holder does not lose if pound sterling falls
    23. 23. Foreign Currency Options (10) Currency option quotations British pound (CME) £62,500; cents per pound Strike Calls-Settle Puts-Settle Price Oct Nov Dec Oct Nov Dec 1430 2.38 . . . . 2.78 0.39 0.61 0.80 1440 1.68 1.94 2.15 0.68 0.94 1.16 1450 1.12 1.39 1.61 1.12 1.39 1.61 1460 0.69 0.95 1.17 1.69 1.94 2.16 1470 0.40 0.62 0.82 2.39 . . . . 2.80
    24. 24. Foreign Currency Options (11) • The time value of an option is the difference between the option’s market value and its intrinsic value if exercised immediately. • The time value of a currency option is a function of the following six determinants: – Underlying exchange rate – Exercise price – Riskless rate of interest in currency d – Riskless rate of interest in currency f – Time to expiration – Volatility in the underlying exchange rate
    25. 25. Foreign Currency Options (12) • Foreign Currency Speculation - Trading on the basis of expectations about future prices • Speculation in Spot Markets • Speculation in Forward Markets – occurs if one believes that the forward rate differs from the future spot rate – if expect Forward < future spot, buy currency forward – if expect Forward > future spot, sell currency forward • Speculation using options – call options – put options • Speculation via Borrowing and Lending: Swaps • Speculation via Not Hedging Trade • Speculation on Exchange-Rate Volatility
    26. 26. An introduction to common derivative productsAn introduction to common derivative products Understanding the key components of the BlackUnderstanding the key components of the Black Scholes pricing methodologyScholes pricing methodology Constructing and using a forward price curveConstructing and using a forward price curve
    27. 27. ““Overview of Black Scholes (1973) , MertonOverview of Black Scholes (1973) , Merton ((1973) and Garman Kohlhagen (1983)”((1973) and Garman Kohlhagen (1983)”
    28. 28. Black, Fischer and Myron S. Scholes (1973).Black, Fischer and Myron S. Scholes (1973). The pricing of options and corporate liabilities,The pricing of options and corporate liabilities, Journal of Political EconomyJournal of Political Economy, 81, 637-654., 81, 637-654. Good Journals
    29. 29. Black Scholes (1973) Options Pricing Formula Values for a call price c or put price p are: where:
    30. 30. The Five Greeks • DELTA measures first order (linear) sensitivity to an underlier; • GAMMA measures second order (quadratic) sensitivity to an underlier; • VEGA measures first order (linear) sensitivity to the implied Volatility of an underlier; • THETA measures first order (linear) sensitivity to the passage of time; RHO measures first order (linear) sensitivity to an applicable interest rate.
    31. 31. The Five Greeks for Black Scholes (1973) Options Pricing Formula for a Call The Greeks—delta, gamma, vega, theta and rho—for a call are: delta = Φ(d1 ) gamma = vega = theta =
    32. 32. The Five Greeks for Black Scholes (1973) Options Pricing Formula for a Put where denotes the standard normal probability density function. For a put, the Greeks are: delta = Φ(d1 ) – 1 gamma = vega = theta =
    33. 33. Good Journals Merton, Robert C. (1973).Merton, Robert C. (1973). Theory of rational option pricing,Theory of rational option pricing, Bell Journal of Economics and Management ScienceBell Journal of Economics and Management Science, 4 (1), 141-183., 4 (1), 141-183.
    34. 34. Merton (1973) Options Pricing Formula Values for a call price c or put price p are: where:
    35. 35. The Five Greeks for Merton (1973) Options Pricing Formula for a Call The Greeks—delta, gamma, vega, theta and rho—for a call are:
    36. 36. The Five Greeks for Merton (1973) Options Pricing Formula for a Put where denotes the standard normal probability density function. For a put, the Greeks are:
    37. 37. Good Journals Garman, Mark B. and Steven W. Kohlhagen (1983).Garman, Mark B. and Steven W. Kohlhagen (1983). Foreign currency option values,Foreign currency option values, Journal of International Money and FinanceJournal of International Money and Finance, 2, 231-237., 2, 231-237.
