Option pricing. School of Business C-thesis in Economics, 10p Course code: EN0270 Supervisor: Johan Lindén

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1 School of Business C-thesis in Economics, 1p Course code: EN27 Supervisor: Johan Lindén Option pricing A Test of the Black & scholes theory using market data By Marlon Gerard Silos & Glyn Grimwade

2 Abstract Date : 3 May 25 Level : Bachelor thesis in Economics Authors : Marlon Gerard Silos & Glyn Grimwade Axel Oxenstiernasgata 15 Gussjö VÄSTERÅS SALBOHED Advisor : Johan Lindén Title : Option Pricing Subtitle : A test of the Black & Scholes theory using market data Problem : With the help of option data performing a test of the Black & Scholes theory. Purpose : To see how well the Black & Scholes theory applies to the real world and how well the market predicts the future development of the stock price through the implied volatility Method : By analysing two years of option data on ABB and Ericsson, choosing three maturity dates of each option and using the Black & Scholes model we try to determine the implied volatility. With the help of stock price data covering the time between first observable day and maturity of each option, try to determine the actual (historical) volatility Results : The assumption of constant volatility does not hold. The implied volatility doesn t come near the actual (historical) volatility Conclusion : The Black & Scholes theory does not hold for these options. An adjusted model of the theory has to be created for these particular options Keywords : Black and Scholes, Implied, Actual (historical) volatility and Options i

3 Table of Content 1. Introduction Background Problem Area Purpose Research Methodology Subject The data Disposition Limitations Theory Options Contracts The value of options Option Pricing The Black & Scholes model The Greeks Option Combinations The Straddle The Strangle The Bull call spread The Bear call spread The Bull and Bear put spreads Working Method Historical Implied Analysis & Conclusion The volatility smile Implied volatility versus Historical volatility (ABB) Implied volatility versus Historical volatility (Ericsson) Acknowledgement Sources...61 ii

4 Table of Figures Figure 1 : Payoff Diagram (Call option) 9 Figure 2 : Payoff Diagram (Put option) 1 Figure 3 : Payoff Diagram for a long Straddle 19 Figure 4 : Payoff Diagram for a Straddle 19 Figure 5 : Payoff Diagram for a long Strangle 21 Figure 6 : Payoff Diagram for a Strangle 21 Figure 7 : Payoff Diagram for a Bull call spread 22 Figure 8 : Payoff Diagram for a Bull call spread 23 Figure 9 : Payoff Diagram for a Bear call spread 24 Figure 1: Payoff Diagram for a Bear call spread 25 Figure 11: Payoff Diagram for a Bull put spread 26 Figure 12: Payoff Diagram Bull put spread 26 Figure 13: A picture representation of the historical volatility calculator 29 Figure 14: Historical volatilities ABB maturing in Figure 15: Historical volatilities ABB maturing in Figure 16: Historical volatilities ABB maturing in Figure 17: Historical volatilities Ericsson maturing in Figure 18: Historical volatilities Ericsson maturing in Figure 19: Historical volatilities Ericsson maturing in Figure 2: Historical volatility graph ABB ( to ) 33 Figure 21: Historical volatility graph ABB ( to ) 34 Figure 22: Historical volatility graph ABB ( to ) 34 Figure 23: Historical volatility graph Ericsson ( to ) 35 Figure 24: Historical volatility graph Ericsson ( to ) 35 Figure 25: Historical volatility graph Ericsson ( to ) 36 Figure 26: Options ABB maturing in Figure 27: Options ABB maturing in and Figure 28: Options Ericsson maturing , and Figure 29: Implied volatility ABB maturing in Figure 3: Implied volatility ABB maturing in Figure 31: Implied volatility ABB maturing in Figure 32: Implied volatility Ericsson maturing in Figure 33: Implied volatility Ericsson maturing in iii

5 Figure 34: Implied volatility Ericsson maturing in Figure 35: Implied volatility ABB Call options on maturing in Figure 36: Implied volatility ABB Put options on maturing in Figure 37: Implied volatility Ericsson Call options on maturing in Figure 38: Implied volatility Ericsson Put options on maturing in Figure 39: Comparison of historical and implied volatility ABB from to Figure 4: Comparison of historical and implied volatility Ericsson from to Figure 41: Implied volatility ABB Call options on maturing in Figure 42: Implied volatility Ericsson Put options on maturing in iv

