Fin 501: Asset PricingFin 501: Asset PricingLecture 3: OneLecture 3: One--period Modelperiod ModelPricingPricingProf. Markus K. BrunnermeierProf. Markus K. BrunnermeierSlide 03Slide 03--11Fin 501: Asset PricingFin 501: Asset PricingOiPiiOiPiiOverview: PricingOverview: Pricing1.1. LOOP, No LOOP, No arbitragearbitrage [L2,3][L2,3]2.2. ForwardsForwards [McD5][McD5]33Options: ParityOptions: Parityrelationshiprelationship3.3.Options: Parity Options: Parity relationshiprelationship[McD6][McD6]4.4. No arbitrage and existence of state No arbitrage and existence of state pricesprices [L2,3,5][L2,3,5]5.5. Market completeness and uniqueness of state pricesMarket completeness and uniqueness of state pricespqppqp6.6. Pricing kernel Pricing kernel qq**7.7. Four pricing formulas:Four pricing formulas:state prices SDF EMM betastate prices SDF EMM betapricingpricing[[L2356]L2356]state prices, SDF, EMM, beta state prices, SDF, EMM, beta pricingpricing[[L2,3,5,6]L2,3,5,6]8.8. Recovering state prices from Recovering state prices from optionsoptions [DD10.6][DD10.6]Slide 03Slide 03--22Fin 501: Asset PricingFin 501: Asset PricingVtNttiVtNttiVector NotationVector Notation•Notation:yx∈Rn•Notation: y,x∈Rn–y ≥ x ⇔ yi≥ xifor each i=1,…,n.–y>x⇔y≥x and y≠x–y > x ⇔y ≥x and y ≠x.– y >> x ⇔ yi> xifor each i=1,…,n.•Inner productInner product–y ≤ x = ∑iyx•Matrix multiplicationMatrix multiplicationSlide 03Slide 03--33Fin 501: Asset PricingFin 501: Asset PricingspecifyPreferences &Technologyobserve/specifyexisting Asset PricesTechnology•evolution of states•risk preferencesNAC/LOOPNAC/LOOPState Prices q(or stochastic discount •aggregationabsolute relativefactor/Martingale measure)asset pricingasset pricingLOOPderiveAtPiderivePi f ( ) tSlide 03Slide 03--44Asset PricesPrice for (new) assetOnly works as long as market completeness doesn’t changeFin 501: Asset PricingFin 501: Asset PricingTh F f NTh F f NARBITRAGEARBITRAGEThree Forms of NoThree Forms of No--ARBITRAGEARBITRAGE1.1. Law of one Price (LOOP) Law of one Price (LOOP) If h’X = k’X then p ≤ h = p ≤ k.pp2.2. No strong arbitrageNo strong arbitrageThere exists no portfoliohwhich is a strongThere exists no portfolio hwhich is a strong arbitrage, that is h’X≥ 0 and p ≤ h < 0.33No arbitrageNo arbitrage3.3.No arbitrage No arbitrage There exists no strong arbitrage nor portfoliokwithk’X> 0 and 0≥p≤kSlide 03Slide 03--55nor portfolio kwith kX> 0 and 0 ≥p ≤kFin 501: Asset PricingFin 501: Asset PricingTh F f NTh F f NARBITR AGEARBITR AGEThree Forms of NoThree Forms of No--ARBITR AGEARBITR AGE• Law of one price is equivalent to every portfolio with zero payoff has zero yp p yprice.•No arbitrage=> no strong arbitrageNo arbitrage > no strong arbitrage No strong arbitrage => law of one price Slide 03Slide 03--66Fin 501: Asset PricingFin 501: Asset PricingOiPiiOiPiiOverview: PricingOverview: Pricing1.1. LOOP, No arbitrageLOOP, No arbitrage2.2. ForwardsForwards33Options: Parity relationshipOptions: Parity relationship3.3.Options: Parity relationshipOptions: Parity relationship4.4. No arbitrage and existence of state pricesNo arbitrage and existence of state prices5.5. Market completeness and uniqueness of state pricesMarket completeness and uniqueness of state pricespqppqp6.6. Pricing kernel Pricing kernel qq**7.7. Four pricing formulas:Four pricing formulas:state prices SDF EMM beta pricingstate prices SDF EMM beta pricingstate prices, SDF, EMM, beta pricingstate prices, SDF, EMM, beta pricing8.8. Recovering state prices from optionsRecovering state prices from optionsSlide 03Slide 03--77Fin 501: Asset PricingFin 501: Asset PricingAlt ti t b t kAlt ti t b t kAlternative ways to buy a stockAlternative ways to buy a stock• Four different payment and receipt timing combinations:–Outright purchase: ordinary transaction– Fully leveraged purchase: investor borrows the full amount– Prepaid forward contract: pay today, receive the share later–Forward contract: agree on price now, pay/receive later• Payments, receipts, and their timing:Slide 03Slide 03--88Fin 501: Asset PricingFin 501: Asset PricingPi i idf dPi i idf dPricing prepaid forwardsPricing prepaid forwards• If we can price the prepaid forward (FP), then we can calculate the price for a forward contract: FFt l fFPF= Future value of FP• Pricing by analogy–In the absence of dividends, the timing of delivery is irrelevantIn the absence of dividends, the timing of delivery is irrelevant– Price of the prepaid forward contract same as current stock price– FP0, T = S0 (where the asset is bought at t = 0, delivered at t = T)Slide 03Slide 03--99Fin 501: Asset PricingFin 501: Asset PricingPi i idf dPi i idf dPricing prepaid forwards Pricing prepaid forwards (cont.)(cont.)• Pricing by arbitragegy g– If at time t=0, the prepaid forward price somehow exceeded the stock price, i.e., FP0, T > S0 , an arbitrageur could do the following:gSlide 03Slide 03--1010Fin 501: Asset PricingFin 501: Asset PricingPi i idf dPi i idf dPricing prepaid forwards Pricing prepaid forwards (cont.)(cont.)• What if there are deterministic* dividends? Is FP0, T = S0 still valid?– No, because the holder of the forward will not receive dividends that will be paid to the holder of the stock Î FP0, T < S00,0FP0, T = S0 – PV(all dividends paid from t=0 to t=T)– For discrete dividends Dtiat times ti, i = 1,…., nii• The prepaid forward price: FP0, T = S0–Σni=1PV0, ti(Dti) (reinvest the dividend at risk-free rate)– For continuous dividends with an annualized yield δ• The prepaid forward price: FP0, T = S0 e−δT(reinvest the dividend in this index. One has to invest only S0 e−δTinitially)Slide 03Slide 03--1111* NB 1: if dividends are stochastic, we cannot apply the one period modelFin 501: Asset PricingFin 501: Asset PricingPi i idf dPi i idf dPricing prepaid forwards Pricing prepaid forwards (cont.)(cont.)• Example 5.1– XYZ stock costs $100 today and will pay a quarterly dividend of $1 25 If the risk-free rate is 10% compounded continuously how$1.25. If the risk-free rate is 10% compounded continuously, how much does a 1-year prepaid forward cost? – FP0, 1= $100 –Σ4i=1$1.25e−0.025i= $95.30• Example 5.2– The index is $125 and the dividend yield is 3% continuously compounded. How much does a 1-year prepaid forward cost? pypp– FP0,1= $125e−0.03= $121.31Slide 03Slide 03--1212Fin 501: Asset PricingFin 501: