Lecture 1/29/09 on closed economies 1. Closed economy is one that does no business with other nations (North Korea); has no interaction and is isolated with the rest of the world o Cannot trade (import or export), there is a productivity problem because they have no idea how other economies operate o They like to keep their economy, military, and even culture shrouded in secrecy o There is no answering what, how, and for whom they'll produce; people have no rights and must like what the government provides and produces, consumer goods are not part of those goods produced o Consumer choice is greatly restricted 2. y = GNP = C + IP + G; the Expenditure Side of GNP o GNP, or Gross National Product, is the sum of all goods and services produced by US citizens in one fiscal year o The higher the GNP, the lower the unemployment o Three components to GNP: C = Consumption, IP = Private Investment, G = Government expenditure 3. GDP is the GNP plus imports and exports 4. yd = Disposable Income = y - T o y = Gross Income, T = Tax 5. Consumption is a function of disposable income (yd), as shown in graph on right o Even with 0 disposable income, through Autonomous Consumption (C0) people can still consume via government coupons; people consume whatever they're told to consume o If yd goes down, C goes down; If T goes up, C will go down and GNP will drop 6. Regulation is always after the fact, and that's the problem; by then it is too late
2/3/09 Lecture on closed v open economies
1. The Expenditure Side of GNP is y = GNP = C + IP + G (Equation 1) o Where C = Consumption, IP = Private Investment, and G = Government expenditure 2. The Income Approach to the GNP is y = GNP = C + Sp + T (Equation 2) o Where C = Government Consumption, SP = Private Savings, and T = Taxes 3. The two approaches together equal equation 3 o C + IP + G = C + SP + T; (Equation 3) or o (SP - IP) = (G - T) (Equation 4) The equilibrium condition in the closed economy indicates that the budget account should be equal to the capital account If G > T, the government is in deficit; If G < T, the government is in surplus (G - T) is the Public Budget Constraint (SP - IP) is The Capital Account 4. Raising taxes is not necessarily the answer o Raising T will lower yd (Disposable Income) o Since C (consumption) is a function of yd, C will drop o If C drops, so too will GNP 5. We need Fiscal Conservatism, lowering G 6. Sp* = Ip + (G - T) (Equation 5) o The optimal Private Savings is equal to the Private Investment plus the Public Budget Constraint o If Ip is negative, and so too is (G - T); then Sp is negative 7. Ip* = Sp* - (G - T) (Equation 6)
o The optimal Private Investment is equal to the optimal Private Savings minus the Public
Budget Constraint 8. In an open economy, you deal with imports and exports o Certain imports we have a demand for characteristics rather than price
2/5/09 Lecture on open economy (closed economy review)
1. 2. 3. 4. 5. 6. 7. Equation 1) y = GNP = C + IP + G Expenditure Approach Equation 2) y = GNP = C + Sp + T Income Approach Equation 3) C + IP + G = C + SP + T Equation 4)(SP - IP) = (G - T) Equation 5) Sp* = Ip + (G - T) Equation 6) Ip* = Sp* - (G - T) In an open economy, you deal with imports and exports o For certain imports we have a Demand for Characteristics rather than price Buying a good based on its characteristics 8. With an open economy, we further modify equation 4 o (G - T) = (Sp - IP) + (X - M) (Equation 7) Where X = Exports, and M = Imports, when they are in equilibrium the result is 0 and you're back at Equation 4 (X - M) is the Current Account; if X > M you have a Trade Surplus, but if X < M you have a Trade Deficit The equilibrium condition in the open economy indicates that the budget account should be equal to the capital account plus the current account KNOW HOW TO DERIVE THE EQUILIBRIUM CONDITION IN A OPEN/CLOSED ECONOMY (FOR THE TEST, GET TO EQUATION 4 USING DIAGRAMS) 9. Where GNP is everything produced by US Citizens, we now have GDP in an open economy (GNP < GDP) o GDP is the sum of all goods and services produced by Americans domestically and abroad for a fiscal year 10. From Equation 7, we modify it further to solve for the optimal Private Savings o Sp* = IP + (G - T) - (X - M) (Equation 8) The optimal Private Savings is equal to the Optimal Private Investment plus the Public Budget Constraint minus the Current Account o IP* = SP* - (G - T) + (X - M) (Equation 9) The optimal Private Investment is equal to the Optimal Private Savings minus the Public Budget Constraint plus the Current Account 11. Tautological (Identity, holds all the time) Relationship with equations 4 and 7, they will always be true