    38. 38. Garman and Kohlhagen (1983) FX Options Pricing Formula Values for a call price c or put price p are: where:
    39. 39. The Five Greeks for Garman and Kohlhagen (1983) FX Options Pricing Formula for a Call The Greeks—delta, gamma, vega, theta and rho—for a call are:
    40. 40. The Five Greeks for Garman and Kohlhagen (1983) FX Options Pricing Formula for a Put where denotes the standard normal probability density function. For a put, the Greeks are:
    41. 41. An introduction to common derivative productsAn introduction to common derivative products Understanding the key components of the Black Scholes pricing methodologyUnderstanding the key components of the Black Scholes pricing methodology Constructing and using a forward price curveConstructing and using a forward price curve
    42. 42. ““Overview of Interest Rate Markets: Bachelior (1901),Overview of Interest Rate Markets: Bachelior (1901), including Single Factor Models like Vasicek (1977) ,including Single Factor Models like Vasicek (1977) , Cox Ingersoll and Ross (1985)”Cox Ingersoll and Ross (1985)”
    43. 43. Stochastic Differential Equation (Wiener processes or Random Walk) Geometric Brownian Motion (Stock Markets) –Geometric Brownian Motion (Stock Markets) – BacheliorBachelior ddistributenormally,incrementstindependen Noise),(WhitemotionBrownian:)( yvolatilitthe: ratelincrementathe: where )( )( )( followsSDEgeometricThe price.assetunderlyingtheas)(Define tW tdWdt tS tdS tS σ µ σµ +=
    44. 44. Models: (Foreign Exchange, Interest Rate and Energy Markets) • Single factor models Vasicek (1977), Cox Ingersoll and Ross (1985), Clewlow and Strickland (2000) • Two factor models Brennan and Schwartz (1982), Kennedy (1997), Pilipovic (1997) and Kerr and Dixon (2002) Price Spikes Long Term Mean Mean Reversion
    45. 45. Interest Rate Markets
    46. 46. Interest Rate Markets (1)
    47. 47. Interest Rate Markets (2)
    48. 48. Interest Rate Markets (3)
    49. 49. Stochastic Differential Equation Single Factor Models (Interest Rate Markets) The Vasicek (1977) ModelThe Vasicek (1977) Model motionBrownian: yvolatilitthe:σ alueinterest vmeanthe:μ ratereverting-meanthe:κ where followsSDEVasicekThe rate.shorttheasDefine W(t) σdW(t)r(t))dtκ(μdr(t) r(t) +−=
    50. 50. Stochastic Differential Equation Single Factor Models (Interest Rate Markets) The Cox, Ingersoll and RossThe Cox, Ingersoll and Ross (1985) Model (CIR)(1985) Model (CIR) motionBrownian:)( yvolatilitthe: alueinterest vmeanthe: ratereverting-meanthe: where )()())(()( followsSDECIRThe rate.shorttheas)(Define tW tdWtrdttrtdr tr σ µ κ σµκ +−=
    51. 51. ““Comparison of Currencies: Australian, US,Comparison of Currencies: Australian, US, Asian , Latin American”Asian , Latin American”
    52. 52. Comparisons of Currencies: Australian Dollar (Daily)
    53. 53. Comparisons of Currencies: Australian Dollar (Monthly)
    54. 54. Comparisons of Currencies: US Dollar (Daily)
    55. 55. Comparisons of Currencies: US Dollar (Monthly)
    56. 56. Comparisons of Asian Currencies (Daily)
    57. 57. Comparisons of Asian Currencies (Monthly)
    58. 58. Comparisons of Latin American Currencies (Daily)
    59. 59. Comparisons of Latin American Currencies (Monthly)
    60. 60. ““Overview of Monte Carlo, ScenarioOverview of Monte Carlo, Scenario Development and Stress Testing”Development and Stress Testing”