6 1. Introduction 1.1 Background The choice of this subject comes from our curiosity of how well economic theory applies to the economic reality. It will give us some idea of how critical we must be when applying the theory into practice. 1.2 Problem Area Traders on the option market evaluate the options traded by using a model. It could be the Black & Scholes model, but also the Binominal model. The writers of options use the models in order to price their options and the buyers of options use the models in order to evaluate whether specific options are worth buying. One trades in the option market for different reasons. One of the reasons could be to hedge an investment portfolio. For a writer of options it is important to price an option correctly, because one is legally bound to buy or sell the underlying asset on which the option is written should the right that is imbedded in the option be exercised. 1.3 Purpose We will focus on one of the models used to valuate options: the Black & Scholes model. Specifically we wish to see how well the Black & Scholes theory applies to the real world. In addition we will also focus on how accurately the option market can predict the future development of the stock price by analysing the implied volatility. In order to do this we will use the Black & Scholes model. 5

7 2. Research Methodology So how did we start? First of all we needed to find a subject for our thesis. After some discussion we decided to write a thesis whereby we could apply our theoretical knowledge to a practical test. 2.1 Subject We had two subjects in mind, fuel hedging for the aviation industry, and application of the Black & Scholes theory. It became clear that we couldn t perform the fuel hedging due to the non-availability of information. This left us with the application of the Black & Scholes theory; here the problem was access to option data. This problem was resolved very quickly due to the willingness of a fellow student Raphael Vides, who shared his data on options. 2.2 The data The data we received from Raphael Vides was compiled through his contact with the OMX-group. He asked for data regarding options traded on the Stockholm Stock Exchange from the beginning of 23 to the end of 24. Due to the vast amount of data received, we decided that the options we would use in our thesis were the options on ABB and Ericsson as they were traded in the option market from the 2 nd of January 23 to the 2th of August 24. The data consists of call and put options with different strike prices, ask & bid prices and closing prices. In other words the data shows the development of the options prices for the two years. We used the Internet and Mälardalen University library for additional information on the Black & Scholes theory, which could be used in our analysis. 2.3 Disposition The report you are about to read is about the Black & Scholes Model and market efficiency. What we will try to do in this report is to give the reader some insight into the Black & Scholes model and with the help of this model see how well the market predicts the future movements of stocks by analyzing prices. We will apply the model to actual options that exist in the market. The build up of the report is as follows; In chapter 3 we will give you some theoretical information on options, option contracts and option pricing. Following this we will explain what the Black & Scholes model is and what the assumptions of the model are. This is supplemented with a description of the Greeks and some option 6

8 strategies. In chapter 4 we will describe the working process we went through in order to make the analysis. In chapter 5 we will analyse our calculations and also draw conclusions based on our findings. 2.4 Limitations A limitation when using the data is that not all options listed in the data had a closing price. Some of the options only have ask and bid prices, but no closing prices. In order to overcome this problem we used weighted averages 1. Another limitation is the time difference between the closing price of the options and the related stock prices. By this we mean that the closing price on options on certain particular days might not reflect the closing price of the underlying stock. It could happen that the closing price on certain options where estimated hours before the closing of the stock market, leading to that the implied volatility (which is an important variable in estimating the price of options) at a certain time on a certain day might reflect the volatility on a stock price on the same time and day that might be different from the closing stock price on that day. 1 See chapter 4.2 for more details 7

9 3. Theory 3.1 Options Contracts The idea of options is certainly not new. Ancient Romans, Grecians, and Phoenicians traded options against outgoing cargoes from their local seaports. When used in relation to financial instruments, options are generally defined as a "contract between two parties in which one party has the right but not the obligation to do something, usually to buy or sell some underlying asset". Having rights without obligations has financial value, so option holders must purchase these rights, making them assets. These assets derive their value from some other asset, so they are called derivative assets. Call options are contracts giving the option holder the right to buy something, while put options entitle the holder to sell something. Payment for call and put options, takes the form of a flat, up-front sum called a premium. 2 The price of a call/put-option is known as the premium. The premium together with the strike or exercise price is the value of a call/put-option. The date on a call/putoption is known as the expiration date or maturity. In this report we focus on European options, which can only be exercised on the maturity date. Options are abbreviated when they are listed at various sources. An example from Dagens Industri : ERICB-4J15 ERICB: The underlying stock, in this case Ericsson B 4: The year the option was written, here 24 J: Letters have two meanings; they tell you which month the option is to be realized, and also what type of option it is; call or put. 2 ( ) 8

10 Letters A to L denotes a call-option, M to X denote a put-option, which expires in: A January M January B February N February C March O March D April P April E May Q May F June R June G July S July H August T August I September U September J October V October K November W November L December X December The value of options In the case of a call-option (if you are a buyer); if the price of the underlying stock is above the exercise price of the call-option it is said to be in-the money. Profit/Loss Figure 1: Payoff diagram (Call option) Profit strike or excercise price of call option Stock Price Premium 9

11 In the case of a put-option (if you are a buyer): if the price of the underlying stock is below the exercise price of the put-option it is said to be in-the money. Figure 2: Payoff diagram (Put option) Profit/Loss Profit strike or excercise price of put option Premium Stock Price 1

