Exotic Options: An Overview

Transcription

1 Exotic Options: An Overview Exotic options are not new to the financial markets. Some came into existence several years before the birth of the Chicago Board of Options Exchange (CBOE) the first organized options exchange in the world, established in Trading volume for standard options was rather thin in the pre-cboe period, and for nonstandard options, it was even thinner. A few years after the establishment of the CBOE, a slow and inconspicuous revolution in option concepts and trading strategy started to take place. Towards the end of the 1970 s and the beginning of the 1980 s, when standard options trading at exchanges became better understood and their trading volume exploded, financial institutions began to search for alternative forms of options to meet their particular needs and to increase their business. All these alternative options are called exotic options. In the late 1980 s and early 1990 s, exotic options became more visible in daily presses and more popular among financial communities. Their trading became more active in the over-the-counter (OTC) market, and their users were big corporations, financial institutions, fund managers, and recently private bankers. Most of these exotic options are traded OTC, although a few have been listed in exchanges recently. The American Stock Exchange, for example, trades quanto options, while the New York Mercantile Exchange (NYMEX) trades spread options. Trading of these options in exchanges represents only a small percentage of all exotic options volume. Because of the opaque nature of the OTC market, exotic options still remain exotic to many investors, professionals, and even many of those who have good knowledge of standard options. These exotic products have been incorporated into general books to such an extent that many financial institutions now feel that they can neither live with nor without them. 1 Institutions Involved in Exotic Options Banks have responded to the loss of some of their best customers in old businesses by embracing new products and taking efforts to create new ones, trying to entice them back with a new concept financial risk management using derivatives. Most leading international financial institutions have paid a lot of attention to product development. Table 1 provides seventeen active product developer banks with established derivatives operations in London. Although product development is not solely for exotic options, exotic options represent a significant portion in product development. It should be noted that Bankers Trust, the major product developer in derivatives business, is not included in Table 1 because its product development for derivatives was predominantly undertaken in the Table 1: Active product developer banks with established derivatives operations in London. Barclays Bank Chase Manhattan Bank Chemical Bank Citicorp Investment Bank Credit Suisse First Boston First National Bank of Chicago Goldman Sachs International Hambros Bank Morgan JP Securities Head office England England Midland Montagu Morgan Stanley National Westminster Bank Noimura Bank International Solomon Brothers International Societe Generale Swiss Bank Corporation Union Bank of Switzerland Head office England England England France Switzerland Switzerland 1

2 Table 2: Leading players in the second-generation derivatives. September 1993 Bankers Trust 216 JP Morgan 43 Credit Suisse Financial Products 42 General Re Financial Products 26 Societe Generale 18 Union Bank of Switzerland 15 Mitsubishi Finance 12 Merrill Lynch 11 Swiss Bank Corporation 11 Solomon Brothers 10 Barclays Bank 8 Morgan Stanley 8 Goldman Sachs 7 September 1994 Bankers Trust 158 Swiss Bank Corporation 86 Goldman Sachs 58 Credit Suisse Financial Products 57 Union Bank of Switzerland 27 Morgan Stanley 25 Merrill Lynch 21 Solomon Brothers 13 JP Morgan 12 Societe Generale 10 Table 2 lists the institutions which are the active players in the second-generation derivatives market. The table is based on the information for risk rankings from the Risk magazine in 1993 and The rankings were done for the ten most popular second-generation derivatives, six of which being exotic options. These exotic options include Asian (average) options, spread options, lookback options, barrier options, quanto options, and compound options. The original risk rankings gave a percentage for the top three institutions which are the most active in one particular product. In order to compare various institutions easily, we provide the sums of percentages for all the ten products for each institution in Table 2. The table clearly shows that Bankers Trust, the New York based investment bank, is by far the leader in the secondgeneration derivatives. The table also shows that the dominant position of Bankers Trust declined significantly from 1993 to 1994, and other houses such as Swiss Bank Corporation, Goldman Sachs and Credit Swiss Financial Products are quickly catching up. 2 User Groups Who are the users of exotic options? Investors and asset managers Derivatives dealers Nondealer financial institutions Nonfinancial institutions (e.g., corporations) Each of these groups uses exotic options in special ways. 2.1 Investors and Asset Managers These people are on the buy side. We can further divide the buy side into asset managers and investment managers, who are close to the market, and retail investors, who are somewhat removed from the market. The investment managers sit by their Reuters screen all day, and they are connected to the market. They know what is going on every day. Retail clients have a much more passive interest in the market. They may read the evening paper and glance at quotes of stocks that they own. In terms of products we can differentiate between active 2

