Exotic Options, Options Engineering, and Credit Risk

17 Exotic Options, Options Engineering, and Credit Risk Chapter Outline Common Credit Derivatives

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Credit Default Swap 432 Credit Spread Option 433

OTC Exotic Options

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Accruals on Basket of Assets 433 American, Bermudan, and European Options with Sensitivities 434 American Call Option on Foreign Exchange 434 American Call Options on Index Futures 435 American Call Option with Dividends 436 Asian Lookback Options Using Arithmetic and Geometric Averages 436 Asset or Nothing Options 436 Barrier Options 437 Binary Digital Options 437 Cash or Nothing Options 438 Chooser Option (Simple Chooser) 438 Chooser Option (Complex Chooser) 438 Commodity Options 439 Currency (Foreign Exchange) Options 439 Double Barrier Options 440 European Call Option with Dividends 440 Exchange Assets Option 440 Extreme Spreads Option 441 Foreign Equity Linked Foreign Exchange Options in Domestic Currency 441 Foreign Equity Struck in Domestic Currency 442 Foreign Equity with Fixed Exchange Rate 442 Foreign Takeover Options 442 Forward Start Options 442 Futures and Forward Options 443 Gap Options 443 Graduated Barrier Options 444 Index Options 444 Inverse Gamma Out-of-the-Money Options 444 Jump Diffusion Options 445 Leptokurtic and Skewed Options 445 Lookback with Fixed Strike (Partial Time) 445 Lookback with Fixed Strike 446 Lookback with Floating Strike (Partial Time) 446

Credit Engineering for Bankers. DOI: 10.1016/B978-0-12-378585-5.10017-X Copyright Ó 2011 Elsevier Inc. All rights reserved.

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Credit Engineering for Bankers Lookback with Floating Strike 446 Min and Max of Two Assets 447 Options on Options 447 Option Collar 447 Perpetual Options 447 Range Accruals (Fairway Options) 448 Simple Chooser 448 Spread on Futures 448 Supershare Options 449 Time Switch Options 449 Trading Day Corrections 449 Two Asset Barrier Options 450 Two Asset Cash or Nothing 450 Two Correlated Assets Option 450 Uneven Dividend Payments Option 451 Writer Extendible Option 451

The market for credit options or credit derivatives is young and wide open for creative engineering and development of exotic instruments. The process of creative engineering and innovation has to start from the fundamental theory of traded and liquid instruments such as exotic options. The only difference is that these credit derivatives use different underlying assets. Traded exotic options use stock prices, reference basket of stocks, indexes, interest rates, foreign exchange rates, and commodity prices. In contrast, credit derivatives use entity or reference credit sources, credit assets, or credit events. For instance, one default or multiple issuers’ defaults will trigger the credit event, or a set of assets is used to determine the credit event (e.g., loans, liquid bonds) or other extreme events (e.g., bankruptcy, failure to pay for a certain maturity threshold and grace period, default, moratorium, repudiation, restructuring, downgrade, changes in credit spread, etc.). Regardless of the underlying asset used, these default conditions can be simulated (see Chapter Seven) and the over-the-counter (OTC) exotic options can be used to value these exotic credit derivatives. This chapter begins by introducing the two most common credit derivatives—credit default swaps and credit spread options—and continues with OTC exotic options. You can access these OTC models through the Modeling Toolkit software.

Common Credit Derivatives Credit Default Swap Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Credit Analysis j Credit Default Swaps and Credit Spread Options Brief Description: A credit hedging derivative that becomes valuable during a default event.

A credit default swap (CDS) allows the holder of the instrument to sell a bond or debt at par value when a credit event or default occurs. This model computes the valuation

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of the CDS spread. A CDS does not protect against movements of the credit spread (only a credit spread option can do that), but it only protects against defaults. Typically, to hedge against defaults and spread movements, both CDSs and credit spread options (CSOs) are used. This CDS model assumes a no-arbitrage argument, and the holder of the CDS makes periodic payments to the seller (similar to a periodic insurance premium) until the maturity of the CDS or until a credit event or default occurs. Because the notional amount to be received in the event of a default is the same as the par value of the bond or debt, and time value of money is used to determine the bond yield, this model does not require these two variables as input assumptions.

Credit Spread Option Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Credit Analysis j Credit Default Swaps and Credit Spread Options Brief Description: A credit hedging derivative that protects against credit spreads and adverse interest rate movements.

In contrast, CSOs are another type of exotic debt option where the payoff depends on a credit spread or the price of the underlying asset that is sensitive to interest rate movements such as floating or inverse floating rate notes and debt. A CSO call option provides a return to the holder if the prevailing reference credit spread exceeds the predetermined strike rate, and the duration input variable is used to translate the percentage spread into a notional currency amount. The CSO expires when there is a credit default event. Again, note that a CSO can only protect against any movements in the reference spread and not a default event (only a CDS can do that). Typically, to hedge against defaults and spread movements, both CDSs and CSOs are used. In some cases, when the CSO covers a reference entity’s underlying asset value and not the spread itself, the credit asset spread options are used instead.

OTC Exotic Options Accruals on Basket of Assets Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Real Options Models j Accruals on Basket of Assets Brief Description: Accruals on basket of assets instruments are essentially financial portfolios of multiple underlying assets where the holder of the instrument receives the maximum of the basket of assets or some prespecified guarantee amount.

