Valuation of structured products.
While the academic literature has offered a variety of approaches for handling these features, no single valuation approach is appropriate for all structured products, and the merits of each approach are not always. obvious. (2) In this article we review the four primary approaches for valuing structured products, and offer our experience with valuing more than 20,000 products released over the past several years. We outline the advantages and disadvantages of each approach and identify the types of products most appropriate for each method. We stress that while similar valuations can often be obtained using several approaches, each approach offers different insight into the composition of the
We especially highlight how each approach handles the ,credit risk of the issuer. The 2008 bankruptcy of Lehman Brothers, one of the largest issuers of structured products, offered .a dramatic demonstration of the importance of incorporating credit risk into product valuations. While it is widely recognized in the academic literature that structured products are sold at a substantial premium, (4) the magnitude of credit risk in those valuations is less appreciated. Investors may continue to purchase structured products despite the known price premium because they do not fully appreciate the inherent credit risk and mispricing of these investments. (5) We describe how credit risk is incorporated into each type of structured product valuation method, allowing practitioners an informed choice of valuation approaches.
In the following section we describe several complexities of structured products that can affect the choice of valuation approach. We then explain the four valuation approaches and how they can handle each of these product features. The next section walks through the implementation of each of these approaches to value a common type of structured product, a Morgan Stanley Buffered PLUS. The last section presents our conclusions.
COMMON STRUCTURED PRODUCT FEATURES
Structured products have varied and complex payoff structures. In this section we describe some of the characteristics that differentiate products.
Structured products are very often tied to the S&P 500, NASDAQ 100, or other broad stock index, but can also be linked to a very wide variety of other asset classes. Exhibit 1 shows the distribution of structured products issued from 2010 to 2012 among various asset classes. (6)
It is important for investors to thoroughly understand the underlying asset or index to which a structured product is linked. Indexes are now available that use extremely complex and sophisticated investment strategies that can lead to unexpected risk and return profiles. For example, a 2008 structured product from Lehman Brothers was linked to an index that tracked a complex bundle of commodities that dynamically allocated between different assets based on conditions in their respective futures markets. (7) While this product had a simple payoff structure, the underlying index involved complex .calculations of the returns to many assets, making this type ofproduct particularly risky and difficult to model. (8) We have noted that more and more structured securities, including structured products and structured certificates of deposit, have been linked to proprietary indexes developed and calculated by issuers.
Valuation is easier when the underlying asset of a product has both an observable daily market price and an actively traded options market, as such assets have readily accessible correlation and implied volatility estimates. Other assets may require complex forward volatility and correlation models which can be difficult to implement and calibrate; in this article, we assume that the underlying assets have reliable implied volatility and correlation information.
Structured products, like common derivatives such as options, generally link to the price of underlying assets, but do not give investors any dividends paid by those assets. For notes linked to assets paying high dividend yields, the payoffs are generally lower than they would be if the structured product were linked to the total return of the underlying asset-rather than its price return.
Principal protection absorbs a portion of the linked security's capital losses. Principal protection always requires that the structured product be held to maturity, and is void if the issuer defaults on the product. If a product is 100% principal-protected, the investor will not lose money on the investment as long as the product is held to maturity and the issuer does not default. A product with more than 0% but less than 100% principal protection is called a "partial principal-protected" or "buffered" product.
Principal protection and buffers are easily modeled with any of the approaches discussed here. Perhaps the most intuitive method is decomposition, where full principal protection on a long position could be modeled as an at-the-money put option. A buffer feature would then correspond to an additional short put option, where the degree of buffering could be thought of as the out-of-the-moneyness of the option. Our discussion of the Buffered PLUS product later in this article includes treatment of buffers using all four approaches.
Some structured products have triggers, or specific return levels that yield contingent outcomes. For example, Citigroup's ELKS are simple bonds unless the trigger is tripped (the underlying security's returns surpass a predefined value), at which point the ELKS becomes a forward contract on the linked security. Other products, like absolute return barrier notes ("ARBNs," issued by UBS, HSBC, Deutsche Bank, and Lehman Brothers) have payoffs that depend on the size--but not the direction--of the linked security's return. (9) As long as the linked security's price does not go up or down by more than a prespecified amount, the structured product pays the absolute value of the return. If the linked security's capital gain or loss exceeds the trigger, the investor earns a 0% return, regardless of the linked security's future price movement. Most contingent claims, such as the ones in Buffered PLUS products, are clearly defined at a particular return level and can therefore be incorporated into any valuation approach.
Relatively few non-interest-rate-linked structured products pay coupons. Of those that do, most also have contingent claims, such as reverse convertibles and auto-callables. Products without coupons instead rely on the linked security's capital gains to provide investors with a positive return. Coupon payments can be incorporated into any valuation approach, though for highly complex coupon payout formulas the simulation approach can often handle this feature most explicitly.
Some structured products can be called by the issuer. Callable products generally guarantee a specified rate of return. Some products are called at the discretion of the issuer, while others are autocallable. Autocalla.ble products are automatically called if a given criteria is satisfied on a predefined call date.
