Financial Stochastic Calculus E-Book Draft 2 Posted On Actuarial Outpost 10/25/08

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1 Financial Stochastic Calculus E-Book Draft Posted On Actuarial Outpost 10/5/08 Written by Colby Schaeffer Dedicated to the students who are sitting for SOA Exam MFE in Nov. 008

2 SOA Exam MFE Fall 008 ebook TABLE OF CONTENTS STOCHASTIC PROCESSES DEVELOPMENT OF BROWNIAN MOTION - - GEOMETRIC BROWNIAN MOTION ITO S LEMMA VOLATILITY & THE RISK-NEUTRAL WORLD STOCK MARKET MODELS & OPTIONS PUT CALL PARITY OPTION CONVEXITY BINOMIAL OPTION PRICING MODEL MULTIPLE PERIOD BINOMIAL TREES SCHRODER S METHOD INTEREST RATE TREES CAPS & FLOORS THE BLACK-SCHOLES FORMULA OPTION GREEKS MARKET MAKING & MAKING THE MARKET EXOTIC OPTIONS EQUATIONS APPENDIX - 8 -

3 SOA Exam MFE Fall 008 ebook Stochastic Processes Stochastic processes are the opposite of the deterministic processes that are found in the world of probability theory. The deterministic system deals with one possible set of end results. The stochastic, or in-deterministic, process models the progressive and continuous change of these deterministic processes. In other words, stochastic processes model changing probability distributions. For instance, a binomial distribution with the same parameters over time may not be the most practical model. How is a stochastic process developed? There are two types of stochastic processes: discrete and continuous. In the domain of discrete time, a stochastic process is a sequence of random variables that amounts to a time series. There are many examples of time series, such as the Markov chain. One method of treating a stochastic process, whether it is discrete or continuous, is to treat it as a function of multiple parameters with the most significant variable being time. The random variables that make up these parameters may have certain probability distributions or may be dependent upon something entirely different. The point is that these variables are simply that - variables. The variables may be completely independent and random, but in most cases they are usually correlated to some degree. Therefore, a stochastic process can be viewed as a random function. The most common example of a stochastic process, specifically a time series, is stock, interest rate and exchange rate fluctuations in their respected markets. This is why stochastic calculus is tied to the financial world so much. Stochastic differential equations Stochastic differential equations, or SDEs, are differential equations in which one, or more terms, is a stochastic process. An equation of stochastic processes, such as the SDE, is therefore a stochastic process itself. It is interesting to note that much of the early development of SDEs was done by Albert Einstein in order to explain Brownian motion. It is a complex topic at first, but stochastic calculus can be broken down into simpler mathematical terms. Brownian motion is a Weiner process and it is ruled as non-differentiable, so it has its own set of rules in Calculus. The more commonly used Calculus is called Ito Calculus.

4 SOA Exam MFE Fall 008 ebook - - Development of Brownian Motion Brownian motion can be thought of as a time series of random walks. It is in fact, a continuous stochastic process of random walks occurring in continuous intervals rather than discrete. As mentioned, the random walk take steps and usually makes one of two decisions go up by one unit or go down by one unit. If the number of steps was increased to infinity within a period, then the process would, in effect, be continuous by this limiting effect. To generate Brownian Motion, the coin flips of the Random Walk can be done infinitely within a period as said. Before actually generating Brownian motion, the goals of the model must be laid down along with some rules. Founder of Brownian motion Botanist Robert Brown developed the model in order to explain the random movements of particles suspended within a liquid or gas state. The stochastic process, or model, that is Brownian motion is simply a method to try and organize what is perceived as chaos. However, particle movement and stock price fluctuations are not completely random and aren t subject to the rigorous mathematics of chaos theory. Rules are what actually guide this stochastic process. With a financial view in mind, these first set of rules make this random world that much more structured. Rules of Brownian motion Like a random walk, Brownian motion starts off at some number, such as zero, but it can be shifted accordingly (to maybe an initial stock price). Movements in Brownian motion are normally distributed and independent. No one movement is dependent upon the prior one. The actual position is not independent, but the moves themselves are. Brownian motion is completely continuous. There are no holes, jumps, asymptotes or gaps in the process. These rules do not suggest any barriers to prevent negative stock prices, but there are certain types of Brownian motion that do. Mathematical Development of Brownian motion Brownian motion is a martingale which basically implies that movements are equally likely to go up as they are to go down. This is the independence rule of Brownian motion. Specifically, it is a conditional expectation for a stochastic process written as a function, f(x):

5 SOA Exam MFE Fall 008 ebook E [ f ( x + t ) f ( x )] = f ( x ) The limit of the sum of infinitely small random movements over a period of time is the accumulation of continuous Random Walks. In other words, that is the bare bones version of Brownian motion. With the rules of Brownian motion set, it can officially be created. Let there be n number of periods of total length T. The length of a period is then = T. Stock price change, the difference of one h n stock price from a stock price the period before can be defined as: St + h St = Yt + h h The Y variable is the Random Walk effect. Let time start at zero, and instead of analyzing the individual periods (that are infinitely small), analyze the entire time period as a whole. Let s be practical here; we re interested in stock price movements over a course of real time, such as a day, week, month, quarter or year, not at a blink of an eye! The sum of these infinitely small periods is simply T, so a series can be developed. T 0 n k = 1 n S S = Y h = h Y k = 1 hk T n = Y n k = 1 In Brownian motion, we want to find the expected stock price change, so some basic statistics have to be applied. hk hk n Yhk k = 1 E [ ST S0] = Ti E n