The accruals on basket of assets instruments are exotic options in which there are several assets in a portfolio whereby the holder of the instrument receives the maximum of either the guaranteed amount or any one of the value of the assets. This instrument can be modeled as either an American option, which can be executed at any time up to and including maturity; a European option, which can be exercised only at maturity; or a Bermudan option, which is exercisable only at certain times.

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Using the multiple assets and multiple phased module of the Real Options SLS software, we can model the value of an accrual option. It is highly recommended that at this point, the reader first become familiar with Chapter 16 before attempting to solve any SLS models. This model, although using the SLS software, is still listed under the exotic options category because basket accruals are considered exotic options and are solved using similar methodologies as other exotics.

American, Bermudan, and European Options with Sensitivities File Name: Exotic Options – American, Bermudan, and European Options with Sensitivities Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j American, Bermudan, and European Options Brief Description: Computation of American and European options with Greek sensitivities.

American options can be exercised at any time up to and including maturity, European options can only be exercised at maturity, and Bermudan options can be exercised only at certain times (i.e., exercisable other than during the vesting blackout period). In most instances, American  Bermudan  European options except for one special case: plain-vanilla call options when there are no dividends In that case, American ¼ Bermudan ¼ European call options, as it is never optimal to exercise early in a plainvanilla call option when there are no dividends. However, once there is a sufficiently high dividend rate paid by the underlying stock, we clearly see that the relationship where American  Bermudan  European options apply. European options can be solved using the Generalized Black-Scholes-Merton model (a closed for equation), as well as using binomial lattices and other numerical methods. However, for American options, we cannot use the Black-Scholes model and must revert to using the binomial lattice approach and some closed-form approximation models. Using the binomial lattice requires an additional input variable: lattice steps. The higher the number of lattice steps, the higher the precision of the results. Typically, 100 to 1,000 steps are sufficient to achieve convergence. Use the Real Options SLS software to solve more advanced options vehicles with a fully customizable lattice model. Only binomial lattices can be used to solve Bermudan options.

American Call Option on Foreign Exchange Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Real Options Models j American Call Option on Foreign Exchange Brief Description: Computation of American and European options on foreign exchange.

A foreign exchange option (FX option or FXO) is a derivative where the owner has the right, but not the obligation, to exchange money denominated in one currency into another currency at a previously agreed on exchange rate on a specified date. The FX options market is the deepest, largest, and most liquid market for options of any kind in the world. The valuation here uses the Garman-Kohlhagen model. You can use the Exotic Options – Currency (Foreign Exchange) Options model in the Modeling Toolkit software to compare the results of this Real Options SLS model. The exotic options model is used to compute the European version using closed-form models; the Real

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Options SLS model—the example showcased here—computes the European and American options using a binomial lattice approach. When using the binomial lattice approach in the Real Options SLS software, remember to set the Dividend Rate as the foreign country’s risk-free rate, PV Underlying Asset as the spot exchange rate, and Implementation Cost as the strike exchange rate. As an example of a foreign exchange option, suppose the British pounds (GBP) versus the U.S. dollar (USD) is USD2/GBP1. Then the spot exchange rate is 2.0. Because the exchange rate is denominated in GBP (the denominator), the domestic risk-free rate is the rate in the United Kingdom, and the foreign rate is the rate in the United States. This means that the foreign exchange contract allows the holder the option to call GBP and put USD. To illustrate, suppose a U.K. firm is getting US$1M in six months, and the spot exchange rate is USD2/GBP1. If the GBP currency strengthens, the U.K. firm loses when it has to repatriate USD back to GBP, but it gains if the GBP currency weakens. If the firm hedges the foreign exchange exposure with an FXO and gets a call on GBP (put on USD), it hedges itself from any foreign exchange fluctuation risks. For discussion purposes, say the timing is short, interest rates are low, and volatility is low. Getting a call option with a strike of 1.90 yields a call value approximately 0.10 (i.e., the firm can execute the option and gain the difference of 2.00 – 1.90, or 0.10, immediately). This means that the rate now becomes USD1.90/GBP1, and it is cheaper to purchase GBP with the same USD, or the U.K. firm gets a higher GBP payoff. In situations where volatility is nonzero and maturity is higher, there is a significant value in an FXO, which can be modeled using the Real Options SLS approach.

American Call Options on Index Futures Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Real Options Models j American Call Option on Index Futures Brief Description: Computes the value of an index option using closed-form models and binomial lattices.

The index option is similar to a regular option, but its underlying asset is a reference stock index such as the Standard & Poor’s 500. The analysis can be solved using a closed-form Generalized Black-Scholes-Merton model. The model used in this chapter is similar to the closed-form model but applies the binomial lattice instead. The difference here is that Black-Scholes can solve only European options, while the binomial lattice model is capable of solving American, European, and Bermudan or mixed and customized options. Instead of using the asset value or current stock price as the input, we use the current index value. All other inputs remain the same, as in other options models. Index futures are an important investment vehicle because stock indexes cannot be traded directly, so futures based on stock indexes are the primary vehicles for trading indexes. Index futures operate in essentially the same way as other futures, and they are traded in the same way. Because indexes are based on many separate stocks, index futures are settled in cash rather than stocks. In addition, index futures allow investors to participate in the entire market without the significant cost of purchasing each underlying stock in an index. Index futures are widely used for hedging market movements and are applied in portfolios for their diversification effects.

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American Call Option with Dividends Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Real Options Models j American Call Option with Dividends Brief Description: Solves the American call option using customizable binomial lattices as well as closed-form approximation models.