3 products and passive products. Active products demand participation by the client. Passive products (after they are sold) do not require active participation. Active products are more suited to asset managers and cannot be sold to retail investors. 2.2 Derivatives Dealers Derivatives dealers are interested in option premiums. The premiums for exotic options are much larger than those for vanilla options. Many banks find that they do 15% of their volume in the foreign exchange options business in exotics, and yet the exotics generate 50% of the profits. Of course, the premiums tend to diminish as the products become more common. In order to profit from these large bid-ask spreads, the dealer establishes an exotic options desk. It is the desk s responsibility to price the options correctly and also to adequately hedge the resulting exposures. Dealers who are selling exotic options must do so carefully and pay special attention to issues of pricing, hedging, and overall risk management. 2.3 Nondealer Financial Institutions Commercial banks or insurance companies frequently have needs that have to do with asset and liability mismatches. These can be handled quite well with exotics. In the case of an insurance company, money from premiums come in, and the company has these potential liabilities later on. For instance, it has to pay out when someone passes away. The insurance company has to perform asset and liability management. Can it generate potentially high-yielding assets? The nondealer financial institutions typically have yield curve risks. A commercial bank takes money from clients and places the funds in demand accounts. It credits its depositors with the interest on short-term deposits. Meanwhile the bank invests the same funds in longterm products. The bank receives proceeds based on the long end of the yield curve. This strategy is called riding the yield curve. The bank borrows short and invests long. As long as the yield curve is upward sloping, things are going to be OK. What if it flattens a little or even inverts? Then the bank will suffer losses. Exotic options may be used to hedge these risks. 2.4 Corporations Corporations use exotic options for two primary purposes: To generate cost-effective funding. 1 To create complex hedge structures to match underlying exposure. 2 Cost-effective funding: The bond originator sells structured notes to the public. When an investor buys the security, the investor is selling an option to the originator. The price of the option is usually very much reduced. The bond originator then strips the embedded options (usually in collaboration with the underwriter). The options will be sold at fair value in the secondary market. The difference between what was paid for the options and what was received for them represents extra cash inflow to the issuer. This in turn equates with sub-libor funding. Many times these structures are done as medium-term notes. Hedging of corporate risks: The science of risk management has become much more sophisticated and a lot more fine-tuned. This is where the banks or dealers can help the corporations. 1 In July 1993, Benetton issued L200 billion of bonds with knock-out warrants attached as part of its debt restructuring. Details of the issue can be found in Bennett [1] and Fisher [5]. 2 Two comprehensive examples on using exotic options in hedging are presented in Turner and Attrill [7]. 3