2/10/09
1. Public Budget Constraint = Capital Account o (G - T) = (Sp - Ip) If Sp > Ip, the government becomes lenders; If Sp < Ip, the government becomes borrowers Paradox of Holding if the government has excess Savings, it's not good to hold Paradox of Thrift if the government has excess Investments, it's not good to just keep saving If G > T, the government is in deficit; If G < T, the government is in surplus 2. Progressive Taxation, t is a tax factor, the more income a person earns, the higher taxes they pay
o0<t<1 o T (Tax paid) = t x Y (the progressive tax factor times income) o Diagram 1
3. Government expenditures are illustrated in Diagram 2 4. yd = y - T; disposable income decreases as taxes increase o yd = y - T + R where R equals Transfer Payments (fixed) R comes from one of four different segments of the market R&D Housing Unemployment Business The government would take resources from one sector and spread them to the others
2/12/09
1. (G - T) = (Sp - Ip) o Public Budget Constraint = Capital Account o Equilibrium in a Closed Economy 2. Ip = f [i] o Private Investments are a function of the interest rate o As the interest rate goes down, Private Investments increase 3. Sp = f [y] o Private Savings are a function of income o As Income increases, so too does Private Savings o When income is 0, there is a negative savings because of credit card debts etc. o The relationship is possible by taking the average marginal propensity to save across the nation With $1, the average nation (taken by combining segments) will spend 40, 50, or 60 4. (G - T) = (Sp - Ip) + (X - M) o Public Budget Account = Capital Account plus Current (Trade) Account o Equilibrium in an Open Economy (by adding imports, M, and exports, X) 5. X = f [y] o Exports are a function of income (or GDP) o As Income increases, the country has more goods to export 6. M = f[y] o Imports are a function of income (or GDP) o As imports are driven down, the income or GDP increases 7. Demand for Characteristics is buying a good based on its characteristics and not necessarily price 8. Lifecycle Hypothesis suggests the in the beginning we have no demand for characteristics because we can't afford anything, in the middle is high, and in the end we lose our demand for characteristics again because we stop caring 9. To determine how strong our currency would be against another nation's, we would compare our variables in the equilibrium equation of an open economy
2/17/09 Lecture on exchange rates
1. Absolute Exchange Rate Concept (know for the exam) o Two countries: US and ROW (Rest of the World, say Canada); domestic and foreign economies respectively that trade with each other
o Domestic Price Level is PDUSA, Foreign Price Level is PFCDN o If the two countries trade with each other, can we determine the level of the exchange rate
between them? o Direct Exchange Rate = ~, where ~ means random, so that ~USA,CDN = PDUSA/PFCDN APPP - The exchange rate is nothing more than the ratio of the two price levels, Domestic over Foreign Exchange Rate is a function of the Price Level Example, if ~USA,CDN = PDUSA/PFCDN = 0.750; that means that $1USA = $0.75CDN o Indirect Exchange Rate = ~ = 1/ ~USA,CDN, how much a foreign currency unit will be worth in the US Example, ~ = 1/ ~USA,CDN, 1/0.75 = 1.333, that means that $1CDN = $1.333USA The Indirect Exchange Rate answers how much a foreign currency unit is worth in terms of the domestic currency unit 2. APPP = Absolute Purchasing Power Parity says the Exchange Rate is the Ratio of the Domestic Price Level over the Foreign Price Level o There may be problems with APPP The index only works when the goods are homogenous, or when countries produce the same goods If one country is missing a good, that good is said to be redundant The taxes must also match up The Demand for Characteristics may also interfere with the trading value between goods Politics may also play a role in determining trading power 3. To find the Price Levels, we use the Las Peyez's Index o PDUSA = PiUSQiUS = TRUS = GDPUS o PFCDN = PiCDNQiCDN = TRCDN = GDPCDN 4. Pure Barter Economy is when countries trade goods for goods o If USA has a comparative advantage, they can trade those goods for goods that they don't have a comparative advantage
2/19/09 Lecture on exchange rates cont.
