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The Return Form of Arbitrage Pricing Primbs, MS&E345 1 Pricing Theory: Linear function form Return form (risk neutral) (pdes) Optimization Primbs, MS&E345 2 Pricing Theory: Returns and Factor Models Return form Profits and Losses (pdes) Absence of Arbitrage Relationships between returns of assets Generalizes the Black-Scholes-Merton argument Market Price of Risk Multiple Factors Futures Primbs, MS&E345 3 Modeling Returns: P1 You put something in. You get something out P0 1 period P P0 r 1 P0 Primbs, MS&E345 4 The time period can be: one year one month one week one day one second one millionth of a second an instantaneous dt. Primbs, MS&E345 5 Example: an instantaneous dt: Assume we model a stock price as a geometric brownian motion. dS Sdt Sdz St+dt St dt What is the return? St dt St r dSt dt dz St St Primbs, MS&E345 6 This is an example of a factor model: dS r dt dz S a bf Where: a dt known at beginning of period b f dz unknown factor Primbs, MS&E345 7 This is an example of a factor model: dS r dt dz S a bf Note that and can depend on S at time t (the beginning of the time period). dSt r ( St , t )dt ( St , t )dz St Primbs, MS&E345 8 The modeling paradigm: We describe the return of a security over a time period dt as a factor model: The factors: •Time: dt (this is for convenience) •Random factors: dz1, dz2, ... These random factors can be increments of Brownian Motion, Poisson Processes, or in general, whatever you want!!! I will write dt, but the time period could be of any length!!! Primbs, MS&E345 9 Pricing Theory: Returns and Factor Models Return form Profits and Losses (pdes) Absence of Arbitrage Relationships between returns of assets Generalizes the Black-Scholes-Merton argument Market Price of Risk Multiple Factors Futures Primbs, MS&E345 10 Profits and Losses Consider an asset, S, with the following return S r dt dz Invest x dollars at initial time St+dt That is, I purchase x Now each share is worth shares St St dt St+dt How much money did I make over dt? x x x Profit: S t dt x St dt St dS x rx St St St S shares price initial amount Primbs, MS&E345 11 Profit/Loss from a Portfolio x1 S1 r1 1dt 1dz We are given the returns on x2 S2 r2 2 dt 2 dz assets which all depend on a common factor, dz. x3 S3 r3 3dt 3dz Let xi be the dollar amount of money invested in asset i: Cost: x1 x2 x3 1T x Profit/Loss: r1 x1 r2 x2 r3 x3 (1dt 1dz) x1 (2 dt 2 dz) x2 (3 dt 3 dz) x3 (1 x1 2 x2 3 x3 )dt (1 x1 2 x2 3 x3 )dz ( T x)dt ( T x)dz 1 1 1 x1 where 1 1 2 2 x x2 1 3 3 x3 Primbs, MS&E345 12 Pricing Theory: Returns and Factor Models Return form Profits and Losses (pdes) Absence of Arbitrage Relationships between returns of assets Generalizes the Black-Scholes-Merton argument Market Price of Risk Multiple Factors Futures Primbs, MS&E345 13 What would an arbitrage portfolio be? An arbitrage is a riskless profit which requires no investment. In the current setting, a portfolio that (1) costs nothing (2) has no risk (3) but makes a profit Primbs, MS&E345 14 Profit/Loss from a Portfolio Cost: x1 x2 x3 1T x Profit/Loss: r1 x1 r2 x2 r3 x3 ( 1dt 1dz) x1 ( 2 dt 2 dz) x2 ( 3dt dz) x3 ( 1 x1 2 x2 3 x3 )dt ( 1 x1 2 x2 x3 )dz ( T x)dt ( T x)dz Consider the following portfolio: No Cost: 1T x 0 No Profit/Loss: T x 0 No Risk: Tx 0 Otherwise, Arbitrage!!! Primbs, MS&E345 15 A necessary condition for No Arbitrage No Cost: 1T x 0 No Profit/Loss: T x 0 No Risk: Tx 0 What are the implications of this? Primbs, MS&E345 16 1T Let’s write the condition as follows: T x 0 T x 0 This truthfulness of this implication is equivalent to saying that is a linear combination of 1 and . Why? Assume: 10 1 1 1T Then: T x T T x 0 Primbs, MS&E345 17 1T Let’s write the condition as follows: T x 0 T x 0 This truthfulness of this implication is equivalent to saying that is a linear combination of 1 and . Another Argument: 1T Let A T then x N (A) and N (A) but N ( A) R( AT ) hence R ( AT ) or AT 1 Primbs, MS&E345 18 A necessary condition for No Arbitrage No Cost: 1T x 0 No Profit/Loss: T x 0 No Risk: Tx 0 There exists a such that: 1 Primbs, MS&E345 19 Pricing Theory: Returns and Factor Models Return form Profits and Losses (pdes) Absence of Arbitrage Relationships between returns of assets Generalizes the Black-Scholes-Merton argument Market Price of Risk Multiple Factors Futures Primbs, MS&E345 20 Our return was given by: r dt dz We have derived a relationship between: 1 Expected Returns and Risk (Actually, this is nothing more that Ross’ 1976 Arbitrage Pricing Theory.) 