Asset pricing Model1

Asset pricing Model: CAPM : equilibrium of all individual investor: its assumptions: individual investors are price takers, rational mean-variance optimisers, single period, investments are limited to traded financial assets, no tax & transaction cost,

info are free and easy to get & homogeneous expectations. E r i=r f+β(E r m?r f) reward to risk ratio: (E

r m?r f)

?m2

=

market risk premium market variance =E r p=w k E r k

k

,βp=w kβk

k, SML: slope= E r m?r f=

market risk premium,SML=E r i=r f+β(E r m?r f),βi= cov r j,r m

?m2

, in market portfolio, β=1, CAPM establishes relationship b/w risk and E(r) for securities, att he equilibrium, that investors form portfolio consisting of riskless assets and market portfolio, combination vary from investor to investor depending upon individual presference for expected risk and return., at equilibrium .E r i=r f+β(E r m?r f). Zero β model, it drops assumption of borrowing at risk free rate: E r i=

r f+(E r m?r f) cov r j,r m

?m2

,Single index Model:r i=αi+r f+βr m?r f+e i Liquidity and the CAPM :Acharya

and Pedersen :E r i=kE Cr f+λ(β+βL1(lqiuidity betas)?βL2?βL3)

E r i expect cost of illiquidity kE Cr f adjustmentfor average olding period over all security,λ:market risk premi average market illiquidity cost,β:measure of systematic market risk,βL1,,βL2,βL3:lqiuidity betas, In real world, CAPM doesn?t capture risk adequately.APT : assumption: not strong as CAPM, single period investment horizon, with no tax, investor can freely borrow or lend at r f, investor can select portfolio based on mean and variance of return, , capital markets are pferfectly competitive and no transaction cost or tax, stochastic process generating set returns can be expressed as a linear function of set of K risk factors or indexes. It assumes that in equilibrium, the return on a zero-investment, zero-systematic-risk portfolio is zero when the unique effects are diversified away, : E(r j) = λ0 + b j1λ1 + b j2λ2 + b j3λ3 + b j4λ4 + ... + b jnλn E(r j) = the

asset's expected rate of return, λ0= the risk-free rate/ expected return on asset with zero systematic risk ,b j = the sensitivity of the asset's return to the particular factor, λ1 = the risk premium associated with the particular factor One Factor Model: linear fuction, Two Factor Model: will calculate each factor separated, e.g: change in inflation is 1%, growth in real GNP is 2%, 3% return on zero-systematic risk asset, b x1 is portfolio x to changes in the rate of inflation= 0.5, b y1=2, b x2 is portfolio x changes in the growth rate of real GNP=1.5, b y2=1.75, E(x)=0.03+0.01(0.5)+(0.02)(1.5)=6.5%, E(y)=8.5%

APT vs. CAPM:? APT applies to well diversified

portfolios and not necessarily to individual stocks ? With

APT it is possible for some individual stocks to be

mispriced inequilibrium- not lie on the SML ? APT is

more general, gets to an expected return and beta

relationship without the assumption of the market portfolio

? APT can be extended to multifactor models ? Some

studies suggest that, APT explains stock returns better;

other studies hardly find any difference between the two.

Implications:? While the CAPM probably provides the best available estimate of risk for most corporate investment decision, managers must recognise that their stock prices ma y fluctuate more than one factor. ? The market is usually smarter than the individual. Hence managers should weight the evidence of the market over the evidence of experts. ? Markets function well

when participants pursue diverse decision rules and their errors are independent. Markets, however, can become very fragile when participants display herd-like behaviour, imitating one another. ? It may be futile to identify the cause of a crash or boom because in a non-linear system small things can cause large scale changes.

