Foundations of Financial Economics Two period financial markets
Paulo Brito
[email protected] University of Lisbon
April 3, 2020
1 / 63 Topics
▶ Financial assets ▶ Financial market ▶ State prices ▶ Portfolios ▶ Absence of arbitrage opportunities ▶ Complete financial markets ▶ Arbitrage pricing ▶ Equity premium ▶ Equity premium puzzle
2 / 63 Information
▶ Assume again that there is complete information regarding time t = 0, but incomplete information regarding time t = 1. ▶ Until now we have (mostly) assumed that the probability distribution is given by nature. ▶ But next we start assuming that information regarding time t = 1 is provided by the financial market. ▶ A financial market is defined by a collection of contingent claims. ▶ The general idea: under some conditions, we can extract an implicit probability distribution of the states of nature from the structure of the financial market.
3 / 63 Financial assets Prices and payoffs
A financial asset or contract or contingent claim: is defined by the price and payoff pair (Sj, Vj) (for asset j): ▶ for an observed price, Sj, paid at time t = 0 ▶ a contingent payoff, Vj, will be received at time t = 1,
⊤ Vj = (Vj1,..., Vjs,..., VjN)
4 / 63 Financial assets Returns and rates of return
▶ Return Rj and rate of return rj of asset j are related as
Rj = 1 + rj,
▶ In the 2-period case: it is the payoff/price ratio Vj,1 Sj 1 + rj,1 ... ... V R = j = Vj,s = 1 + r j Sj j,s Sj ... ... Vj,N 1 + rj,N Sj
5 / 63 Financial assets Timing, information and flow of funds Two alternative ways of representing (net) income flows: ▶ as a price-payoff sequence Vj,1 ... Vj,s ... −Sj Vj,N
0 1
▶ as an investment-return sequence Rj,1 ... Rj,s ... −1 Rj,N
0 1 6 / 63 Financial assets Statistics ▶ From the information, at time t = 0, on the probabilities for the states of nature at time t = 1 we can compute: ▶ Expected return for asset j at time t = 1, from the information at time t = 0 ∑N E[Rj] = πsRj,s = π1Rj,1 + . . . πsRj,s + ... + πNRj,N s=1 ▶ Variance of the return for asset j at time t = 1, from the information at time t = 0 ∑N 2 2 2 V[Rj] = πs(Rjs−E[Rj]) = π1 (Rj,1 − E[Rj]) +. . . πN (Rj,N − E[Rj]) s=1 ▶ Observation: an useful relationship V[R] = E[R2] − (E[R])2
7 / 63 Classification of assets Risk classification
As regards risk: ▶ risk-less or risk-free asset: V⊤ = (v,..., v)⊤ ▶ ⊤ ⊤ risky asset: V = (v1,..., vN) with at least two different elements, i.e. there are at least two elements, vi and vj such that vi ≠ vj for i ≠ j
8 / 63 Classification of assets Types of assets
Particular types of assets as regards the income flows: ▶ one period bonds with unit facial value: 1 S = , V⊤ = (1, 1,..., 1)⊤ 1 + i ▶ deposits:
S = 1, V⊤ = (1 + i, 1 + i,..., 1 + i)⊤
▶ equity: the payoff are dividends
⊤ ⊤ Se, Ve = (d1,..., dN)
9 / 63 Classification of assets Types of assets
Derivatives: ▶ forward contract on an underlying asset with offered price p:
⊤ − − ⊤ Sf, Vf = (v1 p,..., vN p)
▶ european call option with exercise price p:
⊤ { − } { − } ⊤ Sc, Vc = (max v1 p, 0 ,..., max vN p, 0 )
▶ european put option with exercise price p:
⊤ { − } { − } ⊤ Sp, Vp = (max p v1, 0 ,..., max p vN, 0 )