    61. 61. Iterative Procedure (Euler Method) (CIR).1/2or(Vasicek)0where 0atrateinterestinitialwith the )()())(()()( 0 = = ∆+∆−+=∆+ τ σµκ τ tr tWtrttrtrttr 1.0.ofdeviationstandard andzeroofmeanon withdistributinormala fromsamplerandomaiswhere)( εε ttW ∆=∆
    62. 62. Monte Carlo Simulation (Generate Random Numbers) Box Muller: Marsaglia: Note: Box Muller and Marsaglia will generateNote: Box Muller and Marsaglia will generate standard Gaussian random variables basedstandard Gaussian random variables based on two independent uniformly distributedon two independent uniformly distributed random variables from [0, 1].random variables from [0, 1].     = = )2sin()log(2 )2cos()log(2 212 211 XXY XXY π π     =<+= −=−= − V V UXXV UXYXY )log(22 2 2 1 2211 ;1 ;*)12(U;*)12(
    63. 63. Monte Carlo Simulation (Generate Random Numbers from [0, 1]) Pseudo-Random use seed, convergence rate (M is the number of iterations). E.g. Pseudo-Random (400) M 1 Quasi-Random (low discrepancy): use a uniformed sequence, e.g., Van der Corput sequence at every points (k=1,2,…). E.g. Quasi-Random (400) k 2
    64. 64. Using Monte Carlo for FX Market GENERATE 1 RANDOM SAMPLE for FX GENERATE 1 RANDOM SAMPLE for FX FX(5): $A/$US 4:30 pm FX(4): $A/$US 12:30 pm FX(3): $A/$US 8:30 am FX(2): $A/$US 4:30 am FX(1): $A/$US 0:30 am Time $A/ $US
    65. 65. Monte Carlo for FX Market (cont.) GENERATE MULTIPLE RANDOM SAMPLES for FX GENERATE MULTIPLE RANDOM SAMPLES for FX Time $A/ $US
    66. 66. Number of Samples0 $0 1 A$/$US 1A/$ 0.5US STABILISE? -USE STOPPING RULES, I.E. Tolerance- STABILISE? -USE STOPPING RULES, I.E. Tolerance- when the change between two consecutive average monthly fx prices becomes insignificant then the process is said to have stabilised. Estimated Average Monthly Prices In the FX Market The Accuracy of Estimates is related to the number of Simulations
    67. 67. Using Monte Carlo for Sensitivity Analysis on the FX Forward Curve • Construct scenarios – High, medium and low, forecast FX levels • Perform Monte Carlo Simulation – generate fx price paths for each scenario using different sets of sensitivity analysis
    68. 68. Sensitivity Analysis for FX
    69. 69. Scenario Development for FX • Scenario analysis – Is a strategic technique which enables a firm to evaluate the potential impact on its earnings stream of various different eventualities. – It uses multidimensional projections, and helps the firm to assess its longer term strategic vulnerabilities.
    70. 70. Scenario Development for FX (2) • Scenario analysis – Distinguish between scenario analysis and stress testing. – Both are forward looking techniques which seek to quantify the potential loss which might arise as a consequence of unlikely events. – Stress testing is designed to evaluate the short-term impact on a given portfolio of a series of predefined moves, in particular market variables. – Scenario analysis on the other hand seeks to assess the broader impact on the firm of more complex and inter- related developments. Huge losses often occur due to a sequence of several adverse events. Scenario analysis can help to identify such potential problems in advance.
    71. 71. Scenario Development for FX (3) • Scenario analysis – The purpose of scenario analysis is to help the firm’s decision makers think about and understand the impact of unlikely, but catastrophic, events before they happen. A management team that learns its lessons from previous catastrophic situations is more likely to avoid losses in the future. Scenario analysis is an effective tool to assist management in that process.