4 Suppose a company sells in many different countries and now is expanding into new markets in several Eastern European countries and some former Communist countries that it couldn t sell to before. The company must face all kinds of different risks that it was not used to handling before. As it enters new markets, it acquires foreign currency exposures, interest rate exposures, etc. There are also political and other risks to consider. Exotic options are products that can precisely offset the cash flows being hedged Asian options and basket options are good examples. There are also correlation-based hedges that have to do with the correlation between fixed income and equity or between currency and equity. Corporations are especially interested in low-cost hedging strategies. An example of a customized hedging tool is the crack spread option. Consider an oil refinery, a plant that converts raw oil into heating oil. The refinery is not concerned with the price of crude oil or the price of heating oil. The refinery wants to know that it can sell the difference for $2 a barrel. So if crude oil costs $18 per barrel and heating oil costs $20, then the refinery makes $2 per barrel and it is happy. Even if crude is at $25 and heating is at $27, the refinery is still happy. But it is not going to be happy if crude is at $18 and heating is at $19. The NYMEX offers the crack spread options, which are options on the difference. The use of customized hedging tools has to do with increasingly precise dissection and management of corporate risk. Thus a company might use a basket option to hedge its exposure to many different currencies using a single trade. In addition, from a taxation standpoint, companies like to utilize zero-cost or low-cost strategies. 3 Key Applications The key applications of exotic options are: Yield enhancement Proprietary trading/positioning 3 Structured protection Premium reduction strategies We now review these applications in detail. 3.1 Yield Enhancement Interest rates are at an all-time low almost everywhere in the world. This spells trouble for investors who want to achieve reasonable rates of return on their capital. For many years, investors have gotten used to earning respectable yields on highly rated fixed income securities. These high yields have all but disappeared. In addition, many funds have promised their investors a minimal return on their investments. For example, some Japanese funds have promised their investors yields of 5% or better. The yields on highly rated Japanese bonds are about 80 basis points. One way to obtain higher yields is to invest in various structured notes and hope for the best. Such structured notes are bonds with an increased coupon but their principal might be subject to increased market risk. Thus at the maturity of the structured note, the investor may lose part of the principal. For taking on that risk, the investor receives a high coupon during the life of the note. 3 The Benetton bond issue in July 1993 also illustrates how exotic instruments can be used to take advantage of the bullish view of [Benetton s] stock. For details see Bennett [1] and Fisher [5]. 4

5 In equities there is an increased demand for global funds that have the mandate to invest in many different countries. Some Tactical Asset Allocation (TAA) funds may shift their assets from country to country. Each month they determine the best market to be invested in, and then they shift their portfolio to be heavily invested in that market. For example, one month the fund may choose to invest in the index, the next month in Norway, the third month in Thailand, and so on. Such funds rarely shift their portfolios in the cash market. This is due to the high bid-ask spreads and the large transaction costs. Such funds will typically replicate the shift in allocation of their portfolio between the different indices by using equity swaps. Equity swaps have allowed such funds to rapidly shift their portfolios to be heavily weighted in the markets they think will outperform. However, even if a portfolio manager chooses to invest in a particular country, they may not want to be exposed to the currency of that country. In such situations, the quanto and flexo options may be used. These options allows investors to reap the returns from an equity investment while being isolated from the foreign exchange movements. Another situation arises in which the portfolio manager is unable to choose between two market indices. In such cases, the portfolio manager may decide to purchase a best of option whose payout will be tied to the performance of the better index. In this fashion, the portfolio manager has captured the return of that index which will have the better performance at the end of the month. With the current emphasis on emerging markets and international trading, clients are very interested in index swaps, quanto options, best of options, and much more. 3.2 Proprietary Trading/Positioning Exotic options can also be applied to proprietary trading. Assume that a client wants to take a position on the European currencies converging. The client wants to ensure that their riskreward profile is going to be reasonable. If they are correct, they want to make a lot. If they are wrong, they wish to lose little. Here is an example. If the currencies are converging toward each other, the volatility of the exchange rate declines. Assume that the client wants to be short volatility because they think currencies are converging. Traditionally a dealer would advise a client to be short a straddle. The client can sell a call and simultaneously sell a put to be short a straddle. The client has a short position. If the client is correct and the underlying does not move, then they earn both of the premiums. That is great. If the client is wrong and prices do move up or down, then the client will be in really big trouble because they are short options. In our example, the underlying is at $100 and the one-year straddle costs $ If, at the end of the year, the underlying will stay at $100, the client will have earned the entire premium of $ Even if the underlying moves a bit to $105, say, the client will still come out ahead. The client receives $11.58 and pays $5. But what if the underlying ends up at $150? In that case, the client will be obliged to pay $50 a big loss. Now assume that the client wants to be short volatility but they want a different risk-reward profile. Wall Street came up with the idea of the double barrier option. In our case, the client pays $4.30 and receives two double barrier options. The double barrier box will pay the client $112 $88 = $24 if, throughout the entire year, the underlying has not touched either $88 or $112. If the underlying touches either of these values, the double barrier pays nothing. In our example of a short straddle position, a client earns $11.58 to potentially lose an unlimited amount. In a double barrier box trade, the same client spends $4.30 to potentially earn $24. This has a more interesting risk-reward profile. If a client wants to have a specific trading strategy, then some of these exotics really make sense. 5