10 / 63 Financial market Definition:A financial market is a collection of K traded assets. It can be characterized by the structure of prices and payoffs of all K assets : i.e by the pair (S, V): ▶ vector of observed prices, at time t = 0 (we use row vectors for prices) |{z}S = (S1,..., SK), K×1
▶ and a matrix of contingent (i.e., uncertain) payoffs, at time t = 1 V11 ... VK1 . . |{z}V = . . × N K V1N ... VKN
▶ The participants observe S and know V, but they do not know which payoff they will get for every asset (i.e, which line of V will be realized)
11 / 63 Financial market The payoff matrix contains the following information: ▶ each column represents the payoff forasset j = 1,..., K V11 ... Vj1 ... VK1 . . . . . . V = V1s ... Vjs ... VKs . . . . . . V1N ... VjN ... VKN
▶ each row represents the outcomes (in terms of payoffs) for state of nature s = 1,..., N V11 ... Vj1 ... VK1 . . . . . . V = V1s ... Vjs ... VKs . . . . . . V1N ... VjN ... VKN
12 / 63 Financial market ▶ Equivalently, we can charaterize a financial market by the matrix of returns R11 ... RK1 . . |{z}R = . . × N K R1N ... RKN
Vj,s where Rj,s = = 1 + rj,s Sj ▶ Therefore −1 |{z}R = |{z}V |(diag{z(S)) } × × N K N K K×K where S1 ... 0 ... 0 . . . . . . . . . . |diag{z(S}) = 0 ... Sj ... 0 . . . . . K×K . . . . . 0 ... 0 ... SK
13 / 63 State prices
Implicit state valuation: the modern approach to financial economics characterizes asset markets by the prices of the states of nature implicit in the relationship between present prices S and future payoffs V, i.e. Q := {x : S = xV}
▶ if Q is unique it is a vector
|{z}Q = (q1,..., qN) 1×N satisfying ⇔ ⊤ ⊤ ⊤ |{z}S = |{z}Q |{z}V |{z}S = |{z}V |{z}Q 1×K 1×N N×K K×1 K×N N×1
▶ Definition: Q is a state price vector if is it positive , i.e. qs > 0 for all s ∈ {1,..., N}
14 / 63 Arbitrage opportunities
Definition: if there is at least one qs ≤ 0, s = 1,..., N then we say there are arbitrage opportunities Definition: we say there are no arbitrage opportunities if qs > 0, for all s = 1,..., N Intuition: ▶ existence of arbitrage (opportunities) means there are free or negatively valued states of nature ▶ absence of arbitrage means every state of nature is costly and therefore, positively priced.
Proposition 1 Given (S, V), there are no arbitrage opportunities if and only if Q is a vector of state prices.
15 / 63 Completeness of a financial market
Definition: if Q is unique, then we say markets are complete Definition: if Q is not unique then we say markets are incomplete Intuition: ▶ completeness: there is an unique valuation of the states of nature. ▶ incompleteness: there is not an unique valuation of the states of nature.
16 / 63 Completeness of a financial market Conditions for completeness
We can classify completeness of a market by looking at the number of assets, number of states of nature and the independence of payoffs: 1. if K = N and det (V) ≠ 0 then markets are complete and all assets are independent; 2. if K > N and det (V) ≠ 0 then markets are complete and there are N independent and K − N redundant assets; 3. if K < N or K ≥ N and det (V) = 0 then markets are incomplete.
