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# Review Sheet for FINA 8397 with Professor Boulatov at UH

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## Reviews for Review Sheet for FINA 8397 with Professor Boulatov at UH

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Date Created: 02/06/15

Static vs DynaInic Models The main distinction between static and dynamic models is in revelation of uncertainty In a dynamic world uncertainty about future states of the world is resolved gradually and this is captured by the mathematical structure of our models Many of the economic concepts carry over from the static models Things to review 0 State prices and state price density 0 Arbitrage and fundamental theorem of asset pricing Complete markets Arrow Debreu equilibrium CAPM and consumption CAPM Rational expectations equilibrium Also review your probability theory READINGS DISCRETETIME DYNAMIC MODELS Arbitrage and Martingales Huang and Litzenberger Chapters 7 71 72 77 78 and 8 81 87 Applications of Arbitrage Pricing Huang and Litzenberger Chapters 7 711 713 and 8 88 810 813 815 DiscreteTime Models Arbitrage and Martingales 1 The Securities Market Model The securities market model consists of l a probability space 97213 2 a set of trading dates T 0 l7 T 3 a ltration IF 4 N l securities indexed by n 07 N a Security 0 is the riskless security and can be interpreted as the value of a bank account Security 0 does not pay dividends lts price at time 0 is 800 l and its price at time t gt 0 is given by 50 1 Tt71SOt717 where r E E is the short rate process b Securities l7 N are the risky securities Security n has a dividend process 6 and an eX dividend price process S We set 6 606N 6 EN and S 80SN 6 EN In part I we assume that the set of states 9 is nite We also assume that all states have strictly positive probability Dividend and price processes are adapted This is a natural restriction The dividends and prices at time t can only depend on information available up to time t De nition 11 A trading strategy is an adapted process 0 000N 6 EN such that 0T n t represents the number of shares of security n that an agent holds after trading at time t The restriction that trading strategies are adapted is natural The number of shares that an agent holds after trading at time t can only depend on information available up to time t The restriction that 0T 0 means that agents have to close their positions by time T De nition 12 A cash flow is an adapted process 0 E 2 De nition 13 A trading strategy 0 nances a cash flow 0 t for all t E 1 T Ct 71051 6t etst 11 and 00 70080 The term 0t1St it represents the proceeds at time t from the securities that the agent bought at time t 7 l The term QtSt represents the cost of buying new securities at time t The proceeds minus the cost have to nance the cash ow ct This is the dynamic budget constraint De nition 14 A cash flow is marketable t it is nanced by a trading strategy 0 We denote by M the set of marketable cash ows M is a linear subspace of E 2 Motivation Portfolio Choice Our objective here is to show somewhat informally that in order to have a well de ned portfolio choice problem market prices must satisfy certain conditions The most obvious condition is absence of arbitrage otherwise a non satiated agent would take an in nitely large position What is less obvious is that asset prices must satisfy a so called martingale property or equivalently there must exist a linear pricing rule All of the above conditions follow from existence of optimal portfolios Below we will show that in fact absence of arbitrage is equivalent to existence of a linear pricing rule or the martingale property of prices and is also equivalent to existence of a solution to a portfolio choice problem Consider a market without dividends An agent has a continuously differentiable non satiated utility function de ned over terminal consumption Let SUV 81SN and 01N 01 0N ie we drop the zero th asset in our new notation here The wealth process is denoted by Wt so that Wt1 Wt 01Nt81Nt1 7 01NtSt1 23 The objective is to choose an adapted trading strategy 0t to maximize E0UWT subject to WT 2 0 We will explicitly de ne a discounting process Bi 30 Then if we discount the stock price and the wealth process by Bi Wt E VVtBt SUV E B1 811W we nd t Wt1 Wt 01NtS1Nt1 1Nt W0 Z 01Nug1Nu1 3mm u0 Now the optimal portfolio choice problem is to maximize E0UBTWT subject to WT 2 0 Assume there exists an interior solution to the problem 0 Consider a perturbation of the optimal portfolio policy in the direction t so the perturbed policy is 0 6 The wealth process under the perturbed policy is given by til Wt5 E W 5 Z77 1Nu1 3mm u0 Optimality of the policy 0 implies d A EEOlUBTWTElle0 0 24 and