6 3.3 Structured Protection A third key application of exotic options is structured protection. This would be useful for a company like Gillette. Gillette makes shavers in different countries and sells them in other countries. It has foreign currency exposures in thirteen different currencies. How can we give it an umbrella protection against its exposure, not in each currency at a time but over all? A basket option is very useful in this regard and, in fact, is one Gillette used. To generate even more savings, Gillette chose to do an Asian option on the basket. Structured protection is much harder to sell to a client than standard protection. In this case, Gillette does an Asian average rate option on the basket currencies and was able to save a lot money with this scheme. A dealer suggested the same idea to another company, which also has operations in many countries. And, like Gillette, it has foreign currency exposures in different countries. However, this company said no. It turns out that the internal accounting in this company is a bit different. Gillette takes all cash flows, nets them out, and reports their profits in dollars. For accounting purposes, this company separates its units. For example, it has a European unit and a Japanese unit. Suppose the European unit made money and the Japanese unit lost money because of exchange rate movements. The European unit doesn t want to give up its profits in order to compensate the unit in Japan. On the other hand, the European unit still wants foreign currency protection. The challenge is up to the dealer. When he offers an exotic option product to a client, he has to tailor it very carefully to the type of company he is dealing with. The client only cares about risk management, low-cost tailor-made products that fit its needs. 3.4 Premium Reduction Strategies The last application we will deal with here is premium reduction strategies. Many corporations use a continuous hedging process. They are not just buying an option and wanting to double their money. They are constantly hedging and rehedging. The hedging department in a corporation is typically construed as a cost center that doesn t bring in any money but just spends money on option premiums and other hedging costs. How can the company reduce that cost? In many jurisdictions, when a company buys an option, the option premium gets deducted from one account. But when the option pays out, it just gets credited in the general account. So there are tax and accounting issues on paying a premium and receiving the payout. There are tax and regulatory incentives for companies to use options with zero cost. Be aware that many clients do not wish to pay very much for options. This becomes obvious when you examine which of the exotic options have been successful: the barrier, the Asian, and the basket. Note that the lookback doesn t make this list. A lookback option has a very nice return, but the lookback is very expensive. You have to pay such a high premium for it that it scares clients off. What really works for them is something that has a lower premium than a European option. It turns out that premium reduction is a very important theme. 4 Building Blocks of Exotic Structures Exotic options can be distinguished from vanilla ones on one or more of the following six dimensions: (1) time-homogeneity of the structure, (2) continuity of payoff, (3) presence of barriers, (4) number of assets in the structure, (5) order of the option, and (6) path-dependence. 4 4 See Figure 2.1 (The general option) of Taleb [6]. 6

7 We will find it useful to break down exotic-option instruments into a number of categories, namely: 5 Payoff modified options Time/Volatility-dependent options Correlation-dependent/Multifactor options Path-dependent options We list and characterize some important members of each class. 4.1 Payoff Modified Options These are options where the payoff under the contract is modified from the conventional return, which is either zero or the difference between the strike price and the asset price at maturity. Several structures of interest exist, including: Digital options, which pays a fixed amount if the underlying asset is above or below a given level at maturity of the option. Contingent-premium options, which allow the linking of the option premium to be paid to the asset-price performance. Power options, where the payoff under the option is an agreed multiple of the return under a conventional option. 4.2 Time/Volatility-Dependent Options These can be characterized as options where the purchaser has the right to nominate specific characteristics (e.g., type of option put or call) as a function of time. The value of these options is particularly sensitive to volatility over a period which begins not now but in the future. These options are particularly useful when there is some event which occurs in the short term which will then potentially affect outcomes further in the future. Time/volatility-dependent structures include: Chooser (or preference) options, which are not specified as either a call or a put until, at a predetermined date, the purchaser can nominate whether or not the transaction is a call or a put option. Compound options, which are options on options where the holder has the right to buy or sell another predetermined option at a pre-agreed price. Forward start options, which are essentially vanilla options except that the strike price is not set until some date in the future. The strike price will be set as some function of the prevailing asset price on that date. 5 The classification is loosely based on Dehnad, James, and MeVay [4], which also offers examples on using exotic options to solve mundane problems. 7

