INTRODUCTION

Autocallable structured products1, 2 have become increasingly common in recent years. The first autocallable structured product on record in the United States was issued by BNP Paribas on 15 August 2003. Figures 1(a) and (b) plot the number and aggregate face value of autocallable structured products issued between 2003 and 2010. As the figures indicate, the number of issues increased sharply in 2007 and has continued to grow through 2010 at a 40 per cent annual growth rate. In just the first 6 months of 2010 there were more than 2500 autocallable products issued. The aggregate face value of newly issued autocallable structured products follows the same pattern, with a surge in 2007 and continued growth since then.

Figure 1
figure 1

Number and total issue size of autocallable structured products, January 2003–June 2010.

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One reason for the rapid expansion of autocallable structured products is the ease with which the call feature can be attached to existing types of structured products.3, 4, 5, 6 The call feature causes the structured product to be redeemed if the reference asset's price reaches or exceeds a predefined level (the call price) on a call date.

In this article we describe the call feature, explain how to value it and show an example of the valuation methodology. We use this example to discuss the cost this feature can add to a structure product. We value autocallable structured products using a general Partial Differential Equation (PDE) approach. We set up the PDE using the Black-Scholes equation and add boundary conditions representing the product's features, including the autocall feature.7, 8

We divide the autocallable structured products into two categories: products that have discrete call dates (‘discrete autocallables’) and products that have continuous call dates (‘continuous autocallables’). Figures 2(a) and (b) demonstrate graphically the difference between discrete and continuous autocallables. Both figures plot the same underlying stock price over time. The continuous autocallable structured product is called immediately upon crossing the call price C, while the discrete autocallable must wait until t3c before it is called. If the underlying stock price had dropped back below C on t3c, the discrete autocallable structured product would not have been called. Thus, holding all else equal, a continuous autocallable structured product is more likely to be called than a discrete one. Although we only consider constant call prices in the article, the methodologies are expandable to exponentially increasing call prices. Closed-form solutions are also available. The extension is analogous to valuing a barrier option with an exponentially varying barrier.9, 10

Figure 2
figure 2

An autocall event.

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An autocallable structured product is fundamentally similar to a reverse-convertible11 that pays a high coupon in exchange for exposing the investor to the downside risk of the reference security. Although autocallable structured products tend to be issued for longer terms than reverse-convertibles, autocallable structured products can have shorter effective durations because of the embedded call feature. See Arzac,12 Chemmanur et al13 and Chemmanur and Simonyan14 for a discussion of why investment banks issue mandatory convertibles and why investors purchase them.

For example, a common autocallable structured product would have the following payoffs: If the reference asset's price is above the call price on one of the call dates, it is called, and pays a pre-specified fixed-rate return. If the reference asset's price is below the call price on every call date, the product is never called. In such a case, the investors receive the product's face value at maturity, if the final price of the reference asset is above a predetermined threshold. If the final price is below the threshold, investors receive the same negative percentage return as the reference asset.

The article proceeds as follows: In the next section, we explain our valuation framework. The first subsection discusses autocallable structured products with discrete call dates, and the second subsection presents autocallable structured products with continuous call dates. The following section implements our valuation framework for an example of structured product. We conclude in the last section. In the appendix we explain the main features of popular autocallable structured-products.

AUTOCALLABLE STRUCTURED PRODUCT VALUATION MODELS

There are three main characteristics of the call feature that will affect the value of the structured product: the timing of the call dates, the probability of being called on each call date and the determination of the payoff at maturity. In this section we set up the valuation of autocallable structured products as a PDE problem. The PDE problem is general enough to be used on both discrete and continuous autocalls.

Modeling autocallable structured products using PDE

Our valuation model follows the Black-Scholes framework with risk-neutral assumptions. The reference asset's price is a generalized Brownian motion

where r is the risk-free rate, q is the dividend yield, and σ is the volatility of the price process. (Throughout this article, we assume r, q and σ are constant and continuously compounded over the product's term [0, T]. For simplicity, we omit the subscript t from S t .) If we assume the price of the structured product V(S, t) is a function of time t∈[0, T] and the reference asset's price S∈[0, ∞). The Black-Scholes formula implies that a structured product's dynamic value can be expressed as the following PDE:

where CD̄S is the credit default swap (CDS) spread of the issuer. (Structured products are unsecured debt securities, and hence lose value if the issuer defaults. It is therefore essential to include the issuer's credit risk CD̄S in the PDE to calculate the structured product's present value6, 15).