Products can have embedded American or Bermudan call options. American options can be exercised by the issuer any time before the structured product's maturity date. Bermudan options can be exercised on specific dates during the life of the product (e.g., quarterly). The final call date is always the product's maturity date.
Call features can be among the most dificult product features to value. Autocallable products are relatively easier, as the call feature is predictable, but discretionary call features necessitate simulation approaches. A rigorous implementation of a simulation-based valuation for callable interest-rate-linked structured products can be found in Andersen and Piterbarg .
Structured products can give investors leveraged (or deleveraged) exposure to the underlying security's returns. Typically, the leveraged returns are only for a certain subset of possible returns, such as leveraged upside for returns between 0% and 10%. The decomposition approach illustrates this most clearly, as the issuer can just purchase more of the call or put option that generates the relevant return (two long call options, for example, generate 200% participation).
Some products are linked to combinations ("baskets") of two or more linked securities. The payoff of the product is normally linked to the basket as a whole. However, some products allow the issuer to use the worst-performing member of the basket to determine the product's payoff.
When products are linked to baskets, any valuation must also take into account the co-movement among the basket members. There is considerable leeway in calculating this co-movement, but it is typically the correlation of daily returns of the underlying basket members over a reasonably large period of time, perhaps three months. (10) The exact length of time is a subjective judgment, however, as the correlation structure between basket entities can change with time, but a suitable sample size must be achieved.
A structured product may have several or none of these characteristics. For .example, Morgan Stanley's Performance Leveraged Upside Security (also called a PLUS) pays no coupon and is usually not callable. The PLUS. exposes investors to all the downside risk of a single linked security along with levered, but often capped, upside potential.
Some PLUS products offer partial principal protection. This partially principal-protected PLUS has been sold as Buffered PLUS by Morgan Stanley, a Partial Protection Return Optimization Security by UBS and Lehman Brothers, and an Equity Buffer Note by HSBC. For continuity, we will use a Buffered PLUS as the example throughout this article.
Different product types have somewhat different return characteristics. In Deng et al. [2014b] we calculated an index of ex post structured product returns by valuing more than 20,000 products each day from 2007 through 2013. We also calculated sub-indexes based on four common structured product types: autocallables, tracking securities, reverse convertibles, and single-observation reverse convertibles.
Our results for both the aggregate and sub-indexes demonstrate that structured products as a whole have high correlations with equity markets, though lower returns. Those lower returns are in large part attributable to the issue date mispricing.
In this article, we discuss four structured product valuation approaches: (1) simulation, (2) numerical integration, (3) decomposition, and (4) partial differential equations (PDE). While these approaches produce consistent valuations for simple products, not all approaches
arc appropriate for all products, and the approaches can vary in complexity and difficulty of implementation.
In general, structured product payoffs can be described as functions of the underlying security's. level (e.g., stock price) or return. For example, the payoff rule of a Buffered PLUS is a function of the underlying security's level or holding period return at the structured product's maturity. In this article, we use P([S.sub.T]) as the functional form of payoff rules that are a function of the underlying security's ending stock price.
Because the underlying security's holding period return, [R.sub.t], is a function of the security's stock price
[R.sub.t] = ([S.sub.t] - [S.sub.0]) / [S.sub.0]
the functional form of the payoff rule can also be expressed in terms of the underlying security's return. We use f([R.sub.T]) as the functional form of payoff rules that depend on the underlying security's return over the life of the structured product, where P([S.sub.T]) = I * (1 + f([R.sub.T])) and I is the face value of the structured product. The Buffered PLUS' payoff rule, graphed in Exhibit 2, can be expressed algebraically as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
Assumptions Common to all Four Approaches
Each of the four approaches discussed in the following sections relies on a common set of assumptions. In this section we briefly discuss each assumption.
Generalized wiener process of stock prices. We assume stock prices follow a generalized Wiener process with a fixed drift (Glasserman ; Hull ; McLeish ). The Wiener process is denoted as.
d[S.sub.t] = [S.sub.t] [mu]dt +[S.sub.t] [sigma]dZ, (2)
d ln[S.sub.t] = ([mu] - [[sigma].sup.2] / 2)dt + [sigma]d[Z.sub.t] (3)
such that [S.sub.t] is a geometric Brownian motion.
The generalized Wiener process results in an ending stock price [S.sub.T] that is log-normally distributed
[S.sub.T] ~ [S.sub.0] * Log - N (([mu] - [[sigma].sup.2] / 2) T, [sigma][square root of T]) (4)
which is equivalent to
ln([S.sub.T]) ~ ln[S.sub.0] + N(([mu] - [[sigma].sup.2] / 2) T, [sigma][square root of T]) (5)
where N([^[mu]], [^[sigma]]) is a normal distribution with mean [^[mu]] and standard deviation [^[sigma]]. This directly implies that the holding period return [R.sub.T] = ([S.sub.T] - [S.sub.0]) / [S.sub.0] must also be log-normally distributed
[R.sub.T] ~ Log - N (([mu] - [[sigma].sup.2] / 2) T, [sigma][square root of (T)]) - 1 (6)
The underlying security's expected return. As with the Black-Scholes model, we value structured products in a risk-neutral framework (Bjork ; Hull ). Consequently, the expected annualized return for a structured product's underlying security is based on the risk-free rate r. (11) Because structured product payoff rules ignore dividends, we follow the dividend modification of the Black-Scholes model and reduce the
risk-free rate by q, the dividend yield on the underlying security. This means that the underlying security's risk-neutral expected return, [mu], is
[mu] = r - q (7)
The underlying security's implied volatility. The volatility, [sigma], of the linked security's return is typically estimated as the volatility needed to make traded options fairly priced using the Black-Scholes model. Where possible, we match the time to expiration of the option and the structured product. In cases where the structured product's remaining term falls between two option maturities, we use linear or quadratic interpolation to estimate the implied volatility. In the rare case when no options can be found for the underlying security, we use the underlying security's historical volatility.