8 4.3 Correlation-Dependent/Multifactor Options These typically involve a pattern of payoffs based on the relationship between multiple assets as opposed to the price, in the case of a traditional option, for single assets. A variety of structures exist, including: Basket options, where the payout under the contract is related to the cumulative performance of a basket of products. Exchange options, which give the purchaser the right to exchange one asset for another. Quanto options, where the option contract is denominated in a currency other than that of the underlying asset to which exposure is sought or being hedged. Rainbow options, where the payout is based on the relationship between multiple assets as opposed to the price of a single asset. Specific examples include outperformance (or better-of-two-assets) options and spread options. 4.4 Path-Dependent Options Path-dependent options are characterized by payoffs which are a function of the particular continuous path that asset prices follow over the life of the relevant option. The path of the underlying asset price can determine not only the payoff but also the structure of the option. Path-dependent structures include: Average rate options, where the payoff upon settlement is determined by comparing the strike price with the average of the spot asset price over a specific period during the life of the option. Average strike options, where the strike itself is not fixed and the payoff is determined through a comparison of the underlying price of expiration, with the strike price computed as the average of the underlying asset price over a specific period. Barrier options, whereby the option contract is activated or deactivated as a function of the level of the underlying asset price. The most basic forms of barrier options are commonly known as knock-in and knock-out options. One-touch options, where the payoff is a fixed amount if the underlying asset ever trades above or below a given level on any day during the lifetime of the option. A related option is the digital option (see later). Lookback options, where the purchaser has the right at expiration to set the strike price of the option at the most favorable price for the asset that has occured during the specified time. In the case of a lookback call (put), the buyer can choose to purchase (sell) the underlying asset at the lowest (highest) price that has occurred over a specified period (typically the life of the option). Besides the floating strike variety, lookback options also come in a fixed strike variety. 5 Normal Distribution We begin by reviewing the concepts of means and standard deviations. The mean or expectation for a continuous random variable X with probability density function (PDF) f(x) is defined by µ X = E(X) = xf(x)dx. 8

9 The variance of X is defined as the expectation of the variable s squared deviation from its mean, given by { [X E(X) σx 2 ] } 2 = Var(X) =E = (x µ X ) 2 f(x) dx. It follows that variance is positive for any random variable. It is zero for any deterministic variable. The standard deviation of a random variable is the positive square root of its variance. If X has a normal distribution with parameters µ and σ 2, then the PDF of X is f(x) = 1 σ 2π exp [ (x µ)2 2σ 2 ], <x<. It can be shown that E(X) =µand Var(X) =σ 2,soµis the mean of X and σ 2 the variance. Next, let Z =(X µ)/σ ( standardization ). Then Z has a standard normal distribution with zero mean and unit variance. Through a change of variable, the PDF of Z is n(z) = 1 e z2 /2, <z<. 2π Thus, any normal distribution can be related to the standard normal distribution in a simple way. We define the standard normal cumulative distribution function (CDF) by N(z) = z n(t) dt = z 1 2π e t2 /2 dt, <z<. The standard normal CDF N(z) is widely tabulated. 6 Black-Scholes Model Although there are many option pricing models, the Black-Scholes lognormal model is still by far the most popular one. To some degree, the Black-Scholes option pricing model has facilitated option trading and helped the growth of the whole financial derivative market. In a typical Black-Scholes environment, the market is assumed to be frictionless and the underlying asset return is assumed to follow a lognormal random walk. Suppose the underlying asset price S t follows the geometric Brownian motion: ds t = µs t dt + σs t db t, (1) where B t is a standard Brownian motion, 6 and µ and σ are the instantaneous mean and standard deviation of the underlying asset price, respectively. The instantaneous standard deviation σ is more often called the (annualized) volatility of the underlying asset. Solving equation (1) yields ) ] S t = S 0 exp [(µ σ2 t + σb t. (2a) 2 Taking natural logarithm on both sides of (2), we can readily obtain { } ) St X t =ln = (µ σ2 t+σb t. 2 S 0 (2b) Thus, X t is normally distributed with mean (µ σ 2 /2)t and variance σ 2 t. 6 A standard Brownian motion can be thought of as a normal variate with zero mean and variance which equals the length of elapsed time. Thus, B t is normally distributed with zero mean and variance t. 9