Proposition 2 Given (S, V), markets are complete if and only if dim(V) = N
dim(V) = number of linearly independent columns (i.e., assets) 17 / 63 Characterization of a financial market No arbitrage opportunities and completeness
▶ Consider a financial market (S, V), such that K = N and det (V) ≠ 0 ▶ We defined state prices from S = QV ▶ As K = N and det (V) ≠ 0 then V−1 exists and is unique ▶ Therefore, we obtain uniquely −1 |{z}Q = |{z}S |V{z} 1×N 1×K K×N that is −1 V1,1 ... V1,K (q1,..., qN) = (S1,..., SK) ...... VN,1 ... VN,K
▶ If Q ≫ 0 we say there are no arbitrage opportunities ▶ And because Q is unique we say markets are complete
18 / 63 Characterization of a financial market No arbitrage opportunities and completeness: alternative 1
Alternatively, because
⊤ ⊤ ⊤ |{z}V |{z}Q = |{z}S K×N N×1 K×1
If det (V) ≠ 0 then ( )−1 Q⊤ = V⊤ S⊤
that is q −1 S 1 V1,1 ... VN,1 1 . . . = ...... . V ... V qN 1,K N,K SK
19 / 63 Characterization of a financial market No arbitrage opportunities and completeness: alternative 2
As Rjs = Vjs/Sj equivalently, we can write
⊤ |{z}Q |{z}R = |{z}1 1×N N×K 1×K R1,1 ... R1,K ( ) (q1,..., qN) ...... = 1 1 1 RN,1 ... RN,K then Q = 1⊤R−1
20 / 63 Characterization of a financial market No arbitrage opportunities and completeness: alternative 3
Or, alternatively, R⊤Q⊤ = 1 then ⊤ ⊤ −1 |{z}Q = (|R{z) } |{z}1 × N×1 N×K K 1 matricially q −1 1 1 R1,1 ... R1,K . . . = ...... . R ... R qN N,1 N,K 1
21 / 63 Characterization of a financial market Market incompleteness
▶ Consider a financial market (S, V), such that K < N, and in the partition ⊤ ⊤ ⊤ |{z}V = ( |{z}V1 |{z}V2 ) × K N (K×K) (N−K×K)
assume that det (V1) ≠ 0 ▶ We defined state prices from V⊤Q⊤ = S⊤ ▶ But now we have ( ) ( ) ⊤ ⊤ ⊤ Q1 ⊤ V1 V2 ⊤ = S Q2
where Q1 is (1 × K) and Q2 is (1 × N − K).
22 / 63 Characterization of a financial market Market incompleteness
▶ Then ⊤ ⊤ ⊤ ⊤ ⊤ V1 Q1 + V2 Q2 = S
▶ Because det (V1) ≠ 0 we can make ( ) ⊤ ⊤ −1 ⊤ − ⊤ ⊤ Q1 = (V1 ) S V2 Q2
▶ There are only K independent prices: i.e, Q is indeterminate and the degree of indeterminacy is N − K, ▶ As Q is not uniquely determined (i.e, we can fix arbitrarilty N − K state prices) the market is incomplete
23 / 63 Characterization of a financial market Market completeness with redundant assets
▶ Consider a financial market (S, V), such that K > N, partition
|{z}V = ( |{z}V1 |{z}V2 ) N×K (N×N) (K−N×N)
and assume that det (V1) ≠ 0 ▶ We defined state prices from V⊤Q⊤ = S⊤ ▶ But now we have ( ) ( ) ⊤ ( ) ⊤ V1 ⊤ S1 ⊤ Q = ⊤ V2 S2 where S1 is (1 × N) and S2 is (1 × K − N).
24 / 63 Characterization of a financial market Market completeness with redundant assets
▶ Then { ⊤ ⊤ ⊤ V1 Q = S1 ⊤ ⊤ ⊤ V2 Q = S2
▶ Because det (V1) ≠ 0 we can determine Q uniquely ⊤ ⊤ −1 ⊤ Q = (V1 ) S1
▶ Which implies that the prices of the remaining K − N assets can be obtained from the prices S 1 ( ) ⊤ ⊤ ⊤ ⊤ ⊤ −1 ⊤ S2 = V2 Q = V2 (V1 ) S1
▶ There are K − N redundant assets (i.e, they do not add new information on Q) ▶ As Q is uniquely determined the market is complete
25 / 63 Characterization of a financial market Suggestion on how to apply this theory
▶ First: obtain RT which is a (K × N) matrix ▶ Second: check the number of assets, K and the number of states of nature N ▶ if N < K then markets are incomplete ▶ if N ≥ K check the number of independent rowsn of R ▶ if dim (V) < N then markets are incomplete ▶ if dim (V) ≥ N then markets are complete ▶ Third: obtaining the state prices Q ▶ if K = N and det R ≠ 0 compute Q⊤ = (R⊤)−11 ▶ if K > N and R˜ a N × N partition of R has det R˜ ≠ 0 then compute Q⊤ = (R˜ ⊤)−11 ▶ if K = N but det R = 0, then, solve the independent equations in R⊤Q⊤ = 1 ▶ if K < N solve the independent equations in R⊤Q⊤ = 1.
26 / 63 Example 1
( ) 1 1 ▶ Let S = (1, 1) and V = 1 2 ▶ then ( ) ( ) 1 1 1 1 R = , R⊤ = 1 2 1 2
▶ Therefore
( )−1 ( ) ( )( ) ( ) 1 1 1 2 −1 1 1 Q⊤ = = = 1 2 1 −1 1 1 0
then markets are complete but there are arbitrage opportunities.
