Selected Topics in Finance
Selected Topics in Finance FINA 8397
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What is Karma?
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Date Created: 09/19/15
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 Arbitrage and fundamental theorem of asset pricing 0 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 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 ltWgtBT 7 7 gT W7 gt 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 1NkT g1Nktl EtlgT S1NIml Etlng hNJm Etlgr S1NIml gtngm So we established S1Nt Et 1NT EthglN 39l39 26 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 T We K0 2 not 31 t0 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 st1 T St wiKt lt 2 565 l MST 41 t 5 There exists 1h 6 7 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 t wt w st2 Noting that KtltK5XSI KtX5 43 for s 2 s 2 t we can write the last term as T wiKt Km Z 1156 11080 t st2 Equation 41 implies that T Kt1 2 116 080 110180 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 2 1140048 60 7 0080 t0 t1 T71 K0 Z 00t5t w0100601 St1 K0 7JT0TST t0 Equations 42 and 43 imply that K0 00t5t 010t5t1 Stu Koliib tst KtWtHetth St1l KOl btetSt 0thwt16t1 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 10 Kozp rtct 53 t0 Comparing equations 51 and 53 we get pt 54 E The SPD is thus equal to the state price divided by the probability Notice that We 1 since we p0 1 Using equation 54 we can write equation 41 as 1 1 T St i iKt Z 17571 565 pT7TTST 7ft pt st1 The term in parenthesis is simply T Et lt 2 W565 ITTSTgt st1 We thus get T 1 St EEt lt Z 71 565 1 WTSTgt st1 Equation 42 similarly implies that 1 St FEtWt16t1 St1 56 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 In 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 53ng 57 51 and the SPDeadjusted gain process is t o ms 2 m 58 51 Proposition 51 The SPDeadjusted gain process is a martingale Proof Using the de nition of Gquot and equation 56 we get E40234 The martingale property then follows from the law of iterative expectations 6 Equivalent Martingale Measure In section 5 we wrote the function KI as T llc E0 2 m 61 t0 Remember that Ilc 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 Er ET gt 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 d g 7TTBT 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 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 7 lt gt st1 and Q Bt StEt 5t15t1 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 QSt1 Bt t Btu 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 Qi242 mm is a martingale 14 Rerferences Applications of Arbitrage Pricing Merton R 1973 Theory of Rational Option Pricing Bell Journal of Economics and Management Science 4 141 183 Black F and M Scholes 1973 The Pricing of Options and Corporate Liabilities Journal of Political Economy 81 637 654 Cox J and S Ross 1976 The Valuation of Options for Alternative Stochastic Processes Journal of Financial Economics 3 145 166 Ross S 1978 A Simple Approach to the Valuation of Risky Streams Journal of Business 51 No 3 453 475 Merton R 1977 Option Pricing When the Underlying Stock Returns are Discontinuous Journal of Financial Economics 5 125 144 Heston S 1993 A Closed Form Solution for Options with Stochastic Volatility with Ap plications to Bond and Currency Options Reuiew of Financial Studies 6 327 344 No Arbitrage Pricing Applications 1 Redundant Securities We consider a securities market model consisting of l a probability space 9 P 2 a time inverval T 0T 3 a Brownian motion Z Z1 Z1 on 9 P 4 the standard ltration F of Z 5 N 1 securities indexed by n 0 N a Security 0 is the riskless security and its price is given by dBt nBtdt 11 b Securities lN are the risky securities Their prices are given by an lto process 51 d ptdtatdZt 12 SN where M E 1N and 039 E 2NXd We set S 80SN and SUV 81SN We assume that securities pay no dividends and there is no intermediate consumption We denote SO by B Discounted prices follow the process A S M 7 7 S 039 A A dew