12 3.2 Option Pricing The most important concept to understand when it comes to trading options is option pricing. Only by understanding option pricing will you be able to find out if an option is worth buying or not. To understand basic analysis of options we can think of the option price as a function of the stock price of the underlying asset, the exercise price, and the time to maturity. In this report we will deal exclusively with European put- and call-options, and will use the following notation; S t = stock price at time t X = the options exercise price T = the options expiration date C t = the price of the call option at time t P t = the price of the putt option at time t So the value of a call option at time t given a stock price at that time of S t, an exercise price of X, that expires at time T can be written as; C t (S t, X, T-t). Now let s look at the possible profits and losses of a call option at expiration. At expiration the option will either be in-the money, at-the-money, or out-of-the-money. If a call option expires in-the money, this means that the option's value was below the market price of the underlying asset and will yield a profit. If the call option on the other hand expire at- or out-of-the-money, the option s value was at or above the market price of the underlying asset. The notation for the value of a call option at expiration is: C T = MAX {, S T X} This can be read as; the value of the call option at expiration is either zero, or the difference between the stock price and the exercise price. Whichever is the greater. Assume we bought an option with a strike price of 1, written on the ABC share which has a current stock price of 1. Further assume that the option expires 6 months in the future and that the option costs 1 (premium). In order for us to make money on this option the stock price on the day of expiration must be above 11( 1 + the premium of the option). If it is below 11, the option will be out-of-the-money and as such will be worth nothing. 11

13 List of notations for call and put options: Type of Option Notation Long Call C T = MAX{, S T X} Short Call C T = MIN{, X S T } Long Put P T = MAX{, X S T } Short Put P T = MIN{, S T X} Questions that may arise now are: why did that option had a premium of 1? What decides the price of an option on the market? Well, factors we have looked at are the stock price, the exercise price and time remaining till expiration. These factors alone however are not sufficient to actually put a price tag on any specific option. There are two more factors to take into consideration, namely the stock movement between the day of purchase and the day of expiration, and the interest rate. If the option expires in twelve months we know that the stock price will move thousands of times between now and expiration. This stochastic movement has been the subject of many studies in the fields of mathematics and finance. There are several option pricing models available, such as the binomial option pricing model and the Garman and Kohlhagen 3 (1983) option pricing formula. But the option pricing model that is most widely known and used however, is the Black and Scholes option pricing formula. 3 This model is applicable for valuing European call and European put options on foreign exchange. 12

14 3.3 The Black & Scholes model One of the most frequently used options pricing models is the Black & Scholes model. The model was created by Fisher Black and Myron Scholes in the late 6 s and early 7 s. When they first submitted their work to the Journal of Political Economy for publication, it was promptly rejected. After making several revisions based on extensive comments from Merton Miller (Nobel Laureate from the University of Chicago) and Eugene Fama, of the University of Chicago, they resubmitted their paper and were published in Almost simultaneously with the famous Black Scholes paper in the Journal of Political Economy, Robert C. Merton published "The Theory of Rational Option Pricing" in the Bell Journal of Economics. The two approaches to valuation reinforced each other and turned out to be of enormous practical significance. Huge new markets in "derivatives" arose on the basis of them. And in 1997 Robert C. Merton and Scholes shared the Nobel Award in economics (the year after Fischer Black died), for the new method they had devised 4. The Black & Scholes model is relatively simple to use and has a fast mode of calculation. If compared to the binomial model the Black & Scholes model does not rely on calculation by iteration. The Black & Scholes model does however take on some rather steep assumptions. The assumptions of the Black & Scholes model are 5,6 : - The model assumes that the stock pays no dividend 7 during the options life. - The model is limited to the use of only European options, which can only be exercised on expiration date. - The model assumes that markets are efficient. This means that people can t consistently foresee the direction of the market or an individual stock. The model assumes that the market operates continuously following a continuous Ito process. An Ito process is a Markov process ( a process where one observation in time period t depends only on the preceding observation ) 8 in continuous time. - The model assumes that no commissions are charged when people buy or sell options, which is usual in real life Derivative securities, Jarrow & Turnbull, 2, pages For stocks that pay continuous dividends the Merton (1973) model can be used