27 / 63 Example 2
( ) 2 1 ▶ Let S = (1, 1) and V = 1 2 ▶ then ( ) 2 1 R = = R⊤ 1 2
▶ Therefore ( ) ( ) ( )( ) ( ) −1 2 − 1 1 ⊤ 2 1 1 3 3 1 3 Q = = − 1 2 = 1 1 2 1 3 3 1 3 markets are complete and there are no arbitrage opportunities
28 / 63 Example 3 ( ) 2 1 3 ▶ Let S = (1, 1, 2) and V = 1 2 3 ( ) 3 2 1 ▶ 2 1 2 ⊤ then R = 3 and R = 1 2 1 2 2 3 3 2 2 ▶ we pick all the combinations of two assets ( ) ( ) ( )( ) ( ) −1 2 − 1 1 ⊤ 2 1 1 3 3 1 3 Q = = − 1 2 = 1 1 2 1 3 3 1 3 ( ) ( ) ( )( ) ( ) −1 − 2 1 ⊤ 2 1 1 1 3 1 3 Q = 3 3 = − 4 = 1 2 2 1 1 3 1 3 and ( ) ( ) ( )( ) ( ) −1 − 4 1 ⊤ 1 2 1 1 3 1 3 Q = 3 3 = − 2 = 1 2 2 1 1 3 1 3 then markets are complete, there are no arbitrage opportunities and there is one redundant asset
29 / 63 Example 4 ( ) 2 4 ▶ Let S = (1, 2) and V = 1 2 ( ) ( ) 2 2 2 1 ▶ then R = and R⊤ = two assets have the same 1 1 2 1 returns ▶ then R⊤Q⊤ = 1⊤ becomes ( )( ) ( ) 2 1 q 1 1 = 2 1 q2 1
has an infinite number of solutions ( ) (1 − k)/2 Q⊤ = k
for an arbitrary k, then markets are incomplete and if 0 < k < 1 there are no arbitrage opportunities;
30 / 63 Example 5
( ) 2 ▶ Let S = (1) and V = 1 ▶ then ( ) 2 R = 1
▶ then we also have ( ) 2 (q , q ) = 1 1 2 1 or 2q1 + q2 = 1 has an infinite number of solutions Q = ((1 − k)/2, k), for any k, then markets are incomplete and if 0 < k < 1 there are no arbitrage opportunities
31 / 63 Implicit market probabilities: risk-free probabilities ▶ Let Q = (q1,..., qN) be a state-price vector ▶ If there are no arbitrage opportunities then qs > 0, i.e Q ≫ 0 ▶ A Radon-Nikodyn derivative is defined as
Q πs = qs/¯q ∑ N where ¯q = s=1 qs. ▶ Therefore, if there are no arbitrage opportunities then ∑N Q Q 0 < πs < 1, and πs = 1 s=1 Q that is πs is a probability measure ▶ This is a probability measure implicit in the financial market (S, V) ▶ Therefore: ▶ If markets are complete the probability measure is unique ▶ If markets are incomplete the probability measure is not
unique 32 / 63 And asset pricing
▶ Using the definition for qs
∑N ∑N q S = q V = ¯q s V j s sj ¯q sj s=1 s=1 then, for any asset [ ] Q Sj = ¯q E Vj
▶ This means prices are proportional to the expected value of the payoff, using an implicit market probability.
33 / 63 Definition ▶ Let Q = (q1,..., qN) ▶ Assume we have a (objective or subjective) probability distribution for the states of nature
P = (π1, . . . , πN) ∑ N such that 0 < πs < 1 and s=1 πs = 1 ▶ The stochastic discount factor for state s is
qs ms = , s = 1,..., N πs
▶ The stochastic discount factor is the random variable
|{z}M = (m1,..., ms,..., mN) 1×N
34 / 63 And asset pricing
▶ Using the definition for qs
∑N ∑N q S = q V = π s V j s sj s π sj s=1 s=1 s
then, for any asset [ ] Sj = E MVj
▶ This means that the stochastic discount factor combines both market and fundamental information.
35 / 63 The state price approach to asset markets
Summing up: ▶ The state price translates the information structure implicit in financial transactions ▶ The relevant information is related to: ▶ how costly is the insurance against the states of nature ▶ the uniqueness of that insurance cost ▶ suggests a method for pricing new assets: pricing by redundancy
36 / 63 Portfolios
▶ Definition:A portfolio is a vector specifying the positions, θ, in all the assets in the market
⊤ ⊤ K θ = (θ1, . . . , θK) ∈ R
If θj > 0 there is a long position in asset j if θj < 0 there is a short position in asset j ▶
37 / 63 Portfolios
▶ The stream of income generated by position θj is θjVj,1 ... θjVj,s ... −Sjθj θjVj,N
0 1
▶ long position (θj > 0): pay Sjθj at time t = 0 and receive the contingent payoff θjVj at time t = 1 ▶ short position (θj < 0): receive Sjθj at time t = 0 and pay the contingent payoff θjVj at time t = 1
38 / 63 Portfolios
{ θ θ} A portfolio generates a (stochastic) stream of income z0, Z1
∑K θ − − z0 = C(θ, S) = |{zSθ} = Sjθj 1×1 j=1 ∑K θ Z1 = |{z}Vθ = Vjθj N×1 j=1
where ∑ θ K z Vj,1θj 1,1 j=1 ... ... ∑ Zθ = zθ = K 1 1,s j=1 Vj,sθj ... ∑ zθ K 1,N j=1 Vj,Nθj
39 / 63 Portfolios
The flow of income generated by a portfolio θ is: zθ 1,1 ... θ z1,s ... θ θ z0 z1,N
0 1
40 / 63 Portfolios and arbitrage opportunities
Proposition 3 Assume there are arbitrage opportunities. Then there exists at least one portfolio ϑ such that
ϑ ϑ z0 = 0 and Z1 > 0
or ϑ ϑ z0 > 0 and Z1 = 0.
(Obs. A positive vector X > 0 has non-negative elements, and has at least one equal to zero. A strictly positive vector X ≫ 0 has only positive elements ) Intuition with a zero cost we can get a positive income in at least one state of nature. If we have an initial income, we will pay a zero cost in the
future, in every state of nature. 41 / 63 Example 1
( ) 1 1 ▶ We consider Example 1 again: S = (1, 1) and V = 1 2 ▶ ⊤ We can build a portfolio ϑ = (ϑ1, ϑ2) such that ϑ − − z0 = Sϑ = (ϑ1 + ϑ2) = 0, that is
ϑ1 = −x, ϑ2 = x
▶ The return at time t = 1 is ( ) ( ) −x + x 0 Zϑ = Vϑ = = 1 −x + 2x x
then if we take a long position in the risky asset (x > 0) we have a positive income at t = 1 with a zero investment.
42 / 63 Portfolios and arbitrage opportunities
Proposition 4 Assume there are no arbitrage opportunities. A portfolio with ϑ ϑ cost, z0 = 0, generates a nonpositive income at time 1, Z1 .
▶ ϑ Non-positive Z1 means that there is at least one state of nature, θ s such that z1,s < 0. Intuition: with a zero cost although we can get a positive income in several states of nature there is at least one state of nature in which there is a negative income.
43 / 63 Example 2
( ) 2 1 ▶ Consider the previous Example 2: S = (1, 1) and V = 1 2 ▶ ⊤ ϑ − We can build a portfolio (ϑ1, ϑ2) such that z0 = Sϑ = 0:
ϑ1 = −x, ϑ2 = x
▶ The return at time t = 1 is ( ) ( ) −2x + x −x Zϑ = Vϑ = = 1 −x + 2x x
then whatever the position, long (x > 0) or short (x < 0), in the risky asset we will not have positive income at t = 1 (in some states of the nature we win, but we will lose in others).