10 6.1 Historical Volatility Suppose we observe the asset prices at regular intervals of length t (e.g., for daily sampling, t = 1/248 for trading year or 1/365 for calendar year). The sampled asset prices are S 0,S 1,...,S n. Let x j = ln(s j /S j 1 ). 7 Then by (2), x 1,...,x n represent independent realizations from a normal distribution with variance v 2 = σ 2 t. Empirical studies have shown it reasonable to assume zero mean for the distribution. We can now estimate v 2 using ˆv 2 = 1 n n x 2 j. The estimated annualized volatility is therefore given by ˆσ =ˆv/ t. Thus, for daily sampling, we have ˆσ =ˆv 248 or ˆv 365 A variant of the historical volatility is the exponentially weighted (EW) historically volatility. In the latter, the formula assigns importance to recent observations in proportion to their distance away from the present. Using a decay factor of λ<1, the EW estimate of v 2 is ˆv 2 λ = 1 ω n j=1 j=1 λ n j x 2 j = x2 n + λx 2 n 1 + λ2 x 2 n 1 + +λn 1 x 2 1, ω where ω =1+λ+λ 2 + +λ n 1 1/(1 λ) for large n. The EW volatility estimate is therefore ˆσ λ =ˆv λ / t. 6.2 Implied Volatility Implied volatility is that value the volatility parameter in the Black-Scholes formula (5) must take so that observed option price is the same as the Black-Scholes price. By evaluating the implied volatilities using options of different maturities and moneyness, one can obtain a volatility surface for the underlying asset, which represents the implied volatility as a function of time to expiration and the strike price. 7 Valuation of Vanilla Options As a review, we consider the vanilla call option with strike price K, whose payoff at expiration T is (S T K) + := max{0,s T K}. In other words, the holder of the call option receives on the expiration date the amount S T K if it is positive, and zero otherwise. 7.1 Partial Differential Equations Utilizing arbitrage-free arguments, it is possible to derive second-order partial differential equations (PDEs) that need to be satisfied by option prices. Along with the corresponding boundary conditions, we can solve the PDEs to obtain the pricing formulas for the options. For example, by considering the portfolio of a call option and its underlying asset in such proportions that the instantaneous return on the portfolio is non-stochastic, we can show that the value of a European call option is a solution of the following PDE: rc + C t + rs C S σ2 S 2 C S 2 =0, 7 If the interval length t is small, we can equivalently use x j =(S j S j 1)/S j 1. 10