Assumptions about the discount rate.
To estimate the issue date value of a structured product, the expected cash flows must be discounted back to time t= 0. Because the structured product is exposed to both the risk of the underlying security and the issuer's credit risk, we include the issuer's credit default swap spread (CDS) in the discount rate (Hull ). We match the term of the CDS with the term of the structured product.
The Simulation Approach
Simulation has been proven to be a practical, useful tool for valuing financial products (Glasserman 120031; McLeish ). Of the four approaches discussed in this article, the simulation approach is the most "brute force," in. that it relies on computing power to perform Monte Carlo simulations of the linked security's price path. The Monte Carlo results are then input into the product's payoff rule to calculate the exact cash flows that would result from each simulated price path.
The structured product's estimated fair value is the average of the discounted cash flows derived from the simulations. For structured product valuations, we use widely accepted financial models to simulate security levels and returns, interest rates, and exchange rates.
To simulate stock prices following the generalized Wiener process, suppose we track stock prices at discrete time intervals [S.sub.0],[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] , ... , [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], ... , [S.sub.T], and constant interval
[[DELTA].sub.t] = [t.sub.i+1] - [t.sub.i]
At each step, we assume the stock price updates according to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] * [e.sup.[mu][DELTA]t + [sigma][square root of ([DELTA]t)][W.sub.i]] (8)
where [W.sub.i] is a standard normally distributed variable.
To value a structured product, we typically simulate J = 50,000 price paths of the underlying security from t = 0 to t = T. On the jth trajectory, suppose the ending stock price is Sir and the holding period return is [R.sub.T.sup.j]. The structured product's payoff for the jth simulated price path is calculated by inputting [S.sub.T.sup.j]. into the mapping rule P([S.sub.T]) or [R.sub.T.sup.j] into f([R.sub.T]). We repeat the process for each of the 50,000 simulated price paths and calculate' the average payoff. The average payoffis then discounted back to time t = 0 as shown in Equations (9) and (10).
PV([S.sub.T]) = [e.sup.-(r + CDS)T] 1/J [J summation over (j = 1)]P([S.sub.T.sup.j]) (9)
PV([R.sub.T]) = [e.sup.-(r + CDS)T] I [1 + J summation over (j = 1)]f([R.sub.T.sup.j]) (10)
The discounted expected payoff is the issue date value of the structured product.
Except for products with explicit American or Bermuda options, we have been able to value any structured product using the simulation approa-ch. (12) In addition to being versatile, the simulation approach is intuitive and easy to understand. The benefits of the simulation approach are offset, however, by the extensive programming time and computing power needed to perform a simulation, as well as the potentially lower accuracy of the resulting fair values.
The simulation approach allows us to value structured products that are linked to a wide variety of securities, including individual stocks, indexes, interest rates, currency exchange rates, or baskets of securities. The academic literature is constantly making advances in simulation efficiency, such as quasi-Monte Carlo and importance sampling approaches (Glasserman ).
The simulation method can also be applied to calculate "Greeks," the sensitivity measures of derivative financial instruments. The Greeks are useful for traders to hedge risks in a portfolio containing structured products. The general procedure for calculating Greeks using simulations is to introduce an infinitesimal change in a chosen parameter, such as the risk-free rate or volatility, and to measure the resulting change in the fair value of the product. Since simulation introduces random noise in fair value estimations, fixing common random seeds can significantly improve the accuracy of estimating Greeks.
In addition to using the generalized Wiener process to simulate stock prices, (13) we also use it to simulate currency exchange rates, where [S.sub.0] is the spot exchange rate, the "expected return," [mu], is the difference between the risk-free rates of the countries whose currencies are included in the exchange-rate forward contract, and the volatility, [sigma], is the implied volatility of the exchange rate. The generalized Wiener process results in log-normally distributed currency exchange rates, like it does for stock and index returns. To simulate short interest rates we use the Cox, Ingersoll, and Ross model in Cox et al. . To simulate forward interest rates, we use the Heath, Jarrow, and Morton model in Heath et al. . Of course, other interest models (such as the LIBOR Market Model) can be used as well. (14)
The Numerical Integration Approach
The numerical integration approach values structured products more quickly and more accurately than the simulation approach (Glasserman ), but only works for a subset of structured products. Numerical integration directly utilizes the fact that the structured product's payoff rule is a function of a variable with a known distribution, such as [S.sub.T] (see Equation (4)) or [R.sub.T] (see Equation (6)).
