# Pricing a Conversion

One way to conceptualize how to price conversions is to compare them with Treasury bills. Anyone can purchase Treasury bills for an amount less than the face value, that is, at a discount, and receive the face value at maturity. A one-year \$1,000 Treasury bill, for example,might be purchased for \$970. One year later, at maturity, the purchaser
gets back a \$1,000 payment, \$30 of which is interest income and \$970 of which is return of principal.

The interest rate earned in this example is 30 divided by 970, or approximately 3.1 percent.Conceptually, the value of a conversion is the discounted present value (DPV) of its strike price. The DPV is the investment, like the amount paid for a Treasury bill. The strike price is the amount received at expiration, like the Treasury bill maturing at face value.The difference between DPV and strike price is the income earned, like the difference between what was paid for the Treasury bill and the face value received at maturity.

Consider, for example, stock purchased for \$92.20, a 40-day 90 Put purchased for \$2.30, and a 40-day 90 Call sold at \$4.80. This conversion’s net investment totals \$89.70 (92.20  2.30  4.80  89.70),and the profit before costs equals 0.30 (90.00 – 89.70). \$0.30 income in 40 days from an investment of \$89.70 approximately equates to an annual interest rate of 3 percent (0.30  89.70  365/40).Note for mathematicians:

The calculations in the preceding paragraph and the calculations throughout this book are made using simple interest rather than compound interest. Simple interest is used for ease of presentation. The discussions are intended to be conceptual and accessible to non mathematicians. For short periods of time,which is typical for arbitrage positions, the difference is insignificant.After seeing how a conversion is constructed and how its

profitability is calculated, you may reasonably ask, “How much profit is sufficient to justify a conversion position?” The answer depends on three factors: costs (including borrowing costs), the target profit, and the competitive environment.Tables 6-3 through 6-5 provide an example, in three parts, of how a conversion might be priced. Table 6-3 states the 11 initial assumptions.The strike price (1) is 55. The stock price (2) is \$57.70.

The price of the 55 Put (3) is 1.45, and the price of the 55 Call (4) is unknown. The borrowing rate (5) of 5 percent and the 60 days to expiration (6) lead to the DPV of the strike price (7) of 54.55. There are also trading costs (8–10) of 1 cent per share to trade stock (buy and sell), to trade each option, and for option exercise or assignment. These transaction costs lead to total costs of 4 cents per share for opening the position (i.e., buy stock, buy put, and sell call) and for closing (i.e., either the put is exercised or the call is assigned).

Finally, the target profit (11) is 5 cents per share in this example.The 10 known assumptions are used to solve for the unknown one, which is the price of the 55 Call (4). The question is, “What sale price for the 55 Call will yield the target profit?”Table 6-4 contains the second part of pricing a conversion, which calculates the sale price of the 55 Call in two steps. The first step figures the net investment per share.

In the case of a conversion, the net investment per share is the net cost of the position that yields the target profit if held to expiration. The net investment per share equals the DPV of the strike price (line 7 in Table 6-3) minus the sum of costs plus target profit (lines 8–11). The net investment per share, therefore, is \$54.46. Calculations are made on a per-share basis because a market maker cannot know in advance how many options will come into the marketplace and, as a result, how many shares will need to be traded.

Calculating a per-share amount facilitates flexible trades—a trader may trade in small or large quantities but use the same pricing system. Remember that the conceptual value of a conversion is the DPV of its strike price. The strike price is used for the DPV calculation because the strike price is the amount received at expiration after the call is assigned or the put is exercised.

Regardless of whether the stock price is above, below, or at the strike price, when a conversion is established,the net investment always will be less than the strike price because stock plus call minus put must equal the DPV of the strike price in order for the conversion to be profitable. Step 2 in Table 6-4 uses basic algebra to calculate 4.69 as the price of the 55 Call that makes the conversion position (stock plus put minus call) equal the net investment of 54.46.