    72. 72. Scenario Development for FX (4) Risk Political Risk Operational Risk Legal Risk Credit Risk Reputational Risk
    73. 73. Scenario Development for FX (5) • The Scenario analysis process: Step 1: Scenario definition  Description of the starting scenario  Basic assumptions  Definition of the time horizon
    74. 74. Scenario Development for FX (6) Step 2: Scenario-field analysis  Identification of the scenario fields, the risk dimensions and risk factors which are affected and relevant for this scenario analysis
    75. 75. Scenario Development for FX (7) Step 3: Scenario projections  Estimate the likely movements of the identified scenario factors and determine the potential loss in that case
    76. 76. Scenario Development for FX (8) Step 4: Scenario consolidation Consolidate the results Check for consistency errors, doubling counting Independent validation checks
    77. 77. Scenario Development for FX (9) Step 5: Scenario presentation and follow-up  Summarise results  Analyse and evaluate  next steps: eg, put on a hedge
    78. 78. Stress Testing for FX  In financial markets where 4-standard-deviation events happen approximately once per year, the October 1987 crash was a 25-standard deviation event.  Stress testing deals with these “outlier” events. It addresses the large moves in key market variables that lie beyond day-to-day risk monitoring but that could potentially occur.
    79. 79. Stress Testing for FX (2)  Low probability extreme market events;  Hidden assumption in models;  Structural breakdowns in the market environment;  Robustness of risk management systems. Stress Testing is another form of risk management which tests exposure to:
    80. 80. Stress Testing for FX (3) Steps in Stress Testing • Step 1: Picking what to stress  Choice of market variables  Range of stress  Usefulness of stress information vs data overload
    81. 81. Stress Testing for FX (4) Step 2: Identifying assumptions  Will correlations hold or break?  For correlations that break, what are the new assumptions?  Does the underlying financial model still hold?
    82. 82. Stress Testing for FX (5) Step 3: Revaluing the portfolio  Back of the envelope vs  sophisticated modeling  Adjusting for market liquidity Trading Settlements Portfolio Management Contract Management
    83. 83. Stress Testing for FX (6) Step 4: Deciding on action steps  Reporting  Cross-checks on model and pricing validity  Action plan for dealing with actual catastrophe situation
    84. 84. ““Overview of Interest Rate Markets including TwoOverview of Interest Rate Markets including Two Factor Models like Brennan and Schwartz (1982), KerrFactor Models like Brennan and Schwartz (1982), Kerr and Dixon (2003)”and Dixon (2003)”
    85. 85. Stochastic Differential Equations Two Factor Models (Interest Rate Markets) + Monte Carlo Simulation The Brennan and Schwartz (1982) Stochastic VolatilityThe Brennan and Schwartz (1982) Stochastic Volatility ModelModel ( ) ( ) 1/2)or0(motions.Brownian tindependenare)(and)(variance,itsand rateshortebetween thncorrelatiotheiswhere ))(1)(( )()()( )()()()()( variancetheas)(andrateshorttheas)(Define 21 2 2 1 1 =       −+ +−= +−= τ ρ ρρ βνναν νµκ ν τ tWtW tdWtdW tdttmtd tdWtrtdttrtdr ttr Iterative ProcedureIterative Procedure (Euler Method) with(Euler Method) with 1/2)( =τ ( ) ( ) variance.initialtheandrateinitialwith the ))(1)(( )()()()( )()()()()()( 2 2 1 1       ∆−+∆ +∆−+=∆+ ∆+∆−+=∆+ tWtW tttmttt tWtrtttrtrttr ρρ βννανν νµκ τ