d dt dzt E ptdt UtdZt We assume that t and t satisfy the conditions that guarantee existence of an EMM We denote by m the risk premia associated to the Brownian motions These are the solutions to t Tim 13 We denote the EMM by Q and the Brownian motion under Q obtained from Girsanov s theorem by ZQ We can write the dynamics of discounted prices as A s dew d 701sz 14 t We assume that trading strategies are in 18 and are such that the stochastic integral f 05dSSBS is a martingale under De nition 11 A selfe nancing trading strategyd replicates an fTemeasurable random uarie able OT CT QTST De nition 12 A new security with timeNT payo S T is redundant i there exists an ad missible trading strategy 0 that replicates ST Proposition 11 For a redundant security 51 dtSt Proof If N N otst lt St then there is an arbitrage obtained by buying the replicating strategy and shorting the redundant security A similar arbitrage exists if the opposite inequality holds Proposition 12 For a redundant security 5 E exp lt7 fads S T 15 Proof We denote by d the replicating strategy of the redundant security Since 9 is self nancing it is also self nancing wrt the discounted price process SB Therefore its is st y ME 517 TST N A TN A 7 0 05d 5 BT BT t t Z S gt Integrating we get as S EQ 7 Bt t B Proposition 11 implies then that E Q i Bt t BT and thus T 51 EtQ exp 7 rsds 5 t The redundant security is said to be priced by arbitrage QED 3 2 Complete Markets Consider a market in which there exists a bounded market price of risk process nt dint Mt an therefore there exists an EMM Q and there is no arbitrage To apply arbitrage pricing we need to characterize the set of cash ows that can be replicated As always we only consider to the admissible trading strategies Q lt would be easy to demonstrate replication if we allowed for a larger class of trading strategies eg E2 in fact it would appear that all markets are complete But such strategies can produce arbitrage gains We denote by L2Q the set of random variables that are square integrable wrt De nition 21 Markets are complete i all fTemeasarable random variables CT such that CTBT E L2Q can be replicated by a trading strategy in H26 or if CT 2 0 by a trading strategy in S Theorem 21 Markets are complete i the rank of at is equal to d as Proof Here we outline the main ideas of the proof Suppose that the rank of 0 is equal to d as Consider an fT measurable random variable CT such that CTBT E L2Q By the law of iterative expectations the process EtQCTBT is a martingale under Using the martingale representation part of the Girsanov s theorem there exists a process C E 2d such that CT CT t EQ 7 EQ 7 SdZSQ 21 t BT ET 0 ln fact since CTBT E L2Q C E H2d standard result eg Protter 1990 Thm 27 Corollary 3 which says that a local martingale is in fact a square integrable martingale il39l the isometry relation holds Since the rank of at is equal to d we can nd a process le such that t91Ntl339t Ct 22 The process 01 N represents the investment in the risky assets We de ne the investment in the riskless asset by E 6T as 23 Since 01m C E H2d we know that leo39 E E2 Also since lel leo39n C77 and C77 E 2d Cauchy Schwarz inequality implies that lel E E1 Thus 0t is a mathemati cally valid trading strategy ie the corresponding gain process is well de ned We now need to show that the strategy 0 is self nancing replicates CT and in H2653 and in To show that 0 is self nancing we plug equation 22 into equation 21 and get i E 0T EQ 0T 01N5amp5dzg A A EQ CT 01Ndsms 0 EQ 6T t05d 5 0 Combining with equation 23 we get t 05 0050 05015 0 Therefore 0 is self nancing wrt the discounted price process S 33 This implies that 0 is self nancing wrt to the undiscounted price process S Equation 23 for t T implies that 0 replicates CT Finally we need to show that 0 is in H2 and in if CT 2 0 We will not prove the rst result it can be found in Du ie Thm 6L p 118 Note While we didn t prove that 0 E 712SV remember that such condition itself was only su icient for absence of doubling strategies