The numerical integration approach is generally more accurate than the simulation approach because, unlike simulations that rely on the Law of Large Numbers to produce reliable valuations, the numerical integration approach considers the probability of virtually all possible outcomes by integrating the product of the return distribution and the payoff rule. As in the simulation approach, given the structured product's payoff rule and the distribution of the underlying security's return, the present value of the structured product is
PV([R.sub.T]) = [e.sup.(r + CDS)T] I(1+ [[integral].sub.-1.sup.[infinity]] f([R.sub.T]) p df ([R.sub.T)d[R.sub.T]) (11)
Note that Equation (10) converges to Equation (11) as J [vector] [infinity] There are many valid numerical integration methods, such as the adaptive Simpson method using .the Simpson quadrature (McKeeman ) and the adaptive Lobattamethod (Ueberhuber ). Numerical integration approaches differ in how they generate sample points and assign weights to each sample, but should all estimate the same fair value. More advanced numerical integration approaches involve numerical expansions, such as Fourier series expansion (Carr and Madan ) or Fourier cosine series expansion (Fang and Oosterlee ).
When the underlying return has a smooth distribution, such as a normal distribution, the expansion terms could achieve an exponential convergence rate to the integral value. High accuracy results are typically obtained with less than 100 expansion terms.
One drawback to the relatively fast and accurate numerical integration approach is that its use is limited to structured products whose payoffs are a function of variables with known distributions. This can be a problem, for instance, when valuing products whose payoffs depend on the linked security's maximum or minimum price over the life of the structured product. For structured products with these path-dependent payoff functions, (15) the numerical integration approach can be used, but requires transforming the path-dependent payoff function into an equivalent portfolio of path-independent payoff functions. Published papers such as Breeden and Litzenberger  and Carr and Chou  demonstrate how to transform the path-dependent payoff rules.
Using the numerical integration approach, the Greeks are calculated by taking derivatives of the payoff integral with respect to parameters such as [S.sub.0] (the initial price) or r (the risk-free rate).
For example, consider delta, the sensitivity of the structured product's value to the stock price [S.sub.t]:
[[DELTA].sub.t] = [partial derivative]PV / [partial derivative][S.sub.t](12)
A transformation will simplify the. integral. We introduce a new variable W = (ln[S.sub.T] - [^[mu]]) / [sigma], where [^[mu] = ln[S.sub.t] + ([mu] - [[sigma].sup.2] / 2)(T - t) and [^[sigma]] = [sigma][square root of (T - t)] are the parameters of the distribution of [S.sub.T]|[S.sub.t]. The variable W is thus a standard normal variable. It follows that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
If P([S.sub.T]) is a piecewise linear function (as in Buffered PLUS), then P'([S.sub.t]) is a Heaviside step function.
The Decomposition Approach
The decomposition approach can be applied to any product for which the payoff can be expressed as a combination of conventional debt instruments, call and put options, and exotic options such as double-barrier options. After the payoff of a structured product is broken down into an equivalent portfolio of simpler financial instruments, each component of the portfolio is valued using the appropriate formula (such as the Black-Scholes model for valuing call and put options).
This method is faster than the simulation method, uses less computing power, and does not require the product's underlying security to have a continuous, integrable return distribution, as does the numerical integration method. However, because not all structured product payoff rules can be broken down into components with simple formulaic solutions, the decomposition approach only works for a subset of structured products.
A significant advantage of the decomposition approach is its ability to characterize a structured product's payoff rule in terms of other financial products. This can help investors understand the risks involved in the product, and see how they might create a more liquid version of the product by investing in the equivalent portfolio rather than the structured product.
In addition to plain-vanilla options, decompositions can also involve more complex, exotic options such as "down-and-in" single-barrier options (Hernandez et al. ; Szymanowska et al. ). Down-and-in put options are inactive (do not pay out at maturity) unless a lower barrier is breached. If the underlying security ever goes below the lower barrier, the options become active puts. Allowing exotic options like the down-and-in put options in the structured product's equivalent portfolio expands the number of structured products that can be valued with the decomposition approach.
Like the numerical integration approach, the decomposition approach is useful for calculating a structured product's "Greeks." For example, the delta of the structured product is simply the sum of the deltas of each component.
The PDE Approach
The PDE approach can be used to value any structured product, but is by far the most difficult to implement. Like the simulation approach. PDEs capture the dynamic nature of the structured product and linked security's values (Black and Scholes .; Henderson and Pearson ; Wilmott et al. ). However, rather than generating random numbers to predict the movement of the linked security's price, the PDE approach models the relationships between product values and underlying asset prices using partial differential equations.
The various features of a structured product are included in PDE as additional equations or boundary conditions. Since PDEs with several boundary conditions are relatively difficult to solve, only a handful of structured products result in closed-form solutions. The remaining product types rely on numerical methods such as finite difference approximation to obtain an approximate solution.