15 - The model assumes that the interest rates remain constant and known and are the same for all maturity dates. - The model assumes that the stock price follows a lognormal distribution. - The model assumes that the instantaneous expected return and the instantaneous variance of the return on the stock per unit time is a constant. The pricing formula for the Black & Scholes model is: -rt c = S N(d1) - Xe N(d 2 ) rt p = Xe N( d 2) SN( d 1) c = the price of a call option p = the price of the put option S = the current stock price N = cumulative standard normal distribution X = option exercise price e = exponent T = time to maturity r = risk free rate d d 1 S ln r X = σ T σ T 2 2 S σ ln + r T X 2 = = d σ T σ T 2 1 σ 2 = variance of the underlying asset σ = volatility of the underlying asset The SN(-d) 1 part of the Black & Scholes model stands for the expected benefit from -rt acquiring the stock. The Xe N(-d ) part of the model stands for the present value of paying the exercise price on maturity date. By taking the difference between the two we get the fair market value of the call/put option. 2 14

16 3.3.1 The volatility, or standard deviation, measures the movement of a stock s return. The stock s return is the return one expects to earn on an asset in the future. It is an estimate of the value of an investment. If an asset is risky, the stock s return will be the risk-free rate of return plus a certain risk premium. The return investors require from a stock depends on the risk of the stock involved and the level of the interest rate in the economy. The risk-free interest rate is the interest rate to be obtained by investing with no risk. In practice the interest rate on bonds given out by the government are used as the risk free rate. The reason for this is that the probability that the government will not oblige by its payments is extremely low. The risk free rate is an important assumption for constructing portfolios and other financial calculations, such as the Black & Scholes model. To return to the volatility, the volatility is a measure of uncertainty that a stock provides and reflects the size of the random movements of the stock as it moves through time. It is a way to measure statistical risk and as such can be used to determine financial risks. There are two ways to estimate the volatility. Implied volatility 9 Historical volatility 1 For the Black & Scholes model to hold there must be one constant implied volatility regardless of the option s strike price and maturity variations. The implied volatility reflects the sentiments of the market. The historical volatility reflects the actual market fluctuations of the stock price over a specific time period. It shows the realized volatility. According to the Black & Scholes model the implied volatility of the different options must have the same volatility over the same time period. The reason for this is because of the assumption in the model that states that pricing should not permit arbitrage. According to Black & Scholes, their model captures the factors that have influence on the price of an option. The implied volatility depicted in the option prices that are observable in the market should correspond to the movements of the underlying stock itself. The writers and the buyers of options don t only use the Black & Scholes model to depict a future movement of the stock price in order to price the option, they also use the so called Greeks to help them price or evaluate options. 9 How one gets the implied volatility will be discussed in chapter How one gets the historical volatility will be discussed in chapter

17 3.4 The Greeks The Greeks are a set of statistical risk measures that can measure the exposure to risk of an option portfolio. The Greeks give an investor an overall view of how stocks have been performing. The Greeks consists out of different letters of the Greek alphabet, each of these letters shows a measurement of risk. Greeks are first- (Delta, Theta, Vega, Rho) and second (Gamma) order partial derivatives of the Black and Scholes formula. Delta: c Change in the price of the call option Delta = = S Change in the value of the underlying asset Delta is the most commonly used Greek. Delta measures the relationship between the price of an option and the price of the underlying stock and is sometimes referred to as the hedge ratio. Assume a call option on Ericsson that has a delta of.6. If the Ericsson share goes up by 1., then the cost of the option would go up by.6. Put option deltas are, as you may expect, negative. So as the value of the underlying asset goes up, the cost of the put option will go down. As in-the-money options approach expiration, the delta for call (put) options will approach 1( 1). Gamma: 2 dc Change in the value of the call option's delta Gamma = = 2 ds Change in the value of the underlying asset Gamma is a measure of the calculated delta's sensitivity to small changes in the underlying stock price, i.e. the delta of the delta. As a tool, gamma can tell you how "stable" your delta is. A big gamma means that your delta can start changing dramatically for even a small move in the stock price. If we are hedging an option, the gamma tells us how sensitive our hedge is to changes in the underlying asset. For example, assume the Ericsson call has a delta of.6, and the Ericsson put has a delta of -.7, with the price of Ericsson at 3.. Further assume that the gamma for both the Ericsson call and put is.7. If the Ericsson stock goes up by 1. to 4., the delta of the Ericsson call becomes.67 (+.6 + ( 1 *.7), and the delta of the Ericsson put becomes -.63 (-.7 + ( 1 *.7). If Ericsson drops 1. to 2., the delta of the Ericsson call becomes +.53 (.6 + (- 1 *.7), and the delta of the Ericsson put becomes -.77 (-.7 + (-$1 *.7). 16

