Black Scholes Equation Luc Ashwin and Calum Keeley

Transcription

1 Black Scholes Equation Luc Ashwin and Calum Keeley In the world of finance, traders try to take as little risk as possible, to have a safe, but positive return. As George Box famously said, All models are wrong but some models are useful. A theory or a hypothesis only has true value when it can be applied to reality. Take financial modeling, a mathematical model designed to represent the performance and volatility (the change in the value of the stock) of a financial asset, or portfolio, as an example. Unlike many investors, who act on whim or impulse, hedge funds and banks cannot content themselves with this, hence the financial modeling. The Black Scholes model is a mathematical model of a financial market, from which a formula was derived that spurred rapid growth in option trading. Option trading is the trading of one of two types of stock options: a call option gives the buyer the right to buy from the writer of the option at a fixed price called the strike, on certain day called the expiry. The call option only has value if the stock price is above the strike price at expiration. a put option gives the buyer the right to sell from the writer the stock at a strike, and at expiration. The call option only has value if the stock price is below the strike price at expiration. The Black Scholes Equation is widely used in global financial markets by traders and investors to calculate the theoretical price of european options. Both of these terms are defined in the next paragraph. The formula has been demonstrated to yield prices very close to the observed market prices. Although the black scholes formula requires complex mathematics, investors who use the formula can simply plug in five required inputs: the underlying stock s price, the option s strike price, the time to the option s expiry, the volatility of the stock, and the time value of money. However, despite its popularity and its widespread use, the model is built on some non real life assumptions about the market. In finance, an option is a contract that gives the buyer the option to buy or sell a financial asset at a strike price or strike date. For instance consider a stock price currently at $90, and you have got an option to buy the stock for $100, then your option will be worthless unless the option price goes up before it expires. The value will reflect the likelihood of the stock price going above $100. Similarly if you buy an option to sell the stock for $80, then your option will be worthless unless the option price goes down before it expires. To eliminate the risk of a stock, that is the chance of losing money on an investment, we can set up a hedging portfolio. To understand the idea of hedging we must consider what stock options are. There exists two types of stock options: a call option gives the buyer the right to buy from the writer of the option at a fixed price called the strike, on certain day called the expiry. The call option only has value if the stock price is above the strike price at expiration. V=S price -strike a put option gives the buyer the right to sell from the writer the stock at a strike, and at expiration. The call option only has value if the stock price is below the strike price at expiration. V=strike - S price The value of these options can be modelled by the following graphs.

2 Why do we use options? Options allow us to gain larger profits. For example an investor has $450 and is looking to buy IBM stock which is prices at $90. If he thinks the price is going to increase, instead of buying 5 stocks of IBM stock, he will buy month calls prices at If at the end of the month the price is $95, instead of making $25 dollars, he would have made $590 (increase* number of options). Risk is if the stock does not go up, he loses the money. Options can also be used for hedging or shorting a stock. Imagine if an investor owns 1000 shares of Caterpillar priced at $88 per share. To hedge his investment, the investor will buy a 3 month put option with the strike price $85. This gives him the time to bail out at $85 per share over the next 3 months if it goes down (for the price of 1 $1.50 put option, he secures $88,000 worth of investment). Black Scholes Equation: This brings us to the Black Scholes Equation. The elegance of the model is based on the fact that it is derived from the supposition that log return of the stock follows a Brownian motion

3 with drift. It seems a valid assumption since a stock is submitted to a multitude of shocks. This is the basis for the normal distribution. Moreover it gives a model which is not based on the real probabilities of stock prices but on what we call risk neutral probabilities In fact, the model was made to hedge the option by buying and selling the underlying asset in just the right way and, as a consequence, to eliminate risk. This was derived in the late 70s by Fischer Black who was at MIT at the time, (and later went to Goldman Sachs) and by Myron Scholes who is now in San Francisco. This equation can be used either of two ways. Python coding of the Black Scholes Equation: The most direct way to use this equation is to get the price that you think is the right price for an option. Therefore you can decide whether or not you are paying too much or too little for an option. To use this formula, you have to know what the stock price is, (S ) the exercise price (E) the time to maturity(t), the interest rate(r) and have some idea of the volatility (standard deviation) (σ ). The first three parameters are specified in the contract. But you can also turn it around, if you already know what the option is selling for in the market. Options are traded daily and are quoted daily. Then you can infer the variable which is hard to pin down, the volatility σi implied by the market. Remember all the other parameters are known. It gives you a sense of the variability of the stock price. People often use the Black-Scholes Formula to invert it and calculate the implied volatility of stock prices. The latter is based on the fact there must exist some relation between the price of an option and the volatility. For instance, if an out of money call is valuable it must be people think that σ is high. So we solve the equation for σ (numerically!) Meaning that for any given call price (C ), stock price (S ), exercise price (E ), time to maturity (T ) and interest rate (r ), we find the appropriate volatility (σ i ). In other words this implied volatility is the option s market opinion of how variable the stock market will be between now and the exercise date (Assuming that the Black Scholes equation is correct!) In the graph, the blue line is computed by the Chicago Board Options Exchange. Using the Black Scholes equation, based on the front month option, they compute what the options market thought the volatility of the stock market was. The VIX is the σ in the Black Scholes equation so it is in effect the market s expected standard deviation of stock prices. The actual volatility is the red line, meaning the standard deviation of actual stock prices over the preceding year and again annualized, in other words it shows the monthly changes annualized. This actual