1. APPP, or Absolute Purchasing Power Parity o ~USA,CDN = PDUSA/PFCDN o The direct exchange rate is the ratio of the Domestic price level over the Foreign Price Level o PDUSA = PiUSQiUS = TRUS = GDPUS o PFCDN = PiCDNQiCDN = TRCDN = GDPCDN o The Direct Exchange Rate answers how many foreign currency units a domestic currency unit can buy 2. The Law of One Price, LOP, says identical goods should command identical prices given the level of the Exchange Rate o PDUSA = ~ PFCDN o A Deviation in the LOP could be because of taxes (higher in CDN so prices will be higher), patterns (demand for characteristics) , or financial innovations ("know-how") Riskless Arbitrage happens when someone buys in US and sells in CDN to make a profit when there's a price deviation Doesn't exist, because with no risk, there should be no return The prices would level off, and with more people trying to exploit the arbitrage they in fact eliminate it
2/24/09
1. Exchange Rate Equations to know 1.APPP: ~USA,CDN = PDUSA/PFCDN 2.LOP: PDUSA = ~USA,CDN PFCDN o If ~USA,CDN = 1, then PDUSA = PFCDN; Domestic Price Level = Foreign Price Level and Exchange Rate is at Par o If ~USA,CDN > 1, then PDUSA > PFCDN o If ~USA,CDN < 1, then PDUSA < PFCDN 3) RPPP: ^USA,CDN = DUSA - FCDN o Relative Purchasing Power Parity, ^ is an estimated exchange rate (instead of ~ being random) to solve the APPP problem of goods needing to be homogenous o When ~ increases, the PDUSA increases, and also the DUSA increases o DUSA - FCDN is the Inflation Rate Differentiation o If DUSA > FCDN (higher inflation in USA), $CDN goes up and $USA goes down 2. 4) The Domestic Interest Rate: iDUSA = rDUSA + DUSA o The nominal interest rate equals the real interest rate (prime) plus inflation 5) The Foreign Interest Rate: iFCDN = rFCDN + FCDN o The prime rate, r, is the risk-free rate in the sense that it is the same in USA or CDN 6) International Fisherian Equation (IFE): (iDUSA - iFCDN) = (rDUSA - rFCDN) + ( DUSA - FCDN) o Equation 4 - Equation 5, and since rDUSA = rFCDN, they cancel and simplify the equation o iDUSA - iFCDN = DUSA - FCDN o Interest Rate Differentiation is equal to the Inflation Rate Differentiation 7) ^USA,CDN = iDUSA - iFCDN o Since ^USA,CDN = DUSA - FCDN, and iDUSA - iFCDN = DUSA - FCDN 2/26/09 Exchange rate equations 1. APPP: ~USA,CDN = PDUSA/PFCDN 2. LOP: PDUSA = ~USA,CDN PFCDN 3. RPPP: ^USA,CDN = DUSA - FCDN 4. The Domestic Interest Rate: iDUSA = rDUSA + DUSA 5. The Foreign Interest Rate: iFCDN = rFCDN + FCDN 6. International Fisherian Equation (IFE): (iDUSA - iFCDN) = (rDUSA - rFCDN) + ( DUSA - FCDN) 7. RPPP: ^USA,CDN = iDUSA - iFCDN Moving from the IFE iDUSA equation, - iFCDN = DUSA - FCDN, we can attempt to solve for the interest rates and inflation rates iDUSA = iFCDN + ( DUSA - FCDN) o As iFCDN increases, so does iDUSA, and the inflation rate differentiation will rise as well o As the Inflation Rate Differentiation increases, we are able to assume that DUSA > FCDN, and iDUSA will rise If USA doesn't increase the nominal interest after Canada does, then everyone will put their $D in Canada The Current Account, (SP - IP)CDN, would then become positive because of the Flight to Quality o The nominal domestic interest rate is now a function of the Canadian interest rate and foreign inflation rate differential iFCDN = iDUSA - ( DUSA - FCDN) o If iFCDN increases, so does iDUSA, the inflation rate differentiation will drop however because iFCDN > iDUSA
DUSA = FCDN + (iDUSA - iFCDN) o As the domestic inflation rate rises, so too does Canada's, and the interest rate differential