Like the simulation approach, the PDE approach assumes the underlying security price follows a generalized Wiener process (see Equation (2)). If the risk-free interest rate and the volatility are assumed to be constant across time, the PDE satisfies the Black-Scholes equation
[partial derivative]V/[partial derivative]t + 1/2 [[sigma].sup.2][S.sup.2][[partial derivative].sup.2]V/[partial derivative][S.sup.2] + (r - q) S [partial derivative]V/[partial derivative]S - (r + CDS)V = 0 (13)
where where the value of structured product V (S, t) depends on the stock price S [member of] [0, [infinity]) at time t [member of] [0, T]. After solving the equation, the issue date value of the structured product is simply V([S.sub.0], 0).
The various properties of a structured product are included as boundary conditions in the Black-Scholes equation. A few examples:
* At maturity, the market value of the structured product is equal to the product's payoff. The corresponding boundary condition is V (S,T)= P([S.sub.T])
* If the security price hits 0 at time t and the structured product is principal protected, the structured product's value is the present value of its face value. The boundary condition is V(0, t) = I [exp.sup.-(r + CDS) * (T - t)]
* If the product is not principal protected, the structured product's value is the present value of the protected portion of its face value. The boundary condition is V(0, t) = I Buffer * [exp.sup.-(r + CDS) * (T - t)]
* If the structured product can be called by the issuer, the structured product will never be worth more than the call price [[^C].sub.t] on the call date t. The boundary condition is V(S, t) [less than or equal to] [[^C].sub.t]
* If the linked security pays a D dividend on ex-dividend date [t.sub.d], the value of the structured product must be the same just before and after the dividend is paid. The condition is included as V(S, [t.sub.d.sup.-]) = V ( S - D, [t.sub.d.sup.+]) where S remains constant from [t.sub.d.sup.-] through [t.sub.d.sup.+], and [t.sub.d.sup.-] and [t.sub.d.sup.+] are just before and just after the dividend is paid, respectively.
* If the structured product makes a coupon payment C at time [t.sub.c], the value of the structured note is increased by C. At time [t.sub.c], the value is updated V(S,[t.sub.d.sup.-]) = V(S,[t.sub.d.sup.+]) + C
The solution methods for the partial differential equations fall into two categories: closed-form solution and numerical solution. Unfortunately, the majority of the partial differential equations can only be solved numerically.
PDE produces a closed-form solution when the system of equations, including the boundary conditions, is sufficiently simple and can be transformed into standard partial differential equations (e.g., parabolic equation, through a change of variables (16)). We show how this is done when we value a Buffered PLUS using the PDE approach in this article. Fourier transforms and other transforms are also helpful in solving the equation in closed-form, with the solution being expressed in infinite summation form of eigenfunctions (Hui [19961).
Numerical methods are of various types, such as the finite difference method and finite element methods. The finite difference method is the most popular choice and involves approximating the partial differential terms [partial derivative]V/[partial derivative]t, [[partial derivative].sup.2]V/[partial derivative][S.sup.2] and [partial derivative]V/[partial derivative]S with finite difference terms. There are three basic finite difference approximations--explicit, inexplicit, and Crank-Nicolson method. The [theta]-method. uses an interpolating parameter [theta] to transform among these three methods.
COMPARISON OF VALUATION APPROACHES
While many products can be valued using any of the approaches described above, sonic structured product features make certain valuation approaches more difficult than others. Also, some valuation approaches provide a stronger intuition for the underlying features or offer simpler implementation. In Exhibit 3 we outline our experience regarding which valuation approaches are most appropriate for several common product features, including those we describe above. The choice of valuation approach depends on the level of accuracy required, the implementation difficulty (especially the existence of closed-form solutions), and the overall purpose of the valuation. itself.
EXHIBIT 3 Preferred Valuation Approaches for Several Product Features Simulation Numeikal Decomposition PDE Intfegration Callable + + (European) by Issuer Callable + (American) by Issuer Callable + + (Bermudan) by Issuer Maximum + + + + Allowable Return Loss Buffer + + + + Single-barrier + + + (e.g. FLKS) Double-barrier + + (e.g. ARBN) Payoff Depends + + on Average Price Basket of Linked + + + Securities Linked Securily + + + is a Currency Pays Coupons + + + + Minimum of a + Basket
The most important factor affecting the choice of valuation approaches is whether the payoff of the structured product is linked just to the final value of the underlying asset at maturity, or to the specific path of values the asset takes over the life of the note. Path-dependent product features, such as call features, are much less analytically tractable and typically require a simulation-based approach. Structured products that depend only on final values, however, offer more flexibility. For example, principal protection can be implemented in any valuation method, but the decomposition approach most clearly demonstrates the combination of options used to generate the resulting payoff, and can therefore be most directly compared to actual options prices, if such a market exists on the underlying.
Our discussion of these approaches so far has been based on a geometric Wiener process for asset prices (Equation (2)). However, recent developments in financial modeling extend the constant volatility assumption in this model to a stochastic volatility setting. Standard stochastic volatility models include the Heston model, where time-varying volatility is introduced following a CIR (Cox-Ingersoll-Ross) type mean-reversion process (Heston ). More recent models include general Levy-based price models (Schoutons ), where each change in an asset's price is considered as an instantaneous jump. Stochastic volatility models can capture more detailed features of stock returns, such as skewness and kurtosis, than do the traditional constant volatility model.