    86. 86. Stochastic Differential Equations Monte Carlo Simulation (Two Factor Models-CIR) The Brennan and Schwartz (1982) Stochastic VolatilityThe Brennan and Schwartz (1982) Stochastic Volatility ModelModel with Iterative Procedure (Euler Methowith Iterative Procedure (Euler Metho withwith 1/2)( =τ
    87. 87. Foreign Currency  Stochastic Modeling rateinterestforeignthe: rateinterestdomesticthe:where )()())(()( asmodelmotionBrownian geometricafollowsrateexchangespotthat the assumeWerate.exchangespottheas)(Define f f r r tdWtSdtrrtStdS tS σ+−= Stochastic Differential Equations
    88. 88.  Stochastic Interest Rates ( ) rateinterestforeigntheandrate interestdomesticebetween thncorrelatiotheiswhere ))(1)(( )())(()( )()())(()( )()()()()( rateforeigntheas)(andratedomestictheas)(Define 3 2 2 2 1 ρ ρρ βα βα σ         −+ +−= +−= +−= tdWtdW trdttrmtdr tdWtrdttrmtdr tdWtSdttrtrStdS trtr ffffff ft f Stochastic Differential Equations
    89. 89. Monte Carlo Simulation  Iterative Procedure (Euler Method) ( ) ( ) ( )         ∆−+∆ +∆−+=∆+ ∆+∆−+=∆+ ∆+∆−+=∆+ ))(1)(( )()()()( )()()()()( )()()()()()( 3 2 2 2 1 tWtW trttrmtrttr tWtrttrmtrttr tWtSttrtrStSttS fffffff ft ρρ βα βα σ
    90. 90. Do we need a Crystal Ball inDo we need a Crystal Ball in Weather Modelling to see the application forWeather Modelling to see the application for Foreign Exchange Forward CurvesForeign Exchange Forward Curves
    91. 91. Pricing MethodologiesPricing Methodologies • Historical simulation by Hunter (1999), Garman, Blanco and Erickson (2000), Zeng (2000a) • Indirect modeling of the underlying variable’s distribution (via a Monte Carlo technique as this involves simulating a sequence of data), by Pilipovic (1997), Rookley (2000), Garman, Blanco and Erickson (2000), Zeng (2000b) and Dornier and Queruel (2000). • Direct modeling of the underlying variable’s distribution (short and long term forecasting) by Dischel (1999), Torro, Meneu and Valor (2000), Davis (2001), Alaton, Djehiche and Stillberger (2001), Diebold and Campbell (2002), Cao and Wei (2002) and Brody, Syroka and Zervos (2002).
    92. 92. Figure 5.7 Histogram of Sydney Temperature in °C for Whole Season 9.00 11.25 13.50 15.75 18.00 20.25 22.50 24.75 27.00 29.25 31.50 AvgT 0.00 0.02 0.04 0.06 0.08 Figure 5.8 Histogram of Sydney Temperature in °C for Winter Season 9.00 10.64 12.28 13.92 15.56 17.20 18.84 20.48 22.12 23.76 25.40 AvgT 0.00 0.05 0.10 0.15 Figure 5.9 Histogram of Sydney Temperature in °C for Summer Season 13.450 15.255 17.060 18.865 20.670 22.475 24.280 26.085 27.890 29.695 31.500 AvgT 0.00 0.04 0.08 0.12
    93. 93. MMathematicalathematical FFormulation oformulation of MMeanean FFunctionunction ...) 365 6 sin() 365 4 sin() 365 2 sin( 321 2 +++++++++= ϕ π ϕ π ϕ π θ t f t e t dctbtat ...effectyearthirdoneeffectyearhalfeffectyearonetrendmean ++++= Table 5.1 Frequency for Summer Season in Sydney One Year (153 days) Half Year (76.5 days) One-3rd Year (51 days) 0.006535948 0.0130719 0.01960784 Figure 5.10 Periodogram for Summer Spectrum in Sydney Freq Spectrum 0.0 0.1 0.2 0.3 0.4 0.5 -20-100102030 Summer Spectrum Freq Spectrum 0.0 0.005 0.010 0.015 0.020 0.025 0.030 0102030
    94. 94. Further AnalysisFurther Analysis ...) 365 6 sin() 365 4 sin() 365 2 sin( 321 2 +++++++++= ϕ π ϕ π ϕ π θ t f t e t dctbtat ...effectyearthirdoneeffectyearhalfeffectyearonetrendmean ++++= Table 5.2 Frequency for Winter Season in Sydney One Year (212 days) Half Year (106 days) One-3rd Year (70.67 days) 0.004716981 0.009433962 0.01415094 Figure 5.11 Periodogram for Winter Spectrum in Sydney Freq Spectrum 0.0 0.1 0.2 0.3 0.4 0.5 -20-1001020 Winter Spectrum Freq Spectrum 0.0 0.01 0.02 0.03 0.04 510152025