does as well
FCDN = DUSA - (iDUSA - iFCDN) o As the foreign inflation rate rises, so too does USA's, except the interest rate differential
goes down 3/03/09 Lecture on "where to invest" 1. APPP: ~USA,CDN = PDUSA/PFCDN 2. LOP: PDUSA = ~USA,CDN PFCDN 3. RPPP: ^USA,CDN = DUSA - FCDN 4. The Domestic Interest Rate: iDUSA = rDUSA + DUSA 5. The Foreign Interest Rate: iFCDN = rFCDN + FCDN 6. International Fisherian Equation (IFE): (iDUSA - iFCDN) = (rDUSA - rFCDN) + ( DUSA - FCDN) 7. RPPP: ^USA,CDN = iDUSA - iFCDN 8. iDUSA = iFCDN + ^USA,CDN 9. iFCDN = iDUSA - ^USA,CDN 10. DUSA = FCDN + ^USA,CDN 11. FCDN = DUSA - ^USA,CDN A Spot Rate is a direct exchange rate that prevails in the market today, 0 E( 1) = F1, the expected spot rate next year is the forward rate o If 0 > E( 1), there is a Forward Discount o If 0 = E( 1), there is a Forward Par o If 0 < E( 1), there is a Forward Premium At t = 0, the spot exchange rate ( 0) is 0.79 ($1 US = $0.79 CDN ) and your wealth (W) is $10,000 o Wt = W(1 + i)t In US, the i is 10%, so after one year you will have W(1+iDUSA)1, or 10,000(1 + . 10), $11,000 In CDN, you will have W(1 + iFCDN)1 x (1/ 0) [to convert to Canadian now] x E( 1) [to convert back to US, or repatriate, in one year] You would invest in whichever is higher between [W(1+iDUSA)1] and [W(1 + CDN 1 iF ) x (1/ 0) x E( 1)] To avoid arbitrage, we must say that investing in USA is equal to investing in CDN so that there's an equilibrium The Risk Neutrality Condition [W(1+iDUSA)1]= [W(1 + iFCDN)1 x (1/ 0) x E( 1)] Investors are indifferent of where to invest because the returns are identical in both countries Round Tripping is investing in CDN at 0 (spot) and repatriating to US at E( 1) (forward) Currency is a zero-sum game, one currency will gain as the other loses, but they will always balance out to 0 o A forward can be premium when the domestic dollar is 'gaining' (able to purchase more of the foreign currency that's 'losing' the same amount) o FP = [E( 1) - 0] / 0 o Indirect Exchange Rate ~ = 1/ ~, it's reciprocal from the direct exchange rate; whatever America gains, Canada loses 3/05/09 Lecture on exchange rate equations 1. Forward Rate: F1 = E( 1)
2. 3. 4. 5. 6. 7.
8. 9. 10.
11. 12. 13.
Wealth: Wt = W(1 + i)t Conversion: W(1 + iFCDN)1 x (1/ 0) Repatriate: W(1 + iFCDN)1 x (1/ 0) x E( 1) USA 1 Risk Neutrality Condition: [W(1+iD ) ]= [W(1 + iFCDN)1 x (1/ 0) x E( 1)] Forward Premium: FP = [E( 1) - 0] / 0 o Indirect FP is the reciprocal of each value Start with Risk Neutrality Equation and simplify it: o (1+iDUSA) = (1 + iFCDN)1 x [E( 1)/ 0] The Ratio of the Forward to the Spot Rate times the Foreign Interest Rate is equal to the Domestic Interest Rate o [E( 1)/ 0] = [(1+iDUSA) / (1 + iFCDN)] The Ratio of the Forward to the Spot Rate is equal to the Ratio of the Domestic to Foreign Interest Rates in a discrete time horizon o E( 1) = [(1+iDUSA) / (1 + iFCDN)] x 0 F1 = E( 1) = [(1+iDUSA) / (1 + iFCDN)]1 x 0 Fn = E( n) = [(1+iDUSA) / (1 + iFCDN)]n x 0 Sample Question 1: if iDUSA = 8%, iFCDN = 11%, and 0 = 0.93 o With W of 50,000 US, how much CDN do you have? o Calculate E( 1) o Is the US currency trading at a discount? o Show that trading currency is a zero-sum game Sample Question 2: Find F1 through F25 Sample Question 3: Plot F1 through F25 in Excel (graph) Sample Question 4: Show the FP for 25 periods and show it's a zero-sum game