The American call option with dividends model computes the call option using Real Options SLS software by applying the binomial lattice methodology. The results are benchmarked with closed-form approximation models. You can compare the results from this model with the European call option with dividends example. Note that when there are dividends, American options (which can be exercised early) are worth more than Bermudan options (options that can be exercised before termination but not during blackout periods such as vesting or contractual nontrading days), which, in turn, are worth more than European options (options that can be exercised only at maturity). This value relationship is typically true for most options. In contrast, for plain-vanilla basic call options without dividends, American options are worth the same as the Bermudan and European options, since they are never optimal to exercise early. This fact is true only for simple plain-vanilla call options when no dividends exist.

Asian Lookback Options Using Arithmetic and Geometric Averages Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Asian Arithmetic Brief Description: Solves an Asian lookback option using closed-form models, where the lookback is linked to the arithmetic average of past prices.

Asian options (also called average options) are options whose payoffs are linked to the average value of the underlying asset on a specific set of dates during the life of the option. An average rate option is a cash-settled option whose payoff is based on the difference between the average value of the underlying asset during the life of the option and a fixed strike. The arithmetic version means that the prices are simple averages rather than geometric averages. End-users of currency, commodities, or energy trading tend to be exposed to average prices over time, so Asian options are attractive for them. Asian options are also popular with corporations and banks with ongoing currency exposures. These options are also attractive because they tend to be less expensive—sell at lower premiums—than comparable vanilla puts or calls. This is because the volatility in the average value of an underlying asset or stock tends to be lower than the volatility of the actual values of the underlying asset or stock. Also, in situations where the underlying asset is thinly traded or there is the potential for its price to be manipulated, an Asian option offers some protection. It is more difficult to manipulate the average value of an underlying asset over an extended period of time than it is to manipulate it just at the expiration of an option.

Asset or Nothing Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Asset or Nothing

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Brief Description: Computes an asset or nothing option where, as long as the asset is above water, the holder will receive the asset at maturity.

An asset or nothing option is exactly what it implies. At expiration, if the option is in-the-money, regardless of how deep it is in-the-money, the option holder receives the stock or asset. This means that for a call option, as long as the stock or asset price exceeds the strike at expiration, the stock is received. Conversely, for a put option, the stock is received only if the stock or asset value falls below the strike price.

Barrier Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Barrier Options Brief Description: Values various types of barrier options such as up and in, down and in, up and out, and down and out call and put options.

Barrier options become valuable or get knocked in-the-money only if a barrier (upper or lower barrier) is breached (or not), and the payout is in the form of the option on the underlying asset. Sometimes, as remuneration for the risk of not being knocked in, a specified cash rebate is paid at the end of the instrument’s maturity (at expiration) assuming that the option has not been knocked in. As an example, in an up and in call option, the instrument pays the specified cash amount at expiration if and only if the asset value does not breach the upper barrier (the asset value does not go above the upper barrier), thus providing the holder of the instrument with a safety net or a cash insurance. However, if the asset breaches the upper barrier, the option gets knocked in and becomes a live option. An up and out option is live only as long as the asset does not breach the upper barrier, and so forth. Monitoring Periodicities means how often during the life of the option the asset or stock value will be monitored to see if it breaches a barrier. As an example, entering 12 implies monthly monitoring, 52 means weekly monitoring, 252 indicates monitoring for daily trading, 365 means monitoring daily, and 1,000,000 is used for continuous monitoring. In general, barrier options limit the potential of a regular option’s profits and, thus, cost less than regular options without barriers. For instance, if we assume no cash rebates, a Generalized Black-Scholes call option returns $6.50 as opposed to $5.33 for a barrier option (up and in barrier set at $115 with a stock price and strike price value of $100). This is because the call option will only be knocked in if the stock price goes above this $115 barrier, thereby reducing the regular option’s profits between $100 and $115. This reduction effect is even more pronounced in the up and out call option, where all the significant upside profits (above the $115 barrier) are completely truncated.

Binary Digital Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Binary Digital Options Brief Description: Binary digital options are instruments that either get knocked in- or out-of-the-money depending on whether the asset value breaches or does not breach certain barriers.

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Binary exotic options (also known as digital, accrual, or fairway options) become valuable only if a barrier (upper or lower barrier) is breached (or not), and the payout could be in the form of some prespecified cash amount or the underlying asset itself. The cash or asset exchanges hands either at the point when the barrier is breached or at the end of the instrument’s maturity (at expiration), assuming that the barrier is breached at some point prior to maturity. For instance, in the down and in cash at expiration option, the instruments pay the specified cash amount at expiration if and only if the asset value breaches the lower barrier (the asset value goes below the lower barrier), providing the holder of the instrument a safety net or a cash insurance in case the underlying asset does not perform well. With up and in options, the cash or asset is provided if the underlying asset goes above the upper barrier threshold. In up and out or down and out options, the asset or cash is paid as long as the upper or lower barrier is not breached. With at expiration options, cash and assets are paid at maturity, whereas the at hit instruments are payable at the point when the barrier is breached.

Cash or Nothing Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Cash or Nothing Brief Description: Computes the cash or nothing option, where if in-the-money at expiration, the holder receives some prespecified cash amount.

A cash or nothing option is exactly what it implies. At expiration, if the option is in-the-money, regardless of how deep it is in-the-money, the option holder receives a predetermined cash payment. This means that for a call option, as long as the stock or asset price exceeds the strike at expiration, cash is received. Conversely, for a put option, cash is received only if the stock or asset value falls below the strike price.