10 1 1 The ’s tell you how much expected return you are rewarded for taking risk. They are generally referred to as the “Market Price of Risk”. Let’s look at them in a little more depth... Primbs, MS&E345 21 Consequences of the Return Form of AOA. A Risk Free Asset: the risk free rate r0 r f dt Then, from AOA: 0 1 1 0 is the risk free rate! 1 We can think of 0 as the r f 1 0 0 “market price of time”. Primbs, MS&E345 22 Consequences of the Return Form of AOA. A Risk Free Asset: Two other assets: r0 r f dt r1 1dt 1dz r2 2 dt 2 dz Then, from AOA: (1) r f 0 (2) 1 0 11 r f 11 2 0 2 1 rf 2 1 Solve for 1: 1 rf 2 rf 1 1 2 We refer to 1 as the “market price of risk” for dz. Primbs, MS&E345 23 The Market Price of Risk 1 rf 2 rf 1 1 2 (1) It looks like the Sharpe ratio. (2) The market price of risk is associated with the factor dz. (3) All securities that only depend on the random factor dz will have the same market price of risk (instantaneous Sharpe ratio). Primbs, MS&E345 24 The Market Price of Risk 1 rf 2 rf 1 1 2 An important Note: The market price of risk is often a function of St and t because it depends on and which are often functions of St and t. St+dt The market price of risk is known here, just like and . St dt I won’t always make this dependence explicit, but you should keep it in mind! Primbs, MS&E345 25 Connections with CAPM: CAPM says: rf (rM rf ) We just derived that: r f Hence, we should have that: (rM rf ) Example: If returns are uncorrelated with the market: 0 0 Sometimes used to determine market price of risk. Primbs, MS&E345 26 Pricing Theory: Returns and Factor Models Return form Profits and Losses (pdes) Absence of Arbitrage Relationships between returns of assets Generalizes the Black-Scholes-Merton argument Market Price of Risk Multiple Factors Futures Primbs, MS&E345 27 An extension to multiple factors r1 1dt 11dz1 ... 1m dzm x1 r2 2 dt 21dz1 ... 2 m dzm x2 rn n dt n1dz1 ... nm dzm xn Vector notation: r dt Kdz 1 11 1m dz1 dz where n n1 nm dzm No Arbitrage 1 x0 T KT x 0 T x 0 No Cost No Risk No Profit Primbs, MS&E345 28 No Arbitrage 1 x0 T K x0 T T x 0 No Cost No Risk No Profit There exists such that: 1 K Primbs, MS&E345 29 Does any of this make sense? r1 1dt 11dz1 ... 13dz3 Consider: r2 2 dt 21dz1 ... 23dz3 Necessary condition for no arbitrage: 1 0 111 122 133 2 0 211 222 233 In general, these equations do not have a unique solution for the ’s! That is, there could be many ’s that satisfy the equations. Nevertheless, it is still a necessary condition for no arbitrage. Primbs, MS&E345 30 Pricing Theory: Returns and Factor Models Return form Profits and Losses (pdes) Absence of Arbitrage Relationships between returns of assets Generalizes the Black-Scholes-Merton argument Market Price of Risk Multiple Factors Futures Primbs, MS&E345 31 An interesting special case: Futures Contracts When you enter into a futures contract, no money exchanges hands. So, initially, it costs nothing. This makes it a special case. Let, f be the futures price, and assume f follows: df dt dz f But, when I enter into this contract, I pay nothing How does this fit into our factor model framework? Primbs, MS&E345 32 Profit/Loss from a Portfolio that contains Futures We are given the returns on x1 S1 r1 1dt 1dz assets which all depend on a x2 S2 r2 2 dt 2 dz common factor, dz. x3 f r3 3dt 3dz Future Price Let x1, x2 be the amount of money invested in assets 1 and 2: But, x3 is the total future price. You agree to pay this at the delivery date, not now! Cost: x1 x2 1 1 0x Profit/Loss: r1 x1 r2 x2 r3 x3 ( 1dt 1dz) x1 ( 2 dt 2 dz) x2 ( 3dt dz) x3 ( 1 x1 2 x2 3 x3 )dt ( 1 x1 2 x2 x3 )dz ( T x)dt ( T x)dz Primbs, MS&E345 33 A necessary condition for No Arbitrage No Cost: x1 x2 0 No Profit/Loss: No Risk: 1 x1 2 x2 3 x3 0 1 x1 2 x2 3 x3 0 1 1 1 There exist 0, 1 such that: 0 2 1 2 0 1 3 3 Primbs, MS&E345 34 There exist 0, 1 such that: 1 1 1 0 2 1 2 0 1 3 3 Therefore, from the last equation: 3 1 3 I don’t need the risk free rate! I just need to know the dynamics of a futures price and I can calculate the market price of risk! Primbs, MS&E345 35 Some Intuition: Risk free rate: The “market price of time” No risk, only time. T t Futures price: The “market price of risk” No time, only risk. T t Primbs, MS&E345 36 References Ross, S. A., “The arbitrage theory of Capital Asset Pricing”, Journal of Economic Theory, 13, 341-360, 1976. Black, F. and M. Scholes, “The pricing of options and corporate liabilities”, Journal of Political Economy, 81, 637-659, 1973. Merton, R. C., “Theory of rational option pricing”, Bell Journal of Economics and Management Science, 4, 141-183, 1973. Primbs, MS&E345 37