    95. 95. Figure 5.12 Mean and Variance over Time for Sydney Year Temperature 0.0 0.5 1.0 1.5 2.0 2.5 3.0 10152025 Year Varaince 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0204060
    96. 96. ModelModel: 2 Factor Mean-Reverting Diffusion Process with Stochastic Volatility: 2 Factor Mean-Reverting Diffusion Process with Stochastic Volatility Kerr Q. and G. Dixon (2002) ~ 2FMRDwithSVKerr Q. and G. Dixon (2002) ~ 2FMRDwithSV where (kappa) and (alpha) are two constant mean-reverting rates and (beta) is a constant volatility of the stochastic volatility process for simplicity. Time varying volatility (nu) based on the observed temperature . (e.g. high temperature then high volatility) The mean (mean temperature – theta) and (mean of volatility) are periodical functions which contain sine and cosine functions. and are two correlated Wiener processes, i.e., . So is the temperature model and is the volatility model for temperature. tX t κ α β tθ tm 1 tW 2 tW dtdWdWcor tt ρ=),( 21 Let denote the daily average temperature at time . The daily average temperature is the arithmetic average of the maximum and minimum temperature recorded on a day from mid-night to mid-night basis. Taking into account the seasonality and stochastic volatility, a temperature model can be given as     +−= +−= 2 1 )( )( ttttt tttttt dWdtmd dWXdtXdX νβναν νθκ γ tν X tdX tdν
    97. 97. Markov Chain Monte Carlo MethodMarkov Chain Monte Carlo Method In our SV model, we have to estimate the parameter set and its time varying volatility based on the observed temperature ,that is a complete joint distribution . By Bayes Rule, we could possibly decompose the joint distribution to . This theorem implies that knowing the marginal distributions of and would completely characterise the joint distribution . Furthermore, the likelihood functions can be obtained as and ),,,,,( ρβαθκ tttttt m=Θ tν X )|,( Xp νΘ )|,( Xp νΘ )()|(),|()|,( ΘΘΘ∝Θ ppXpXp ννν ),|( νΘXp )|( Θνp )|,( Xp νΘ ∏ − = ∆+ Θ=Θ=Θ 1 0 0 ),,|(),|,...,(),|( T t tttttT XXpXXpXp ννν ∏ − = ∆+ Θ=Θ 1 0 ),|()|( T t tttpp ννν
    98. 98. Gibbs Sampling AlgorithmGibbs Sampling Algorithm The iterative estimating procedure is defined by the following algorithm 1. Given initial values 2. Simulate based on the given distribution where we can choose the prior distributions for different parameters such as normal distribution or inverse gamma distribution. 3. Simulate based on the given distribution . (Steps 2 and 3 will be repeated until it converges) )(),|(),|( )0()0( ΘΘ∝Θ pXpXp νν )(Θp )1( ν ),|( )0( Xp Θν ),( )0()0( νΘ )1( Θ
    99. 99. Monte Carlo Simulations (Euler Method)Monte Carlo Simulations (Euler Method) Given a real pay off function , we can define the derivative price as A Monte Carlo approximation of can be expressed as where is the number of simulations. The discrete version of the dynamic process can be written as     ∆+∆−+= ∆+∆−+= ∆+ ∆+ 2 1 )( )( ttttttt tttttttt Wtm WXtXXX νβνανν νθκ γ N ),( xtu ∑= ≈ N i i TX N xtu 1 )( )( 1 ),( φ ),( xtφ )|)((),( 00 xXXExtu T == φ
    100. 100. UsingUsing ) 365 2 sin(2 ϕ π θ ++++= t dctbtat ) 365 2 sin(and φ π ++= t vumt Table 5.3 Parameter Estimation for Winter in Sydney: the Mean Functions for Temperature and Volatility processes in our fitting Prior PosteriorParameter List Mean Std Mean Std a 10 20 13.93033 0.02291 b 1 1 0.6902098 0.3422 c 1 1 -0.1012509 0.1883 d 1 2 2.259497 0.2131 t θ ϕ 1 2 -1.570796 0.2360 κ 100 25 110.0321 0.1532 α 80 25 68.0563 0.1001 β 0.5 1 0.3243 0.0490 u 5 10 11.99307 0.0232 v 1 2 2.105039 0.1673 tm φ 1 2 1.560437 0.0669 ρ 0.5 1 0.09213 0.0115
