Help Session 7. David Sovich. Washington University in St. Louis

Transcription

1 Help Session 7 Davi Sovich Washington University in St. Louis

2 TODAY S AGENDA Toay we will learn how to price using Arrow securities We will then erive Q using Arrow securities

3 ARROW SECURITIES IN THE BINOMIAL TREE Suppose we are in a one-perio, two-state Ω = {u,} bimomial tree Suppose also that an Arrow security is trae for each state, u an An Arrow security for state u () is a security that pays off 1 if state u () is realize, an zero otherwise Denote the time t price of the u Arrow security as φ u t The time T payoff vectors of the Arrow securities are φ u T = [1,0] an φ T = [0,1]

4 ARROW SECURITES AND THE FTAP FTAP: The absence of arbitrage implies the existence of a set of strictly positive Arrrow security prices φt u an φt In complete markets, this is an if an only if statement This also implies that we can extract a unique risk-neutral measure Q from the Arrow security prices

5 PRICING CLAIMS WITH ARROW SECURITIES Suppose there is NA an we are given φt u an φt. Consier a security with price X t an T payoff vector X T = [X u,x ] Replication of the payoffs is simple using Arrow securities - we simply hol X u of the u security an X of the security Therefore, by the Law of One Price, we have that X t = X u φ u t + X φ t Observation: pricing claims is easy given Arrow security prices

6 EXAMPLE: MIN OPTION Suppose we want to price an option that pays V T = min{x T,Y T } on two assets X an Y The price using Arrow securities is V t = φ u t min{x u T,Y u T} + φ t min{x T,Y T} Example: φ u t = φ t = 1 2, X T = [1 0], Y T = [0 1]

7 BACKING OUT ARROW SECURITY PRICES Now suppose that the Arrow security prices {φ u t,φ t } are unknown We are given two asset prices, X t an Y t, with linearly inepenent payoff vectors ( ) ( ) X u T Y u XT an T Y T How o we solve for the Arrow security prices?

8 BACKING OUT ARROW SECURITY PRICES We coul fin the Arrow prices by replication using the X an Y securities Or we coul notice that the absence of arbitrage implies that the following system must hol X t = XTφ u t u + XTφ t Y t = YTφ u t u + YTφ t This is two linear equations in two unknowns, an thus a unique solution exists

9 EXAMPLE BACKING OUT ARROW SECURITY PRICES Suppose the payoff vectors are X T = [11] an Y T = [24] with X t = 1 an Y t = 3 The absence of arbitrage implies that the following system must hol 1 = 1φt u + 1φt 3 = 2φt u + 4φt Solving the system yiels φ u t = φ t = 1 2

10 RELATING ARROW SECURITIES TO Q By replication, we foun that the price of any claim X t satisfies X t = X u φt u + X φt But we also know the FTAP must hol in this setting, so that [ ] X t = E Q XT = q ux u + q X 1 + r 1 + r Is there any way to relate the two expressions above?

11 ONE APPROACH: CONSTRUCT Q Note that we can re-write our replication formula as X t = [ φt u + φt ] [ φt u φt u + φt X u + φ t ] φt u + φt X Also, φt u + φt = B t = (1 + r) 1, so we have X t = [ 1 ][ φt u 1 + r φt u + φt X u + φ t ] φt u + φt X

12 ONE APPROACH: CONSTRUCT Q Now efine q u = φ u t /(φ u t + φ t ) an q = φ t /(φ u t + φ t ) Note that these terms sum to one, an thus form a vali probability measure Q = {q u,q } q u + q = φ t u + φt φt u + φt = 1 Therefore, we have erive an FTAP formula for the price in terms of the Arrow securities [ ] [ ] 1 X t = [q u X u + q X ] = E Q XT 1 + r 1 + r

13 NEW INTUITION ABOUT Q Note that the q ω probabilities reflect the relative value of a payoff of one in state ω Thus, the risk-neutral measure assigns higher weight to states where a payoff of one is more valuable How o we relate this to physical probabilities an risk ajustments?

14 NEW INTUITION ABOUT Q Note that we coul have equivalently price X t with some ajuste physical probabilities φt u φt X t = p u X u + p X = E P [MX T ] p u p where M ω = φ t ω p ω factor is the state-price ensity or stochastic iscount The state price ensity reflects the price of a state relative to its physical likelihoo of occurring

15 NEW INTUITION ABOUT Q Notice that q probabilities are higher than p probabilities when it is more valuable to receive payoffs in specific states q ω p ω = M ω (1 + r) where M w is higher in states with lower consumption - e.g. recession states