or for existence of EMM to imply absence of arbitrage it was not necessary In fact a weaker condition would su ice that 0 must be martingale generating under Q ie that f Qtdgt is a martingale under It is easy to see that in our case we obtained such a 0 f0 05 85 is a martingale under Q since t 050155 E CT 7 EQ CT 24 0 The results that 0 E is easier to show Eq 23 implies that 5 EtQ 2 0 since CT 2 0 For the converse implication see Du ie 6l The condition for market completeness has a similar avor to the condition in discrete time In discrete time markets are complete iff the number of linearly independent securities is equal to the number of nodes one period ahead In continuous time the number of nodes one period ahead is replaced by the number of Brownian motions Moreover linear independence of securities is measured by the rank of the diffusion matrix at and thus corresponds to an in nitesimal time interval In continuous time we can replicate an in nite dimensional set of cash ows with a nite number of securities To some extent this is not surprising since trading can take place in nitely often As in discrete time markets are complete iff the EMM is unique Theorem 22 Markets are complete i the EMM is unique Proof Suppose that markets are complete Then the rank of 0 is equal to d as Consider an EMM Q denote its Radon Nikodym derivative wrt P by T and set 5 Et T By the law of iterative expectations the process g is a martingale under P The martingale representation theorem implies that there exists a process p E 2d such that t t g go 7 p SdZS 1 7 pgdzs 25 0 0 Therefore d5 iplet and lto s lemma implies that I 1 2 bggn4 Ze ngm t t t I 1 t 2 gtexp 7 ampdZ577 ds 0 55 2 0 55 Since 5 is a martingale Girsanov s theorem implies that under Q discounted security prices follow the process S dlt1wltmimampma Bt Q where ZQ is a Brownian motion under Since Q is an EMM discounted security prices are a martingale and thus lntegrating we get mimamp0 Since the rank of 0 is equal to d this equation uniquely determines ptgt Therefore Q is unique The converse is left as an exercise 3 The BlackScholes model 31 The Model We assume that there are 2 securities The price of the rst security is given by BO l and dBt TBtdt 31 The short rate is thus constant We refer to the rst security as the bond The price of the second security is given by w umag g Q2 where Z is an one dimensional Brownian motion The drift and the diffusion are thus proportional to the price Such an lto process is called a Geometric Brownian Motion We refer to the second security as a stock and denote its price by St lto s lemma implies that 1 dlogSt M 7 502 dt UdZt Therefore 1 St SO exp Kg 7 502 t UZt The discounted stock price follows the process St St St d 7 7 7dt 7dZ B u m3 03 t This is also a Geometric Brownian Motion The conditions that guarantee existence of an EMM are satis ed The risk premium for the Brownian motion is M 7 7 739 The Radon Nikodym derivative of the EMM Q wrt P is 77 1 T exp lt777ZT 7 577271 The Brownian motion under Q obtained from Girsanov s theorem is Z Z m Since 039 gt 0 markets are complete and the EMM is unique 32 Pricing The Martingale Approach Consider now a new security with a time T payoff S E L1 This security is redundant and its time t price is given by S E exp77 T 7 mg If the time T payoff is a function of ST ie S GST we have 3 E expcrc 7 mow 33 To compute the expectation 33 we note that the discounted stock price follows the process St St Q d 7 7dZ Bt UBt t Therefore 7 l 2 7 Q 7 Q ST 7 Stexp r 2039 T t 0ZT Zt Since Z is a Brownian motion under Q the expectation 33 is 1 exp7rT 7 tEyG st exp r 7 502 T 7 t oxT 7 t y 34 where y is a standardized normal variable ie a normal variable with mean 0 and variance 1 Notice that the expectation 34 depends only on St and t When the new security is a European call or a European put we get the Black Scholes prices We denote the price of a European call by CStt and that of a European put by IquotSt7 t Proposition 31 We have CSt7 t StN21 7 exp7rT 7 tKN227 35 and PSt7 t exp7rT 7 tKN722 7 StN721 36 where is the eumulatiue distribution function of the standard normal distribution Z log e r as T 7 t 1 am 3397 22 21 7 oxT 7 t 38 Proof We compute the price of the call The price of the put follows by a similar argument or by put call parity PStt CStt Kexp7rT 7 t 7 St The time T payoff of the call is GST maxST 7 K 0 max Stexp Kr 7 02 T 7 t oxT 7 t y 7 K0 This is equal to 0 for Stexp r 7 02 T7t ox717703 K 7 Z 710gr702ltT7tgt lt7y 2 am i 8 and to St exp K7 7 02 T 7t ax717703 7 K otherwise The call price is CStt exp77 T 7 tEy max St exp lt7 7 02 T 7 t OW717703 7 K 0 01 027 where 01 exp77 T 7 2 00 St exp r 7 172 T 7 t am y exp 7 dy 1 W L and 00 02 7exp77 T 7122 Kexp 7 dy The term C2 is equal to C2 7 exp77 T 7 tK1 7 N722 7 exp77 T 7 tKNZg Therefore it is equal to the second term in the Black Scholes equation 35 The term C1 is equal to 01 exp77 T 7 tSt exp7 T 7 exp 7 y 7 Ix717702 dy 1 0 1 S 7 exp 7732 dy t V 27139 7zg7lTx7Tt 2 Setting 21 22 IxT 7 t we can write C1 as 01 NZl Therefore C1 is equal to the rst term in the Black Scholes equation 35 33 Pricing The PDE Approach So far we priced redundant securities by following the martingale approach ie by using the fact that the price of a redundant security is the expectation under the EMM of the secu rity s discounted payoff The original approach to arbitrage pricing was the PDE approach It consists in deriving a Partial Differential Equation PDE for the price of the redundant security We now present the PDE approach in the context of the Black Scholes model We also explain the relation between the martingale and the PDE approaches 9 We assume that the time T payoff of the redundant security is CST and denote the time t price of the security by 9St t The martingale approach implies that 9St7t EtQ 16Xp7 T 1tGST1 We can write this as exp77 tgStt E exp77 TGST 39 By the law of iterative expectations the process exp77 tgSt t is a martingale under Suppose that the function gSt is twice continuously differentiable Noting that dSt rStdt astdzfl and applying 1to s lemma we get dexp7 t95mt exp77 t 7gStt7 9595 2 94313 dt 955 0173101253 310 where 1 0593 2 95SttrSt 5955Stt028t2 311 Since the process exp77 tgSt t is a martingale under Q the drift is equal to 0 Therefore the function gSt solves the PDE 79097071 1 DSgS7 t 1 gtSvt 039 The PDE approach consists in solving the PDE 312 with the terminal condition gST GS Conversely suppose that the function gSt solves the PDE 312 with the terminal con dition gST Then equation 39 follows from the Feynman Kac theorem The Feynman Kac theorem concerns the general PDE 1 M570 9097071570 95SytMSyt 5953Sta8t2 945 07 313 with the terminal condition 9S T To this PDE is associated a stochastic differ ential equation SDE dSt MSttdt 0SttdZt 314 Under regularity conditions there exists an 1to process S that solves the SDE 314 with the initial condition St x The Feynman Kac theorem is that under regularity conditions a solution to the PDE 313 is given by 9w E 1T gtltnsgthlt83 W wmwwm ts exp rsudu 10 where 34 Replication The pricing of redundant securities rests on the fact that they can be replicated So far we used the martingale representation theorem to ensure that a replicating strategy exists However we have not determined the replicating strategy We now determine the replicating strategy in the Black Scholes model We consider a security with time T payoff equal to GST and denote the security s time t price by 965213 The replicating strategy consists in holding 0 E 95Stt shares of the stock and eth E 909M 95St7t5t dollars in the bond We need to show that the strategy 03 052 is self nancing replicates GST is in B S and is such that the stochastic integral f0 fdSSBS is a martingale under We can show these properties directly but we can also simply notice that the strategy 0305 is constructed exactly as in theorem 21 lndeed equations 39 310 and 312 imply that Efrexpwmw EQrexplt7rTgtGltsTgti gong with S c exp77 tgs8t tUSt 03 t Moreover 033 055 E expmrmw t 11