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# Advanced Financial Topics ESTM 60202

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This 0 page Class Notes was uploaded by Ms. Hermann Spinka on Sunday November 1, 2015. The Class Notes belongs to ESTM 60202 at University of Notre Dame taught by Alex Himonas in Fall. Since its upload, it has received 28 views. For similar materials see /class/232715/estm-60202-university-of-notre-dame in Customer Services at University of Notre Dame.

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Date Created: 11/01/15

ESTM 60202 Financial Mathematics Aler Himzmas 02Lecture Notes1 October 5 2009 1 Introduction to Assessing Risk Important Question How do we assess the risk an investor faces when choosing among assets In this discussion we examine how an investor would assess the risk associated with investments This assessment is carried out in terms of how an individual is affected by taking on additional risk In Figure 1 we plot the monthly returns on stocks left graph and short term government bonds right graph from 1926 7 2008 The smooth line is a standard normal distribution with a standard deviation which matches the data for stock and bond returns respectively The stock return is the value of the weighted average of all stocks traded in the United States The arithmetic geometric average return on stocks is 094 080 or 1128 96 per year The in ation rate over this same period was 025 per month or 30 per year As a result the return on stocks adjusted for in ation was 828 per year The short term government rate is the 3 month Treasury bill rate The average short term rate is 03 36 per year per month or a rate of return after adjusting for in ation of 025 per month the real return on short term government bonds over this period was 06 per year As a result the real return on stocks was greater than the return on short term bonds by 768 per year Given such an advantage to holding stocks why do people hold short term bonds Momhly Smck Reiums 19264008 Monthly Risk Free Returns 192672008 Gmmmncwmnwm GmmvertManmnw mm Man 02 Amhmelt Mean am 8 70 6 70 4 70 2 0 0 2 0 4 7 0 720 710 10 20 0 We m Retum Figure 1 Monthly Returns on Stocks and Bonds In this lecture we will show that the excess return on stocks stems from the fact that stocks are riskier assets than bonds and so investors must be paid an additional amount known as a risk premium to compensate for the added uncertainty and volatility One way to see the risk is to look 1Based on joint notes with Tom Cosimano and with the assistance of Kyle Dempsey 33 so that P O i O 21 O 22H Suppose we use the actuarially fair gamble in example 1 with initial consumption 0 507 000 The risk premium becomes p 50000 7 55000 5 4500005 25063 This means that this investor would pay 25063 to avoid the actuarially fair gamble with standard deviation of 57 000 Exercise 2 Given the actuarially fair gamble in exercise 17 nd the risk premium for the investor in example 2 Why do you think the risk premium changed 4 Relating Stocks and Bonds to an Actuarially Fair Gamble We can relate an actuarially fair gamble to the uncertainty in stocks relative to investing in risk free bonds Remember that the payoff from investing in either stocks or bonds are given by the decision trees in Figure 4 s1H usO Figure 4 Payoffs from Stocks and Bonds Ir Bo 1 117 SIT dSO 1 Given these payoffs from stocks and bonds we can compare the utility of the investor when she buys bonds and stocks If she purchases bonds7 then her utility is certain7 and is measured at the point B in Figure 5 Her utility at this point is uB u1 rX0 where X0 is the amount invested If she uses all her wealth to buy stocks7 such that SO X07 then her utility is uncertain Before the random event occurs7 her expected utility from buying stocks is E5uX0 puuX0 1 7 pudX0 Warning The symbol u inside the utility function u stand for up value77 of the stock In order for the trade off between stocks and bonds to be an actuarially fair gamble7 we require that7 under the conditions described above7 so that the expected utilities of the value of the investors wealth when investing in the stock and the value of the wealth after accruing interest for one period are the same This is demonstrated in Example 3 below In Figure 57 we take p i so the expected utility is given by the point S Thus7 the investor is expected to lose utility when the stock is an actuarially fair gamble We