3/10/09 Lecture on exchange rates and long memory effects 1. Forward Premium: FP = [E( 1) - 0] / 0 2. Risk Neutrality Condition: [W(1+iDUSA)1]= [W(1 + iFCDN)1 x (1/ 0) x E( 1)] 3. (1+iDUSA) = (1 + iFCDN)1 x [E( 1)/ 0] 4. [E( 1)/ 0] = [(1+iDUSA) / (1 + iFCDN)] 5. E( 1) = [(1+iDUSA) / (1 + iFCDN)] x 0 6. Long Memory Effects refers to the fact that the Forward Premium and the Forward Discount do not cancel out when the length is longer than 1 period 7. [E( 1)/ 0] - 1 = [(1+iDUSA) / (1 + iFCDN)] -1; subtract 1 in a common denominator o [E( 1) - 0] / 0 = (1+iDUSA - 1 - iFCDN) / (1 + iFCDN) o (iDUSA - iFCDN) / (1 + iFCDN) = [E( 1) - 0] / 0 8. Long Memory Effect: (iDUSA - iFCDN) / (1 + iFCDN) = [E( 1) - 0] / 0 o The ratio of the RPPP to the Foreign Interest rate is equal to the Forward Premium o Some of the difference goes to the RPPP (interest rate differential) or the Foreign interest rate because the governments are always playing with the interest rates 9. If we assume the foreigners are dropping the interest rate is going to 0, then we continue to modify the Long Memory Effect o (iDUSA - iFCDN) = [E( 1) - 0] / 0 o iDUSA = iFCDN + [E( 1) - 0 / 0] o iFCDN = iDUSA - [E( 1) - 0 / 0] 10. Question 1) Do all equations from the beginning 3/12/09
1. House selling example for test: where V0 = $275m, Appreciating at 35%, Collecting Rent of $25m, iDUS= 4%, DUS= 3%, FGR= 5%, 0 = 1.27, iFGR= 4-(3-5) = 6% o Scenario 1: Sell the house today for $275m, then convert to Euros 275m x (1/1.27) = 216,535,433.01Euro o Scenario 2: Sell the house in 1 year, + appreciation + rent, then convert 275m + (35%)(275m) + 25m = 396,250,000Euro E( 1) = (1+4%)/(1+6%) x (1/1.27) = 0.772545 396,250,000 x 0.772545 = 306,120,956.30Euro o Translation Gain = 306,120,956.30 - 216,535,433.01 = 89,585,523.29Euro Gain by selling one year later 3/19/09 Options 1. V0 = P0 x N o Our Value is the Price per share times the number of shares we have bought 2. An option is a contract between two parties at a specified strike price and a specified horizon o A Put Option assumes that the stock will go down, and a profit will ensue if the stock does in fact go down 4/02/09 1. A Swiss woman is worth 100 million Swiss Francs, and the spot exchange rate is 1 US = 1.12 SF; Scenario 1 is to sell the house now (t=0) and Scenario 2 is sell at t=1 o In either case: you sell, convert, invest o Scenario 1: Sell house now, receive $100 m SF. Converting at the spot ( x 1/1.12) = $89,285,714.29 US; then investing that for a year at 6% = $94,642,857.15 o Scenario 2: Rent the house for a year at 10 m SF, then sell at 100m, and gain the appreciation at 15% of 15m, giving a total of $125m SF Convert at the expected Forward rate (.8984) and that'll give you 112,300,000; the Translation Gain is 112,300,000 - 94,642,857.15, take Scenario 2 4/14/09 Stock Options 1. Stock Option Example Question (KNOW FOR FINAL) o Suppose there's a stock where PS = $118, where we call the stock the "underlying asset" o V0 = p0 x N = $118 x 100 shares = $11,800 o V1 = p1 x N = $70 x 100 shares = $7,000 o Gain/Loss = V1 - V0 = $7,000 - $11,800 = ($4,800), could we have avoided the loss? Yes, with an option 2. An option is a contract between two parties, one with the long position and one with short o Given the strike price, given the volatility, given the time horizon o Long position is where you stand to gain o Short position is where you stand to lose o You gain the difference between the terminal price and strike price, but lose nothing 3. Put option gives you the right to sell at a given price o Purchased when you think Ps is going down o The writer/seller of a put earns the premium the buyer pays, but loses as the buyer gains o The payoff, or PE = max {x - ST; 0} The superscript 'E' is for European option, where it can be exercised after 333 (3 mos, 3rd Fri, 3 PM) American option can be exercised immediately, before maturity 4. Call option gives you the right to buy at a given price o Purchased when you think Ps is going up