44 / 63 Replicating portfolios
Definition: Replicating porfolio: Assume we have a ⊤ contingent income W = (w1,..., wN) . To the portfolio, ϑ, build using the assets in the market, generating the same (contingent) income ϑ ≡ {θ : Z1 = Vθ = W} we call replicating portfolio. The cost of the replicating portfolio is C(ϑ, S) = Sϑ
45 / 63 Arbitrage asset pricing theory (APT)
▶ Deals with the determination of asset prices through replication. ▶ APT vs GEAP (general equilibrium asset pricing): ▶ in APT we take (S, V) as given (S is exogenous) ▶ in GEAP we take V as given and want to determine S from the fundamentals (S is endogenous) ▶ Both theories have three equivalent formulations: ▶ using state prices Q ▶ using the stochastic discount factor M ▶ using market or risk-neutral probabilities πQ
46 / 63 Arbitrage asset pricing Using state prices
Proposition 5 Assume there is a financial market (S, V) such that there are no arbitrage opportunities. Then, given a new asset with payoff Vk its price can be determined by redundancy by using the expression
Sk = QVk
where Q = SV−1 ≫ 0 is determined from the market pair (S, V).
That is ∑N Sk = qsVks s=1
47 / 63 Arbitrage asset pricing Using state prices From what we have previously seen there are two important results
Proposition 6
If markets are complete then the value, Sk, is unique. If markets are incomplete then Sk is not unique.
Proposition 7 Assume a financial market (S, V) is complete and there are K − N redundant assets. Then the price of any asset is equal to the cost of their replicating portfolios build with N independent assets.
48 / 63 Arbitrage asset pricing Using stochastic discount factors
▶ If Q is a state price vector and π is the vector or probabilities, we define the stochastic discount factor (SDF) for state s by
qs ms ≡ πs
Proposition 8 Assume there is a financial market (S, V) such that there are no arbitrage opportunities. Then, given a new asset with payoff Vk its price can be determined by redundancy by using the expression
Sk = E[MVk]
⊤ where M = (m1,..., mN) is the stochastic discount factor.
49 / 63 Arbitrage asset pricing Using stochastic discount factors
∑ ▶ N Proof: as Sk = s=1 qsVks and using the definition of SDF we get ∑N ∑N Sk = qsVks = πsmsVks s=1 s=1 ▶ Intuition: if there are no arbitrage opportunities the price of an asset is the expected value of the future payoffs discounted by the stochastic discount factor (which is the market discount for uncertain payoffs)
50 / 63 Arbitrage asset pricing Using risk-neutral probabilities
▶ Therefore,
∑N ∑N Q EQ Sk = qsVks = ¯q πs Vks = ¯q [Vk] s=1 s=1 or compactly Q Sk = ¯qE [Vk]
▶ Intuition: if there are no arbitrage opportunities there is an equivalent probability measure such that the price of an asset is proportional of the expected value of the future payoffs ▶ Although ¯q is mysterious we can calculate it a intuitive way.
51 / 63 Arbitrage asset pricing Existence of risk-free asset
Proposition 9 Assume there are no arbitrage opportunities and there is a risk-free bond with price 1/(1 + i) and face value 1. Then the price of any asset verifies the relationship 1 S = EQ[V ] j 1 + i j
Exercise: prove this.
52 / 63 Arbitrage asset pricing Existence of risk-free asset
Proposition 10 Assume there are no arbitrage opportunities and there is a risk-free asset. Then there is a (market) probability measure such that the expected returns of every asset is equal to the return of the risk-free asset.
▶ Proof. From Proposition 8 we have
Q 1 + i = E [Rj], for any, j = 1,..., K As this holds for any asset, therefore πQ has the property
Q Q 1 + i = E [R1] = ... = E [RK]
53 / 63 Application Arbitrage and completeness with two assets
▶ Assume that N = 2 and there is a risky and a risk-free asset ( ) ( ) 1 1 d S = , p , V = 1 1 + i 1 d2
▶ and that r1 < i < r2. Then markets are complete and there are no arbitrage opportunities. The return matrix is ( ) 1 d1 ( ) 1 p 1+i 1 + i 1 + r1 R = 1 d2 = 1 p 1 + i 1 + r2 1+i
54 / 63 Application Arbitrage asset pricing
▶ If there are risk-free assets and absence of arbitrage opportunities, then any asset can be priced by redundancy as
Sk = q1Vk,1 + q2Vk,2 where
i − r2 r1 − i q1 = , q2 = . (1 + i)(r1 − r2) (1 + i)(r1 − r2)
▶ Proof: assume there is a risk-free and a risky asset such that QR = 1⊤ becomes
( ) ( )−1 ( ) q 1 + i 1 + i 1 1 = q2 1 + r1 1 + r2 1
and if there are no arbitrage opportunities r1 < i < r2 implying q1 > 0 and q2 > 0.