6 SOA Exam MFE Fall 008 ebook The expected value of the Y variable in the series is zero as it is equally likely to go up as it is to go down. Therefore, the expected stock price change is zero! This doesn t mean that the model predicts no change in the stock. The dynamics of variance must be considered. The variance of the Y variable is simply one unit as the random walk will either go up 1 or go down 1. n Yhk k = 1 Var [ ST S0] = Var n n 1 k = 1 = Var n 1 = Var * n n = n = 1 n This is why the square root of the time period is taken. It lets the sum of independent random variables approach a standard normal distribution! This is the Central Limit Theorem at work. This is the structure of Brownian motion. The rate of return and the historical volatility of the stock need to be considered. After all, we are trying to develop an accurate model. Arithmetic Brownian motion An equation is developed with a nonzero mean (the vertical shift that was mentioned before) and an arbitrary variance based on the normal distribution that was used above. X ( t + h) X ( t ) = αh + σy ( t + h) h As n approaches infinity, and consequently h approaches zero, the normal distribution term is developed. X ( T ) X (0) = αt + σz ( T )

7 SOA Exam MFE Fall 008 ebook Differentiating this equation leads us to the accepted form of arithmetic Brownian motion. dx ( t ) = αdt + σdz ( t ) The equation is differential and stochastic in this form. The vertical shift of the mean is the drift term, α dt. Note that the difference in stock prices is normally distributed ~ N ( αt, σ T ) as it is the standard normal distribution (Z(T)) changed by the drift and variance variables.

8 SOA Exam MFE Fall 008 ebook Geometric Brownian Motion One issue with arithmetic Brownian motion is the effect of the constants. A better model may involve the drift and volatility to vary with the asset, or stock, price itself. In fact, the equation can easily be transformed from Arithmetic to Geometric by a simple right side multiplication of the X(t) function. dx ( t ) = αx ( t ) dt + σ X ( t ) dz ( t ) Dividing by X(t) gives us the accepted form of Geometric Brownian motion. dx ( t ) X ( t ) = αdt + σdz ( t ) This model is more acceptable than its Arithmetic counterpart for several reasons: 1) Asset price can never go negative ) Mean (drift) and variance (volatility) are proportional to the asset price One change should be made to the formula when applying Geometric Brownian motion to stocks. ds ( t ) = ( α δ) S ( t ) dt + σs ( t ) dz ( t ) The continuous dividend yield (delta) must be included as a reduction in the continuous rate of return (alpha) on the stock. Discussions about mean and variance warrant the use of probability. After all, we want to predict stock prices, so what is the probability that a stock will be a given price at some time? P ( S as ) = P (ln( S S ) > a) t t 0 Notice that the mean and variance are in terms of stock returns, or change in the stock price. The price of the stock is assumed to be lognormally distributed, so the return would be the natural log of the variable stock price over the initial stock price. The mathematics behind this is quite simple

9 SOA Exam MFE Fall 008 ebook Since the price of stock is assumed to be lognormally distributed, the stock price then has parameters of µ S = ( α 0.5 σ ) t and σs = σ t. This assumption makes our probability equation much simpler, and more importantly, it keeps all variables in the same measurement (expected returns as a percent). P Z > ln( a) µ σ Remember, we are measuring how feasible some stock price in the future is. Therefore, we are comparing that possible gain (or loss) to the expected return while taking into account standard deviations (σ ) in the expected return. Example Geometric Brownian motion models the price of a stock that has an initial price of 1. Its expected rate of return is 8%, the stock has no dividends, and the volatility is 5%. If the stock price rises to 15 one year from now, what is the likelihood that it will continue to rise to at least 18 two years from now? Conditional expectation has no effect on the Brownian motion model since it is a martingale, so the probability can be measured from the stock price of 15 to a possible stock price of 18 a year later. Therefore, the effective time of measurement here is one year (not two years). Mean: µ = ( (0.5) ) = S Standard Deviation: σ S = = 0.5 P ( S as ) = P (ln( S S ) > ln( a)) t = 1 P (ln( S S ) < ln(18 1)) t = 1 P (ln( S S ) < ln(1.5)) t 0 0 ln(1.5) = 1 P Z < 0.5 ( ) t 0 = 1 P Z < 1.43 = 1 N (1.43) = =

10 SOA Exam MFE Fall 008 ebook There is 7.64% chance that the stock will shoot up 50% in two years, given it has a 5% gain in the first year. That doesn t seem like something that you should bet on. Ito Process A stochastic process is considered an Ito process when it follows this basic form: ds ( t ) = α( S, t ) dt + σ( S, t ) dz Both the mean and variance can vary upon the asset price and time. This is simply a variable version of Arithmetic Brownian motion. It also can be Geometric Brownian motion if the asset, or stock, price is a coefficient multiplier upon the mean and variance. Example An Ito process of the form ds ( t ) = 0.3 S ( t ) dt S ( t ) dz models a given stock. Calculate the probability that S(t) is at least 10% more than S(0) 6 months from now. The mean expected return is µ S = ( (0.1) ) t = 0.95t and the volatility is S 0.1 t would be about 15% with an approximate 7% swing in volatility. σ =. Therefore, in a half year, the expected return p = P ( S 1.1 S ) = 1 N (ln( S / S ) < ln(1.1)) t 0 t 0 ln(1.1) = 1 N Z < 1 N ( ) = = N (0.74) = The conclusion of this problem is that the stock is fairly likely to make a 10% gain. The other.96% of the time, the stock may still have a gain, just not a 10% gain or better.