Asset Allocation expected return: E r=P S r(S)

s p(s): % of a state, r(s) : return if a state occurs, Variance/ dispersion of

return : ?2=P(S)[r s?E9r)]2

s , s.d: ?, cov of two random return: cov(r1,r2)=P s r1s?E r1[r2s?

n

i=1

E r2]Diversification: Pooling of uncorrelated events, as the No. of asset in the portfolio increases, the s.d falls, Most people are risk averse, principle of insurance is based on concept of “ diversification”.Role of Uncorrelated Securities: If two securities have low correlation, the interactive risk will be small, If two securities are uncorrelated, the interactive risk drops out ,If two securities are negatively correlated, interactive risk would be negative and would reduce total risk. Concept of Dominacne and Efficiency: A portfolio dominates all others if: For its level of expected return, there is no other portfolio with less risk. For its level of risk, there is no other portfolio with a higher expected return. All rational investors will clearly prefer one alternative. Those portfolios that are not dominated constitute the efficient frontier. Mean-Variance Efficient Frontier: Investors : prefer a higher expected return to lower returns r A≥ r B , Dislike risk s2(r A) ≤ s2(r B) , or σ(r A) ≤ σ(r B) Covariance and correlation : Cov(r A, r B), r12, Objective: Minimise portfolio variance. Formula for portfolio variance: s2p= ∑ w i2 s i2+ ∑ ∑ w i w j s, Formula with the following assumptions imposed: s2p = (1/n) s2 + ((n-1)/n) rs2, If n is large (1/n) is small and ((n-1)/n) is close to 1. Hence : s2p rs2, Portfolio risk is …covariance risk?. Assumptions: Assets have same variance : s i2 = s2, covariance s ij = rs2, and Invest equally in each asset (i.e. 1/n) , Two-Fund Theorem: aw1 + (1-a)w2 , If there are two efficient portfolios, then any other efficient portfolio can be constructed using those two. One Fund Theorem: There is a single fund M of risky assets, so that any efficient portfolio can be constructed as a combination of this fund and the risk free rate. Expected return:

E(r C ) = (1 – w)r f + wE(r p), Riskiness: σ2C = w2σ2p or σC= wσp, maximise utility function of investing to get optimum portfolio weights: W=[E(r p)-r f]/(A σ2p), maximize: U= E(r C)-0.5A σ2c, Capital Market Line and the Market Portfolio: When the security universe includes all possible investments, point M is the market portfolio, The tangent line passing from the risk-free rate through point M is the CML., CML constains: It contains every risky asset in the proportion of its market value to the aggregate market value of all assets, and it is the only risky asset risk-averse investors will hold .

State Preference Theory and Expected Utility Von Neumann and Morgenstern?s (vNM) have presented a model that allows the use of an expected utility under some conditions. This paradox arises because individuals do not make decisions based

purely on wealth, but rather on the utility of their expected wealth. E(x)= a i x i n i =1 Exp Utility = E[U(W)]Difference between maximising wealth vs. maximising utility? Utility: An economic term referring to the total satisfaction received from

consuming a good or service. Wealth: A measure of the value of all of the assets of worth owned by a person, community, company or country. Fair gamble: it means: cost to play the gambles= expect value of the outcome., unfair gambles: means outcome less the cost to play. St. Petersburg Paradox: if the E(X)= 1=∞n i =1 one would pay a “ fair” price to play a fair game., paradox arises because individuals do not make decisions based purely on wealth, but rather on the utility of their

expected wealth, Marginal utility of wealth declines as we get more wealth, expected value of a game is finite. Expected Utility Theory: Five Axioms of utility and one assumption provide set of conditions for consistent and rational behaviour. Axioms mean all individuals make rational (reasonable) decisions and they are made among thousands of alternatives that they (hard) have. Extra assumption: People are greedy, prefer more wealth than less. Five axioms and this assumption is all we need to develop an expected utility theorem and apply the rule of max E[U(W)] =max ΣiαiU(Wi ) Utility Functions:1. Order preserving: if U(x) > U(y) => x > y2. To rank combinations of risky alternatives: U[G(x,y:α)] = αU(x) + (1‐α) U(y)

U[E(W)] Vs. E[U(W)]: U[(E(W)] is the utility associated with the known level of expected wealth (although there is uncertainty around what the level of wealth will be, there is no such uncertainty about its expected value). E[U(W)] is the expected utility of wealth, utility associated with level of wealth that may be obtained. The relationship between U[E(W)] and E[U(W)] is very important. Certainty equivalent and risk premium CE:A guaranteed return that someone would accept, rather than taking a chance on a higher and uncertain, return. If you've ever thought about leaving your job to start your own business, and potentially make more money, but decided to stay and continue drawing a salary instead, then the amount of your salary is your certainty equivalent. You might need to come up with a business idea with a higher potential payoff to be convinced to leave the security of your existing job. Risk Premium :The return in excess of the risk-free rate of return that an investment is expected to yield or the amount would be paid to avoid risk( gamble). The Expected wealth is 10 ? The E[U(W)] = 1.97 ?