55 / 63 Application Simple Black and Scholes (1973) equation
▶ Consider an european call option with exercise price p on an underlying asset with payoff ⊤ Vunderlying = (d1, d2)
▶ Then the contingent payoff of the option is ⊤ { − } { − } Voption = (max d1 p, 0 , max d2 p, 0 )
▶ Question: assuming there are no arbitrage opportunities and there is a risk-free asset what is the market price of the option ? ▶ Answer: the price for a call option with exercise price p is
(r2 − i) max{d1 − p, 0} + (r1 − i) max{d2 − p, 0} Soption = (1 + i)(r1 − r2) (prove this)
56 / 63 Equity premium
▶ We call equity premium to the difference in the rates of return between the risky and a risk-free asset ( ) r − i r − i = 1 r2 − i
▶ The Sharpe index is defined as ∑ E − 2 − [r i] s=1 πs(rs i) = √∑ σ[r] 2 − − E − 2 s=1 πs((rs i) [r i]) √ where σ[r − i] = V[r − i] is the standard deviation of the equity premium (prove that σ[r − i] = σ[r]).
57 / 63 Risk neutral probabilities
Proposition 11 If there are no arbitrage opportunities then there is a (market) risk neutral probability distribution such that the expected value of the equity risk premium is zero,
EQ[r − i] = 0
Q This is the reason why πs are called risk neutral probabilities: PQ Q Q = (π1 , . . . πN ) is a probability measure such that the expected value of the equity premium is zero.
58 / 63 Risk neutral probabilities
▶ Proof: Let ( ) 1 + i 1 + i R⊤ = 1 + r1 1 + r2
▶ Then, because R⊤Q⊤ = 1 { (1 + i)q1 + (1 + i)q2 = 1
(1 + r1)q1 + (1 + r2)q2 = 1
▶ Then (1 + r1)q1 + (1 + r2)q2 = (1 + i)q1 + (1 + i)q2. ▶ Therefore q1(r1 − i) + q2(r2 − i) = 0
59 / 63 Risk neutral probabilities
▶ Proof (cont)Define ¯q = q1 + q2 q q 1 (r − i) + 1 (r − i) = 0 ¯q 1 ¯q 2
▶ Define again πQ = qs . If there are no arbitrage opportunities, s ¯q ∑ Q 2 Q Q then qs > 0 and πs > 0. Because s=1 πs = 1 then πs are probabilities. ▶ At last Q − Q − π1 (r1 i) + π2 (r2 i) = 0
60 / 63 The Sharpe index and the SDF
▶
Proposition 12 Assume there are no arbitrage opportunities and there is a risk-free asset. Then the Sharpe index verifies the relationship
( )− E[r − i] E[M] 1 = −ρ − σ[r] r i,m σ[M]
where ρr−i,m is the coefficient of correlation between the equity premium and the stochastic discount factor.
▶
61 / 63 The Sharpe index and the SDF
▶ Proof: Using our previous definition of the stochastic discount factor ms = qs/πs then
E[M(r − i)] = 0
▶ But E[M(r − i)] = Cov[M(r − i)] + E[M] E[r − i] ▶ Using the definition of correlation
Cov[M(r − i)] ρ − = ∈ (−1, 1) r i,m σ[M] σ[r − i]
▶ Then ρr−i,m σ[M] σ[r − i] + E[M] E[r − i] = 0
62 / 63 The equity premium and DGE models
▶ Intuition: the Sharpe index is equal symmetric to the product between coefficient of variation of the stochastic discount factor and correlation coefficient between the equity premium and the stochastic discount factor, ρr−i,m ▶ The coefficient of variation of the stochastic discount factor contains the aggregate market value of risk. ▶ E[M]/σ[M] can be derived from simple DSGE models (as we will see next) ▶ Observe that the Sharpe index satisfies
E[R − Rf] σ[R]
where Rf = 1 + i is the risk-free return and R can be taken as a return on a market index. (see http://web.stanford.edu/~wfsharpe/art/sr/SR.htm
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