18 Theta: c Change in the price of the call option Theta = = t Decrease in the time to expiration Theta measures the sensitivity of the option value to small changes in time untill expiration. Specifically, it tells you how rapidly a portfolio's market value will change with time, assuming that all market variables the underlying asset, implied volatilities, interest rates, etc. do not change 11. But theta doesn't reduce an option's value at an even rate. Theta has much more impact on an option with fewer days to expiration than an option with more days to expiration. For example, assume an Ericsson put expiring in October that is worth 3., has 2 days until expiration and has a theta of Further assume an Ericsson put expiring in December is worth 4.75, has 8 days until expiration and has a theta of -.3. If one day passes, and the price of the Ericsson stock doesn't change, and there is no change in the implied volatility of either option, the value of the Ericsson put expiring in October will drop by.15 to 2.85, and the value of the Ericsson put expiring in December will drop by.3 to Vega: c Change in the price of the call option Vega = = σ Change in the volatility Vega measures the calculated option value's sensitivity to small changes in volatility. Vega, along with gamma and delta, is often used in hedging. Higher volatility results in higher option prices. This is because a higher volatility means that larger price swings in the price of the underlying asset is expected, which implies that there is an increasing likelihood that an option will make money at expiration. Assume an Ericsson call again. Let s say it has a value of 2. and a Vega of +.2 and with a volatility of 3.%. If the volatility of Ericsson was to rises to 31.%, the value of the Ericsson call would rise to 2.2. If the volatility of Ericsson falls to 29.%, the value of the Ericsson call would drop to 1.8. Rho: c Change in the price of the call option Rho = = r Change in the value of the interest rate Rho measures the calculated options value s sensitivity to changes in the risk free interest rate. Rho together with Theta is one of the least used Greeks, because the

19 interest rates are relatively stable and the chance that the value of an option position will change dramatically because of a drop or rise in the interest rates is quite low. Assume an Ericsson call with a value of 1. and a Rho of +.2 with the Ericsson stock at 3. and interest rate at 5.%; If the interest rate went up to 6.%, the value of the Ericsson call would increase to 1.2. If on the other hand the interest rate went down to 4.%, the value of the Ericsson call would decrease to.98. Together with the Black & Scholes model and the Greeks investors try price their option or apply an option strategy or strategies depending on their calculated historical and implied volatility. 3.5 Option Combinations By trading option combinations we can change the return/risk characteristics of our option position. We will show how we can profit from using option combinations when stock prices move up or down or even when stock prices stand still The Straddle A straddle consists of a call option and a put option with the same strike price (usually at the money), the same expiration, and on the same underlying asset. With a long straddle we will be able to take advantage of a move of the stock in either direction, up or down. A long straddle involves buying the two options while a short straddle involves selling these same options. The short straddle should be used when you believe the price of the underlying asset will stay essentially unchanged. Assume that we buy a call and a put, both with a strike price of 1. Further assume the call costs 2 and the put costs 1 and the current stock price is 1. The profits and losses for buying this long straddle are nothing but the combined profits and losses from buying both options. Let C t denote the current price of the call option and let C T denote the value of the call option at expiration. S T denotes the stock price at expiration and X denotes the strike price. The cost of the long straddle is: C t + P t and the value of the long straddle upon expiration will be: C T + P T = MAX{, S T X} + MAX{, X S T } The cost of the short straddle is: -C t - P t (which is a positive cash inflow as the short trader receives the premium payment for accepting the short straddle position.) 18

20 and the value of the short straddle upon expiration will be: -C T - P T = - MAX{, S T X} - MAX{, X S T } Profit/Loss Figure 3: Payoff diagram for a Long Straddle Stock price Figure 4: Payoff diagram for a Straddle Long Put Long Call Profit/Loss Stock price Long Straddle Short straddle Viewing the second graph it becomes clear that option speculation is a zero-sum game, the profits and losses for the buyer mirrors that of the seller. The maximum loss for the buyer is the cost of the two options, 3, which is also the maximum profit for the seller. As regards the profits for the buyer they are almost unlimited. The maximum profit for the seller if the stock falls (to zero) is 7. The other side of the payoff, if the stock rises in value, is potentially unlimited. 19

21 3.5.2 The Strangle The option strangle is very similar to the straddle. A strangle consists of a put option and a call option with the same expiration date and written on the same underlying asset. The difference between the straddle and the strangle is that with the strangle the call option has an exercise price above the stock price and the put option has an exercise price below the stock price. A long strangle is cheaper than a long straddle because of this, i.e. both options have a strike that is out of the money. On the other hand a strangle requires a larger move in the underlying asset to be profitable. Let C t,1 denote the cost of the call option with exercise price X 1 at time t, and let P t,2 denote the cost of the put option with exercise price X 2 at time t, with the constraint that X 1 > X 2. The cost of the long strangle is: C t,1 (S t, X 1, T) + P t,2 (S t, X 2, T) And the value of the strangle at expiration will be: C T,1 (S t, X 1, T) + P T,2 (S t, X 2, T) = MAX{, S T X 1 } + MAX{, X 2 S T } The cost of the short strangle is: -C t,1 (S t, X 1, T) - P t,2 (S t, X 2, T) At expiration the value of the short strangle will be: -C T,1 (S t, X 1, T) P T,2 (S t, X 2, T) = - MAX{, S T X 1 } - MAX{, X 2 S T } 2