Many different structured product features can be modeled as variations on equation (2). For example, when the structured product is not called, the payoff at maturity f(S T ) is typically a function of the value of the reference asset at maturity:

For simplicity and without loss of generality, we assume the initial principal of a structured product is equal to the reference asset's initial value S0. Embedded call and put options and the autocall feature can all be modeled as boundary conditions.6 The autocall feature's boundary condition is

where C is the time-independent call price, P t is the final payoff if the note is called, and is a set of discrete or continuous call dates. Once the autocall is triggered, the structured product matures immediately and the final payout is P t . Called structured products typically pay out a fixed rate of return. Therefore, the payoff follows

where B is the rate of return, and H is a constant.

Valuing autocallable structured products with discrete call dates

Most discrete autocallables do not have a closed-form solution. Instead, the PDE is solved and the products are valued via numerical methods such as the finite-difference method.16

For discrete autocallable structured products, the boundary conditions of the PDE are the equations

The first condition requires that the product's value never exceeds the autocall payout on a call date. The second condition guarantees that if the reference asset's price hits 0 it will remain 0. For tractability, we define it using a general function f(0)=0. This boundary condition is necessary as it guarantees that the structured product cannot ever be called if the reference asset becomes worthless.

The first step in solving the PDE is to simplify the complex notation and transform the equation into a standard heat equation. Using a ‘dimensionless’ change of variables similar to Wilmott et al8 and Hui,17 we transform the variables {S, t, V(S, t)} into {x, τ, u(x, τ)} as follows

where the constants are

After the change of variables, the Black-Scholes equation is reduced to a heat equation

the boundary conditions become

and the initial condition becomes (the change of variables converts the final condition into an initial condition)

To further simplify notation we denote

reducing the boundary conditions and initial condition to

The finite difference method allows us to discretize the domain of the function u(x, τ), which is a plane (x, τ)∈(−∞, 0] × [0, Tσ2/2]. We discretize the plane into an N × M grid, where the size of each grid block is δx × δτ. Because x has no lower bound, we can assign x an arbitrarily large minimum value of −Nδx. The bounds on τ require that δτ satisfy

Generally speaking, the accuracy of the valuation increases as δx and δτ get smaller. δτ is typically set to correspond to one trading day, such that of a year.

There are three finite difference methods: the explicit finite difference method, the implicit finite difference method and the Crank-Nicolson method. The methods differ in how they approximate the derivatives ∂u/τ and ∂2u/x2. In this example, we use the explicit finite difference method, which approximates the derivatives as

Solving equation (4) with the conditions in equation (5) is equivalent to solving

with the conditions

The formula updating mδτ to (m+1)δτ is therefore

The solution is derived iteratively from m=0 → M, which corresponds to t=T → 0. For the convergence and stability of the explicit finite difference method, we require that δτ/(δx)2⩽1/2. Once all of the u n M, for n=1, 2,…, N are derived, we can approximate u(x, Tσ2/2) for every x. By reversing the change of variables, we can use u(x, Tσ2/2) to finally solve the original function V(S, t) at t=0.

An alternative probability approach to valuing discrete autocallables

Another way to estimate the value of a discrete autocallable structured product is by calculating the probability of the autocall being exercised on each call date, and then use the probability at each date to value the structured product. Let p i , i =1,…, n be the probability of the call being exercised at time t i c. The probability of the call never being exercised is then 1−∑i=1np i , where each p i is conditional on the call not being exercised at any previous call date (t1c, …, ti−1c). Recall that the conditional distribution of follows a lognormal distribution

where Δt i c is the time between call dates Δt i c=t i c−ti−1c and W i , i=1,…,n are i.i.d. standard normal variables. To simplify notation, we use to represent the continuously compounded return from ti−1c to t i c. This means the ending stock price S T can be written as

Because of the price's Markov property, the X i 's are pairwise independent. Furthermore, if Δt i c is a constant, the X i 's are i.i.d. normal variables. The probability of the call being exercised at time t i c can now be written as

where g(x1,…, x n ) is the joint probability density function (PDF) of X1,…, X n . Because the X i 's are independent, the joint PDF can be expressed as the product of each X i 's individual PDF.