Chooser Option (Simple Chooser) Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Simple Chooser Brief Description: Computes the value of an option that can become either a call or a put by a specific chooser time.

A simple chooser option allows the holder to choose if the option is a call or a put within the chooser time. Regardless of the choice, the option has the same contractual strike price and maturity. Typically, a chooser option is cheaper than purchasing both a call and a put together, but it provides the same level of hedge at a lower cost. The strike prices for both options are identical. The complex chooser option (see next) allows for different strike prices and maturities.

Chooser Option (Complex Chooser) Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Complex Chooser Brief Description: The complex chooser option allows the holder to choose if it becomes a call or a put option by a specific chooser time, while the maturity and strike price of the call and put are allowed to be different.

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A complex chooser option allows the holder to choose if the option is a call or a put within the chooser time. The complex chooser option allows for different strike prices and maturities. Typically, a chooser option is cheaper than purchasing both a call and a put together. It provides the same level of hedge at a lower cost, while the strike price for both options can be different.

Commodity Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Commodity Options Brief Description: Models and values a commodity option where the commodity’s spot and future values are used to value the option, while the forward rates and convenience yields are assumed to be mean-reverting and volatile.

This model computes the values of commodity-based European call and put options, where the convenience yield and forward rates are assumed to be mean-reverting, and each has its own volatilities and cross-correlations. This is a complex multifactor model with interrelationships among all variables.

Currency (Foreign Exchange) Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Currency Options Brief Description: Values a foreign exchange currency option, typically used in hedging foreign exchange fluctuations, where the key inputs are the spot exchange rate and the contractual purchase or sale price of the foreign exchange currency for delivery in the future.

A foreign exchange option (FX option or FXO) is a derivative where the owner has the right, but not the obligation, to exchange money denominated in one currency into another currency at a previously agreed on exchange rate on a specified date. The FX options market is the deepest, largest, and most liquid market for options of any kind in the world. The valuation here uses the Garman-Kohlhagen model. As an example, suppose the British pound (GBP) versus the U.S. dollar (USD) is USD2/GBP1. Then the spot exchange rate is 2.0. Because the exchange rate is denominated in GBP (the denominator), the domestic risk-free rate is the rate in the United Kingdom, and the foreign rate is the rate in the United States. This means that the foreign exchange contract allows the holder the option to call GBP and put USD. To illustrate, suppose a U.K. firm is getting US$1M in six months, and the spot exchange rate is USD2/ GBP1. If the GBP currency strengthens, the U.K. firm loses if it has to repatriate USD back to GBP, but it gains if the GBP currency weakens. If the firm hedges the foreign exchange exposure with an FXO and gets a call on GBP (put on USD), it hedges itself from any foreign exchange fluctuation risks. For discussion purposes, say the timing is short, interest rates are low, and volatility is low. Getting a call option with a strike of 1.90 yields a call value approximately 0.10 (i.e., the firm can execute the option and gain the difference of 2.00 – 1.90, or 0.10, immediately). This means that the rate now becomes USD1.90/GBP1, and it is cheaper to purchase GBP with the same USD, or the U.K. firm gets a higher GBP payoff.

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Double Barrier Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Double Barriers Brief Description: Various double barrier options are valued in this model, including up and in, up and out, with down and in, and down and out combinations, where a call or put option is knocked in- or out-of-the-money, depending on if it breaches an upper or lower barrier.

Barrier options become valuable or get knocked in-the-money only if a barrier (upper or lower barrier) is breached (or not), and the payout is in the form of the option on the underlying asset. Double barrier options have two barriers, one above the current asset value (upper barrier) and one below it (lower barrier). Either barrier has to be breached for a knock-in or knock-out event to occur. In general, barrier options limit the potential of a regular option’s profits and, thus, cost less than regular options without barriers. As an example, in an up and out, down and out call option, the instrument is knocked out if the asset breaches either the upper or lower barriers, but it remains in effect, and the option is live at the end of maturity if the asset prices remain between these two barriers.

European Call Option with Dividends Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Real Options Models j European Call Option with Dividends Brief Description: This model uses the customized binomial lattice approach to solve a European call option with dividends, where the holder of the option can exercise only at maturity and not before.

A European call option allows the holder to exercise the option only at maturity. An American option can be exercised at any time before, as well as up to and including, maturity. A Bermudan option is like an American and European option mixed: At certain vesting and blackout periods, the option cannot be executed until the end of the blackout period, when it then becomes exercisable at any time until its maturity. In a simple plain-vanilla call option, all three varieties have the same value, since it is never optimal to exercise early, making all three options revert to the value of a simple European option. However, this qualification does not hold true when dividends exist. When dividends are high enough, it is typically optimal to exercise a call option early, particularly before the ex-dividend date hits, reducing the value of the asset and, consequently, the value of the call option. The American call option typically is worth more than a Bermudan option, which is typically worth more than a European option, except in the special case of a simple plain-vanilla call option.

Exchange Assets Option Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Exchange Assets Brief Description: This European option allows the holder to swap out one asset for another, with predetermined quantities of each asset to be swapped at expiration.

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The exchange of assets option provides the option holder the right at expiration (European version) to swap and give away Asset 2 and, in return, receive Asset 1, with predetermined quantities of the first and second assets. The American option allows the swap to occur at any time before, as well as up to and including, maturity. Clearly, the more negative the correlation between the assets, the larger the risk reduction and diversification effects and, thus, the higher the value of the option. Sometimes in a real options world, where the assets swapped are not financial instruments but real physical assets, this option is called a switching option.