o The writer/seller of a call earns the premium the buyer pays, but loses as the buyer gains o The payoff, or CE = max {ST - x; o} ST > X, "in the money"; ST < X, "out of the money"
4/21/09 1. If S = 20, x = 20, rf = 12%, t = 3/12, 2 = 0.16, and = 0.16(1/2) = 0.4 o CE = SN(d1) - Xe-r t N (d2) {Equation 22} 2. Greek letter options: o If Stock Price, S, increases, it's (Delta) = CE/S o If Strike Price, X, increases, it's (Kappa) o If Risk-Free Rate, rf, increases, it's (Rho) o If Time Horizon, t, increases, it's (Theta) o If Risk, 2, increases, it's (Lambda) 3. If F = 50, X = 50, rf = 5%, t = 4/12, 2 = 0.16 ( = 0.4) o Max [F - X; 0] (intrinsic value = 0 , equilibrium)(d1 and d2 will be symmetrical) o Use Equations 14a-d 4. If 1E = $1.29, rDUSA = 5%, rFE = 4%, 2 = 0.49, X = Spot at 1 E = $1.29 4/23/09 1. Example Problem, Question 4 of Exam o Spot 1E = $1.34 o Exercise 1E = $1.29 o rDUSA = 6% o rfE = 4% o T - t = 4/12 o 2 = 0.49 CE = SN(d1) - Xe-(rd-rf)(T-t) N(d2) CE = 1.34 N(d1) - 1.29e-(6%-4%)(4/12) N(d2) d1 = ln(S/X) + [(rdUSA - rfE) + ( 2/2)](T-t) / (T-t) d1 = ln (1.34/1.29) + [(6%-4%) + (0.49/2)](4/12) / 0.7(4/12) d1 = 0.3127, or 0.31, or 0.3 + 0.01, Z table is 0.6217 (62.17% chance that price will be above quoted/actual price) d2 = d1 - (T-t) d2 = 0.3127 - 0.7(4/12) d2 = -0.0915, or -0.09, or -(0.0 + 0.09), Z table is 0.4641 (46.41% chance that price will be below quoted/actual price) CE = 1.34(0.6217) - 1.29e-(2%)(4/12)(0.4641) CE = 0.833 - .5947 = .2383, if this is lower than quoted price (overvalued), sell; if it's higher (undervalued), buy it PE = CE + Xe-(rd - rf)(T-t) - S PE = 0.2383 + 1.29e-(2%)(4/12) - 1.34 PE = 0.1797 2. Homework Problem: Change Spot to $1.44 and Exercise to $1.19, then to $1.66 and $1.38 respectively 5/05/09 Perfect Hedging Example 1. S1 = $100; 50% chance to be in Good State, S+ = $120; in Bad State, S- = $70; x = 100, CE = $10 o Multiplying Factor, u = S+ / S, 120/100 = 1.2 o Multiplying Factor, d = S- / S = 70/100 = 0.7
o So C+= CU = 10 x 1.2 = 12 (value increases by 12, 112-100) and C- = CD = 10 x 0.7 = 7
(value decreases by 30, 70-100) 2. Optimum Hedge Ratio, h* = (S+ - S-)/(C+ - C-) 5/07/09 CAPM 1. E(R~MNE) = RF + MNE(Rm - RF) o As the increases, so does the Market Risk Premium o The E(R~MNE) is the Theoretical Return 5/12/09 Perfect Hedge review 1. Question 1) Stock Options o Equation 22 o Solve an option, tell if it's overvalued or undervalued o Know the 4 diagrams 2. Question 2) Parameter Changes (Greek Symbols) o Know the table 3. Question 3) Oil "Futures" Options o Equations (14a-d) 4. Question 4) Currency Options o Equation 5. Question 5) Perfect Hedge, Good State/Bad State o S+ - hC+ = S- - hC h* = (S+ - S-)/(C+ - C-) = Perfect Hedge o If S = 120, X = 120, u = 1.6, d = 0.9, rf = 5%, CE = 12 S+ = Su = 120 (1.6) = $192 S- = Sd = 120 (0.9) = $108 C+ = Cu = 12 (1.6) = 19.20 C- = Cd = 12 (0.9) = $10.8 = 0 because you would not exercise the call if the value is less than C h* = [(120x1.6)-(120x0.9)]/[(12x1.6)-(12x0.9)] = (192-108)/(19.2-10.8) = 84/8.4 = 10:1 o CValue = (1+rf-d)/(u-d) * (C+/1+r) If the Value is less than the price, sell the option because it's overvalued If the Value is more than the price, buy the option because it's undervalued