11 SOA Exam MFE Fall 008 ebook Ito s Lemma In the last lesson, the idea of an Ito process was introduced. Ito s lemma simply evaluates a function of an Ito process. Ito s lemma is interested in the process followed by a function of a stock (or another asset). Options, such as calls and puts, are dependent upon a stock s price. In fact, the function of a stock is often called a claim. The notation C(s, t) is used for claims and it may vary upon the stock price and the time. Multiplication Rules We are interested in the change of the claim with respect to the stock price and/or the time. Therefore, we need a multi-variable derivative. We are interested in the derivative of the claim from a stochastic Calculus perspective. Before that can be done, some multiplication rules have to be stated within the framework of Brownian motion. ( dt ) = 0 ( dz ) = dt dz dz ' = ρdt dt dz = 0 The reason for these rules is complex, but they can be summarized in a fairly simple way ( ) 0 dt = The infinitesimal nature of dt makes it very small, so when raised to a power, it effectively becomes zero (at least very close to). ( ) dz = dt The infinitesimal change in Z(t) may be expressed as the following: ( dz ) = [ Z ( k + 1) Z ( k )] ~ t N (0,1)

12 SOA Exam MFE Fall 008 ebook = 1 ( dz ) ( t ) x s = ( t ) x n n n E [ s ] = ( t ) E [ x ] = ( t ) n n In effect, the change in the standard normal distribution, squared, acts like the change in time. dz dz ' = ρdt The multiplication of two standard Brownian motion processes together (dz) will then result in a change in time with a correlation factor ( ρ ). dt dz = 0 This multiplication can be transformed using the previous rules dt dz = dt dt = ( dt ) = ( dt ) ( dt ) = 0( dt ) = 0 Putting all of these rules together leads us to the development of Ito s Lemma Ito s Lemma Using Taylor s approximation as an assistant to stochastic differentiation, we have the following approximation of the change in the claim: C C C C C dc = ds + dt ds dt + dsdt S t S t S t The first two terms come from the chain rule in Newton s Calculus. The second two terms are simply nd derivatives. The last term is the nd derivative of the claim with respect to both time and stock price. Our multiplication rules will help simplify this equation. Note that the changes in time are linear, so going beyond the 1 st derivative will give us zero in this model. Also, the change in asset price with the change in time will effectively be zero. We now arrive at Ito s lemma. C C C dc = ds + dt ds S t S

13 SOA Exam MFE Fall 008 ebook The stock price terms are determined by the given Brownian motion model that is being used. The claim on the stock then follows Ito s Lemma.

14 SOA Exam MFE Fall 008 ebook Short Rate Bond Pricing Models These lessons start off from a strict mathematical perspective with stochastic processes. It then progresses to stochastic Calculus in detail (Brownian motion) whose primary goal is to measure future stock price changes. Ito s Lemma bridges the gap between the stochastic models of stock price change to functions based on stock prices. These functions are known as financial derivatives (not actual derivatives in Calculus). Before talking about the variety of options, and how to price them, much more simple financial derivatives should be introduced. This is where we get to bond pricing Bonds are basically a prepaid value, or series of pre-paid values, based on some interest rate. Any bond can be modeled as a sum of zerocoupon bonds at the time of each payment on the bond. Ignoring credit risk, all bonds are simply T-bills and will not default on the investor. Modeling bond prices directly can be difficult as the value of the bond at maturity must be a given value. What is then modeled is the actual interest rate used. Variable rate bonds must be valued some how. They can be immediately valued on the spot rate (interest rate at time of purchase) or on the forward rate (some interest rate in the future, perhaps, at time of maturity). Valuing a spot rate bond is simple all the necessary information is there. Valuing a forward rate bond requires some prediction using stochastic Calculus. Bonds & Coupons The basics of bonds are fairly elementary, but the terminology must first be understood before it s used in the subject of stochastic calculus. Par value, also known as the face or principal value, is how much the investor will receive at maturity. For example, a $1,000 par value bond will be worth $1,000 when it matures. The coupon is actually the interest rate the bond pays. It is called the coupon rate because bonds, once upon a time, came with a book of coupons, which the holder had to clip and send in to receive an interest payment. Bond investors are still referred to sometimes as coupon clippers. This interest rate does not usually vary over the life of the bond, although there are some bonds, which have a variable interest rate tied to an external index.