Extreme Spreads Option Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Extreme Spreads Brief Description: Computes extreme spread option values, where the vehicle is divided into two segments, and the option pays off the difference between the extreme values (min or max) of the asset during the two time segments.

Extreme spread options have their maturities divided into two segments, starting from time zero to the first time period (first segment) and from the first time period to maturity (second segment). An extreme spread call option pays the difference between the maximum asset value from the second segment and the maximum value of the first segment. Conversely, the put pays the difference between the minimum of the second segment’s asset value and the minimum of the first segment’s asset value. A reverse call pays the minimum from the first less the minimum of the second segment, whereas a reverse put pays the maximum of the first less the maximum of the second segments.

Foreign Equity Linked Foreign Exchange Options in Domestic Currency Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Foreign Equity Linked Forex Brief Description: Computes the option where the underlying asset is in a foreign market, the exchange rate is fixed in advance to hedge the exposure risk, and the strike price is set as a foreign exchange rate rather than a price.

Foreign equity linked foreign exchange options are options whose underlying asset is in a foreign equity market. The option holder can hedge the fluctuations of the foreign exchange risk by having a strike price on the foreign exchange rate. The resulting valuation is in the domestic currency. There are three closely related models in this and the following two sections (foreign equity linked foreign exchange option, foreign equity struck in domestic currency, and foreign equity with fixed exchange rate). Their similarities and differences can be summarized as follows: l

l

l

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The underlying asset is denominated in a foreign currency. The foreign exchange rate is domestic currency to foreign currency. The option is valued in domestic currency. The strike prices are different where: The exchange rate is the strike for the foreign equity linked foreign exchange option. The domestic currency is the strike for the foreign equity struck in domestic currency option. The foreign currency is the strike for the foreign equity with fixed exchange rate option. l

l

l

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Foreign Equity Struck in Domestic Currency Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Foreign Equity Domestic Currency Brief Description: Values the options on foreign equities denominated in foreign exchange currency while the strike price is in domestic currency.

Foreign equity struck in domestic currency is an option on foreign equities in a foreign currency but with the strike price in domestic currency. At expiration, assuming the option is in-the-money, its value will be translated back into the domestic currency. The exchange rate is the spot rate for domestic currency to foreign currency, the asset price is denominated in a foreign currency, and the strike price is in domestic currency.

Foreign Equity with Fixed Exchange Rate Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Foreign Equity Fixed Forex Brief Description: Values foreign equity options where the option is in a currency foreign to that of the underlying asset but with a risk hedging on the exchange rate.

Quanto options, also known as foreign equity options with fixed exchange rate, are traded on exchanges around the world. These options are denominated in a currency different from that of the underlying asset. They have an expanding or contracting coverage of the foreign exchange value of the underlying asset. The valuation of these options depends on the volatilities of the underlying assets and the currency exchange rate, as well as the correlation between the currency and the asset value.

Foreign Takeover Options File Name: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Exotic Options – Foreign Takeover Options Brief Description: Computes a foreign takeover option, where the holder has the right to purchase some foreign exchange at a prespecified rate for the purposes of a takeover or acquisition of a foreign firm.

In a foreign takeover option with a foreign exchange element, if a successful takeover ensues (if the value of the foreign firm denominated in foreign currency is less than the foreign currency units required), the option holder has the right to purchase the number of foreign units at the predetermined strike price (denominated in exchange rates of the domestic currency to the foreign currency), at the option expiration date.

Forward Start Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Forward Start Brief Description: Computes the value of an option that technically starts in the future, with its strike price being a percentage of the asset price in the future when the option starts.

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Forward start options start at-the-money or proportionally in- or out-of-the-money after some time in the future (the time to forward start). These options sometimes are used in employee stock options, where a grant is provided now, but the strike price depends on the asset or stock price at some time in the future and is proportional to the stock price. The Alpha variable measures the proportionality of the option being in- or out-of-the-money. Alpha ¼ 1 means the option starts at-the-money or that the strike is set exactly as the asset price at the end of the time to forward start. Alpha < 1 means that the call option starts (1 – Alpha) percent in-the-money or (1 – Alpha) out-of-the-money for puts. Conversely, for Alpha > 1, the call option starts (Alpha – 1) out-of-the-money and puts start (Alpha – 1) in-the-money.

Futures and Forward Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Futures and Forward Options Brief Description: This model applies the same generalities as the Black-Scholes model, but the underlying asset is a futures or forward contract, not a stock.

The futures option is similar to a regular option, but the underlying asset is a futures or forward contract. Be careful here, as the analysis cannot be solved using a Generalized Black-Scholes-Merton Model. In many cases, options are traded on futures. A put is the option to sell a futures contract, and a call is the option to buy a futures contract. For both, the option strike price is the specified futures price at which the future is traded if the option is exercised. A futures contract is a standardized contract, typically traded on a futures exchange, to buy or sell a certain underlying instrument at a certain date in the future at a prespecified price. The future date is called the delivery date or final settlement date. The preset price is called the futures price. The price of the underlying asset on the delivery date is called the settlement price. The settlement price normally converges toward the futures price on the delivery date. A futures contract gives the holder the obligation to buy or sell, which differs from an options contract, which gives the holder the right, but not the obligation, to buy or sell. In other words, the owner of an options contract may exercise the contract. If it is an American-style option, it can be exercised on or before the expiration date, but a European option can be exercised only at expiration. Thus, a futures contract is more like a European option. Both parties of a futures contract must fulfill the contract on the settlement date. The seller delivers the commodity to the buyer, or if it is a cash-settled future, cash is transferred from the futures trader who sustained a loss to the one who made a profit. To exit the commitment prior to the settlement date, the holder of a futures position has to offset the position either by selling a long position or by buying back a short position, effectively closing out the futures position and its contract obligations.