22 To illustrate the long strangle, assume the costs are 1 for the call option and.5 for the put option. Further assume the call option has a strike price of 11 and the put option has a strike price of 9. Profit/Loss Figure 5: Payoff diagram for a Long Strangle Stock price Long Put Long Call Profit/Loss Figure 6: Payoff diagram for a Strangle Stock price Long Strangle Short Strangle As can be seen from the above diagram, the long strangle has a positive cash flow for any expiration stock price above 12.5 or below 7.5. If the stock price at expiration is above 11 the call option will be exercised but not until it has reached 12.5 will you have a positive cash flow due to the cost of the two options. Likewise the put will be exercised if the stock price at expiration is below 9, but will not generate a positive cash flow until it has dropped below

23 3.5.3 The Bull call spread If you believe there will be an upward move in the underlying asset, but you are uncertain about the extent of the move, the bull spread utilizing call options could be a good strategy. A call option bull spread is made up of two call options with the same expiration date and written on the same underlying asset, but with different exercise prices. The buyer of a bull spread purchases a call option with a strike price below the stock price and sells a call option with a strike price above the stock price. Using this strategy, the potential loss is limited to the premium you paid for the long call less the premium you collected for the call you sold. Let C t,1 denote the cost of the call option with exercise price X 1 at time t, and let C t,2 denote the cost of the call option with exercise price X 2 at time t. Add to this the constraint that X 1 < X 2 and you have a bull spread. The cost of the bull call spread is: C t,1 (S t, X 1, T) - C t,2 (S t, X 2, T) At expiration the value of the bull call spread will be: C T,1 (S t, X 1, T) C T,2 (S t, X 2, T) = MAX{, S T X 1 } - MAX{, S T X 2 } Assume that the underlying asset is trading at 2 per share and you buy a call that has an exercise price of 16 and costs 5. Now sell a call option with an exercise price of 24 for 3. The total cost of this position is 2. Figure 7: Payoff diagram for a Bull call spread Profit/Loss Stock price Long Call Short Call 22

24 Figure 8: Payoff diagram for a Bull call spread Profit/Loss Stock price Bull call spread 23

25 3.5.4 The Bear call spread A bear call spread is used if you believe there will be a downward trend in the underlying asset. A bear call spread is nothing more than the short position to the bull call spread. A call option bear spread is made up of two call options with the same expiration date and written on the same underlying asset, but with different exercise prices. The buyer of a bear spread sells a call option with a strike price below the stock price and purchases a call option with a strike price above the stock price. Using this strategy, the potential loss is again limited to the premium you paid for the long call less the premium you collected for the call you sold. Let C t,1 denote the cost of the call option with exercise price X 1 at time t, and let C t,2 denote the cost of the call option with exercise price X 2 at time t. Again add to this the constraint that X 1 < X 2 and you have a bear spread. The cost of the bear call spread is: - C t,1 (S t, X 1, T) + C t,2 (S t, X 2, T) At expiration the value of the bear call spread will be: - C T,1 (S t, X 1, T) + C T,2 (S t, X 2, T) = - MAX{, S T X 1 } + MAX{, S T X 2 } For simplicity we can assume that the figures are the same as in the previous example, the underlying asset is trading at 2 per share and you buy a call that has an exercise price of 24 and costs 3. Now sell a call option with an exercise price of 16 for 5. The total cost of this position is - 2. Now you can see from the graphs that the profits and losses form the bear call spread is a mirror image of the bull call spread. Profit/Loss Figure 9: Payoff diagram for a bear call spread Stock price -1 Long Call 24 Short Call

26 Figure 1: Payoff diagram for a Bear call spread Profit/Loss Stock price Bear call spread The Bull and Bear put spreads One can also create bull and bear spreads using put options. Again the bull put spread is executed if you believe there will be an upward trend in the underlying asset and the bear is executed if you believe there will be a downward trend. To illustrate how this would work assume that we have two put options with the same expiration date and written on the same underlying asset, currently trading for 2. One put option has an exercise price of 16 and costs 3, while the other has an exercise price of 24 and costs 5. To create a bull put spread you would buy the put option with strike price 16 and sell the put option with strike price 24. To create a bear put spread you take the opposite side in these two options. Let P t,1 denote the price of the put option with strike price X 1 at time t, and let P t,2 denote the price of the put option with strike price X 2 at time t, where X 1 < X 2. The cost of the bull put spread is: P t,1 (S t, X 1, T) P t,2 (S t, X 2, T) The value of the bull put spread at expiration will be: P T,1 (S T, X 1, T) P T,2 (S T, X 2, T) = MAX{, X 1 - S T } - MAX{, X 2 S T } The cost of the bear put spread is : -P t,1 (S t, X 1, T) + P t,2 (S t, X 2, T) And the value of the bear put spread at expiration will be: -P T,1 (S T, X 1, T) + P T,2 (S T, X 2, T) = - MAX{, X 1 - S T } + MAX{, X 2 S T } 25