15 SOA Exam MFE Fall 008 ebook Maturity is the length of time before the par value is returned to the investor. At maturity, the bondholder receives the par value of the bond. Ito process of interest rates Interest rates are assumed to follow an Ito process: ds ( t ) = α( S, t ) dt + σ( S, t ) dz That Ito process is based on stock prices and time. When talking about the short term interest rate, all instances of stock price are replaced with the rate. dr = a( r ) dt + σ( r ) dz Not much has changed outside of variable names; a(r) is still the mean, or drift. Rendelman-Bartter model This model follows the basic Ito process to the T via geometric Brownian motion. dr = ardt + σrdz The asset in this case is the interest rate (r). It cannot go negative and the volatility is proportional to the interest rate. However, the rate can increase beyond what is considered a practical rate. Mean reversion can be used to limit the rate Vasicek model dr = a( b r ) dt + σdz The problem with the Vasicek model is that the resultant interest rate (mean = a( b r ) ) can be negative. Also, the model is not of the geometric Brownian motion mold and the volatility is constant and unaffected by the mean rate. Cox-Ingersoll-Ross model This model is one that works. Mean reversion is the characteristic of interest rates that sets itself apart from most other assets. In the long run, the interest rate will revert to its mean. This must be included in

16 SOA Exam MFE Fall 008 ebook the model as high rates will ruin economic activity and low rates can not decrease indefinitely. dr = a( b r ) dt + σ r dz

17 SOA Exam MFE Fall 008 ebook Volatility & The Risk-Neutral World Volatility has already been discussed. It creates the up and down movements in the stock market. It is the reason why Stochastic Calculus is so important in the financial world. Volatility is what drives the price of options more than the current stock price, strike price, time to maturity and risk-free rate. The call option may be very valuable if the stock price is high, strike price is low and time to maturity is short. However, the higher the volatility is, the greater chance that the stock can go up OR down causing the stock to be riskier. With a more risky stock, the option becomes riskier and it may not be such a safe bet. How Can Volatility Be Measured? A student doesn t need to know Stochastic Calculus, interest rate models, Put-Call Parity or even option pricing techniques. An assumed volatility can be based off of past data. Volatility computed from historical stock returns is the historical volatility. It becomes a more reasonable assumption for future volatility with more data. Volatility is the variations in price compared to the average price itself. It is the standard deviation of exponential changes in stock prices. The best way to show this is through examples. Example.5-A ln exponential change in stock prices. Let St be the stock price at some given time, and S t S t St 1 St ln St be the Let s assume this entire period was one year and that these stock prices were taken every two months. Over the course of the year, the stock went up increasing by 0%, a quality return on the investment. Like most stocks, the price wasn t increasing steadily, and was rather

18 SOA Exam MFE Fall 008 ebook volatile in the middle of the year. There are a couple ways to find standard deviation, and with the information in the table, the easiest way is the following x = = ˆ σ = = The difference x ˆ σ is the variance. To make the variance unbiased, the number needs to be divided by one less than the sample which is 4 instead of 5, so multiplying by 5 4 will make the variance unbiased. 5 ( s = ˆ σ x ) = We want the standard deviation which is the square root of the variance s = = This is the volatility on a two month basis. To find the annual volatility, multiply that number by the square root of six, the amount of two month periods in a year. 1 1 σ = s = = = h 1 6 Using stochastic calculus and Option & Stock price models such as the Binomial Model or Black-Scholes Model can let us predict values of the stock price in the future, and therefore forecast the volatility. The Risk-Neutral World So far, the rate of discount has been some given interest rate. What does it mean to discount at some rate of interest? If you discount something according to how much you could save by instead having that money in a savings account, then you should discount it by the interest rate that your bank gives you. Is it correct to assume some interest rate for an option? If so, what should be the assumed interest

19 SOA Exam MFE Fall 008 ebook rate? We do not know what the rate of return on an asset will be in the future as there is some volatility in the ups and downs of the stock market. This is why options are purchased, to begin with, to hedge such losses. A standard discounted cash flow calculation would require computing some expected value of an option with the probability that there will be a payoff or not. The expected payoff may depend on the investor themselves. The risk-averse investor would take the sure thing over a risky bet that has a potentially higher reward. The risk-neutral investor sees no difference between the two if the expected payoff is the same between two types of investments. In pricing options, a risk-neutral probability that stocks will go up or down are taken. Having risk-neutral chances of payoffs means that there can be assumed some rate of return, the risk-free rate. The risk-free rate is exactly that; it is an expected positive rate assuming there will be a payoff and that the investment is not risky. We are now one step closer to pricing the premium of an option (claim). Discounting a claim payoff back to the present date of measurement (present value) determines the value of the premium. rt C = e E [ C ( t )] The expected value may be determined by true probabilities or riskneutral probabilities. With true probabilities, there is more risk involved in the investment. Financial stochastic calculus is used to model future prices so the present value can be determined. The risk involved can be determined by the risk premium and Sharpe ratio. Risk premium = α r Sharpe ratio α r = σ Alpha (α ) is the expected rate of return for the asset, and r is the risk-free rate. Sigma (σ ) is the volatility (standard deviation of the returns) associated with the asset. Therefore, if risk-neutral probabilities are being used, and then there is no risk premium or Sharpe ratio, hence there being no risk (risk-neutral)