Gap Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Gap Options

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Brief Description: Gap options are valued with this model, where there are two strike prices with respect to one underlying asset, and where the first strike acts like a barrier that, when breached, brings the second strike price into play.

Gap options are similar to barrier options and two asset correlated options in the sense that the call option is knocked in when the underlying asset exceeds the reference Strike Price 1, making the option payoff the asset price less Strike Price 2 for the underlying asset. Similarly, the put option is knocked in only if the underlying asset is less than the reference Strike Price 1, providing a payoff of Strike Price 2 less the underlying asset.

Graduated Barrier Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Graduated Barriers Brief Description: Graduated barrier models are barrier options with flexible and graduated payoffs, depending on how far above or below a barrier the asset ends up at maturity.

Graduated, or soft, barrier options are similar to standard barrier options except that the barriers are no longer static values but a graduated range between the lower and upper barriers. The option is knocked in- or out-of-the-money proportionally. Both upper and lower barriers should be either above (for up and in or up and out options) or below (for down and in or down and out options) the starting stock price or asset value. For instance, in the down and in call option, the instruments become knocked-in, or live, at expiration if and only if the asset or stock value breaches the lower barrier (asset value goes below the barriers). If the option to be valued is a down and in call, then both the upper and lower barriers should be lower than the starting stock price or asset value, providing a collar of graduated prices. For instance, if the upper and lower barriers are $90 and $80, and if the asset price ends up being $89, a down and out option will be knocked out 10% of its value. Standard barrier options are more difficult to delta hedge when the asset values and barriers are close to each other. Graduated barrier options are more appropriate for delta-hedges, providing less delta risk and gamma risk.

Index Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Index Options Brief Description: Index options are similar to regular plain-vanilla options and can be solved using the Black-Scholes model, with the only difference being that the underlying asset is not a stock but an index.

The index option is similar to a regular option, but the underlying asset is a reference stock index such as the Standard & Poor’s 500. The analysis can be solved using a Generalized Black-Scholes-Merton Model as well.

Inverse Gamma Out-of-the-Money Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Inverse Gamma Out-ofthe-Money Options

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Brief Description: Analyzes options using an inverse gamma distribution rather than the typical normal-lognormal assumptions; this type of analysis is important for valuing extreme in- or out-of-the-money options.

This model computes the value of European call and put options using an inverse gamma distribution, as opposed to the standard normal distribution. This distribution accounts for the peaked distributions of asset returns and provides better estimates for deep out-of-the-money options. The traditional Generalized Black-Scholes-Merton Model is typically used as benchmark.

Jump Diffusion Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Jump Diffusion Brief Description: Sometimes the underlying asset in an option is assumed to follow a Poisson jump-diffusion process instead of a random walk Brownian motion, and applicable for underlying assets such as oil and gas commodities and price of electricity.

A jump diffusion option is similar to a regular option except that instead of assuming that the underlying asset follows a lognormal Brownian motion process, the process here follows a Poisson jump-diffusion process. That is, stock or asset prices follow jumps, which occur several times per year (observed from history). Cumulatively, these jumps explain a certain percentage of the total volatility of the asset.

Leptokurtic and Skewed Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Leptokurtic and Skewed Options Brief Description: Computes options where the underlying assets are assumed to have returns that are skewed and leptokurtic or have fat tails and are leaning on one end of the distribution rather than having symmetrical in returns.

This model is used to compute the European call and put options using the binomial lattice approach when the underlying distribution of stock returns is not normally distributed, is not symmetrical, and has additional slight kurtosis and skew. Be careful when using this model to account for a high or low skew and kurtosis because certain combinations of these two coefficients actually yield unsolvable results. The BlackScholes results are typically used to benchmark the effects of a high kurtosis and positive or negatively skewed distributions compared to the normal distribution assumptions on asset returns.

Lookback with Fixed Strike (Partial Time) Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Lookback Fixed Strike Partial Time Brief Description: Computing an option where the strike price is predetermined but the payoff on the option is the difference between the highest or the lowest attained asset price against the strike.

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In a fixed strike option with lookback feature (partial time), the strike price is predetermined, while at expiration, the payoff on the call option is the difference between the maximum asset price less the strike price during the time between the starting period of the lookback to the maturity of the option. Conversely, the put will pay the maximum difference between the lowest observed asset price less the strike price during the time between the starting period of the lookback to the maturity of the option.

Lookback with Fixed Strike Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Lookback Fixed Strike Brief Description: Computes the value of an option where the strike price is fixed but the value at expiration is based on the value of the underlying asset’s maximum and minimum values during the option’s lifetime.

In a fixed strike option with lookback feature, the strike price is predetermined, while at expiration, the payoff on the call option is the difference between the maximum asset price less the strike price during the lifetime of the option. Conversely, the put will pay the maximum difference between the lowest observed asset price less the strike price during the lifetime of the option.

Lookback with Floating Strike (Partial Time) Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Lookback Floating Strike Partial Time Brief Description: Computes the value of an option where the strike price is not fixed but floating and the value at expiration is based on the value of the underlying asset’s maximum and minimum values starting from the lookback inception time to maturity as the purchase or sale price.