The simulation approach offers the most flexibility, and has been practically used in all varieties of stochastic volatility models to simulate asset prices (Korn .et al. ). However, this approach can be computationally intensive. For example, when simulating the Heston model, two processes need to be simulated (the asset price process and the volatility process).
Stochastic volatility has also been .applied to the numerical integration approach in recent years. Since all asset return information: is defined in a characteristic function, the typical integral form involving a return density function (Equation (11)) can be transformed into a series of summation terms using simple characteristic function valuations (Fang and Oosterlee ). Therefore, switching between different return models involves simply plugging in corresponding characteristic functions. The decomposition approach retains its natural simplicity, since valuations of vanilla options as well as simple exotic options are readily available in closed form using stochastic volatility.
While the first three approaches can easily be extended to incorporate stochastic volatility, the PDE approach is more restrictive. The only stochastic volatility model that can be expressed in the PDE approach is the Heston model (Heston ).
An Example with All Four Approaches
In this section we demonstrate how to use each of the four approaches to value a Buffered PLUS issued by Morgan Stanley on December 31, 2008 (CUSIP: 617483797). The Buffered PLUS has a two-year term (T = 2) and is based on the S&P 500 index. The return mapping function for this product is presented algebraically in Equation (1) and graphically in Exhibit 2.
The product has a leverage ratio of 2 ([alpha] = 2), a maximum allowed return of 60%, and a loss buffer of 10%. This means that investors in this product lose money if the S&P 500 index drops more than 10% during the two-year period; earn a 0% return if the S&P 500 index loses between 0% and 10%; and make money if the S&P 500 index increases. (17)
We collect the necessary variables from Bloomberg. On the pricing date, the S&P 500 index level was [S.sub.0] = 863.16 and the index's 24-month implied volatility was [sigma] = 37.75%. We assume the dividend yield will, remain constant over the two years, and use the annual dividend yield q = 3.714% for .each year. The continuously compounded two-year Treasury spot rate is r= 0.850%, and the two-year CDS quote for Morgan Stanley is CDS = 5.209%.
Using the simulation approach. To apply the simulation approach,. we simulate monthly index levels [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], ...[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] using the discretized. updating formula in Equation (8), where [DELTA]t = 1/12. We simulate the index's price path J = 50; 000 times and use them to calculate [R.sub.T.sup.j], j = 1,2, ... , 50,000. We input each kir into the mapping function (Equation (1)) to obtain the structured product's return f([R.sub.T.sup.j]).
The Buffered PLUS's fair value is the present value .of the average f([R.sub.T.sup.j]). Depending on the random number seed we use, the value of the product ranges from $87.30 to $87.70.
Using the numerical integration approach. To apply the numerical integration approach, we first define the function we want to integrate. In this case, the function is
[[integral].sub.-1.sup.[infinity]]f([R.sub.T])p df([R.sub.T] d[R.sub.T] (14)
f([R.sub.T]) = min ([R.sub.T] + Buffer,O) + min([alpha][R.sub.T] , Cap) - min([alpha][R.sub.T], 0)
and pdf([R.sub.T]) is the distribution from Equation (6).
We use a numerical integration method with adaptive Lobatto quadrature provided in Matlab. We require a tolerance level of 10e-10 and estimate the fair value of the product to be $87.52.
Using the decomposition approach. As described above, we apply the decomposition approach by calculating the fair value of each of the product's components. The Buffered PLUS payoff rule, described previously in Equation (1), can be rewritten in terms of equity derivatives and a zero-coupon bond as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
By recalling that the payoff rule of a put option has the form P([S.sub.T]) = max(K - [S.sub.T], 0), we can decompose the Buffered PLUS payoff into four components:
1. A zero-coupon bond with a face value of I(1 + Cap).
2. I/[S.sub.0] short put options with a strike price K of [S.sub.O] - [S.sub.0] Buffer.
3.[alpha]I/[S.sub.0] short put options with a strike price of [S.sub.0] + [S.sub.0]Cap/[alpha].
4. [alpha]I/[S.sub.0] long put options with a strike price of [S.sub.0].
For this Buffered PLUS, the zero-coupon bond has a face value of $160 (I(1 + Cap) = 100(1 + 60%) = $160). The fair value of this component is
Fair Value = [e.sup.(-CDS+r)T] I(1 + Cap) = [e.sup.-(5.209$ + 0.8496%)2] * $160 = $141.74 (16)
The second component is 0.1159 shares (I/[S.sub.0] = 100/863.16 = 0.1159) of short put options with a strike price of 776.84 ([S.sub.0 - [S.sub.0] Buffer = 863.1.6-863.16 *10% = 776.84). We then use the Black-Scholes formula to calculate the option's fair value. The fair value of one put option with these parameters is $131.71, so the fair value of 0.1159 put options is $15.26.
The third component is 0.2317 shares (2 I/[S.sub.0] = 2 * 0.1159 = 0.2317) of short put options with a strike price of 1122.11 (K = [S.sub.0] + [S.sub.t]Cap/[alpha] = 863.16 + 863.16 * 60% / 2 = 1122.11). Using the Black-Scholes formula, we calculate the fair value of one put option with these parameters to be $346.35 meaning the fair value of this component is $80.25.