therefore T71 E0U W ET 2 77tSlN 1 7 SlN t 0 for any trading strategy nt 771 nN t t0 De ne a new probability measure Q with Radon Nikodym derivative U KW BT EolU WBTl T 5t Etlng Elngftl 25 We now have the following characterization of asset prices T71 EOlgT Z 77t 1Nt1 SlNtl 0 for any trading Strategy 77t 771 77Nt t0 Consider a particular example of 77mg Choose a particular event A E E and assume that all values of 77m are zero except for n k s E 13 739 and the nodes following the event A in which case 7719 s 1 More formally 77w 1A 1set7nk We see that A A E0 T81Nk 7 7 SlN k t 1A 0 for any event A E E 5 which is equivalent to 0 Etl T 1Nl T 1Nktl EtlgT S1N1ml Etlngngm EtlgT S1NIml t 1Nkt So we established S1Nt Et QS1NJ EthglN 39l39 26 t Measure Q is called an Equivalent Martingale Measure It is equivalent to P because both Q and P assign zero probabilities to the same events It is called a martingale measure because the discounted price process of any stock is a martingale under Remembering the de nition of discounted price processes B 71 ST Bt The above equality holds trivially for the zero th asset Note that 27 also de nes a linear pricing rule stock price today is a linear function of future cash ows St E 27 Why is the new characterization in 27 useful First our derivation links optimal portfolio choices to asset prices remember that the martingale measure Q is related to marginal utilities evaluated at the optimal wealth process This is a very useful property and is a basis of many empirical tests Second remember that 26 must hold for all asset prices If the set of stock price processes is su iciently rich markets are dynamically complete see below then we have many restrictions on T and in fact we can show that T is unique independent of individual preferences As soon as we know that we can invert 25 and express optimal wealth state by state as a function of the density of the EMM T which is known This provides a powerful tool for solving portfolio choice problems So far we proceeded under the assumption that there exists an interior solution to the portfolio choice problem This assumption is too strong and not necessary Below we will show that absence of arbitrage alone implies existence of an equivalent martingale measure 3 The FTAP Not any dividend and price process 6 S is consistent with equilibrium in nancial markets For 6 S to be consistent with equilibrium it must admit no arbitrage The FTAP funda mental theorem of asset pricing provides a necessary and su icient condition for 6 S to admit no arbitrage De nition 31 An adapted process X is positive i Xt 2 0 for all t An adaptedpi oeess X is strictly positive i Xt gt 0 for all t We denote by 3 the set of positive adapted processes by 30 the set of positive adapted processes that are not equal to the zero process and by the set of strictly positive adapted processes De nition 32 An arbitrage is a marketable cash flow in 23 De nition 33 A function Kl I a R is strictly increasing i for any two processes X a Y such that Xt 2 Yt for all t we haue llX gt Theorem 31 Fundamental Theorem of Asset Pricing There exists no arbitrage i there exists a strictly increasing linear function Kl I a R such that llc 0 for all 0 E M Proof Suppose that there exists no arbitrage Then we have 30 N M The separating hyperplane theorem for cones see Duf e appendix B implies that there exists a non zero linear function KI such that lt Ily for each x E M and each y 6 13 Since M is a linear subspace 0 for each x E M lndeed suppose that 0 for some x E M Then we can choose A E R and y 6 30 such that 000 AKIJx gt Ily a contradiction Therefore Ily gt 0 for each y 6 13 This together with the linearity of 11 implies that KI is strictly increasing Conversely suppose that there exists a strictly increasing linear function KI E gt R such that Ilc 0 for all 0 E M If there exists an arbitrage c then Ilc gt 0 since KI is strictly increasing and c 6 13 However since 0 E M Ilc 0 a contradiction To better understand the FTAP we write the function KI as We K0 2 ie 31 ie as the sum over all nodes of the product of ct times an adapted process wt Since KI is strictly increasing w E 74 We normalize w by setting we 1 Example 31 Suppose that Q 7 T and F are as in example 7 Using the notation of example 7 for the process we we haue Mcl 00 f Widcm f 12012 24021 22022 23026 240241 We refer to the process w as a state price process wt can be interpreted as the price we need to pay at time 0 in order to obtain one unit of consumption at the node lw t Likewise Ilc can be interpreted as the value of the cash ow 0 where the component of 0 corresponding 7 to a given node is evaluated at the state price corresponding to that node The FTAP states that absence of arbitrage is equivalent to