Stochastic Processes Inc.

Forward Foreign Exchange Contracts

From the perspective of the Canadian investment banking industry, the derivatives industry started with forward foreign exchange contracts, whereby clients of the banks/dealers could meet future commitments for foreign exchange, namely the Canadian Dollar, by entering into a contract to sell their USD dollar forward. Historically, this was advantageous to Canadian exporters, since the Canadian dollar forward market was at a premium to spot.

Although a futures market existed where currencies could be sold forward, there were several advantages in dealing over-the-counter. In contrast with the futures exchange, there were no variation margin requirements. The contract agreement was based on the corporation's line of credit with the dealer. In addition, the amounts traded were large, and the dealers were able to provide the necessary liquidity drawing on their large corporate client base. In addition, due to the size of the contracts, dealers were willing to provide customized contract sizes and delivery dates.

The market in short term forwards (2 years and under) was also supported by money market arbitrage or swaps. If the interest rates were higher in Canada than in the United States, then a U.S. depositor could borrow U.S. dollars, sell them spot for Canadian dollars, and then invest in a Canadian Dollar instrument. The Canadian dollars would then be sold in the Foreign Exchange Forward market to avoid exchange rate risk, and the proceeds used to pay the U.S. borrowing. This arbitrage would be carried out until there was no gain to be made between the proceeds of the Foreign Exchange Forward and principal interest required on the U.S. borrowing.

Longer term forward foreign exchange contracts were approached more cautiously due to credit risk. If a party to a forward contract defaulted when the other party had a market value gain on the contract, then the non-defaulting party, assumed to be in a hedged position with another counter-party, would have the obligation to pay the market value gain to the hedge counter-party, but would have a low likelihood of receiving the market value from the defaulting party. The further out the term of the contract, the more time there would be for the contract rate to diverge from the market rate and thus create the possibility for a higher loss.

Another factor that resulted in caution when entering long-dated markets at that time, was the lack of liquidity for foreign exchange forward rates beyond 2 years. An arbitrage market also developed for long dated forward rates using Canadian and U.S. Treasury bonds.

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Short-term contracts were traded then, and still now, on the basis of forward points. Forward points are the difference between the Spot rate for a currency, and the Forward rate as determined through interest rate arbitrage. If the Canadian Dollar , for example, was trading at 1.60 for delivery 'spot' and one year forward at 1.5960 then the forward points are a discount of -40. If the trader expected the points to narrow to -20, then the trader would enter into a swap, selling US dollars spot and buying them forward. If spot moved to 1.59 and the forward to 1.5880, the trader would make 100 points on the spot trade and lose 80 points on the forward for a net gain of 20 points or US$ 10,000,000 x .0020 = C$20,000 on a $10 Million dollar contract.

From a valuation perspective, the quotation methods were simplified, mainly for ease of quoting. The above valuations did not take into account the time value of money, ie. the present value effect. The pricing components of a currency swap, are the spot rate and the forward points which provide the forward outright rate. The total value of the swap is based on the movement of the spot rate and the outright forward rate. When a portfolio of such swaps resulted in unrealized gains in the future offset by realized losses at present, without close attention to the funding of the realized losses serious leakages could occur. Use of time value of money became more structured with the mathematical formulae developed for Interest Rate Swaps.

Interest Rate Swaps

The Interest Rate Swaps market began in the early eighties. The market started with a source of low interest rate funding in Japan. The low fixed rate was swapped by a dealer with another dealer that had issued a fixed rate loan to a U.S. counter-party. The U.S. borrower achieved a lower all-in-rate and the dealer counter-parties to the swaps collected fee income. The attractiveness of interest rate swaps was related to the low credit risk. The credit risk in a loan is the entire principal amount plus future interest rate payments whereas the credit risk on an interest rate swap is the market value of the fixed vs floating interest rate payments.

Initially, the dealer played an intermediary role and collected fee income from bringing the counter parties together for the structure. Valuation for accounting purposes was relatively simple, particularly if the total fee was received up-front. Fee income at inception of this business was lucrative and brought in competitors coincidental with the creation of complex structures. Dealers became more willing to take on market and credit risk, and replaced up-front fee income with spread income. The original valuation method was based on the underlying loan, with an even accrual of interest rate spreads over the life of the swap. The difficulty with this method was the requirement to match income on unusual cashflows. In a large portfolio, this requirement was virtually impossible to achieve.

Valuation for pricing purposes, however, had evolved to an interesting discounting method using the zero coupon curve. This can most likely be attributed to Lee Wakeman, an early pioneer in the interest rate swaps market. A precedent to the zero coupon curve were earlier studies on the term structure of interest rates and forward interest rates. Using the results of those studies it was possible to forecast future cashflows and to value them in present terms. This feature, perhaps, singularly defines valuations of derivative instruments including forward contracts, and options.