27 Profit/Loss Figure 11: Payoff diagram for a bull put spread Stock price -14 Long Put Short Put Figure 12: Payoff diagram for a Bull put spread 4 2 Profit/Loss Stock price Bull put spread 26

28 4. Working Method Now that we have introduced the theoretical thinking behind the Black & Scholes model, we will continue to show the process of how to calculate the historical and implied volatilities. In this paper we have chosen to focus on options written on the ABB and Ericsson stocks. To be more specific we will focus on ABB options that have a maturity in February, April, and June 24. Regarding Ericsson we focused on the options that have a maturity in February, May, and August Historical The price volatility of an asset can be defined as the standard deviation of the lognormal returns of the asset price or the standard deviation of the percentage change in price. Mathematically this can be expressed as: Where: = σ s = s τ Σu n 2 i 2 ( Σui ) n( n 1) 1 τ = trading days u i S i = ln Si 1 n = number of observations u i τ = time intervals in years S = stock price on day i S i = daily return i-1 = stock price on day i -1 s = standard deviation of the daily return σ = volatility For the options under investigation we calculated the historical volatility from each day the option was traded until the maturity of the option. What we wanted to do was to compare the historical volatility from a certain date until expiration with the volatility implied by the price of the option on the market. The reason for doing this is to see how well the market estimates the volatility of the underlying asset in question. When an option trader wants to derive the value of an option he has to assume a volatility which he believes will adequately depict the future movement of the underlying asset. Since we are looking back in time, we can compare these assumed volatilities with the true volatilities derived from the actual fluctuations in the price of the underlying asset itself. To calculate the historical volatility of the stock prices we first need to select the stock prices over the relevant intervals. For this we chose the closing price of the stock over the entire interval, from the first day the option was traded until the day when the option expired. The next step was to calculate the daily percentual 27

29 S changes in price (the price relatives), S price relatives we calculated the daily returns, i i 1. On the basis of the results we got from the u i S = ln Si i 1. Then we took the sum of the daily returns and the sums of the daily returns squared and used it to estimate the standard deviation of the daily return, s = Σu n 2 i 2 ( Σui ). We looked up how many n( n 1) trading days per year there are in Sweden and use this information to calculate the length of time intervals in years, 1 τ =. With the help of the standard trading days deviation of the daily return and the length of time intervals per year, we calculated the historical volatility, σ = s. We have now completed the historical volatility τ calculation for one single time interval. Since the options we were investigating had up to 283 trading days until expiration we needed to repeat this procedure for every interval, i.e. 282 observations. In order to simplify the calculations we made a spread sheet in Excel that could calculate the historical volatility automatically for any given time period. 28

30 Figure 13: A picture representation of the historical volatility calculator 12 Next we needed to compile this information in a separate worksheet so we could make a graphical representation of the observed historical volatilities as we moved towards expiration. 12 See excel file for more details 29

31 A picture representation of the worksheet 13 : Figure 14: Historical volatilities ABB maturing in Options maturing in Days to Time span % Maturity , ,71% , ,82% , ,91% , ,87% , ,98% , ,74% , ,82% , ,94% , ,99% , ,11% , ,97% , ,8% , ,2% , ,14% , ,26% , ,27% , ,3% , ,14% , ,13% , ,17% , ,28% 263 Figure 15: Historical volatilities ABB maturing in Options maturing in Days to Time span % Maturity , ,82% , ,1% , ,22% , ,43% , ,57% , ,43% , ,54% , ,39% , ,85% , ,1% , ,9% , ,87% , ,98% , ,2% See excel file for more details 3

32 Figure 16: Historical volatilities ABB maturing in Options maturing in Days to Time span % Maturity , ,86% , ,94% , ,1% , ,23% , ,38% , ,54% , ,7% , ,85% , ,84% , ,9% , ,7% , ,6% , ,76% , ,89% , ,33% , ,6% , ,18% , ,13% , ,11% , ,28% , ,4% , ,56% , ,67% , ,51% , ,41% , ,6% , ,7% , ,86% 96 Figure 17: Historical volatilities Ericsson maturing in Options maturing in Days to Time span % Maturity , ,99% , ,2% , ,11% , ,76% , ,83% , ,9% , , ,6% , ,17% , ,85% , ,9% , ,96% , ,86% , ,97%