show in subsequent lectures that this leads to a higher expected return on stocks to convince risk averse investors to hold the stock Stocks l l l l l l l l l l l l l l C1rX0 Cu uX0 Figure 5 All stocks versus all bonds Example 3 Suppose that the the price of the stock at the end of the rst period year is 51H 100 and SAT 25 Also let the risk free interest rate be r 005 and the probability of the good state p 05 Suppose the investors wealth is X0 5952 After one year the investors wealth is 6250 If SO 59527 then the stock price is a fair gamble To see this calculate the expected stock price pSlH17pSlT 05 gtlt 100 05 gtlt 25 625 The proceeds from the bond is 1 rX0 105 gtlt 5952 625 As a result7 the investor ends up with 6250 on average if she invests in either the stock or the bond However7 the expected utility from the stock as illustrated in Figure 5 is less than the expected utility from investing in the risk free bond Thus for any investor to buy stock at time t 0 its price must be less than 5952 giving the investor a reward for taking risk Exercise 3 Suppose the probability of the good state is 097 what should the stock price be for the stock to be an actuarially fair gamble relative to the risk free bond Let the individual investor have logarithmic preferences What does the stock price have to be so that the investor receives the same utility from the stock and bond 5 Measuring the Risk Premium We now show how to relate the risk premium to the shape of the utility function following the argument by Kenneth Arrow7 the 1972 winner of the Nobel Prize in Economics Theorem 51 A risk averse investor with utility function uC and consumption 0 has a risk premium p which is approccimated by uO 7 0 ma gt p m 7 Varz for an actuarially fair gamble Proof To start7 we can de ne the risk premium at the speci c consumption 0 according to De nition 47 ie7 u0 7 p pu0 2117pu0 22 54 The proof applies a rst order Taylor approximation to the left hand side of this de nition near the point 0 Second7 we take a second order Taylor approximation of the right hand side near the point 0 We then equate these two results and solve for the unknown risk premium p Recall that the notation Tquot07 s refers to the n th Taylor polynomial of z estimated at the point zo which has known values Take a rst order Taylor7s polynomial expansion of the left hand side of 54 near the point 0 to nd ulto 7 p Tlt1gtltoio 7 p 7 um u ltogtltlto 7 p 7 0 so that ulto 7 p um 7 IL0 55 Next7 nd the second order Taylor7s polynomial expansion of the right hand side of 54 near the point 0 to obtain pu0 21 1 i pu0 22 m T2O 0 21 T2O 0 22 p u0 u021 u 0z lt1 7 p um urea22 awn2 u0 u 0 p21 1 p221u 0 192 1 i 1923 Note that in the rst step above we took the Taylor expansion of only the parts of the formula containing 21 and 22 resepectively Now recognize that the random variable is an actuarially fair gamble so that p21 17p22 0 and Varz p212 1 7 As a result the approximation of the right hand side of 54 is pu0 2117pu0 22 u0 UH0Varz 56 Thus7 the approximation of 54 is given by comparing 55 with 56 140 7 Mi0 u0 u 0Varz The nal step is to solve for the risk premium p 7Varz Lastly7 note that by the de nition of the utility function7 we have u 0 lt 0 and u 0 gt 07 so the formula implies that uO 7 0 ma gt 1 p m 7 Varz 38 This completes the proof Exercise 5 asks you to employ an argument similar to the one used in this proof to nd more precise approximations of the risk premium using higher order Taylor polynomials D Example 4 Redo Example 2 using the Arrow measure of the risk premium Solution From Example 1 the variance is 25000 000 The derivatives of the logarithmic utility function are dlnO 1 lenO i 1 7 i d 7 7 do 0 an do2 02 As a result the risk premium is given by 1 u 0 1 1 1 1 m 77V 72 500 0007 72 500 0007 250 p 2 2 1110 2 O 2 50 000 Exercise 4 Suppose the payoffs from the stock and the risk free bond are as in Example 3 Let the investors preferences be given by uC lnC How much does the stock price have to fall from SO 5952 so that this investor purchases the stock Exercise 5 Suppose we have a utility function with derivatives of order n1 Find a more accurate formula for the risk premium in terms of the higher order derivatives of the utility function Exercise 6 a Suppose the utility function is linear in Figure 5 What happens to the risk premium in Theorem 11 Why do we call this person risk neutral b Now suppose the utility function in Figure 5 is