Interest Rate Options

Forward Contracts and Interest Rate Swaps imply a one way view on the market. The purpose of these instruments for the corporate hedger, is to obtain a locked-in cost for an underlying exposure. Thus, if a U.S. corporate has European exposures, a locked-in hedge is to sell Euro's forward. This type of hedging is most publicly evident with gold producers who sell their production several years ahead. Options, on the other hand, provide the buyer of the option a means to participate in some of the gain, if the underlying exposure in fact turns out to be a windfall.

Interest Rate Caps (protection against interest rates going over a certain rate) and Floors (the opposite) provided the client with protection, although at a cost, and were a natural match to the interest rate swap market. In terms of market liquidity however, interest rate options were a small component of the major instrument propelling the swap market, that is, the U.S. Long Bond. The lack of liquidity, coupled with the volatility of the underlying instrument, created difficulty for the dealer in hedging client structures particularly when the volatility underwent periods of entrenchment or escalation. In Canada, this was more pronounced since the market is correspondingly smaller than in the U.S.

Currency Options

Currency options first appeared in the London exchanges around 1974. Initially, it is believed, these options were traded on market price, similar to futures. The Black-Scholes formula, published by Fischer Black and Myron Scholes in 1973 was not widely known to futures speculators. In addition, there were few methods available to use the pricing formula, with the exception perhaps of a novel program derived by Mark Rubenstein to be run on a handheld calculator, where the program was stored on a narrow strips.

Interbank currency options suffered some set backs in the early stages of the market, and still continue to produce problems from time to time as market players find themselves on the wrong side of volatility. However, currency options have some favourable features compared to interest rate options. The most evident is the average turnover in foreign exchange contracts, which in April 1998 was estimated at $1.5 trillion per day providing an enormous amount of liquidity in the underlying contracts. Second, much of the hedging that is required by corporate clients is for short term receivables/ payables to cover immediate cashflow requirements. Therefore, the expiry dates of currency options is generally only several months ahead, reducing valuation/operational/credit risk.

Exotic Options

"Options, Futures and Other Derivatives" was written by John Hull in 1989. This book provided a basis for valuations of exotic options, along with articles published in trade journals, notably Risk Magazine. A particular article by Mark Rubenstein on barrier options, ca 1993, became a benchmark for knock-out options. Whereas a non-Exotic Option, or 'vanilla' option, had a fixed pay-out structure, the shape of which is referred to as a 'hockey-stick', the Exotic Option tended to reduce the pay-out based on the probability of certain events occurring, thereby reducing the premium that a purchase had to pay to obtain the option. For these reasons, a purchaser with a certain view of the direction of a currency, could obtain protection and pay less premium, although subject to increased risk. Combinations of event driven options created a number of descriptive names for various strategies.

Valuations were for the most part based on analytical formulae and their variations , such as those that exist for Barrier Options and Average Rate Options. An analytical approach made for faster processing time in the risk management processes. Another method was the use of the Cox, Ross and Rubenstein binomial tree formula. The Monte Carlo approach was flexible and could accommodate most models, however, the number of iterations required to solve for the price and the various risk parameters, was a disadvantage.

Again, liquidity limited the usage of exotic options in most markets. In order to have a reasonably secure market to trade exotic options, it was necessary to have a reasonably liquid market in the vanilla options that underly the market. Amongst the foreign exchange, interest rate and equity markets, vanilla options are again, most liquid in the foreign exchange market. As a result, the market for exotic options, tended to be in the currency markets. Out of this market, came a wide variety of products that were based on barrier options, average rate options and digital options, with varying underlying features, such as timing, frequency of observation, method of observation, ranges, pay-out ratios etc. For the most part, these products were tailored for the corporate hedger, in order to help firm the forecasting process in the corporate user's reporting currency.

Exotic Options - non-vanilla

Where there is very little or no market liquidity, the exotic option can be considered a 'non-vanilla' exotic option. An easy example, is an option that uses two currencies that are not traded much in the vanilla market, e.g. the Canadian dollar against the British Pound or against the Japanese Yen. Although each of these currencies are liquid against the U.S. dollar, there is very little volume traded in the Canadian dollar against the Yen or British Pound. More difficult examples are 2 asset options, such as a Put on Euro that knocks-out when Oil is at a certain level. Both of the examples could be priced better if a market existed for correlation between the 2 underlying assets. Such problems are amplified in a 'basket' option where currencies in a 'basket' have no underlying correlation market. Having said that however, in absence of market correlations, if the number of transactions carried out were such that the 'stochastic processes' would find a way to prove the price, e.g. as in the actuary business, then historical correlations would be used.