The final component is 0.2317 shares ([alpha]I/[S.sub.0]) of long put options at strike price of $863.16. The fair value of one such option is $178.20, indicating the fair value of this component is $41.29.
We combine the fair values of the four components to calculate the fair value of the Buffered PLUS.
$141.74 - $15.26 - $80.25 + $41.29 = $87.52
Presenting the Buffered PLUS payoff as a combination of options and a simple debt instrument can help investors understand that the Buffered PLUS is heavily exposed to the downside risk of the underlying security and the default risk of the issuer. The decomposition also shows the investor that a portfolio of a zero-coupon bond and put options on the S&P 500 Index would mimic the payoffs and stock market exposures of the product, but with less counterparty risk and more liquidity. The same decomposition is represented graphically in Exhibit Al, included in the Appendix.
In order to calculate the delta, we need to assume that the investment is I = [S.sub.0], which means comparing the structured product investment to the investment in the underlying securities. The delta for a first component is 0, since the first component is a zero-coupon bond. Other components include one share of short in-the-money put option, two shares of short at-the-money put options, and two shares of long out-the-money put options.
The deltas for the four components are 0, -0.30, -1.05, and -0.73, respectively. The delta for the structured product is 0 - (-0.30) - (-1.05) + (-0.73) = 0.63.
Using the PDE Approach. The Black-Scholes equation is the core equation used to value structured products, with boundary conditions
V(S, T)= P([S.sub.T]),
V(0, t) = I Buffer [e.sup.-(r + CDS) * (T - t)],
V(S, t) ~ I (1 + Cps) as S [right arrow] [infinity]
We adopt the "dimensionless" approaches in Wilmott et al. (1994) to simplify the equation and boundary conditions. Transforming the variable set (S, t) to a new variable set (x, [tau]) by the following equations:
S = [S.sub.0][e.sup.x], [tau] = T - 2[tau]/[[sigma].sup.2],
V(S, t) = [S.sub.0][e.sup.[alpha]x + [beta][tau]]u(x, [tau]) + I Buffer [e.sup.-(r + CDS) * (T - t)]
where the constants
[k.sub.1] = 2(r - d)/[[sigma].sup.2], [alpha] = -1/2([k.sub.1] - 1), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The Black-Scholes equation after the change of variables becomes
[partial derivative]u/[partial derivative][tau] = [[partial derivative].sup.2]u/[partial derivative][x.sup.2], for -[infinity] < x < [infinity], [tau] > 0
The initial condition becomes
u(x, 0)= [u.sub.0](x) (17)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
This is a standard heat equation with solution:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The product value at issuance, plugging back [tau] = 1/2T[[sigma].sup.2] ,is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Calculating the integral inside u(0, 1/2T[[sigma].sup.2]), the value matches that derived using the decomposition approach.
Notice that the fair value is exactly the same using the numerical integration approach, the decomposition approach, or the PDE approach. The fair value calculated using the simulation approach is different because it varies depending on the number of simulated price paths, J, and the random seed. As J [right arrow] [infinity], the range of fair values provided by the simulation approach will converge to $87.52.
Valuation without default risk. In the example above, all four approaches yield a price of about $88. If we recalculate the price assuming the probability of default is zero, we get a price closer to $97. This highlights the potentially important component that default risk has in the proper valuation of these investments. Given the level of complexity of the valuation and the nonlinear aspect of default, it is possible that investors seem willing to purchase these investments at a substantial premium because they do not fully comprehend the cost associated with default risk.
Another way to highlight this possibility is by comparing identical or almost identical products, issued by two different issuers, that have substantially different default risk. For example, on December 31, 2007, both UBS and Morgan Stanley issued an essentially identical structured product, an Absolute Return Barrier Note (ARBN). (18)
ARBNs are structured products that guarantee to return principal to the investor, and a participation if the reference index stays between two barriers until maturity, as long as the issuer does not default on the note. (19) The two above notes have very similar parameters: They were issued on December 31, 2007, and matured on June 30, 2009. The underlying security is the S&P 500 index.
They differ in two ways: First, the UBS note has a maximum return of 25%, and the Morgan Stanley Note has a maximum return of21%. Second, and most important, UBS had a CDS spread of only 0.216%, whereas Morgan Stanley had a CDS spread of 1.27%. Given those two differences, one might expect the two notes to have a different price. However, they both were priced at an identical $10 per note. Using our pricing algorithms, we calculate that the value of the UBS note is $9.65 per note, and the Morgan Stanley is $9.37 per note.
This example--like many similar examples ofiden-tical notes with significantly different default risks that were issued at the same price by different underwriters--high lights the possibility that of the many parameters that set the value of these complex products, default risk may not be properly priced by investors. This may partially explain the popularity of these products, which tend to sell at a premium.