curved toward the origin rather than away from the origin What happens to the risk premium in Theorem 11 What would you call an individual with this type of preferences Given Theorem 11 Arrow introduced the following de nitions of risk aversion De nition 5 The measure of an investor7s absolute risk aversion is AO JAG where C is the investors consumption without undertaking the actuarially fair gamble As a result according to Theorem 11 the risk premium is given by 1 p m EVarzAC De nition 6 The measure of an investor7s relative risk aversion is u O R O 7 O u O 7 where C is the investors consumption without undertaking the actuarially fair gamble Remark This measure of risk is called relative risk aversion since a multiplicative random variable 610 rather than a additive random variable 539 2 leads to the risk premium formula p VarzRC 39 Exercise 7 The example in Figure 5 is a multiplicative random shock where e u and 622 As a result consumption is given by Cu 6110 in the good state and Cd 6120 in the bad state Prove Theorem 11 for the case of these multiplicative shocks Example 5 Suppose the investors absolute risk aversion is a decreasing function of consumption In particular7 let AC 7 Suppose an average investor has 50000 to consume per year For the actuarially fair gamble in Example 17 the investor is willing to pay ON 1 1 2 w 7v A 0 7725 000 000 500 p 2 2 250000 to avoid this gamble We can use this result to explain why a risk averse individual purchases various types of insurance such as health and life insurance Suppose you have a car worth 5000 and you have a 50 chance that you would have an accident in which your car will be destroyed over the next year7 since you are a bad driver If your absolute risk aversion is like example 57 then you are willing to pay an insurance company 500 per year for an auto insurance policy7 which pays for the automobile if it is destroyed Thus7 insurance company makes money by reducing risk for individuals in exchange for an insurance premium 6 Risk Aversion and the Investor s Utility Function We now want to investigate the relation between risk aversion and the functional form of the utility function of an investor We want to know how risk aversion changes with the standard of living of a person in order to gain some insight into investor behavior Since an investor has less absolute risk aversion as his consumption increases7 AC must be a decreasing function7 that is AO lt 0 67 Taking the derivative of the formula for absolute risk aversion using of the quotient rule we get we 7 WNW0 A O Therefore A C lt 0 is equivalent to we 7 WNW0 lt 0 68 Individuals with a higher standard of living are more willing to take on gambles7 and so their utility function should satisfy condition 68 Example 6 If A C lt 0 for people7 then investors like Warren Buffett or Bill Gates would be willing to take on more risk than the average person This helps to explain why Berkshire Hathaway is willing to insure other insurance companies against catastrophes such as the fall of the World Trade Center It also helps to explain why Hedge funds7 who invest for wealthy investors7 would take on more risk Exercise 8 For the logarithmic utility function nd A C lt 0 Does this utility function lead to increasing or decreasing absolute risk aversion We now want to consider a class of utility functions which would allow for decreasing absolute risk aversion Below we use the simplest functional form for absolute risk aversion such that the rst derivative is negative We consider the following general form 7b 7 so that A C i a b02 AW 69 where a and b are constants As a result b gt 0 would correspond to decreasing absolute risk aversion Theorem 61 An investor with absolute risk aversion given by 69 has a utility function given by one of the following three formulae where K and L are any constants Case 1 Ifb7 0 andb7 1 and y then uC K a bor 7 L 1 177 Case 2 Ifb 1 then uC Klna C L Case 3 Ifb 0 then uC iKae g Remark Note that we use the general constant k in each ofthe three formulae since multiplication by any constant will not change the optimal behavior of the investor Case 3 is called constant absolute risk aversion since AC is constant AC Also the case a 0 is called constant relative risk aversion since RC is constant In this case AC and so 1 1 RO AOO EC 3 Note that we cannot have a 0 bRemember also that we are considering the case of decreasing absolute risk aversion so b 2 0 in all the cases