		Exotic Options - Historical Perspective Covering Valuation Issues






	Forward Foreign Exchange Contracts From the perspective of the Canadian investment banking industry, the derivatives industry started with forward foreign exchange contracts, whereby clients of the banks/dealers could meet future commitments for foreign exchange, namely the Canadian Dollar, by entering into a contract to sell their USD dollar forward. Historically, this was advantageous to Canadian exporters, since the Canadian dollar forward market was at a premium to spot. Although a futures market existed where currencies could be sold forward, there were several advantages in dealing over-the-counter. In contrast with the futures exchange, there were no variation margin requirements. The contract agreement was based on the corporation's line of credit with the dealer. In addition, the amounts traded were large, and the dealers were able to provide the necessary liquidity drawing on their large corporate client base. In addition, due to the size of the contracts, dealers were willing to provide customized contract sizes and delivery dates. The market in short term forwards (2 years and under) was also supported by money market arbitrage or swaps. If the interest rates were higher in Canada than in the United States, then a U.S. depositor could borrow U.S. dollars, sell them spot for Canadian dollars, and then invest in a Canadian Dollar instrument. The Canadian dollars would then be sold in the Foreign Exchange Forward market to avoid exchange rate risk, and the proceeds used to pay the U.S. borrowing. This arbitrage would be carried out until there was no gain to be made between the proceeds of the Foreign Exchange Forward and principal interest required on the U.S. borrowing. Longer term forward foreign exchange contracts were approached more cautiously due to credit risk. If a party to a forward contract defaulted when the other party had a market value gain on the contract, then the non-defaulting party, assumed to be in a hedged position with another counter-party, would have the obligation to pay the market value gain to the hedge counter-party, but would have a low likelihood of receiving the market value from the defaulting party. The further out the term of the contract, the more time there would be for the contract rate to diverge from the market rate and thus create the possibility for a higher loss. Another factor that resulted in caution when entering long-dated markets at that time, was the lack of liquidity for foreign exchange forward rates beyond 2 years. An arbitrage market also developed for long dated forward rates using Canadian and U.S. Treasury bonds. * * * * * * * * * * * * * * * * * * * * * Short-term contracts were traded then, and still now, on the basis of forward points. Forward points are the difference between the Spot rate for a currency, and the Forward rate as determined through interest rate arbitrage. If the Canadian Dollar , for example, was trading at 1.60 for delivery 'spot' and one year forward at 1.5960 then the forward points are a discount of -40. If the trader expected the points to narrow to -20, then the trader would enter into a swap, selling US dollars spot and buying them forward. If spot moved to 1.59 and the forward to 1.5880, the trader would make 100 points on the spot trade and lose 80 points on the forward for a net gain of 20 points or US$ 10,000,000 x .0020 = C$20,000 on a $10 Million dollar contract. From a valuation perspective, the quotation methods were simplified, mainly for ease of quoting. The above valuations did not take into account the time value of money, ie. the present value effect. The pricing components of a currency swap, are the spot rate and the forward points which provide the forward outright rate. The total value of the swap is based on the movement of the spot rate and the outright forward rate. When a portfolio of such swaps resulted in unrealized gains in the future offset by realized losses at present, without close attention to the funding of the realized losses serious leakages could occur. Use of time value of money became more structured with the mathematical formulae developed for Interest Rate Swaps. Interest Rate Swaps The Interest Rate Swaps market began in the early eighties. The market started with a source of low interest rate funding in Japan. The low fixed rate was swapped by a dealer with another dealer that had issued a fixed rate loan to a U.S. counter-party. The U.S. borrower achieved a lower all-in-rate and the dealer counter-parties to the swaps collected fee income. The attractiveness of interest rate swaps was related to the low credit risk. The credit risk in a loan is the entire principal amount plus future interest rate payments whereas the credit risk on an interest rate swap is the market value of the fixed vs floating interest rate payments. Initially, the dealer played an intermediary role and collected fee income from bringing the counter parties together for the structure. Valuation for accounting purposes was relatively simple, particularly if the total fee was received up-front. Fee income at inception of this business was lucrative and brought in competitors coincidental with the creation of complex structures. Dealers became more willing to take on market and credit risk, and replaced up-front fee income with spread income. The original valuation method was based on the underlying loan, with an even accrual of interest rate spreads over the life of the swap. The difficulty with this method was the requirement to match income on unusual cashflows. In a large portfolio, this requirement was virtually impossible to achieve. Valuation for pricing purposes, however, had evolved to an interesting discounting method using the zero coupon curve. This can most likely be attributed to Lee Wakeman, an early pioneer in the interest rate swaps market. A precedent to the zero coupon curve were earlier studies on the term structure of interest rates and forward interest rates. Using the results of those studies it was possible to forecast future cashflows and to value them in present