In a floating strike option with lookback feature (partial time), the strike price is floating. At expiration, the payoff on the call option is being able to purchase the underlying asset at the minimum observed price from inception to the end of the lookback time. Conversely, the put will allow the option holder to sell at the maximum observed asset price from inception to the end of the lookback time.

Lookback with Floating Strike Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Lookback Floating Strike Brief Description: Computes the value of an option where the strike price is not fixed but floating and the value at expiration is based on the value of the underlying asset’s maximum and minimum values during the option’s lifetime as the purchase or sale price.

In a floating strike option with lookback feature, the strike price is floating. At expiration, the payoff on the call option is being able to purchase the underlying asset at the minimum observed price during the life of the option. Conversely, the put will allow the option holder to sell at the maximum observed asset price during the life of the option.

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Min and Max of Two Assets Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Min and Max of Two Assets Brief Description: Computes the value of an option where there are two underlying assets that are correlated with different volatilities and the differences between the assets’ values are used as the benchmark for determining the value of the payoff at expiration.

Options on minimum or maximum are used when there are two assets with different volatilities. Either the maximum or the minimum value at expiration of both assets is used in the option’s exercise. For instance, a call option on the minimum implies that the payoff at expiration is such that the minimum price between Asset 1 and Asset 2 is used against the strike price of the option.

Options on Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Options on Options Brief Description: Computes the value of an option on another option, or a compound option, where the option provides the holder the right to buy or sell a subsequent option at the expiration of the first option.

Options on options, sometimes known as compound options, allow the holder to call or buy versus put or sell an option in the future. For instance, a put on call option means that the holder has the right to sell a call option in some future period for a specified strike price (strike price for the option on option). The time for this right to sell is called the maturity of the option on option. The maturity of the underlying refers to the maturity of the option to be bought or sold in the future, starting from now.

Option Collar Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Options Collar Brief Description: Computes the call-put collar strategy—that is, to short a call and long a put at different strike prices such that the hedge is costless and effective.

The call and put collar strategy requires that one stock be purchased, one call be sold, and one put be purchased. The idea is that the proceeds from the call sold are sufficient to cover the proceeds of the put bought. Therefore, given a specific set of stock price, option maturity, risk-free rate, volatility, and dividend of a stock, you can impute the required strike price of a call if you know what put to purchase (and its relevant strike price) or the strike price of a put if you know what call to sell (and its relevant strike price).

Perpetual Options File Name: Exotic Options – Perpetual Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Perpetual Options

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Brief Description: Computes the value of an American option that has a perpetual life where the underlying is a dividend-paying asset.

The perpetual call and put options are American options with continuous dividends that can be executed at any time and have an infinite life. Clearly a European option (only exercisable at termination) has a zero value, Hence, only American options are viable perpetual options. American closed-form approximations with 100-year maturities are typically used to benchmark the results.

Range Accruals (Fairway Options) Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Real Options Models j Range Accruals Brief Description: Computes the value of fairway options, or range accrual options, where the option pays a specified return if the underlying asset is within a range but pays something else if it is outside the range at any time during its maturity.

A range accrual option is also called a fairway option. Here, the option pays a certain return if the asset value stays within a certain range (between the upper and lower barriers) but pays a different amount or return if the asset value falls outside this range during any time before and up to maturity. The name fairway option is sometimes used because the option is similar to the condition in the game of golf, where if the ball stays within the fairway (a narrow path), it is in play, and if it goes outside, a penalty might be imposed (in this case, a lower return).

Simple Chooser Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Simple Chooser Brief Description: Computes the value of an option where the holder has the ability to decide if it is a call or a put at some future time period; this option is similar to purchasing a call and a put together but costs less as a simple chooser option. Computes the value of an option that can become either a call or a put by a specific chooser time

A simple chooser option allows the holder to choose if the option is a call or a put within the chooser time. Regardless of the choice, the option has the same contractual strike price and maturity. Typically, a chooser option is cheaper than purchasing both a call and a put together but provides the same level of hedge at a lower cost. The strike prices for both options are identical. The complex chooser option (see “Chooser Option (Complex Chooser)”) allows for different strike prices and maturities.

Spread on Futures Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Spread on Futures Brief Description: Computes the value of an option where the underlying assets are two different futures contracts, and their spreads (the difference between the futures) are used as a benchmark to determine the value of the option.

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A spread on futures options is an option where the payoff is the difference between the two futures values at expiration. That is, the spread is Futures 1 – Futures 2, while the call payoff is the value of Spread – Strike, and the put payoff is Strike – Spread.

Supershare Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Supershares Brief Description: Computes the value of an option where it is a hybrid of a double barrier fairway option that pays a percentage proportional to the level it is in-the-money within the barriers.

Typically, supershare options are traded or embedded in supershare funds. These options are related to down and out, up and out double barrier options, where the option has value only if the stock or asset price is between the upper and lower barriers, and, at expiration, the option provides a payoff equivalent to the stock or asset price divided by the lower strike price (S/X Lower).

Time Switch Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Time Switch Brief Description: Computes the value of an option where its value depends on how many times a barrier is breached during the life of the option.

In a time switch option, the holder receives the Accumulated Amount  Time-Steps each time the asset price exceeds the strike price for a call option (or falls below the strike price for a put option). The time-steps are how often the asset price is checked if the strike threshold has been breached (typically, for a one-year option with 252 trading days, set DT as 1/252). Sometimes, the option has already accumulated past amounts or as agreed to in the option as a minimum guaranteed payment as measured by the number of time units fulfilled (which is typically set as 0).