4 volatility is then computed thanks to past data. The blue line overstates the volatility of the market. Financial markets are very stable for a long time, but there is a risk of anomalies like The Great Depression. Outliers or black swan events are big disruptors of economic theory. But the Black-Scholes theory is not a black swan theory. It assumes normality of distributions so it is not always reliable. The option pricing theory (The Black Scholes theory) is very elegant and is especially useful when things behave normally. However, the probability of anomalies is not well computed. Some of the criticism of the Black Scholes model includes volatility: a measure of how much a stock can be expected to move in the near term is a constant over time. Being mathematically elegant does not guarantee accuracy and as a matter of fact it might be an incentive to lead to the opposite. For starters, the volatility is not constant as it is supposed in the model. According to the model, σ must remain constant over time. However, the standard deviation is clearly not constant. In fact, as the term volatility assumes, it is a very volatile measure. While volatility can be relatively constant in very short term, it is never constant in longer term. Large price changes tend to be followed by large price changes, and vice versa leading to a property called volatility clustering. σ is just the standard deviation, so σ can be viewed as a measure of the volatility of the stock price. Another reason that explains why the Black-Scholes model is not truly reliable is linked to the more general idea of prediction in the social sciences. Indeed, in the social sciences unlike the formal sciences, systems are contingent and not random. This results from the fact that there is a human side to the social sciences and that, subsequently, one has to account for anticipation. We cannot predict the anticipation of the anticipation of the anticipation. For example, in the classic game of rock, paper scissors, the first player will probably try to anticipate the move of his opponent. Let s say he thinks that his opponent will play rock, he will then, obviously, choose to play paper. However, what if his opponent had anticipated such an anticipation? Then, the first player would have to play rock in order to beat the scissors of his opponent, and, so on and so forth we spiral down into a never ending successions of predictions of predictions of predictions Thus, the fact that such a system is contingent and, hence, overdetermined means that there is almost no certainty in any social science prediction, unlike in the formal science where the fact that the system is random means that it is less determined and that the certainty is higher. Indeed, more a system is predictive, less it is determined. For example, in quantum science, the position of an electron is not at all determined, but we can predict such a position through

5 probabilities that are very precise. It may seem paradoxical but to make any predictions in the social sciences, one has to delete the human side. Certain models try to take this into account. There are time series techniques that try to model the evolution with time of models, relying more on factually based events, than raw hypotheses. There are also Bayesian techniques that are meant to tune in your prior distribution and give you an answer that is supposed to be more accurate with data coming. The continuous flow of information into these models is intended to give a more accurate answer. Finally, there are also models that assume uncertainty for pricing options, and factor that into their calculations. In today s financial world, people are willing to try anything to get it right. Even though the Black Scholes can be useful, and does offer estimates for prices of European-style options, its results do not always reflect reality. The flawed foundation of the model leads to this difference. Although the fault in the assumption about volatility could be corrected, (one can include volatility time dependency in the model or even stochastic volatility to account for randomness, etc ) the fallibility of predictions prevents the possibility of ever having an exact model when it comes to stocks and options. A major solution to the fault in the volatility is in calibration, finding the volatility day to day, as opposed to assuming it remains constant. Here, we can solve the equation with volatility as the variable, as opposed to the option price itself, something that we know everyday. However, this causes heavy reliance on the information that gives the volatility, which as we saw earlier on, overstates reality. Nowadays people use other techniques such as statistical learning and Big Data for more accurate models, but the question remains, will we ever have a truly infallible financial model?