11 subject to the boundary condition on the final payoff: C(S, T )= S K for S K, and 0 otherwise. PDEs like this can then be solved with either analytical or numerical methods. Although analytical solutions are beautiful and convenient, this method is very limited in practical use. From the rapid progress in computer technology, a lot of studies have been done to solve PDEs numerically. We present here a finite-difference method to solve PDEs to obtain option prices. This method has been widely used in pricing various kinds of complex derivatives. Finite-difference approximation: The basic idea is simply to replace partial derivatives used in any PDEs with their corresponding finite-difference approximations. Specifically, we write f(x, y) y 2 f(x, y) y 2 f(x, y + δy/2) f(x, y δy/2), δy f(x, y + δy) 2f(x, y)+f(x, y δy) (δy) 2. We solve the resulting finite-difference equations by building a lattice on which approximate values of the desired variables can be obtained. Although the finite-difference method is powerful enough to solve most of the problems in pricing derivatives products, it suffers from the lack of intuition. Unlike the tree-based methods (see later), intuition is somewhat buried in the mathematics or computer programs. The method is like a black box which is supposed to give correct answers to some specific problems. Yet whenever problems arise, they are not as easily fixed as with the tree-based methods. 7.2 Risk-Neutral Valuation Relationship From the structure of our option valuation model, it follows that whenever a portfolio can be constructed, which includes a contingent claim and its underlying asset in such proportions that the instantaneous return on the portfolio is non-stochastic, the resulting valuation relationship is risk-neutral. Under risk neutrality, values of any contingent claim do not involve any parameters of investor s preferences such as risk aversion. In fact, the expected return of the underlying asset µ in equations (1) and (2) must be the same as r q if the underlying asset pays dividends at the rate q. Essentially, the values of European options can be obtained by discounting the expected payoffs of the options at maturity by the risk-free rate of return r, i.e., initial value = e rt E [ payoff T ]. (3) As an illustration, the expected payoff of the European call option is E [ (S T K) +] = ln(k/s 0 ) (S 0 e x K)f(x) dx, (4) where f(x) is the PDF of the normal distribution with mean (r q σ 2 /2)T and variance σ 2 T. The integral in (4) can be evaluated via standardization (i.e, using the change of variable z =(x (r q σ 2 /2)T )/σ T ) to give S 0 e (r q)t N(d 1 ) KN(d 2 ), where d 2 = ln(s 0/K)+(r q σ 2 /2)T σ T Upon discounting, we recover the Black-Scholes formula: and d 1 = d 2 + σ T. C = S 0 e qt N(d 1 ) Ke rt N(d 2 ). (5) 11

12 7.3 Monte Carlo Simulations Monte Carlo methods involve generating large numbers of numerically simulated realizations of random walks followed by the underlying asset prices, consistent with equation (2), and these simulated realizations are used to price derivative products through a modification of equation (3): by averaging over all simulated values of payoff T. Monte Carlo simulation is simple and flexible in that it can be easily modified to accommodate different processes governing the underlying instrument movement. The use of Monte Carlo simulations to price path-dependent derivatives has increased because products have become more complex in nature and it is difficult to obtain closed-form solutions for many of these complicated products. Another obvious advantage of the Monte Carlo method over other procedures is that it can value derivative products with several underlying assets more efficiently. However, the potential drawback of the Monte Carlo method is that the standard error of the estimate is inversely proportional to the square root of the number of simulation trials. Although any desired level of accuracy can be obtained by increasing the number of simulated trials, there are more efficient ways to reduce the standard error. There are two techniques often used in simulations which can reduce variances quickly, namely the control-variate method and the antithetical variate method. These techniques are normally called variance-reduction techniques. 7.4 Lattice- and Tree-Based Methods The binomial model was originally developed to price standard options. It was then extended to the trinomial tree model in which the underlying asset is assumed to follow three different paths in each following period. Since the late 1970 s, the lattice- and tree-based methods have been widely used in pricing essentially all kinds of derivative products, especially path-dependent and other complicated products such as interest-rate derivatives involving the term structure of interest rate. It has become a powerful and efficient method because of its intuitive nature. We concentrate on the recombining tree model, where the total number of upward moves and that of downward moves determine a path completely (as illustrated in Figure 1). A popular choice of tree parameters is given by u = e σ, d =1/u, and p = probability of an upward move = er d u d, where = T/n in an n-step tree. The recombining tree is often used to price many derivatives. Specifically, we evaluate payoff T at the terminal nodes and work backwards through the tree to get the values of intermediate nodes as p value u +(1 p) value d, where value u is the value at the next up-node and value d is the value at the next down-node. 8 Hedging Vanilla Options An option s delta measures how fast an option price changes with the price of the underlying asset. Mathematically, an option s delta is the first-order partial derivative of the option price with respect to the price of its underlying asset. The European call option s delta can be obtained from equation (5) as follows: = C S = e qt N(d 1 )+Se qt n(d 1 ) d 1 S Ke rt n(d 2 ) d 2 S = e qt N(d 1 ), 12