Trading Day Corrections Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Trading Day Corrections Brief Description: Computes the value of plain-vanilla call and put options corrected for the number of trading days and calendar days left in their maturity.

An option with a trading day correction uses a typical option and corrects it for the varying volatilities. Specifically, volatility tends to be higher on trading days than on nontrading days. The trading days ratio is simply the number of trading days left until maturity divided by the total number of trading days per year (typically between 250 and 252). The calendar days ratio is the number of calendar days left until maturity divided by the total number of days per year (365). Typically, with the adjustments, the option value is lower. In addition, if the trading days ratio and calendar days ratio are identical, the results of the adjustment is zero, and the option value reverts back to the generalized Black-Scholes results. This is because the days left are assumed to be all full trading days.

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Two Asset Barrier Options Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Two Asset Barrier Brief Description: Computes the value of an option with two underlying assets, one of which is a reference benchmark and the other is used to compute the value of the option payoff.

Two asset barrier options become valuable or get knocked in-the-money only if a barrier (upper or lower barrier) is breached (or not) and the payout is in the form of the option on the first underlying asset. In general, a barrier option limits the potential of a regular option’s profits and, thus, costs less than regular options without barriers. A barrier option is thus a better bet for companies that are trying to hedge a specific price movement, since it tends to be cheaper. As an example, in the up and in call option, the instrument is knocked in if the asset breaches the upper barrier but is worthless if this barrier is not breached. The payoff on this option is based on the value of the first underlying asset (Asset 1) less the Strike price, while whether the barrier is breached or not depends on whether the second reference asset (Asset 2) breaches the Barrier. Monitoring Periodicities means how often during the life of the option the asset or stock value will be monitored to see if it breaches a barrier. As an example, entering 12 implies monthly monitoring, 52 means weekly monitoring, 252 indicates monitoring for daily trading, 365 means monitoring daily, and 1,000,000 for continuous monitoring.

Two Asset Cash or Nothing Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Two Asset Cash Brief Description: Computes the value of an option that pays a prespecified amount of cash as long as the option stays in-the-money at expiration, regardless of how valuable the intrinsic value is at maturity.

Cash or nothing options pay out a prespecified amount of cash at expiration as long as the option is in-the-money, without regard to how much it is in-the-money. The two asset cash or nothing option means that both assets must be in-the-money before cash is paid out (for call options, both asset values must be above their respective strike prices, and for puts, both assets must be below their respective strike prices). For the up-down option, this implies that the first asset must be above the first strike price, and the second asset must be below the second strike price. Conversely, the down-up option implies that cash will be paid out only if at expiration, the first asset is below the first strike, and the second asset is above the second strike.

Two Correlated Assets Option Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Two Asset Correlated Brief Description: Computes the value of an option that depends on two assets: a benchmark asset that determines if the option is in-the-money and a second asset that determines the payoff on the option at expiration.

The two correlated asset options use two underlying assets: Asset 1 and Asset 2. Typically, Asset 1 is the benchmark asset (e.g., a stock index or market comparable

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stock), whereby if at expiration, Asset 1’s values exceed Strike 1’s value, then the option is knocked in-the-money, meaning that the payoff on the call option is Asset 2 – Strike 2 and Strike 2 – Asset 2 for the put option (for the put, Asset 1 must be less than Strike 1). A higher positive correlation between the reference and the underlying asset implies that the option value increases because the comovement of both assets is highly positive, meaning that the higher the price movement of the reference asset, the higher the chances of it being in-the-money and, similarly, the higher the chances of the underlying asset’s payoff being in-the-money. Negative correlations reduce the option value as even if the reference asset is in-the-money and the value of the option of the underlying is very little in-the-money or completely out-of-the-money. Correlation is typically restricted between –0.9999 and þ0.9999. If there is a perfect correlation (either positive or negative), then there is no point in issuing such an asset, since a regular option model will suffice. Two correlated asset options sometimes are used as a performance-based payoff option. For instance, the first or reference asset can be a company’s revenues, profit margin, or an external index such as the Standard & Poor’s 500, or some reference price in the market such as the price of gold. Therefore, the option is live only if the benchmark value exceeds a prespecified threshold. This European option is exercisable only at expiration. To model an American (exercisable at any time up to and including expiration) or Bermudan option (exercisable at specific periods only, with blackout and vesting periods where the option cannot be exercised), use the Real Options SLS software to model the exotic option.

Uneven Dividend Payments Option Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Uneven Dividends Brief Description: Computes the value of plain-vanilla call and put options when the dividend stream of the underlying asset comes in uneven payments over time.

Sometimes dividends are paid in various lump sums, and sometimes these values are not consistent throughout the life of the option. Accounting for the average dividend yield and not the uneven cash flows will yield incorrect valuations in the BlackScholes paradigm.

Writer Extendible Option Location: www.ElsevierDirect.com (search for this book, click Companion Site button, click link in Additional Models section) Modeling Toolkit j Exotic Options j Writer Extendible Brief Description: Computes the value of various options that can be extended beyond the original maturity date if the option is out-of-the-money, providing an insurance policy for the option holder, and thereby typically costing more than conventional options.

Writer extendible options can be seen as insurance policies in case the option becomes worthless at maturity. Specifically, the call or put option can be extended automatically beyond the initial maturity date to an extended date with a new extended strike price assuming that, at maturity, the option is out-of-the-money and worthless. This extendibility provides a safety net of time for the holder of the option.