the existence of strictly positive state prices Moreover the value of any marketable cash ow measured at these state prices has to be zero Consider a cash ow that is nanced by a trading strategy 0 We have T T llc K0 2th 0 gt 0050 700 KOZW t1 t0 The term 0080 is the cost of buying securities at time 0 to nance the cash ow from time 1 on This cost is equal to the value of that cash ow measured at the state prices 4 Security Prices and State Prices What does the absence of arbitrage existence of state prices imply for security prices Theorem 41 The following are equivalent 1 There exists no arbitrage 2 There exists 1h 6 such that for all t lt T T St iKt lt 2 565 MST 41 wt st1 5 There exists 1h 6 such that for all t lt T l S in 9thth St1 42 Proof 1 2 Suppose that there exists no arbitrage Then there exists a state price process iJ E 74 Consider the trading strategy that consists of buying one share of security n at the node lwt holding it until time T and selling it at T This corresponds to n t l for the node lwt 0m l for all T gt s gt t and all subsequent nodes and 05 0 otherwise The trading strategy nances the cash ow 0L 7 mt for the node lw t 05 5m for all T gt s gt t and all subsequent nodes 0T SET SET for all subsequent time T nodes and c5 0 otherwise For this cash ow we have T llc K0 Ease 0 50 8 Since 0 is zero except for the node lw7 t and all subsequent nodes we have T T K0 Z we K0 Z we 50 st Equation 41 follows immediately 2 3 Equation 41 implies that T 1 1 St iKt t16t1 Kt lt Z 74565 wTSTgt wt t st2 Noting that KtK5X5 KtX5 43 for s 2 s 2 t we can write the last term as 1 T 4K Km 2 56 wTST wt st2 Equation 41 implies that T Kt1 Z 60 080 must st2 Equation 42 follows immediately 3 1 Consider a cash ow 0 that is nanced by a trading strategy 0 Using equations 11 and 12 we get T T K0 2m K0 0080 Z w007180 60 7 0080 t0 t1 T71 K0 Z 00t5t 010t5t1 St1 K0 T0TST t0 Equations 42 and 43 imply that K0 00t5t 010t5t1 Stu Koliib tst KtWtHetth St1l Kol wtetst ethwt16t1 St1l K0 7Jt0tst 00 0 Since in addition 0T 0 Since 1b 6 11 0 cannot be an arbitrage Equation 41 states that the price of a security is equal to the value of all future dividends and the time T price The value is measured using the state prices Moreover since the state prices are relevant for time 0 and we are at time t we need to divide by the time t state price Equation 42 states that the price is equal to the value of the dividends and the price in the next period Equations 41 and 42 are very intuitive They link the price of the security to its future cash ows Moreover cash ows are weighted using the state prices The state prices incorporate three elements First investors subjective probability be liefs Second discounting ie the time value of money Third investors risk aversion In order to separate the three elements we de ne two concepts The rst concept is the state price density SPD This is the state price divided by the probability The SPD thus incorporates only discounting and risk aversion The second concept is the equivalent martingale measure The EMM incorporates only investors probability beliefs and risk aversion 5 StatePrice Density The FTAP states that no arbitrage implies the existence of a strictly increasing linear func tion 11 E gt R such that 110 0 for all 0 E M In section 3 we wrote this function as T We K0 2 m 51 t0 We can also write it as T We E0 2 not 52 t0 ie as the expectation as of time 0 of the sum over periods of ct times an adapted process 71 Since 11 is strictly increasing 7139 6 11 We refer to 7139 as a state price density SPD process Remember that 110 is the time 0 value of the cash ow 0 measured at the state prices This value can be written as the expectation of the cash ow adjusted by the SPD Adjusting by the SPD adjusts for discounting and risk aversion The difference between equations 51 and 52 is the following In equation 51 probability beliefs discounting and risk aversion all work through the state prices while in equation 52 probability beliefs work through the expectation and discounting and risk aversion through the SPD Comparing equations 51 and 52 we can link 7139 to the state price process 1b and to investors probability beliefs lnvestors probability beliefs over the states in Q are given by the proba bility measure P We denote by pw t the probability of the node lw t We suppress the 10 dependence on w and use pt instead of pwt Using this notation we can write equation 52 as T 110 K0 Zp rtct 53 t0 Comparing equations 51 and 53 we get 1 pt 54 E The SPD is thus equal to the state price divided by the probability Notice that We 1 since 0 p0 Using equation 54 we can write equation 41 as T 1 1 87 7K 5565 S t LR tltZp7T pT7TT Tgt st1 The term in parenthesis is simply T Et lt Z W5657TTSTgt st1 We