# OPTION REPLICATION

Options are prevalent in the world of finance and serve many functions.

INTRODUCTION

Options are prevalent in the world of finance and serve many functions. Investors and traders use options to gain or reduce exposure, change the portfolio structure,(ie, convexity), enhance returns and enact arbitrage strategies. For some, the price of an option is considered expensive and synthetically creating the option may be more cost effective. Because option pricing theory is constructed on the idea of dynamically creating a "hedge" or replicating portfolio, understanding option replication is fundamental to understanding the intricacies of option pricing and hedging.

There are many reasons an investor would want a portfolio that contains "optionality." One is the investor gains unlimited upside potential at a fixed downside cost. However, some investors cannot or will not use derivative products, either because of investment restrictions or lack of sophistication. Second, because option implied volatilities are often higher than actual volatilities, options can be overpriced. A replicating strategy can be a viable alternative for investors wishing to obtain a fairly priced option like asset. An asset manager can use an option replication strategy in conjunction with their own "value added" strategies to provide clients guaranteed returns and option-like payoff profiles.

This article provides an overview of option replication, discusses a delta replication strategy, and provides examples of how an asset manager would use a replicating strategy.

WHAT IS OPTION REPLICATION

Option pricing theory is based on the creation of a dynamically hedged portfolio designed to replicate the option payoff during the life of the option. Each time we rebalance our replicating portfolio, we ask the question--how will the option price change if the price of the underlying asset, or other factor such as volatility changes? The exact answer to this is given by the option model, and an appropriate hedge is transacted based up on that answer.

A portfolio that perfectly replicates an option will exactly match the option's payout and the hedging cost of creating this portfolio will exactly match the initial option price. A call that initially costs USD10 would theoretically cost USD10 to hedge. Because the price path of the underlying asset is impossible to know in advance, the original option cost and the actual hedging costs are often different. Unlike the case when an option is purchased and the cost is known upfront, replicating an option can lead to higher, or lower, costs.

TYPES OF REPLICATION

There are many ways to replicate or hedge an option, depending on which option characteristic you are concerned about. The three most common types of replication are delta, gamma, and vega replication. Delta replication focuses on the option's price sensitivity to changes in the underlying asset, gamma replication on the option delta's sensitivity to changes in the underlying, and vega replication on the option price's sensitivity to changes in implied volatility.

DELTA REPLICATION

The replicating portfolio should mimic the price behavior of the option everyday through expiration. The option model provides a hedge ratio or delta, which tells how much the option price will change as the underlying asset changes. For example, a delta of 0.5 means that for a USD1 change in the underlying, the option price will change by USD0.50. Delta, which ranges between 0 and 1, is used directly to allocate the investment in the underlying and risk-less investment--Treasury bills. The higher the delta, the more money is invested in the underlying asset and the less invested in the T-bill.

Unlike an option purchase where the up-front cost is known, the hedger can not be sure, until the option expires, how much it will cost to replicate the option. Following a delta (or other) hedging strategy does not guarantee hedging costs will equal the option cost. Some reasons that hedging costs will not equal the option cost include:

*
actual asset volatility is different than the option

implied volatility;

* transaction costs from rebalancing the portfolio;

* mis-specification of the option model; and

*
discontinuous asset prices leading to poor;

hedging performance.

FIXED INCOME ASSET MANAGER EXAMPLE

Consider a six months call option on a 30 year Treasury bond that is trading at par with no accrued interest. The investor wants a guaranteed minimum return of 0% (full return of principal). The notional amount of the strategy is USD100 million. Table I shows the program details. Lets compare two strategies, buying a call option and replicating a call option. We use a Black-Scholes based option model for the option calculations.

BUYING THE CALL

The asset manager invests the T-bill discount amount in the six month T-bill. For a bill priced at 97.34, USD97.34 million is invested in the year bill, which will accrue back to USD100 million at the end of one year, hence, guaranteeing full principal return. Here, the implied option cost is USD2.66 (100-97.34).

The option cost of USD2.66 million is invested in bond call options. At worst, this option expires worthless at a total cost up-front of USD2.66 million. The choice of option strike is unlimited as the investor can buy less of a lower strike call, or more of a higher strike call as long as they do not spend more than USD2.66 million. The strike can also be solved for by an iterative procedure assuming the investor wants to buy one call option for each amount of initial investment.

REPLICATING STRATEGY

Given the option chosen from above, calculate the option delta and invest (USD100 million x delta) in the bond and (USD100 million x (1-delta)) in the T-bill. For a delta of .52, USD52 million will be invested in the bond and USD48 million will be invested in T-bills. Each day, the portfolio is re-balanced to reflect changes in the option delta. Table II shows how the bond position will change as the bond price changes as of the first day of the program. Figure 1 shows the same relationship but also shows how the position changes with respect to time. As you can see, as the price of the bond increases (decreases), the replicating strategy requires additional purchases (sales) of the bond. Also notice that sensitivity is enhanced as the program nears it's expiration date.

A pitfall of replication is unexpected hedging costs. Consider what would happen if the bond traded at 100 and 103 on consecutive days. When the price moved to 103, you would buy about USD15 million more bonds, and then be forced to sell those same bonds at a 3 point loss. Just a few days of such volatility could cause the hedging costs to exceed the anticipated cost of the option. If one had a perfect forecast for volatility, then the initial cost of the option would have been estimated using a higher volatility, and the actual cost would more closely match the anticipated cost.

ASSET MANAGER CONSIDERATIONS

An asset manager must consider some specific issues when developing an option replication strategy. Often, the manager's strategy provides upside participation with a "guaranteed" minimum return. The manager has the opportunity to "add value" by making curve and convexity bets in the replicating portfolio. In effect the manager is deviating from the option model based on their view of the market. The problem is how to offer the minimum return guarantee. It is impossible to know initially what the actual bond volatility will be for the life of the program, nor is it possible to know for sure whether the option will expire in or out of the money. By making the model more sensitive to market conditions, it may be possible to increase the likelihood the guarantee will be met.

Back-testing and simulation will allow the manager to understand the types of markets that will cause the strategy to fail. One such proprietary modification was developed by the author for a large asset manager and incorporated into an integrated bond option replication module. The modification works well under difficult market environments without giving up upside participation. The module generates model deltas and other risk measures, keeps track of trading profit and loss, and allows for intense back-testing and simulation.

*This week's Learning Curve was written by Steve Pelletier, vice president of Theoretics, a Park City, Utah based financial*

software company.