of Theorem 12 by assumption Proof Recall that O 1 AC 7 m where a and b are xed parameters With this in mind let us prove each of the cases of the theorem individually Case 1 In this case b 31 0 b 31 1 and y Also we have the following two equations u C 1 lnabC 7 dO71 o K d 7d0 K W nlu l an a0 where K is the constant of integration However both integrals are representations of fACdC so lna b0 7 Ink0 b K 41 where we have altered the constant of integration K without losing equality After some rnanipula tion of this equation and once again simplifying the constant of integration we arrive at we Ka boy Integrating both sides of this equation with respect to C we get 1 1 uC K abO1 L D h A where L is another constant of integration Then substituting in y we arrive at the correct formula for Case 1 uC K a bot7 L 1 7 7 Case 2 In this case we have b 1 so clearly substituting this into the equation from the rst case would yield an unde ned solution Therefore we must start again with the rst step So we have AC mic giving us the following two representative integral equations 71110 7 7 nu an 1 n a u0d07 1 OK daOdO 1OK These equations can be combined to give lnluC 7 lna C K which simpli es by formal manipulation to 1 O K7 u a 0 Integrating both sides of this equation we get the nal formula for Case 2 uC Klna C L Case 3 This case is the simplest Since b 0 we have AC Therefore by taking a step similar to the rst step in the proofs of Cases 1 and 2 we get two integral equations of forms of AC u O 1 O 77d07l O K d idOi K Mm nmltn w a 0 Accepting a change in the constant of integration K these equations can be combined as mWW 7K which by rnanipulation sirnpli es to C uC K67 Then through a simple integration of this equation we arrive at the following formula for the third case uC iKae g This completes the proof of Theorem 12 D Remark Note that it is traditional to set L 0 in Cases 1 and 2 above since this additive constant of integration does not effect optimal investment decisions Also it is standard to set K 1 in all cases for simplicity However we have chosen to be explicit in this lecture by including them References Arrow Kenneth J 1971 Essays in the theory of risk bearing North Holland Amsterdam Cvitanic J and F Zapetatero Introduction to the economics and mathematics of nancial markets MIT Press Boston 2004 pp104 110 2 Has a zero probability of losing money 3 Has positive probability of making money No Arbitrage Axiom NA It is impossible to have a trading strategy which can turn nothing into something without running the risk of a loss That is there is no arbitrage Lemma 11 If there is no arbitrage NA then 12 Proof By contradiction First we prove that d lt 1 r If the opposite is true that is d 2 1 r then at t 0 we could borrow from the bank SO dollars to buy one share of stock Then no matter what is the outcome of the coin toss at time t 1 our stock will be worth 1 2 dSO while we will need 1 rSO to pay off the loan from the bank Therefore with a certain probability of 1 we will be making a pro t of 81 7 TSO Z dSO 7 TSO d 7 1 rSo 13 20 sinced21r Also with probability p gt 0 we will make a pro t equal to 7 TSO USO 7 TSO 2 u 7 dSO 14 gt 0 since u gt d Thus we have produced a strategy which begins with zero money has a zero probability of losing money and positive probability of making money This violates the NA axiom Therefore inequality d 2 17 is false Next we prove that 1 r lt u again by contradiction lf 1 r 2 u then at t 0 we could borrow a share of stock this is called short selling a stock sell it for SO dollars and invest the proceeds in a bank paying interest r At t 1 this deposit at the bank would grow to 1 rSO At this time to replace the stock borrowed at time t 0 in the worst situation it would cost USO Therefore with a certain probability of 1 this strategy produces a gain of 1 TSO 7 81 Z TSO 7 USO 1r 7ulSo 15 2 0 since 17 Eu Also with probability 1 7 p gt 0 we will make a pro t equal to 1 TSO 7 TSO 7 dSO 2 u7dSO 16 gt 0 since u gt d Thus again we have produced a strategy which violates the NA axiom D Remark The reason why the NA axiom comes about follows from this proof If you had the inequality d 2 1 r then companies would nd it pro table to buy lots of stock at time 0 with price SO As this occurs the price SO would be bid up As a result both u and d would go down Thus the pro t motive or the fact that people and corporations are greedy leads to the NA axiom in nance Replicating the