    101. 101. Average Temperature in New York and Philadelphia for 22 years Figure 5.13 Average Daily Temperature in New York (LGA) from 1980-2002 (22 years) in °F Figure 5.14 Average Daily Temperature in Philadelphia (PHI) from 1980-2002 (22 years) in °F
    102. 102. Mean Fitting Curves in New York and Philadelphia for 3 years Figure 5.15 Mean Fitting Curve in New York (LGA) from 1998-2000 (3 years) in °F Figure 5.16 Mean Fitting Curve in Philadelphia (PHI) from 1998-2000 (3 years) in °F
    103. 103. Figure 5.17 Standard Deviation Fitting of Temperature in New York (LGA) for 2000 (1 year) in °F
    104. 104. Figure 5.18 Standard Deviation Fitting of Temperature in Philadelphia (PHI) for 2000 (1 year) in °F
    105. 105. Figure 5.19 Average Temperature Simulation vs Average Observed Temperature in New York (LGA) from 1998-2000 (3 years) in °F
    106. 106. Figure 5.20 Average Temperature Simulation vs Average Observed Temperature in Philadelphia (PHI) from 1998-2000 (3 years) in °F
    107. 107. Energy Derivative Price Comparison (Using 2002 Calender Year – Weekly) 2002 Weeklyswaps for QLD 0 50 100 150 200 250 300 350 1 5 9 13 17 21 25 29 33 37 41 45 49 MRJD BS MCLP Actual Price 2002 Cap with the strike price $50 for QLD 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 1 5 9 13 17 21 25 29 33 37 41 45 49 BS MRJD MCLP Actual Payoff Average Price - BS (47.91) MRJD ($46.08) MCLP ($38.58) Actual ($40.78) Average Price - BS ($9840.1) MRJD ($7804.5) MCLP ($6419.78.58) Actual ($6767.63) Swap Price ComparisonSwap Price Comparison Cap Price ComparisonCap Price Comparison
    108. 108. Good Industry Books Baird, Allen J. (1993).Baird, Allen J. (1993). Option Market MakingOption Market Making should be theshould be the secondsecond book you read on options trading.book you read on options trading. Boyle, Phelim and Feidhlim Boyle (2001).Boyle, Phelim and Feidhlim Boyle (2001). DerivativesDerivatives containscontains intriguing details about the historical origins of the Black-Scholesintriguing details about the historical origins of the Black-Scholes formula.formula. Chriss, Neil A. (1997).Chriss, Neil A. (1997). Black-Scholes and BeyondBlack-Scholes and Beyond is the definitiveis the definitive non-technical introduction to option pricing theory and financialnon-technical introduction to option pricing theory and financial engineering.engineering. Haug, Espen G. (1997).Haug, Espen G. (1997). Option Pricing FormulasOption Pricing Formulas is an encyclopediais an encyclopedia of published option pricing formulas.of published option pricing formulas. Hull, John C. (2002).Hull, John C. (2002). Options, Futures and Other DerivativesOptions, Futures and Other Derivatives is theis the standard introduction to financial engineering.standard introduction to financial engineering. Merton, Robert C. (1992).Merton, Robert C. (1992). Continuous Time FinanceContinuous Time Finance is an editedis an edited collection of Merton's most important papers. It includes Mertoncollection of Merton's most important papers. It includes Merton (1973).(1973). Natenberg, Sheldon (1994).Natenberg, Sheldon (1994). Option Volatility and PricingOption Volatility and Pricing. Most. Most introductions to options trading are brief.introductions to options trading are brief. This one isn't.
    109. 109. Thank You - Mr Glen DixonThank You - Mr Glen Dixon Email: g.dixon@griffith.edu.auEmail: g.dixon@griffith.edu.au “Foreign ExchangeForeign Exchange Markets are Key Research Areas for Griffith University”Markets are Key Research Areas for Griffith University”

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