4.2 Analysis of hedging strategies using the Black-Scholes framework

The purpose of this project is to examine hedging strategies for options. I will assume that the reader is familiar with options basics such as the definition of a put and a call and how to calculate the payoff for an option, and also has an understanding of put-call parity and similar no-arbitrage arguments.

Throughout this project I utilize the Black-Scholes framework for valuing options. This framework is a simplification of the option markets and equity markets, which relies on a specific set of assumptions. After examining the outcomes of hedging in this framework, I alter the Black-Scholes assumption of the equation that governs the change in the asset price as well as the assumption of constant asset volatility. Specifically, this project examines a situation in which an option dealer buys or sells a large volume of options and, as a result, the asset price changes in a semi-deterministic manner in the time immediately following the trade.

Individuals and firms that make a living buying and selling options are forced to take on risk in order to do business. This risk arises because by entering into an options contract one is forced into a world where his/her financial future is uncertain. For example, what happens when one buys a European call option contract? The buyer has left a world where he/she invests the call option premium and receives some "sure" risk-free return over the course of the option's life. The buyer enters into a world where he/she loses that call option premium amount today for some payoff in the future that depends on the future price of the asset.

If one wants to make a living buying and selling these contracts (and do so for an extended period of time), it becomes essential to have some ability to minimize the risk. This situation is similar to what is known as the "gambler’s ruin” problemⁱ. A modified version of the problem is: A gambler starts off with $10, flips a coin, and every time the coin comes up heads the gambler loses $1, while every time the coin comes up tails the gambler wins $1. The gambler continues to play this game until he/she goes broke or has $20. A natural question to ask is: What is the probability that one will go broke before one reaches $20?

To analyze the “gambler’s ruin” problem, take the general situation where one lets C0 equal the gambler’s cash position at the beginning and a > C0 is the upper value the gambler stops at if/when a is reached. Let p be the probability that the gambler wins $1, and q=1-p is the probability that the gambler loses $1 on any toss. Then, notice that the probability that one will go broke given that one has $C at any time is

The purpose of introducing the “gambler’s ruin” is to show how one can manage risk as a function of the amount that one wagers. From the analysis above we know that when a relatively large amount of one’s total wealth is being wagered, one has a higher chance of going broke at any point. Thus, if one wants to be in the business of buying and selling options (i.e., flipping coins and wagering on the outcomes), then it is best to make smaller wagers rather than large ones. Of course, if one is in the options business, hopefully it is because of some advantage that will help one make a profit (i.e., flipping a biased coin that is weighted in one’s favor). Here are just a few reasons why the odds might be tipped in an option dealer’s favor:

• The dealer has access to private information that will impact the price of options once it is made public.

• The dealer controls fast technology that allows you to quote updated prices on an exchange one step ahead of your competitors.

• The dealer is experienced and employs smart traders with superior decision making skills.

• The dealer charges a commission to act as a counterparty in an option contract with a customer.

• The dealer buys options for a price below the "true" value and sells options for a price above the "true" value. In other words, he/she collects the bid/ask spread by being a market maker.

• The dealer has the ability to consistently price options at their true theoretical value.

• The dealer has the ability to take on a greater amount of risk since he/she benefits from economies of scale of trading a large volume of options.

So, what happens when one has the odds tipped in his/her favor? Now, one has a reason to take a little risk, since one will be getting paid for it more often than not. Take the extreme example: there is a probability of one of winning on each toss; of course, one will bet as much as possible. When one has an edge, there is a tradeoff between taking risk to maximize wealth and reducing risk to ensure one will be able to play the game for a long time into the future. Here is an analysis of the “gambler’s ruin” when the gambler has a large edge versus when he/she does not (note that these probabilities are survival probabilities or one minus the probability of ruin):

Table 18

Probability of Survival With and Without Edge ( a  )

S=C0/bet per toss	p = .75	p = 0.51
1	66.667%	3.922%
5	99.588%	18.129%
10	99.998%	32.972%
20	99.999%	55.072%
30	100%	69.885%
40	100%	79.815%
50	100%	86.470%
100	100%	98.169%

As you can see, having an edge drastically improves the chances of staying in the game indefinitely. But, as S, the proportion of the total initial wealth bet on each toss, increases, the probability of survival depends less on the amount of edge one has. Once one has an edge and knows that there is a very small chance of going broke, one might be inclined to bet a larger amount of his/her total wealth on each coin flip. In other words, because one enjoys having more money now rather than later one will take some risk by betting a larger amount and hoping to get more money with fewer coin flips.

In the 1973 issue of the Journal of Political Economy, Fischer Black and Myron Scholes published a seminal research paper on the pricing of options, “The Pricing of Options and Corporate Liabilities.” In this paper, Black and Scholes derived a closed form solution for the price of an option. They concluded that the value of an option could be determined as a function of time and the price of the underlying asset as well as the parameters: risk-free rate, volatility of the underlying asset, time to expiration, and strike price of the option. In order to arrive at their solution, Black and Scholes laid out some assumptions that I will refer to as the Black-Scholes Framework. Their assumptions are:

• The risk-free interest rate and the volatility of the underlying asset are known and stay constant through time.

• The stock price follows a random walk in continuous time, and the distribution of stock prices is log-normal.

• The stock does not pay any dividends during the life of the option.

• The option is European style, meaning it can only be exercised at the maturity date.

• There are no transactions costs involved in buying or selling assets or options.

• Assets are divisible, in other words you are allowed to buy a fraction of an asset.

• Short selling is allowed, and there are no restrictions on short selling.

One assumption that the Black-Scholes Framework does not explicitly include, but must be included implicitly, is that the market has infinite liquidity. In other words, if I want to buy or sell an asset at any time, there will always be a counterparty who will allow me to make the trade. This assumption is required in order for the hedging strategy, which involves buying and selling shares of the underlying asset, to work.

When you place the Black-Scholes Framework into a real options market, the assumptions undoubtedly begin to be violated. In particular, the liquidity of an option has a profound impact on how it is priced in a real market. In options markets, this liquidity comes with a cost. For instance, an options market maker will sell ten at-the-money options for less than he will sell one thousand at-the-money options. Obviously, one thousand contracts has more risk, and if the market maker is wrong on his price he’ll have a much larger loss with the larger order size.

In the Black-Scholes framework the change in the asset price is given by the following stochastic differential equation:

dS    S  dX    S  dt ,^ix (34)

where S is the asset price,  is the volatility (or standard deviation of returns of the asset),  is a deterministic average growth rate of the asset, dt is an infinitesimal change in time, and dX is a Normal random variable with zero mean and variance dt . First, it was necessary to implement this model for the asset price to generate a (random) sample path of the asset price. Below shows three sample paths for an asset that starts at an initial price S₀ of $50 and has a volatility of 0.04 with different time steps dt .