terms. This feature, perhaps, singularly defines valuations of derivative instruments including forward contracts, and options. Interest Rate Options Forward Contracts and Interest Rate Swaps imply a one way view on the market. The purpose of these instruments for the corporate hedger, is to obtain a locked-in cost for an underlying exposure. Thus, if a U.S. corporate has European exposures, a locked-in hedge is to sell Euro's forward. This type of hedging is most publicly evident with gold producers who sell their production several years ahead. Options, on the other hand, provide the buyer of the option a means to participate in some of the gain, if the underlying exposure in fact turns out to be a windfall. Interest Rate Caps (protection against interest rates going over a certain rate) and Floors (the opposite) provided the client with protection, although at a cost, and were a natural match to the interest rate swap market. In terms of market liquidity however, interest rate options were a small component of the major instrument propelling the swap market, that is, the U.S. Long Bond. The lack of liquidity, coupled with the volatility of the underlying instrument, created difficulty for the dealer in hedging client structures particularly when the volatility underwent periods of entrenchment or escalation. In Canada, this was more pronounced since the market is correspondingly smaller than in the U.S. Currency Options Currency options first appeared in the London exchanges around 1974. Initially, it is believed, these options were traded on market price, similar to futures. The Black-Scholes formula, published by Fischer Black and Myron Scholes in 1973 was not widely known to futures speculators. In addition, there were few methods available to use the pricing formula, with the exception perhaps of a novel program derived by Mark Rubenstein to be run on a handheld calculator, where the program was stored on a narrow strips. Interbank currency options suffered some set backs in the early stages of the market, and still continue to produce problems from time to time as market players find themselves on the wrong side of volatility. However, currency options have some favourable features compared to interest rate options. The most evident is the average turnover in foreign exchange contracts, which in April 1998 was estimated at $1.5 trillion per day providing an enormous amount of liquidity in the underlying contracts. Second, much of the hedging that is required by corporate clients is for short term receivables/ payables to cover immediate cashflow requirements. Therefore, the expiry dates of currency options is generally only several months ahead, reducing valuation/operational/credit risk. Exotic Options "Options, Futures and Other Derivatives" was written by John Hull in 1989. This book provided a basis for valuations of exotic options, along with articles published in trade journals, notably Risk Magazine. A particular article by Mark Rubenstein on barrier options, ca 1993, became a benchmark for knock-out options. Whereas a non-Exotic Option, or 'vanilla' option, had a fixed pay-out structure, the shape of which is referred to as a 'hockey-stick', the Exotic Option tended to reduce the pay-out based on the probability of certain events occurring, thereby reducing the premium that a purchase had to pay to obtain the option. For these reasons, a purchaser with a certain view of the direction of a currency, could obtain protection and pay less premium, although subject to increased risk. Combinations of event driven options created a number of descriptive names for various strategies. Valuations were for the most part based on analytical formulae and their variations , such as those that exist for Barrier Options and Average Rate Options. An analytical approach made for faster processing time in the risk management processes. Another method was the use of the Cox, Ross and Rubenstein binomial tree formula. The Monte Carlo approach was flexible and could accommodate most models, however, the number of iterations required to solve for the price and the various risk parameters, was a disadvantage. Again, liquidity limited the usage of exotic options in most markets. In order to have a reasonably secure market to trade exotic options, it was necessary to have a reasonably liquid market in the vanilla options that underly the market. Amongst the foreign exchange, interest rate and equity markets, vanilla options are again, most liquid in the foreign exchange market. As a result, the market for exotic options, tended to be in the currency markets. Out of this market, came a wide variety of products that were based on barrier options, average rate options and digital options, with varying underlying features, such as timing, frequency of observation, method of observation, ranges, pay-out ratios etc. For the most part, these products were tailored for the corporate hedger, in order to help firm the forecasting process in the corporate user's reporting currency. Exotic Options - non-vanilla Where there is very little or no market liquidity, the exotic option can be considered a 'non-vanilla' exotic option. An easy example, is an option that uses two currencies that are not traded much in the vanilla market, e.g. the Canadian dollar against the British Pound or against the Japanese Yen. Although each of these currencies are liquid against the U.S. dollar, there is very little volume traded in the Canadian dollar against the Yen or British Pound. More difficult examples are 2 asset options, such as a Put on Euro that knocks-out when Oil is at a certain level. Both of the examples could be priced better if a market existed for correlation between the 2 underlying assets. Such problems are amplified in a 'basket' option where currencies in a 'basket' have no underlying correlation market. Having said that however, in absence of market correlations, if the number of transactions carried out were such that the 'stochastic processes' would find a way to prove the price, e.g. as in the actuary business, then historical correlations would be used. contact: sales@stocpro.com