Ln(CE) = 1.97? Exp(ln(CE)) = CE = 7.17 RP:10-7.17=2.83 In general, if U[E(W)] > E[U(W)] then risk averse individual (RP > 0),if U[E(W)] = E[U(W)] then risk neutral individual (RP = 0) if U[E(W)] < E[U(W)] then risk loving individual (RP < 0), risk aversion occurs when the utility function is strictly concave, risk neutrality occurs when the utility function is linear, risk loving occurs when the utility function is convex RP (π)= an individual's expected wealth, given he plays the gamble- level of wealth the individual would accept with certainty if the gamble were removed. The Arrow ‐Pratt Premium :The risk premium can be defined as the value that satisfies the following equation: E[U(W + Z)] = U[ W + E(Z) ‐ ( W , Z)] (*) on the LHS: expected utility of the current level of wealth given by gamble, RHS: utility of the current level of wealth plus the expected value of the gamble less the risk premium. Use a Taylor series expansion to (*) to derive an expression for the risk premium (W,Z) Absolute Risk Aversion (ARA) ? Arrow ‐Pratt Measure of a Local Risk Premium (derived from (*)above) π=0.5 σ2z [(-

U "(W )U ′(W )] ARA=[(-U "(W )U ′(W )] , We define the coefficient of Absolute Risk Aversion (ARA) as a local measure of the degree that an agents dislike risk. Its useful properties: ? It is invariant under an affine transformation or if U and V are two vNM utility functions then ARA of U and V are the same. ? We can use the ARA then for interpersonal comparisons. ?

Suppose Jim and Jack have the same wealth but different preferences. Jim?s utility function V is more concave than Jack?s, so Jim always demands a higher risk premium for a given level of risk. ? Here then ARA for V (w) is larger than the ARA for U (w). CARA ‐DARA ‐IARA ? A utility function U exhibits constant absolute risk aversion (CARA) if ARA does not depend on wealth or ARA (w) = 0. ? U exhibits decreasing absolute risk aversion or DARA if richer people are less absolutely risk averse than poorer ones or ARA?(w)<0. ? U exhibits increasing absolute risk aversion or IARA if ARA?(w) > 0. What do these mean in finanical terms? Consider a simple binary lottery ‐ you cannot win anything but can loose $10 with 50% probability. ? CARA millionaire requires the same payment to enter this lottery as a beggar would. ? IARA millionaire requires a larger payment than the beggar! ? DARA M illionaire takes it for a smaller payment than a beggar ‐ most realistic case. Relative Risk Aversion ? Coefficient of RRA : RRA=(-W ?[U "(W )U ′(W )] , ? If R is independent of wealth, we have constant RRA (CRRA ). Empirical Evidence ? Empirical evidence (Friend and Blume (1975)) indicates that individuals have decreasing ARA and constant RRA = 2 ? Power Utility Function U(W)=‐W ‐1, U'(W) = W ‐2 > 0, U"(W) = ‐2W ‐3 < 0, ARA = 2/W => dARA/dW < 0, RRA = 2W/W = 2 => dRRA/dW = 017 Differences in Two Approaches ? Markowitz premium is an exact measures whereas the AP measure is approximate ? AP assumes symmetry payoffs across good or bad states, as well as relatively small payoff changes. ? It is n ot always easy or even possible to invert a utility function, in w hich case it is easier to calculate the AP measure ? The accuracy of the AP measures decreases in the size of the gamble and its asymmetry. Mean &Variance : they?re the choice variables investors

concern about in order to max E[U(w)]. Mean‐Variance Utility ? Many researchers (Markowitz ,Sharpe etc.) used mean variance utility functions. But is it compatible with vNM theory?? The answer is yes, approximately, under some conditions.? What are these conditions?– U is quadratic e.g. when U =aw‐bw2, belongs to the linear distribution class, risk is small, asset returns are joint normal.? We will not go into the details of these issues here.? The most relevant justification for