We can now estimate the product's present value as the discounted expected cash flows, where the cash flow probabilities are the p i we just calculated.

If the structured product's payoff at maturity is constant f(S T )=P T , the equation can be further reduced to

Valuing autocallable structured products with continuous call dates

For a continuous autocallable structured product, the boundary conditions of the PDE are continuous equations

We apply the change of variables and simplifications from the section ‘Valuing autocallable structured products with discrete call dates’, yielding the heat equation

with the boundary conditions

and the initial condition

The next step is to convert the two boundary conditions so that they are both zero boundaries (homogenous boundaries). To do this we introduce the transformation v(x, τ)=u(x, τ)−y(x, τ), where y(x, τ)=exh1(τ). Using this transformation, the Black-Scholes equation becomes

The new, homogenous boundary conditions are

and the new initial condition is

The transformed continuous autocall PDE problem is thus a standard inhomogeneous PDE problem with homogeneous boundary conditions. By further simplifying notation, the PDE problem can be solved using methods in Evans.18 Specifically, let h3(x)=h2(x)−exh1(0) and h4(x, τ)=ex(h1(τ)−h1′(τ)). The PDE problem is then the following general form

The solution to this PDE is

Once v(x, τ) is solved, we can now solve our transformation u(x, τ)=v(x,τ)+exh1(x). Once this function is solved we can fold back and find the value of V(S, t).

EXAMPLE OF AN AUTOCALLABLE STRUCTURED PRODUCT

As an example of our valuation methodology we describe a simple autocallable structured product. (This example is similar in its features to one of the more popular brands, the ‘Autocallable Optimization Securities with Contingent Protection’. More than US$1.4 billion in face value of these products were issued in 2009. See the Appendix for more details of the different brands of autocallable structured products and their main features.) If the reference asset has a cumulative positive return on any autocall date, the structured product is called and investors will receive a positive, pre-specified yield. If the product is not called, at maturity the payoff will be:

where I is the structured product's face value, S0 is the reference asset's initial value, S is the reference asset's final value and L is the threshold price.

If the structured product is not called, investors will receive a 0 per cent or a negative return. Figure 3 illustrates the autocallable structured product's payoff at maturity if it is not called.

Figure 3
figure 3

Maturity payoff if the autocallable structured product is not called.

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To demonstrate the application of our models, we value three stylized types of autocallable structured products. The first example, our benchmark case, does not have an autocall feature, but has a constant coupon payment. The payoff structure resembles a plain vanilla reverse convertible structured product. The second type has an autocall feature with monthly call dates, and the third type has an autocall feature with continuous call dates.

For all three examples we assume that the reference asset's initial stock price S0 and the face value of the note I are both $100, the call price is $102, the risk-free rate r is 5 per cent, the volatility σ of the reference asset is 20 per cent, the dividend yield q of the reference asset is 1 per cent, the issuer's CDS spread CD̄S is 1 per cent, the contract length T is 1 year, and the threshold L is $80. If the reference asset's price is over the call price on an call date (that is, S t ⩾C=102), the product will be called and will pay a 9.2 per cent annualized return (that is, P t =HeBt=100e0.092t). (This case is our benchmark case, hence we use a 9.2 per cent coupon rate that makes this example first type non-autocallable note a par value note, that is, principal=$100.) Many autocallable products have a call price identical to the price of the stock (C=S0); however, our assumption C>S0 is without loss of generality. (In a continuous case, if the call price were identical to the stock price the product would likely be immediately called at issuance, defeating the point of such a call provision).

Case 1: Benchmark – Not autocallable

In this case, the valuation of autocallable structured product is relatively straightforward. Because the reference asset's final price follows a lognormal distribution

the value of the structured product is the discounted expected cash flow

where g( ) is the PDF of S T . We set the product's issue date value to be $100.00 per $100.00 face value by our choice of parameters. As many have shown (see for example Henderson and Pearson5) reverse convertible structured products tend to be overpriced, that is, that they are issued on average at a price that exceeds the present value of their expected future cash-flows. We use this as a benchmark example and hence set it artificially to be priced at face value.