Until recently it has been difficult to estimate the fair value of many types of structured products, due to the complex nature of payoff rules. In this article we have reviewed and demonstrated four approaches that can be used to estimate the fair value of a wide variety of structured products, highlighting the benefits and limitations of each approach. This work is based on and complements our daily valuation of over 20,000 U.S. structured products described in Deng et al. [2014b], as well as our product-specific valuation experience in Deng et al. , Deng et al. [2011b], Deng et al. [2011a], Deng et al. , and Deng et al. .
Like many other researchers, we have documented significant issue date mispricing of many types of structured products. (20) We have also demonstrated that this issue date mispricing is largely responsible for the poor ex post performance of structured products in the aggregate. While the mispricing could be due to transaction costs, underwriting fees, or differences in cost of capital between issuers, it is likely that issuers price their products to secure a gross margin sufficient to cover these costs and leave a net profit. The approaches described here, complemented by the SEC's new fair value disclosures for structured products, could detail the components of issue date mispricing.
Currently, the numerical integration and decomposition approaches are able to value only a subset of structured products, while the PDE and simulation approaches can value many more kinds of products. We will be better able to use the numerical integration approach as we derive the probability density functions of increasingly complex payoff rules. Similarly, as closed-form valuations become available for more derivative instruments, the number of structured product types that can be valued using the decomposition approach will increase.
In the meantime, simulation approaches are becoming ever more sophisticated and can take advantage of improvements in computational efficiency, especially in parallel environments. While valuation must keep up with the enormous growth and innovation in the structured product market, the approaches outlined here offer a powerful set of tools to investors and industry practitioners.
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(1.) Bloomberg Structured Notes Brief, January 6, 2011; January 5, 2012; 2012 Review and 2013 Outlook (January .3, 201.3); and 2013 Year End Review (January 9, 2014).
(2.) See for example Doebeli and Vanini , Bernard .and Boyle , Bergstresser , Bethel and Ferrel , and Alexander and Venkatramana.n .
(3.) The approaches described here can also be used to value structured certificates of deposit. The market for structured certificates of deposit is estimated to be in the tens of billions of dollars and growing; see Deng et al.  for details. Variable annuity issuers are also releasing products based on structured product--like payoffs, which can be valued using similar approaches, as we describe in Deng et al. [2014a].
(4.) See especially Deng et al.  and Henderson and Pearson .
(5.) See for example Deng et al. [2011b], Deng et al. [2011a], Deng et al. , and Deng et al. .
(6.) Source: Bloomberg Structured Notes Brief, January 5, 2012 (page 9); January 6, 2011 (page 11); and 2012 Review and 2013 Outlook (January 3, 2013, page 7). Reverse convertibles can be linked to a variety of underlying assets, but are listed separately because their payout can be in the form of the asset itself.
(7.) Lehman Brothers Medium-Term Notes, Series I, 100% Principal Protected Notes Linked to ComBATS I due August 7, 2012. CUSIP: 5252M0GJ0
(8.) For a more recent example, consider the $235 million JPMorgan Return Notes Linked to the J.P. Morgan Enhanced Beta Select Backwardation Alternative Benchmark Total Return Index due November 27, 20.18, issued on November 22, 2013 (CUSIP: 48126NTE7).
(9.) See Deng et al. [2011a].
(10.) One approach to approximating the value of basket options is presented in Alexander and Venkatramanan .
(11.) For simplicity, all returns in this article are annualized and continuously compounded.
(12.) The American and Bermuda options mentioned here are explicitly part of the payoff rule. For example, sonic structured products allow the issuer to call the product at various dates prior to maturity.
(13.) When the linked security is a basket of securities, we use a multivariate generalized Wiener process and include the entire variance-covariance matrix of the securities in the simulation.
(14.) For an extensive treatment ofinterest rate models, see Andersen and Piterbarg 
(15.) A path-dependent payoff function depends on one or more historical prices over the life of the structured product and can be written as P([[bar.S].sub.t]).t = [0,[t.sub.1], [t.sub.2], ... , T]. A path-independent payoff function depends only on the final price, [S.sub.T] of the underlying security.
(16.) The process is called "dimensionless" in Wilmott .et al. .
(17.) Summary information for this product is available at http://www.sec.gov/Archives/edgar/data./895421/000095010308003056/dp12147_424b2-psll.htm.
(18.) For the UBS prospectus see: http://www.sec.gov/Archives/edgar/data/1114446/000139340107000374/v098098_69075-424b2.htm, and for the Morgan Stanley prospectus see: http://www.sec.gov/Archives/edgar/data/895421/000095010307003121/dp08060_424b2-ps443.htm.
(19.) See, for example, Deng et al. [201la] for a discussion of these notes and their pricing.
(20.) See especially Henderson and Pearson .
GENG DENG is the director of research at Securities Litigation and Consulting Group in Fairfax, VA. firstname.lastname@example.org TIM HUSSON is a senior financial economist at Securities Litigation and .Consulting Group in Fairfax, VA. email@example.com CRAIG MCCANN is the president of Securities Litigation and Consulting Group in Fairfax., VA. craigmccann@Mcg.com
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|Author:||Deng, Geng; Husson, Tim; Mccann, Craig|
|Publication:||The Journal of the American Oriental Society|
|Date:||Oct 1, 2013|
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