13 S u 4 S u 3 S u 2 S u 3 ds us u 2 ds uds u ud ds d 2 d 2 S 2 S ud 3 S 2 S d 3 S d 4 S Figure 1: Price movements in a 4-period binomial model. since Se qt n(d 1 )=Ke rt n(d 2 ) and d 1 / S = d 2 / S. Gamma is another important sensitivity measure for an option. It measures how fast the option s delta changes with the price of its underlying asset. For a European call option, the gamma is Γ= S = e qt n(d 1 ) d 1 S = e qt n(d 1 ) Sσ T. We now introduce two important concepts in option trading: delta hedging and gamma hedging. Delta hedging is a trading strategy to make the delta of a portfolio neutral to fluctuations of the underlying asset price. For example, the portfolio which is long N(d 1 ) units of the underlying asset and short one unit of a European call option on the underlying asset is a well-known example of delta hedging. Here, N(d 1 ) is evaluated at the prevailing price of the underlying asset. If a portfolio has a positive (negative) delta δ, to carry out delta hedging, we can simply sell (buy) δ/ =δ/n(d 1 ) units of call option so that the delta of the portfolio will be zero. Whereas delta hedging is to make the delta of a portfolio neutral to the fluctuation of the underlying asset price, gamma hedging is to neutralize the gamma of a portfolio or to make the gamma of the portfolio zero. In the above example of delta hedging, the portfolio is always delta-hedged, yet it is not gamma-hedged because its gamma is e qt n(d 1 ) Sσ > 0. T If a portfolio has a positive (negative) gamma γ, to carry out gamma hedging, we sell (buy) γ/γ=γsσ T/n(d 1 ) units of call option so that the gamma of the new portfolio will be zero. A portfolio may not be gamma-hedged when it is delta-hedged, as our example showed, or it may not be delta-hedged when it is gamma-hedged. This is because when we change the composition of the portfolio to achieve the goal of either delta hedging or gamma hedging, the 13

14 other is changed at the same time. However, this is not a serious problem because the need for one hedge often dominates the other, so it is alright to consider the more important issue and hedge it consequently. 9 Summary The emergence of exotic-option products has added an extra dimension to derivatives markets. However, a number of barriers to the further development of these markets exist: A lack of understanding and knowledge about these products, in particular about the valuation, hedging and trading of these instruments. The market for most of these products is characterized by few market-makers and low levels of liquidity. The lack of standardization of: (i) terminology, (ii) terms of quoting and trading conventions, (iii) pricing models and methodology. The relatively underdeveloped stage of application of these products to the management of financial risks encountered by corporations and investors. Further development of the market would be assisted by addressing these deficiencies. 8 The market for exotic-option products has an importance in capital and derivatives markets out of proportion to its size. Not only do these products offer insights into the trading, valuation and risk management of more traditional derivatives, they also inherently isolate and facilitate trading of additional levels of exposure to risk. This process of risk dissection and its contribution to the identification and analysis of various types of risk, such as correlation risk, has the potential to significantly enhance our current understanding of risk and its management. References [1] Bennett, R. (1993). Benetton Clothes Naked Exposure. Euromoney (Jul 93), [2] Brady, S. (1994). Handle Exotics with Care. Corporate Finance (Mar 94), [3] Brady, S. and H. Murphy (1993). Exotic Cures for Currency Headaches. Corporate Finance (Oct 93), [4] Dehnad, K., M. S. James, and J. C. F. MeVay (1992). Mundane Problems, Exotic Solutions. Euromoney (Aug 92), [5] Fisher, M. (1993). Benetton Styles a New Lira Bond. Corporate Finance (Aug 93), 8. [6] Taleb, N. (1997). Dynamic Hedging: Managing Vanilla and Exotic Options. New York: John Wiley & Sons. [7] Turner, M. and R. Attrill (1993). Exotic Currency Options. Corporate Finance Derivatives Supplement (May 93), For a survey of some of these and other problems when dealing with exotic options, the reader is referred to Brady and Murphy [3] and Brady [2]. 14