Case 2: Autocallable at discrete call dates

Generally, autocallable structured products have discrete autocall dates. We assume that the product in this example is callable monthly.

We first implement the explicit finite difference method to calculate the product value. We set the range of x to be [−5, 0] and the range of τ to be [0, (Tσ2/2]. The resulting n × m grid has 1000 × 500=500 000 blocks. Following equation (6), the value of the product is $98.39 per $100.00 face value. We also calculate the monthly probabilities of the call being exercised p i , and show the results in Table 1.

Table 1 The probability of the product being called on each monthly call date, conditional on not being called at an earlier date
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Case 3: Continuously autocallable

If the call dates are continuous, we can follow the steps in the section ‘Valuing autocallable structured products with continuous call dates’ to get the closed-form solution.

After the first phase of change of variables, we get a homogeneous heat equation

Using our notation, h1(τ)=C−1e−βτP t and h2(x)=C−1e−αxf(Cex). Applying the second phase of change of variables to make the boundary conditions equal zero. Let y(x, τ)=exh1(τ)=C−1ex−βτP t and a new function v(x, τ)=u(x, τ)+y(x, τ), then the PDE changes to an inhomogeneous equation

Here and Applying equation (10), the solution is

where the parameters are

The value of the of the product is V(S, t) valued at (S0, 0), where the form is

The value V(S,0) is $99.54.

Comparing the three cases

We can now compare the values in the three different cases: $100.00, $98.39 and $99.54, respectively. Investments with the autocall feature are worth less than their non-callable benchmark. The reason for it is fairly intuitive. A non-callable investment essentially guarantees a coupon payment until maturity. With an autocall feature, the coupon may be paid for a shorter period or may not be paid at all. Since both investments share the same downside risk, adding the call feature (without adjusting the price or the coupon) lowers the value of the investment.

In this example, the continuously autocallable structured product is more valuable than the discrete autocallable structured product. Although this is not necessarily always true. Once the coupon payment of a plain vanilla reverse convertible is replaced with an autocallable feature, the investment has a higher value if it is called and the longer it takes to get called. A discrete autocallable feature is less likely to be called, but holding all else equal may be called later if it is called. Hence, it is more likely for a continuous feature to be more valuable than a discrete one but this does not have to be always the case.

Real-life example

We calculate the product value of a real ‘Autocallable Optimization Securities with Contingent Protection’ note issued by UBS. (The CUSIP for the product is 90267C136. See the product's pricing supplement at http://www.sec.gov/Archives/edgar/data/1114446/000139340110000136/c178916_690465-424b2.htm.) The note is linked to the stock of Bank of America. It was issued on 26 March 2010 and had a maturity of 1 year. The reference asset's price on the issue date was S0=$17.90. The dividend yield q and implied volatility of the underlying stock σ were 0.2235 per cent and 35.21 per cent, respectively. UBS's 1-year CDS spread was 0.4531 per cent. On the issue date, the 1-year continuously compounded risk-free rate was 0.4951 per cent. The call price C equaled the initial price S0. If the note were called, investor would receive a return of 16.1 per cent, and if it were not called, the contingent protection level was L=0.7S0. Applying our methods, we get a product value of $97.73 per $100.00 invested.

CONCLUSION

An autocallable structured product is called by the issuers if the reference asset's price exceeds the call price on a call date. The feature has been embedded in many different types of structured products, including Absolute Return Barrier Notes and Optimization Securities with Contingent Protection.

We provide a general PDE framework to model autocallable structured products. We solve the PDE for autocallable structured products with discrete call dates, for which there is typically not a closed-form solution, by using the finite difference method. For continuous autocallables, we derive the closed-form solution. We illustrate our modeling approaches with an example. We then quantify the incremental cost of adding an autocall feature to a plain-vanilla reverse-convertible. We also show the difference between the value of an autocall feature with continuous call dates and one with discrete call dates. The Securities and Exchange Commission, as a matter of policy, disclaims responsibility for any private publication or statement by any of its employees.