thus get 1 T St 7E lt 2 ms MST 55 m st1 Equation 42 similarly implies that 1 St iEt 39tJrlthrl St1 5 6 t Equation 55 states that the price of a security is equal to the conditional expectation of all future dividends and the time T price adjusted by the SPD To adjust a cash ow at time s we multiply by the SPD at time s and divide by the SPD at time t This adjusts for discounting and risk aversion Equation 56 has a similar interpretation Equations 55 and 56 differ from equations 41 and 42 in that they give prices as conditional expectations When there are no dividends we can write equation 56 as WtSt E TtHStH This means that the process 7119 is a martingale This process is the price times the SPD ie the price adjusted for discounting and risk aversion 1n order to extend the martingale property in the presence of dividends we de ne the gain process 11 De nition 51 The gain process is t Gt St 65 57 51 and the SPDeadjusted gain process is t o ms 2 ms 58 51 Proposition 51 The SPDeadjusted gain process is a martingale Proof Using the de nition of Gquot and equation 56 we get E40211 G The martingale property then follows from the law of iterative expectations 6 Equivalent Martingale Measure In section 5 we wrote the function Kl as T llc E0 2 m 61 t0 Remember that llc is the time 0 value of the cash ow 0 This time 0 value is the expectation of the cash ow adjusted by the SPD Adjusting by the SPD adjusts for discounting and risk aversion lntuitively the adjustment for risk aversion works by assigning a large SPD in states where investors value consumption more and vice versa We will show that in equilibrium investors value consumption more in states where their consumption is low We will now adjust for risk aversion not through the SPD but through the probabilities That is we will de ne a probability measure Q such that we wig 62 where Bi 8 Notice that we adjust for discounting by dividing by the price of the riskless security The probability measure Q is the equivalent martingale measure It is also called the risk neutral probability This is because it corresponds to the probability beliefs that risk neutral investors should have in order to support the dividend and price processes in equilibrium 12 The difference between equations 61 and 62 is the following In equation 61 probability beliefs work through the expectation and discounting and risk aversion through the SPD while in equation 62 probability beliefs and risk aversion work through the EMM and discounting through the riskless security To obtain equation 62 we need to de ne the probability measure Proceeding as in section 5 we denote by 1a t the probability of the node lw t as of time 0 Moreover we suppress the dependence on w and use qt instead of 1a t We can write equation 62 as T ct KI K 7 63 lt0 0 Z tht lt gt t0 Comparing equations 51 and 63 and using equation 54 we get 1 11th gtqtthtpt7rtBt 64 t Equation 64 is a necessary condition on Proposition 61 There exists a unique probability measure Q satisfying equation 64 Proof Equation 64 implies that the Radon Nikodym derivative of Q wrt P is dQ dP To show that WTBT is indeed a Radon Nikodym derivative we need to show that it has expectation equal to 1 Writing equation 56 for the riskless asset we get 7TTBT WtBt EtWt1BtL 66 Using equation 66 and the law of iterative expectations we get E07TTBT 7TOB0 Therefore WTBT is indeed a Radon Nikodym derivative and Q is well de ned We nally need to show that Q satis es equation 64 For a state w denote by At the set of states that are at the same node as to at time t Equation 7 implies that It QUE PAtEt WTBTlAt ptEt WTBTlAt Using equation 66 and the law of iterative expectations we get EthTBTl WtBt 67 Therefore Q satis es equation 64 Therefore we have de ned a probability measure Q such that equation 62 holds 13 Using equation 64 and the de nition of conditional expectation we can write equations 41 and 42 as T B B SW 4 is w t t Z 35 BT T lt gt st1 and Q Bt StEt 75t15t1 7 69 Bm respectively Equations 68 and 69 give prices as conditional expectations of future cash ows Cash ows are discounted by the riskless rate and conditional expectations are under the EMM Q and not under the original probability measure When there are no dividends we can write equation 69 as E EtQ St1 Bt Bt1 This means that the process 83 is a martingale under the EMM This process is the price discounted by the price of the riskless security lntuitively the martingale property means that the return on an asset cannot be higher than the riskless rate with probability 1 or lower with probability 1 otherwise there would be an arbitrage When there are dividends the discounted gain process A S 65 QJZ mm is a martingale 14

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