20 SOA Exam MFE Fall 008 ebook Stock Market Models & Options Stochastic Calculus can be used to develop a short-rate bond model, but it can also be used to model stocks and the financial derivatives known as options that can be based on stock prices. The mathematics of Stochastic Calculus will not help someone successfully forecast what the stock market or a specific stock price will do, but it will model all of the possible random changes that a stock could make. This is extremely useful in option pricing. What is an option? Options are financial derivatives that give the buyer the option to buy or sell the stock at a given point. The buyer pays a premium initially, and they lose the risk of the stock price going down. Optionally, if the stock price goes up, they may buy it, and then sell it if they wish, and make money on the stock. Since stock prices can be erratic, this may be an attractive option to the investor. There are two basic types of options. The call option is a contract where the buyer has the right to buy, but is not obligated to buy, the stock (or another asset) at the time of maturity of the option. The buyer pays a premium at the onset of the contract; remember, there is no such thing as a free lunch! The put option is a contract where the put option buyer has the right to sell, but is not obligated to sell, the asset at the time of maturity of the option. The owner of the put option is betting that the stock price will go down. If they are wrong, and the stock price goes up, then they have the option to not sell. The idea of buying and selling can be confused here. The investor who bought the put is effectively a seller of the stock index. These options simply do not mean the investor is looking for an insured bet on if the stock will go up or down. Rather, they are betting whether or not they will go above (call) or below (put) a value known as the Strike Price. Letting the strike price equal the initial stock price makes these bets into a simple up or down situation. Payoff The mathematics behind the payoff is simply algebraic. Let S be the stock price at a given time t, T the time of maturity, and K be the strike price. The payoff when the option is exercised at maturity is equal to the following for both calls and puts.

21 SOA Exam MFE Fall 008 ebook Call Put Payoff Max(0, S K) Max(0, K S) Payoff can easily be shown through a few examples: Example.1-A A call option has a strike price of 40, and the stock price at maturity of this option is 60. What is the payoff, if any, of this call? Payoff = Max(0, S K) = Max(0, 60 40) = 0 Example.1-B A put option has a strike price of 35, a maturity period of 1 year, and the initial stock price is 33. Is it worthwhile to exercise a put option (take the stock) a year from now if the stock price is modeled by the following function? S( t) = S(0) e.06t First, let s find the price of the stock one year from now. S(1) = 33e.06 S(1) = Payoff = Max(0, K S) = Max(0, ) = 0 It is not optimal to exercise the put option and take the stock as there would be a loss since the stock price has become greater than the strike price. Option Premium The price of the option premium is dependent upon several factors, and Stochastic Calculus is used to price such premiums. The profit from an option is equal to the payoff minus the option premium. This is why options can be viewed as insurance against a loss (decrease in stock price for a call / increase in stock price for a put) Option Style There are many styles of options that are different based on how they are exercised. The two most common types of options that depend upon exercise time are European and American options. The examples

22 SOA Exam MFE Fall 008 ebook used thus far have dealt exclusively with European options which can only be exercised at the time of maturity. American options may be exercise any time up to the time of maturity. This choice of early exercise can be a lucrative decision if the price of the stock is right.

23 SOA Exam MFE Fall 008 ebook Put Call Parity It is reasonable to conclude that the premium of an option is dependent upon the time to maturity, strike price and initial stock price. For if the initial stock price is very low, the strike price is very high, and the time to maturity is a short period, a call option would be considered worthless. For the call option to produce a payout, the stock price would have to increase very quickly. Likewise, the put option would be valuable for the likelihood of a high payout is very probable. This common sense view tells us that call and put options are related and probably dependent upon each other. This intuition is correct! Puts are simply the mirror image of a call. The owner of the call option wants the price to go up whereas the owner of the put option wants the price to go down. Calls and puts are related, and the premium prices are dependent upon each other. This idea forms the balance that Financial Economists call Put Call Parity. As the name says, there must be parity between a put and a call. If one is high then the other is low and vice versa. The premium prices of puts and calls simply do not add up to be some constant number. A put and a call with the same parameters (strike price, time to maturity) are equal to the present value difference of the stock minus the strike. C P = PV ( S K ) This should make sense to the student reading this. If the current stock price, and present value of future dividends, is more than the discounted strike price, then a call option is worthwhile. The algebra proves this since the right side of the Put-Call Parity equation will be positive, then so should the right meaning the call is worth more than the put. Likewise, if the stock price ends up being lower than the strike, then there will be negative results on both sides meaning that the put costs more than the call. Present Value The student should have a background of basic interest theory before studying this material. The present value of the stock is the value of the stock at the onset of the option contract. There is no modeling of future stock prices here, but, rather, it takes into account discounted dividends to be paid in the period of the option.

24 SOA Exam MFE Fall 008 ebook - - Example -A If the initial stock price is 100, a 1-year option is purchased on this stock and the stock pays dividends of 5 every 6 months starting 3 months after the stock is bought, and the rate of discount is 6%, what is the present value of the stock price? Each dividend should be discounted back and added to the initial stock price like so S 0 PV ( Div) The present value of the stock price is , much lower than the initial stock price. Not taking into account dividends can inflate the option price. Why are dividends accounted for? The option is always compared to having the stock outright. If the investor owned a stock, then they would be owed dividends as promised. The owner of the call option does not receive such dividends and the investor loses out on such dividends. The effect of this deflates the price of the call option. Stocks could theoretically pay continuous dividends at a rate of discounted using the exponential function. So we can write Put-Call Parity as S PV ( Div ) = S e δt 0 0 δt C P = S0 e PV ( K ) The strike price is discounted back as well, at the rate of discount that was used already for the individual dividends. At this point, we can now solve for parameters of options or premium prices of options with the right information.

