Market-Consistent Scenario Generation

Market-Consistent Scenario Generation

by Sam Nandi

Consider one of the following situations: You're a valuation actuary of a company that sells guaranteed living benefits embedded within its Variable Annuity products, and you've been assigned the responsibility of developing a methodology for fair valuation of the guarantees under FAS 133. Or, you're a pricing actuary of a company selling similar benefits, and your company wants to dynamically hedge the guarantees and price them appropriately. Or, maybe your company sells Equity Index Annuities with exotic crediting designs and you're in charge of evaluating the quotes your company is receiving from OTC dealers.

In each case, your job involves taking an insurance benefit or guarantee and valuing it as if it were a financial derivative. Given the preponderance of such benefits within a wide range of life insurance products, actuaries increasingly find themselves needing flexible derivative pricing tools. Specifically, what's necessary in all the tasks described above is a market-consistent scenario generator.

Scenario Generators
                                               
An economic scenario generator is something with which actuaries have become very familiar. It's simply a tool that produces random sequences of market variables, like interest rates and equity returns, assuming that those variables follow some pre-defined stochastic process. The processes may be simple or complex. They may be isolated to a single market variable that directly drives the value of the instruments in question (for instance, the S&P 500 index) or they may jointly simulate combinations of many different variables (multiple equity indices, several points on the yield curve, exchange rates, inflation, equity indices and yield curves in other countries – the list can be endless).

Market consistency simply means that the scenario generator is specified and parameterized such that if we used the scenario generator and Monte Carlo simulation to price a set of derivatives, we would get the same answers as the prices at which the instruments trade in the market.

Sometimes, market consistency is equated with the term "arbitrage-free" or "risk-neutral." We won't get into the technical differences between all these terms in this article. For the purpose of this article, they mean the same thing–a way to produce scenarios that are consistent with market prices.

Use of market consistent scenarios for pricing purposes involves discounting cash flows to obtain an expected present value. Typically, market consistent scenarios are generated assuming that cash flows will be discounted under "risk-neutrality" (i.e., using discounted factors appropriate for valuing bonds with no cash flow uncertainty). This article concerns itself exclusively with such "risk-neutral" scenarios, which can be contrasted with the so-called "real-world" scenario generators that are applied to problems like capital adequacy, where consistency with market prices is not a goal.

Pricing With Scenario Generators:  An Illustration

Pricing derivatives using Monte Carlo paths produced by a scenario generator is simple in concept. First, we generate a large number of market scenarios that include the market variable that drives the payout of the derivative in question. Next, we evaluate the payout of the derivative over each scenario. Finally, we take the expected (sample average) present value of the payoffs across the scenarios. This result represents the value of the derivative according to our model.

Let's try a simple illustration. We'll value a one-year at-the-money European put option on a stock that has a current value of 100 and does not pay a dividend. Let's assume that we use a lognormal scenario generator for simulating stock prices at the expiration date of the option. That is, our stock price one year from today is given by the following equation:

S[1]= S[0]*exp((mu - (sigma^2)/2)*dt + (sigma*(dt)^0.5)*eps)

where S[1] is the stock price after a year, S[0] is the current stock price, mu and sigma are the drift and volatility parameters for the lognormal process, and eps is a standard normal random draw for each scenario. For this exercise we set dt=1, representing the one year left until option maturity. This is the assumed stochastic process underlying the Black-Scholes option pricing equation.

How do we set mu? If we are not concerned about market consistency, we can set it any way we want. But, if we are, we'll set mu equal to the prevailing continuously compounded risk-free rate (in this case, we'll assume 4 percent). This is an example of an application of the principle of risk-neutral valuation, which states that we can assume that all investors are risk-neutral when valuing options.

How do we set sigma? Again, if we're not concerned about market consistency, we can choose a volatility assumption based on the stock's historical volatility or our subjective expectation about its future volatility. But, if we are, we'll set sigma equal to the implied volatility for a one-year at-the-money put option (let's assume 15 percent). The implied volatility merely represents the volatility parameter we would have to plug into the Black-Scholes equation in order to reproduce the market price (at this stage, our exercise is a bit circular). It also, in some sense, represents the market's expectation for the stock's future volatility.

We generate 20 scenarios, as shown below, and we calculate the payout of the option in the table below.

Using the data produced by our simulation above, we calculate an Expected Present Value of payoff equal to 4.85. This is the value of the option using the 20 scenarios produced by our generator. This compares to our Black Scholes price for a one-year at-the-money option of 4.11. If we ran enough scenarios, we would reproduce our Black Scholes price (try it).

What have we done so far? We have developed a scenario generator that correctly prices (if we run enough paths) a one-year put option. Suppose there is some other option on the same stock (perhaps some kind of insurance guarantee) whose value depends on the price of the stock one year from now; however, a traded market does not exist for that option. We can then use the same paths (or paths that were produced by the same generator parameterized in the same way) to price this new option. This illustrates the concept of using a market-consistent scenario generator to value a non-traded instrument.

Of course, the example above is very simplistic. In reality, the guarantees or benefits or options we have to value are far more complicated. For instance the payouts associated with a Variable Annuity guarantee don't just depend on the price of a single stock one year from now. They might depend on the levels of several equity indices (like the S&P 500, Russell 2000, NASDAQ) every year in the future for 30 to 50 years. They might depend on interest rate levels in the future. They might depend on the future performance of foreign equity indices. A good market-consistent scenario generator for valuing such a guarantee must be consistent with the prices of derivatives on a wide range of underlying investments.

Extending the Illustration: Volatility Term Structure and Skew

In our example, the single instrument we used to calibrate our model was a one-year at-the-money put. In practice, even if we are only concerned about that one stock or index, there are many options of different maturities with which we want to demonstrate consistency. In the case of Variable Annuity guarantees, which are long-dated by nature, we will want to include options in our calibration set that have the longest available maturities.

Let's extend our example. Suppose we now have to value an exotic derivative whose payout is dependent on the price of the stock, not just one year from today, but on the sequence of anniversary prices over the next 10 years. We'll assume that the yield curve is flat, with a continuously compounded interest rate of 4 percent. Suppose we have prices (and, thus, implied volatilities) of put options on the hypothetical stock for maturities out to 10 years, as follows:

The sequence of volatilities in the table above is called the implied volatility term structure. With this information, we now have the ability to develop a scenario generator that achieves consistency with the market prices of options of one to 10 years in maturity. We do this by solving for the forward volatilities (effectively the market's expectation of volatility for years in the future) implied by all of these prices.

We adjust the specification of our stock price process to:

S[t] = S[t-1]*exp((mu - (sigma[t]^2)/2)*dt + (sigma[t]*(dt)^0.5)*eps[t])

where sigma[t] is the forward volatility for each year, and can change with each time-step.

If we generate enough 10-year scenarios and use the scenarios to price European put options of the various maturities, we will see that they are consistent with market prices.  We can then use these scenarios to value our exotic derivative consistently with the prices of the market-traded options.

We can further extend this concept to develop our scenario generator to be consistent not just with the prices of options of different maturities, but different strikes as well.  Volatility smile (or skew) refers to the property of options of different strikes to price back to different implied volatilities. For instance, using the approach above, our generator is able to price a five-year option with a strike of 100 consistently with the market, but not necessarily a five-year option with a strike of 120.

There are two types of approaches for developing scenario generators that reflect implied volatility skew, which we will describe generally in this article, but not illustrate in detail.  The first is known as the Local Volatility approach. Under this methodology, the "local" volatility used in the Monte Carlo simulation at each time step along each scenario will not just vary with time (as our forward volatilities do), but also as a function of the path-wise stock price realizations. The "local" volatility is still deterministic (rather than stochastic); but it is a deterministic function of two variables rather than one, and its path-wise values depend on the evolution of the stock price. The benefit to this model is that it makes it possible (in theory) to fit all option prices (the entire volatility surface) exactly.

Another approach to modeling volatility skew is the use of stochastic volatility models.  Under this approach, the volatility parameter (sigma) itself follows a stochastic process. A robust stochastic volatility framework will model the underlying volatility with its own volatility (volatility of volatility) and as a mean-reverting process with some long-term target. A stochastic volatility model will have several parameters. Generally, optimization techniques must be used to find the parameters that best fit the set of market prices. Stochastic volatility models will not be able to fit all option prices exactly. Suppose, for instance, we have a set of options of five different strikes at each of 10 different maturities that we want our generator to re-price. If our stochastic volatility model has five parameters, we can never fit all 50 option prices exactly. We'll simply have to settle for the five parameters that best minimize some measure of the tracking error of our model prices relative to market prices.

Other Complexities

In practice, market consistent valuation of insurance guarantees is much more complicated than the illustrative examples we've discussed. Guarantees embedded within insurance products are often long-term in nature; the effective maturity of the guarantee might very well be longer than any option priced in the marketplace. It will often be left up to the actuary's judgment as to how to combine information on market prices with other data (perhaps historical) or subjective views to develop and parameterize the generator. As mentioned above, the payouts of an insurance guarantee may depend on not just the realizations of a single equity index, but several. The practitioner is left with the challenge of ensuring that all of the indices or sub-accounts are modeled reflecting appropriate mutual return correlations, as well as consistently with prices of derivatives on each index. In addition, depending on the specific nature of the guarantee or benefit being valued, it may be necessary to use a market-consistent generator that produces scenarios of stochastic yield curve movements, as well as equity index prices. In this case, we would have to ensure the consistency of our yield curve paths with market prices of bonds of different maturities, as well as calibrate our interest rate model against interest rate derivative prices (caps, floors, swaptions), not to mention the equity derivative calibrations described previously.

The challenges are formidable. It will take actuaries with knowledge of the insurance products being valued, understanding of derivative pricing and computational know-how to develop scenario generators that will be up to the task.