## Bryant Angelos and Dr. Jeffrey Humpherys, Math Department

In recent years, capital markets have experience remarkable growth in both the total volume of trades and in the complexity of the financial instruments being traded. Investors are becoming increasingly sophisticated as they search for newer and more effective ways of managing and protecting themselves from risk. Retail markets have undergone a similar transformation as retailers seek to maximize profit while minimizing risk. Supply chains are becoming more interconnected, and the contracts governing them have become more complex in recent years.

In this research project, we investigate the use of a financial instrument called a put option as a means of managing and controlling inventory levels. A put option is a financial instrument that one party purchases from an option writer. The option writer agrees to buy one unit of the underlying good for a fixed price called the strike price at a future expiration time. The option holder however, is not required to sell the underlying security. In other words, the option holder purchases the right, but not the obligation, to sell the underlying security for a fixed strike price at a future time.

As put options have been extensively studied in mathematical finance, we turn our attention to a different type of put option, which we call a retail option. A retail option is a type of theoretical put option, where the underlying security is the inventory of a retail store. Thus a retailer can purchase a retail option from an option writer, which gives him or her the right to sell any leftover inventory at expiration time for an agreed upon amount. Throughout this project, we assumed that the underlying inventory good was a perishable, and that the option expiration time coincided with the expiration date of the underlying good.

Retail options would be useful for a variety of reasons. First, they could be used to reduce the profit variance for retailers. Many economists assume that people are risk-averse, meaning that they prefer safer investments, as measured by variance. Another possible use for retail options is as a management tool. Options could be bought and sold between retail outlets, suppliers, and corporate headquarters as a means of controlling the flow of inventory. Finally, retailers could purchase retail options to lock in profit at certain levels. In this case, retail options behave much like an insurance policy.

The principle question we wished to investigate in this project was how much should a retail option cost? Under a risk-neutral paradigm, the option cost is the expected value of the option payout. The payout of a retail option is given by multiplying the strike price by the total number of inventory remaining at expiration time. Therefore to price the option, we had to determine the expected value of the inventory at a specified future time.

One factor which complicates this task is that the rate at which inventory is sold is highly dependent on the price at which it is being sold. In our model of a retail environment, we assumed that demand follows a non-homogenous Poisson process, where the Poisson parameter is a function of sale price. By making this assumption, we were able to describe the probability of making sales during certain time intervals. However, using this model means that the prices set by the retailer will have an effect on the expected remaining inventory. Therefore before we could calculate the expected remaining revenue, we had to determine the optimal sale prices at each moment of the sale period.

To determine the optimal sale price at each time, we first discretized the sale period into equal intervals. Using the probabilities given by the Poisson process, we created a dynamic program which calculated the sale price during each interval which maximized expected revenue. By increasing the number of intervals, the optimal sale price can be found at more and more times. Finally, by taking the limit as the number of intervals approaches infinity, we were able to find a differential equation which can be solved to give the optimal sale price at each time and for each inventory level.

Using this, we were able to follow a similar process to determine the probability of having any inventory level at any time. First, we assumed that some probability distribution on the initial inventory was known. Then we once again discretized the sale period into equal intervals. By conditioning on the number of items sold during each interval, we were able to determine the probability of being at a certain inventory level during any period. Then we once again took the limit as the number of periods approached infinity, and found a differential equation characterizing the probability of being at any inventory level at any time. The solution to this differential equation evaluated at expiration time gave us enough information to calculate the expected remaining inventory at expiration time, and hence the option price.

In addition to answering our original question, this research has prompted several more research ideas. The biggest unsolved question to be answered is to find the optimal course for a retailer given information about demand and prices. For example, what is the optimal initial inventory when an option is allowed to be purchased? In what situations is it advantageous for a retailer to purchase an option, and if so, at what strike price? Another direction of research we wish to pursue is to look at the rate of return of an option. How does it compare with rates of return in capital markets? Can we incorporate economic theory such as the capital asset pricing model in our discussion of retail options? Finally, there are many types of options in financial markets, some of which may be applicable to retail. Perhaps some of the “exotic” options could be reformatted for a retail environment.

Aspects of this research project have been presented at two conferences. In June, we presented our work at the 2009 American Controls Conference in St. Louis . One month later we presented at the SIAM 2009 Annual Meeting, as part of a special session on undergraduate research. We are currently preparing a journal paper for publication, and will submit it within the next few months.