25 SOA Exam MFE Fall 008 ebook Example.-A Suppose that a price of a stock is $50, the continuous rate of dividends is 6%, the continuously compounded interest rate is 8%, and the options have 6 months to expiration. A 50-strike European call sells for $3.00. How much does the put, with the same parameters, cost? δt Using Put-Call Parity, C P = S0 e PV ( K ) P = 50e 50e P = 50( e e ) P = 3 50( e e ) P = 3 50( ) = The price of the put premium cost 5 cents more than the call premium. This should make sense as the present value of the stock is less than present value of the strike. When the stock price is less than the strike price then the put is favored. Put-Call Parity for American Options Parity generally fails for American options because the American option will be at least as valuable as the European option. The American option can do everything the European option and more, hence, making it more valuable. So, for American options, Put-Call Parity is not an equation of equality, but instead it is an inequality because both the prices of the American Put and Call will be greater than or equal their European counterparts. When is it optimal to exercise a call or put early? How high, or low, should the stock price be? The golden rule is that if dividends are not being paid, American style options should never be exercised on a non-dividend paying stock. Early exercise is optimal for a call if, C < S(t) K, and for a put, P > S(t) K. Putting these two equations together gives us, C P < S(t) K. This notion violates Put-Call Parity for American options, so Put-Call Parity must be fine tuned to: C P >= PV(S K) The effect of dividends is what differentiates the American options from the European options. Early exercise is optimal for an American option when

26 SOA Exam MFE Fall 008 ebook S rk >, if σ = 0 δ σ is the variability factor in stock prices that stochastic calculus models. If no dividends are paid from the stock, then early exercise is not optimal. Options on Other Assets General Put-Call Parity is not based on the stock, but, rather, it is based on a given asset and its forward price. Let F denote the prepaid forward price of the underlying asset. The strike price is simply the opposing asset price. It is what the options are betting over (call) or under (on). The strike price and forward price are based on similar assets. Put-Call Parity can then be refined to: C P = PV(F K) The variable K is the strike price or opposing asset price. The four most common asset types that Put-Call Parity can be applied to are Stock Prices, Bond Prices, Currency Exchange Rates and Two Different Commodities such as two company stock prices (making bets against each other) or commodities such as oil or corn. Stock: F = S 0 PV 0,T (DIV) Bond: F = B 0 PV 0,T (Coupons) Currency: F = Currency Exchange Rate 1, K = Currency Exchange Rate Different Assets: F = Asset1, K = Asset, discounted accordingly Put-Call Parity, like any equation, must have the same units for each variable, so the currency version must equate everything in one currency or another.

27 SOA Exam MFE Fall 008 ebook Option Convexity Option prices are dependent upon the asset prices (forward and strike), the discount rate, dividend rate if applicable and time to maturity. Specifically, the strike price and time to maturity have identifiable trends as they increase or decrease. The notion of premium prices decreasing or increasing at given rates because of strike prices is known as Option Convexity. Let there be a situation we are comparing multiple options to each other to verify their premium prices. Remember, there is a price balance in the financial world, that if broken permits arbitrage the effect of profiting on no risk, the figurative free lunch if you will call it. There are no free lunches, prices of options must go up or down at a relative relate compared to the strike prices. Now, say that the strike prices of these three options are K1 K K 3 The middle strike price can be equated as a weighted average of the other two strike prices such that K = λk1 + (1 λ) K 3 where λ is between zero and one. Call and premium prices, with the same time to maturity, are related in a similar equation. C ( K ) λc ( K ) + (1 λ) C ( K ) 1 3 P ( K ) λp ( K ) + (1 λ) P ( K ) 1 3 The convexity can best be shown through a graph such as the one in Figure.3-A. Arbitrage Inequalities For both American and European options, there is no free lunch! K1 K K 3 0 C ( K ) C ( K ) K K P ( K ) P ( K ) K K 1 1

28 SOA Exam MFE Fall 008 ebook Notice that the equation for Put Options has the order of the options reversed compared to the equation for the Call Options. If options are European, then the difference in option premiums must be less than the present value of the difference in strike prices. Premiums decline at a decreasing rate as we consider calls with progressively higher strike prices. Premiums also decline for puts but when the strike price monotonically decreases. This is known as Option Convexity. One way to look at this is through equations based on strike price. C ( K 3) C ( K ) C ( K ) C ( K1) K K K K 3 1 P ( K 3) P ( K ) P ( K ) P ( K1) K K K K 3 1 However, it may be easier to see this through an illustration. Figure.3-A Option Trends American options become more valuable as time to expiration increases, but the value of European options may go up or down. As the strike price increases for calls or decreases for puts, the options become less valuable with their price decreasing at a decreasing rate. With dividends, longer term European options may be less valuable than shorter term European options.

29 SOA Exam MFE Fall 008 ebook - 7 -

30 SOA Exam MFE Fall 008 ebook Binomial Option Pricing Model The development of the Binomial Option Pricing Model is based on a fairly elementary idea. Stock prices have the nature of following a Random Walk. The Random Walk models completely random up and down movements of one unit. This ends up being a cumulative sum of unit increases and decreases that are dependent upon a coin flip. Let Z denote value of a random walk after the nth event, or nth coin flip n if you want to view it that way. Z n n = Y, i = 1 Where Y i = ± 1 It turns out that as more events accrue, that, on average, the absolute value of Z will increase. However, one would think that the expected value would be zero. The student is correct to think such an idea because for every positive value of Z, there is the equal likelihood of a negative value of Z. In the random walk model, there becomes more and more variance in the stock as time goes on. This can be shown best through a graph like the one in Figure 3.1-A. Figure 3.1-A i

