DOC PREVIEW
Chicago Booth BUSF 35150 - Week 3 Asset Pricing Theory Extras

This preview shows page 1-2-23-24 out of 24 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

10 Week 3 Asset Pricing Theory Extras1. From p = E(mx) to all of asset pricing. Everything we do is just special cases, that are usefulin various circumstances.(a) In most of finance we do not use consumption data. We instead use other tricks to comeup with an m that works better in practical applications.(b) Theorem: If there are no arbitrage opportunities, then we can find an m with which wecan represent prices and payoffsbyp = E(mx)(c) Thus, the m structure allo ws us to do “no arbitrage” asset pricing.(d) From E(Ri)=Rf+ βi,∆cλ∆cto CAPM, multifactor models, APT, etc.i. Basic idea: We can’t see ∆c.So,weproxy∆c = a + bRm, consumption goes downwhen the market goes down. → CAPM.ii. Is that it? Do other things drive changes in consumption? (The CAPM isn’t justthe statement that consumption goes down when the market goes down; it’s thestatement that consumption only goes down when the market goes down.) ∆c =a + b1Rm+ b2X → E(R) depends on cov(R, Rm)andcov(R, X) This leads tomultifactor models(e) Bond pricesP(1)t= Et(mt+1× 1)P(2)t= Et(mt+1mt+2× 1).Term structure models (Cox Ingersoll, Ross, etc.): model mt+1,(model∆ct+1),i. For examplemt+1= φmt+ εt+1ThenP(1)t= φmtP(2)t= φ2mt= φP(1)tP(3)t= φ3mt= φ2P(1)tP(N)t= φNmt= φN−1P(1)tLook! We have a “one-factor arbitrage-free” model of the term structure. We candraw a smooth curve through bond prices (and then yields) in a way that we knowdoes not allow arbitrage. (Week 8)(f) Option pricing (Black-Scholes). Rather than price options from consumption, find mthat prices stock and bond, then use that m to price option. (Asset Pricing derivationof Black-Scholes)2. Quadratic utility is very popular (it lies behind mean-variance frontiers). It’s only an approx-imation, though easy to work with.u(c)=−12(c∗− c)2→ u0(c)=(c∗− c)(c<c∗)147−2 −1 0 1 2 3 4−6−5−4−3−2−101234c*Quadratic utility functionu(c)u’(c)Quadratic utility makes deriving the CAPM easy, and mean-variance portfolio theory.3. An example of why covariance is important. Suppose there are two states u, d tomorrowwith probability 1/2 (As in binomial option pricing.)pt= E(mx)=12muxu+12mdxd.u is “good times” with high c,lowm. Thus, suppose mu=0.5,md= 1. Now, suppose xpays off well in “good times”, If xu=2,xd=1.pt= E(mx)=12× 0.5 × 2+12× 1 × 1=1.But suppose we switch — same volatility but x pa ys off well in bad times and badly in goodtimes. xu=1,xd=2.pt= E(mx)=12× 0.5 × 1+12× 1 × 2=1.25.udm0.51x21P=1udm0.51x12P=1.25148Note E(x), σ(x) is the same. The pa yoff is worth more if the good outcome happens whenm is high (hungry) rather than when m is low (full). The same m and the same x deliverdifferent risk adjustments depending on cov(m, x).4. “Risk-neutral pricing”. How p = E(mx) is the same as what you learned in options/fixedincome classes.(a) Our formulap = E(mx)=SXs=1πsmsxs(b) “Risk-neutral probabilities” (Veronesi, options pricing) Definep =Xsπsmsxs=ÃXsπsms!Xsπsms(Psπsms)xs=1RfSXs=1π∗sxsp =1RfE∗(x)if we defineRf≡1E(m)=1Psπsms.andπ∗s=πsmsPsπsms= Rfπsms(we’ll see Rf=1/E(m) below; for now just use it as a definition)i. NoteXsπ∗s=1so they could be probabilities10.ii. Interpretation of p =1RfE∗(x) : price equals risk-neutral expected value using special“risk-neutral probabilities” π∗(c) A discount factor m is the same thing as a set of “risk neutral probabilities”i. Option 1: find “probabilities” π∗that price stock and bond using p =1RfE∗(x). Usethose probabilities to price option using the same formulaii. Option 2: find m that prices stock and bond using p = E(mx). Use that m to priceoption using p = E(mx). (Using true probabilities)iii. These are exactly the same thing!5. Beta model (CAPM) reminders:(a) The steps of running a beta model:10Also since m comes from u0(c)andu0(c) > 0, π∗s> 0 which probabilities have to obey149i. Run time series regression to find betasRit+1= ai+ βi,∆c∆ct+1+ εit+1t = −1, 2,...T for each iii. Average returns should be linearly related to betas,E(Ri)=Rf+ βi,∆cλ∆cβ is the right hand variable (x), λ is the slope coefficient (β)iβ()iERAll asset returns should lie on the lineSlopeλ(b) i in Rito emphasizei. The answ er to FF question: This is about why average returns of one asset arehigher than of another (cross section). NOT about fluctuationinex-postreturn(why did the market go up yesterday?) or predicting returns (will the market go uptomorrow?)ii. ERi,βi, vary across assets; “quantity of risk”. λ is common to all. “price of risk.”(c) Is high E (Re) good or bad?i. Neither. An asset must offer high E (Re)(good)tocompensateinvestorsforhighrisk (bad).ii.Thisisaboutequilibrium, after the market has settled down, after everyone hasmade all their trades. It’s about E(R)thatwill last, not disappear as soon asinvestors spot it.iii. Example: what if we all want to short? The price must fall until w e’re happy tohold the mark et portfolio again. How must price and E(R) adjust so that peopleare happy to hold assets?6. Long lived securities, the explicit derivation:U = Et∞Xj=0βju(ct+j)150pay ptξ,getξdt+1,ξdt+2...ptu0(ct)=Et∞Xj=1βju0(ct+j)dt+jpt= Et∞Xj=1βju0(ct+j)u0(ct)dt+jpt= Et∞Xj=1mt,t+jdt+j= Et∞Xj=1(mt+1mt+2...mt+j)dt+jThis is the present value formula with stochastic discount factor.7. Question: What if people have different γ, β,ordifferent utilities? Then we get differentprices depending on who we ask?Answer: Yes if we’re asking about genuinely new securities (then sell to the highest value guy).But no if we are talking about market prices. In a market everyone adjusts until they valuethings at the margin the same way. Example: One is patient, prefers consumption later. Oneinvestor is impatient, prefers consumption now. At their starting point, the patient investorimplies a lower interest rate, as you would expect. But facing the same market rate, P savesmore and I borrows more, until at the margin they are willing to substitute over time at thesame interest rate, as shown.C tC t+1$1$RPatient starting pointImpatient starting pointPatient saves, ends up hereImpatient borrows, ends up here8. Question: you slipped in to talking about economy-wide average consumption, not individualconsumption. What’s up with that?Answer: Right. There is a “theory of aggregation” that lets us do this. Here’s what needsto be proved: that the average consumption across people


View Full Document

Chicago Booth BUSF 35150 - Week 3 Asset Pricing Theory Extras

Documents in this Course
CLONES

CLONES

8 pages

Load more
Download Week 3 Asset Pricing Theory Extras
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Week 3 Asset Pricing Theory Extras and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Week 3 Asset Pricing Theory Extras 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?