33 Figure 18: Historical volatilities Ericsson maturing in Options maturing in Days to Time span % Maturity , ,68% , ,41% , ,24% , ,34% , ,41% , ,5% , ,57% , ,65% , ,63% , ,72% , ,78% , ,58% , ,66% , ,67% , ,25% , ,15% , ,22% , ,29% , ,8% , ,16% , ,2% , ,28% , ,34% , ,37% , ,31% , ,38% , ,36% , ,36% , ,47% 225 Figure 19: Historical volatilities Ericsson maturing in Options maturing in Days to Time span % Maturity , ,6% , ,63% , ,66% , ,75% , ,83% , ,75% , ,6% , ,63% , ,56% , ,65% , ,68% , ,77% , ,84% , ,91% , ,94%

34 The resulting average historical volatilities are: ABB ( to ): 57,3 % ABB ( to ): 37,56 % ABB ( to ): 34,7 % Ericsson ( to ): 51,56 % Ericsson ( to ): 48,92 % Ericsson ( to ): 45,24 % Now we have all the information we need to graph the movement of the historical volatility as the option moves through time towards expiration. A picture representation of the historical volatility as it moves towards expiration: Figure 2: Historical volatility graph ABB ( to ) 1,% Historical (ABB) to ,% 8,% 7,% 4,%

35 Figure 21: Historical volatility graph ABB ( to ) Historical (ABB) to ,% 3,% 2,% 1,%,% Figure 22: Historical volatility graph ABB ( to ) 45,% Historical (ABB) to ,% 35,% 3,% 25,% 2,% 15,% 1,% 5,%,%

36 Figure 23: Historical volatility graph Ericsson ( to ) 7,% Historical (ERIC B) to ,% 55,% 45,% 4,% 35,% 3,% Figure 24: Historical volatility graph Ericsson ( to ) Historical (ERIC B) to ,% 45,% 4,% 35,% 3,%

37 Figure 25: Historical volatility graph Ericsson ( to ) Historical (ERIC B) to ,% 3,% 2,% 1,%,% After we performed the calculations necessary to come to the historical volatilities, we focused on the calculation of the implied volatilities. 36

38 4.2 Implied 14 To calculate the implied volatilities we used the Black and Scholes formula and the daily options prices from to for ABB and Ericsson. To calculate the daily implied volatility we need: The closing prices of the options The stock prices The strike prices The times to maturity The annual interest rate on a 2-year government bond on particular days as the risk-free rate, i.e. we used the daily price (=annual interest rate as of a certain day) on the bond as the risk-free rate. We inserted these values into the Black & Scholes model to give us daily implied volatilities for the different options. For example an ABB option: ABB4B2, on the 2 nd of January 23. We used the closing price quoted for this option on that day (13,15 SEK) and also the stock price of ABB on the same day (27,6 SEK). Next we calculate the time to maturity, in this case 1, years, and the annual interest rate on the 2 nd of January for the government bond (3,79%). Following this we insert the collected information into the Black & Scholes model, in this case the model for call options: -rt c = SN(d1) - Xe N(d 2 ), and by trail & error we determined the implied volatility. In this case the resulting implied volatility was 87,%. This process was repeated for all options on all the days that they were traded. On days when the market could not agree on a closing price (i.e. only ask and bid prices were given), we used the weighted average of the ask and the bid of the last known price on these options in order to come to a estimated closing price. The weight we re referring to is calculated by using ask and bid prices on the options together with their closing price: Closing price = Weight * Ask + (1 Weight) * Bid. To give an example: on the 9 th of January 23 option ABB4B2 had: An ask price of 12,75 SEK A bid price of 11,25 SEK A closing price of 11,5 SEK Putting this into the above formula we get a weight of,17. On the 1 th of January the same option only had an ask price (14,25 SEK) and a bid price (11,25 SEK). In 14 See excel file for more details 37

39 order to get a closing price we use the weight of,17, use this together with the ask & bid price and put this into the above mentioned formula to get a closing price of 11,75 SEK. Those options that did not have a closing price at all (i.e. they had never had an agreed closing price but only ask and bid prices) were not taken into consideration when calculating the daily implied volatility. From the option data that we have, we selected & analysed 15 : Figure 26: Options ABB maturing in Option Maturity Average daily Implied ABB4B ,58% ABB4B ,56% ABB4B4(X) ,2% ABB4B45(X) ,51% ABB4B5(X) ,94% ABB4B ,42% ABB4B ,37% ABB4N ,7% ABB4N ,6% ABB4N ,14% ABB4N35(X) ,1% ABB4N4(X) ,99% ABB4N45(X) ,51% ABB4N5(X) ,1% ABB4N ,88% 15 For information on these options and other options not mentioned here, see the excel file 38