31 SOA Exam MFE Fall 008 ebook There are three practical problems with the random walk model. 1) It permits negative values which are unreal for a stock as prices can never become negative. ) The magnitude of the move should depend upon how quickly the coin flips occur and the value of the stock price. If we flip coins once a minute, then a $1 change may be too much, and that $1 change in the stock price may be too much for a $10 stock. 3) On average, the stock should have a positive return. The random walk model does not allow this. This is why the binomial distribution is used as a variant of the random walk model. It is the numerically discrete method to price options. First off, the binomial distribution can be used to approximate the lognormal distribution, which assumes that continuously compounded returns on stock are normally distributed. What is the Binomial Model? Binomial option pricing is developed from the random walk. It assumes that over time, the price of the underlying asset moves up or down by a given amount. This is why the binomial distribution is used as it only allows two choices with given probabilities. Figure 3.1-A Let this be the scenario for a stock price; it can go up to 65 or down to 40 from an initial stock price of 50. We want to price a call option with a strike price of 55. This can be done by pricing a replicating portfolio. In other words, buy and sell stocks and bonds that would have the same result as buying a call option. The law of one price dictates that positions that have the same payoff should have the same cost, or else there is arbitrage. To do

32 SOA Exam MFE Fall 008 ebook this, we should value a portfolio that buys shares of stock and lends a dollar amount B. Assuming that the stock has a continuous dividend yield of, let it automatically be reinvested in the stock. Therefore, if one share is bought at time t, then at time t + 1, there should be r e shares. This is the binomial model of stock prices. It either has a gain at a rate of u or a loss at a rate of d. The call, or put, option has a payoff dependent upon the strike and stock prices, so the stock price tree has a corresponding option value tree: The variable C represents a call option, but a P can be used in the same manner for a put. Let this tree of two different results be one rh period of length h. The interest factor is then equal to e. The value of the replicating portfolio at time h, with stock price S h, is then: S + Be h The money that was lent has earned interest at the risk-free rate during the period, and shares of stock were purchased that has change in total price due to the stock price change. The stock price either goes up to a value S*u or down to a value S*d. We are now ready to price the option! rh

33 SOA Exam MFE Fall 008 ebook u d δh C = Sue + Be δh C = Sde + Be These are known as the replicating portfolio system of equations, and the system can be combined to form one option pricing equation. ( r δ ) h ( r δ ) h rh e d u e u d S + B = e C + C u d u d The cost of the option is S + B. The given stock price movements at a rate of u and should present arbitrage, so the following inequality must hold. ( r δ ) h u > e > d Example 3.1-A Using the stock price tree from Figure 3.1-A, how would a one year call option with strike price of 55 be priced, given that the stock pays no dividends, at a risk free rate of 6%? The up and down movements can be factored out of the tree itself. C C u d 65 u = = d = = rh rh = Max (0, 65 55) = 10 = Max (0, 40 55) = 0 (0.06 0)1 (0.06 0) e e S + B = e 10 * + 0 * S + B = * =

34 SOA Exam MFE Fall 008 ebook The price of the call option is $4.93. It is the premium that the investor must pay to safe guard themselves from a fall in the stock price. Risk-Neutral Pricing If we let e p* = ( r δ ) h d u d Then the price of the option can be written as rh * * u C = e [ p C + (1 p ) C ] This falls in line with the notion of Risk-Neutral probabilities and expected values of option prices. The variable p* is not the actual, or true, probability that the stock will go up, but it is the risk-neutral probability of an increase in the stock price. Construction of the Binomial Tree The nature that the stock price is assumed to go up and down must account for some variance as the Random Walk model surely shows! The binomial tree, based on forward prices, includes the risk-free rate, the dividend yield and the volatility. u = e d = e ( r δ ) h + σ h ( r δ ) h σ h To construct a stock price tree, all that is needed is the initial stock price and the rate of up and down movements. To price the option, the risk-free rate, dividend yield, volatility, time to maturity and strike price need to be known. Example 3.1-B Let a one period binomial tree be used to model a one year put option where 0 = 100, = 150, = 0.15, = 0, = 0.07 S K σ δ r d u = e = d = e =

35 SOA Exam MFE Fall 008 ebook Now, let s map out the one period stock price tree with its corresponding put option payoffs. p * e = ( r δ ) h 0.07 d u d e = = = There is about a 46% risk-neutral chance of an increase in the stock price. The put premium is simply an expected value equation. rt P = e E [ P ( T )] rh * * u This is just the same as C = e [ p C + (1 p ) C ] for a put! P = e 0.07 [ * ( ) * 57.69] = [ ] = d That may seem like a high price for a put, but considering the strike price is 150% of the initial stock price, a $50 difference, then this

36 SOA Exam MFE Fall 008 ebook amount seems viable. The investor may be better off short selling the stock, but the put hedges the risk of the stock price increasing too much, such is the benefit of using options.

37 SOA Exam MFE Fall 008 ebook Multiple Period Binomial Trees Pricing options based on the binomial pricing model is most commonly done with more periods. The reason for this is that more periods will give a more accurate result of the option price, just like a large sample is a better indicator of the population it represents. Let s expand on Example 3.1-B by adding one more period, and one more year as well, so now it is a -year put option that is being priced. We should expect the put premium to be lower since the time to maturity is longer. The extra time gives way to an increased chance in the stock price going up. Figure 3.-A The two period binomial tree in Figure 3.-A is a recombining tree. The only trees that do not recombine are stocks that pay discrete dividends.