option would create the same payoffs as the option As a result we want to choose X0 and A0 such that X1H V1H and X1T V1T Using the formula for X1 these two equations take the form A081H 1 rX0 7 A050 V1H A051T1 7 X0 A050 V1T After separating the Xo from the A0 terms we have X0 l 51H Sol A0 viH7 X0 51T 50 A0 VNTl Subtracting the second from the rst equation gives 1 1 iSHiST A7VH7VT W1 1ltgt 11o Wm 1ltgt 1lt 1 Solving for A0 we obtain the so called deltahedging formula V1HV1T A0 7 51H i 51T39 113 Next substituting in the rst equation of system 112 formula 113 gives 1 781H 750 XML v1ltHgt7v1ltTgt 1mm 39 51H i 51T 1 Then letting 51H USO and 81T dSO we obtain 1 1 r Eliminating SO and solving for X0 gives 1 X 0 17 WHH 17 1 u v1ltHgt7v1ltTgt l 1 V1ltHgtlt1reugt 1L WI mm 1L 39WTgt 114 This value of X0 must be equal to V5 the desired price of the European option This completes the proof of the theorem D OF X0 V1H Example 3 Continued 1n the case of Example 3 using delta hedging formula 113 we nd that 7 30 i 0 7 2 7 100 i 25 7 339 That is to replicate this option we must buy 25 shares of stock This will cost A050 25 50 20 dollars Therefore our cash position is X0 7 A080 11105 7 20 m 7952 which means we need to borrow 20 7 11105 dollars from the bank at the interest rate of r 005 A0 Exercise 3 Find the price of a European option for a stoch whose value today is 40 a share its up value is 60 and its down value is 30 its strike price a year from now is 50 the annual interest rate is 4 and the probability of the good state is 06 Exercise 4 For a stock in a binomial world assume that SO 100 51H 140 81T 75 and r 008 What is the price of a novel77 call option for this stock which at t 1 has payoffs V1H 25 and V1T 5 Exercise 5 Derive a formula for the price of a European put option in the case of the one period binomial option pricing model Exercise 6 Find the price of a European put option for a stock whose value today is 100 a share its up value is 150 and its down value is 75 its strike price a year from now is 125 and the annual interest rate is 5 Unresolved Issues 1 1n the Theorem the formula for the option price is independent ofthe probabilities ofthe ups and down movements in stock This result is puzzling since the purpose of the option is to insure the purchaser against events they do not want to occur We will see that the probabilities are related to the variables u 155 and d The reason for this is that the stock price today SO is dependent on the probability of the good and the bad state For example if there is a 100 chance of the high price next period then the stock price today would be much higher relative to a 100 chance of the low price next period Consequently taking the stock price today as given implicitly assumes a given probability of an up and down move In future classes we will show this explicitly 2 We have not discussed why someone would want to buy or sell a European call option A buyer would want to use the call option to insure against a particular change in prices For example suppose you run an Airline in which your cost of operation is dependent on oil prices since jet fuel prices go up whenever the price of oil goes up Now suppose the call option the airline bought pays off when the price of oil goes up For example it could be a call option on BP stock Now when the price of oil goes up your airline would face higher jet fuel cost On the other hand they make a bigger payoff on the option written on BP Thus the call option reduces uctuations in the pro ts of the airline company when the price of oil goes up If the airlines do not like uncertainty then they are made better off This opens the question as to why someone would sell a call option It turns out that an oil drilling company may also want to avoid uctuations in oil prices since they make their money mainly from producing oil and they wish to avoid uncertainty As a result the oil drilling company say BP would sell a call option on its oil When the price of oil goes down below a certain level the call option on BP stock is not exercised and so BP make a pro t Vb per share As a result the decreased value of oil for the oil drilling company is canceled partially by the pro t of the call options sold Thus7 the call option also insures the oil drilling company against uctuations in the price of oil Why this works will be discussed in more detail as we go through the semester

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