38 SOA Exam MFE Fall 008 ebook Valuing a two period binomial tree follows the same step as the one period tree, except that step has to be done for each split in tree. Notice, in Figure 3.-A, that the tree has three splits, so three expected values will have to be taken. We start from the end of the tree like so: Example 3.-A Recall that Example 3.1-B had parameters of S0 = 100, K = 150, σ = 0.15, δ = 0, r = We must discount the put option payoffs from the upper node and the middle notes at the two year mark. Each period represents a period of time, and in this example, each period is one year, so there are zero, one and two yearmarks..07 * * P = e [ p (0) + (1 p )(34.97)] u = 0.934[ * 34.97] = 17.5 This is different than the payoff value at the upper node of the one year mark. It is actually the same as the premium of a one-year put option with an initial stock price of and all of the same parameters as the option we are currently valuing. The same process must be done for the bottom of the tree..07 * * P = e [ p (34.97) + (1 p )(64.79)] d = 0.934[ * * 64.79] = 0.934[ ] = The last step is discounting these prices using the risk-neutral probability of an increase in the stock price with the risk-free rate..07 * * P = e [ p (17.5) + (1 p )(47.09)] d = 0.934[ * * 47.09] = 0.934[ ] =

39 SOA Exam MFE Fall 008 ebook The price of the two-year put, based on a two-period binomial tree, is $31.15 Notice that the tree does not necessarily need to be drawn out to solve the problem. It is merely an illustrative tool to help the student see how the option payoffs are being discounted. Alternative Trees Binomial trees do not necessarily have to be based on forward prices. The binomial model is an approximation of the lognormal distribution. The standard deviation of a sum of n independent random variables is n times the standard deviation of each one. This is where annual volatility of a stock price, σ, is calculated for a period h: Forward Tree ( r δ ) h Nodes of the binomial tree are centered at S0e with the upper node higher than this price and the lower node lower than this price. This assures arbitrage-free pricing. It is the standard tree used for binomial pricing. u = e d = e ( r δ ) h + σ h ( r δ ) h σ h Cox-Ross-Rubinstein Tree The risk-free rate and dividend yield are not even used in this variation. This will (usually) only work for small values of h. u = e d = e Lognormal Tree This is also known as the Jarrow-Rudd binomial model. u = e d = e σ σ h h ( r δ 0.5 σ ) h + σ h ( r δ 0.5 σ ) h σ h Subtracting σ 0.5 is done because the stock price is assumed to have a lognormal distribution. The mean of a lognormal distribution is σ h

40 SOA Exam MFE Fall 008 ebook ( µ σ )h e where µ = r δ. Therefore, by subtracting 0.5σ h from µh such that µ = ( r δ 0.5 σ ) h, the stock price will then have a ( r δ ) h mean of e which is the same center for the forward tree. Example 3.-B 0 = 100, = 0., = 0.01, = 0.05, = 0.5 For S σ δ r h, determine S du using: A) Forward prices B) Cos-Ross-Rubinstein tree C) Lognormal tree u = e = e = e ( r δ ) h + σ h.04(0.5) = d = e = e = e ( r δ ) h σ h.04(0.5) = The stock price after a down and an up move using forward prices is S du = 100(1.1163)(0.9139) = 10.0 The Cox-Ross-Rubinstein model only looks at the volatility and ignores the risk-free rate and dividend yield. The rate of a down movement is an inverse of the rate of an up movement, so we can expect the stock to remain the same!

41 SOA Exam MFE Fall 008 ebook u = e = e = e σ = d = e = e = e σ h = h The stock price, S = 100(0.9048)(1.105) = 100, remains the same du because the Cox-Ross-Rubinstein tree is centered at the initial price of the stock! Lastly, in addition to the risk-free rate and dividend yield, the center of the lognormal tree is dependent upon volatility unlike forward priced binomial trees. u = e = e = e = e ( r δ 0.5 σ ) h + σ h ( (0.) ) (.0) (0.5) = d = e = e = e = e ( r δ 0.5 σ ) h σ h ( (0.) ) (.0)0.5 0.(0.5) =

42 SOA Exam MFE Fall 008 ebook The price of the stock, in a lognormal tree, after a down and up movement is equal to S du = 100(0.9094)(1.1107) = Example 3.-C Using a Cox-Ross-Rubinstein binomial tree, what is the payoff of a 1- year call option with monthly periods, after 8 up movements and 4 down movements, such that S0 = 5, σ = 0., r = 0.05, K = 5? We want to find Payoff = S u d S Note that for this model, the down movement is simply the inverse of an up movement, so the payoff equation can be simplified. Payoff = S 0 ( u 1) The rate of an up movement is easy to calculate in the Cox-Ross- Rubinstein model. 4 u = e = e σ h 0. 1/1 = Payoff = S ( u 1) = 5( ) = The payoff is approximately 1.30 which is a 6% rate of return on the stock, but since the call option removes some risk from owning the stock, it also removes some of the return. If the premium for the call option was 1, then the profit would only be 0.30 which is a 6% return on investment. The owner of the stock had a return four times as great as the call option owner. The best way to learn binomial option pricing is by practicing many different scenarios. American options, such as the American put in Example 3.-B, also need to be considered in the Binomial model