mean‐variance is probably the case of small risks. Then, we may use a second order Taylor approximation of the vNM utility function.? In other words, any risk‐averse vNM utility function can locally be approximated with a quadratic function.? The expectation of a quadratic utility function can be evaluated with the mean and variance. Stochastic Dominance & Utility? Stochastic “denotes the process of selecting from among a group of theoretically possible alternatives those elements or fact ors whose combination will most closely approximate a desired result” ? The most general efficien cy criteria relies only on the assumption that utility is non-decreasing in return, or investors prefers more of at least one good to less. ? Portfolios are efficient if they are not dominated by other portfolios, are inefficient if at least one other port folio dominates them. ? Rational investors prefer efficient investments. Concept of Dominance & Efficiency: a portfolio dominates all others if: for its lvl of expected return, there?s no other portfolio with less risk., for its lvl of risk, no others has higher expected return, all rational investor will clearly prefer one alternative, and those portfolios that are not dominated constitute the efficient frontier. Futures pricing & hedging: S0: Spot price today F0: Futures or forward price today T: Time until delivery date r: Risk-free interest rate for maturity T ? Long & Short Hedges: A long futures (short) hedge is appropriate when you know you will purchase (sell) an asset in the future and want to lock in the price. Short Selling: involes sellin g securities you don?t own.-your broker borrows the securities from another client and sells them in the market in the usual way.- at some stage you must buy the securities to be replaced in the account of the client. Basis Risk: Basis is usually defined as the spot price-the futures price. Forward vs Future Price: when the maturity & asset price are the same, forward & futures prices are usually assumed to be equal (Euro $ futures are an exception), when interest rates are uncertain they are, in theory, slightly different: a strong +ve correlation b/w interest rates and the asset price implies the future price is slightly higher than the forward price, a strong –ve correlation implies the reverse. Forward Price: F0=S0e rT,Assuption: theoretical no-arbitrage relationship b/w F0 and S0 must hold., when investment gives a known income: F0=(S0?I)e rT, if income is unknow F0=S0e(r?q)T where q is the average exp. yield during the life of the contract (continuously compounding)/ average dividend yield. Forward contract: it is worth zero( except for bid-offer spread effects) when is 1st negotiated. Later it may have a +ve/-ve value, long forward contract:

F0?K e?rT, short forward contract is:K?F0e?rT Hedging? Hedging: eliminating unwanted risk ?Arguments in favor: Companies should focus on their main business and minimize risks arising from interest rates, exchange rates, etc. ? Arguments against: Shareholders are usually well diversified & can make their own hedging decisions.? Ex plaining a situation where there is a loss on the hedge and a gain on the underlying can be difficult. Long Hedge for Purchasing of an Asset:? Define F1 : Futures price at time hedge is set up, F2 : Futures price at time asset is purchased, S2 : Asset price at time of purchase, b2 : Basis at time of purchase, Cost of asset: S2, Gain on Futures: F2-F1, net amount pai: S2 -( F2-F1)=F1 + b2 , Short Hedge for Sale of an Asset: Define: F1 : Futures price at time hedge is set up, F2 : Futures price at time asset is sold,S2 : Asset price at time of sale, b2 : Basis at time of sale, Price of asset: S2, Gain on Futures:F1?F2, net amount pai: S2 +( F1?F2)=F1 + b2 , Choice of Contract ? Choose a delivery month that is as close as possible to, but later than, the end of the life of the hedge? When there is no futures contract on the asset being hedged, choose the contract whose futures price is most highly correlated with the asset price. This is known as cross hedging. Optimal Hedge Ratio: ?=ρ?S

?F

?S is the standard deviation of ΔS, the change in thespot price during the hedging period, ?F is the standard deviation of ΔF, the change in thefutures price during the hedging period, ρ is the coefficient of corrrlation b/w ΔS & ΔF, Optimal No. of Contract :Q A Size of position being hedged (units), Q F Size of one futures contract (units), V A= Value of position being hedged (=spot price time QA), V F =Value of one futures contract (=futures price times QF), Tailing the hedge Adjusting the number of futures so the present market exposure of

the hedge offsets the underlying exposure. Optimal No. of contracts if no tailing adjustment: = ?Qa

Qf , Optimal after tailing: = ?Va

Vf

,

Imperfect Hedging (Cross hedging) asset being hedged and the asset underlying the futures contract are not identical, the futures contract usd to hedge is closed out before the delivery date. Hedging a stock portfolio using index futures: βV a

V F

ADV of Hedging:want to be out of the mkt for a while avoids the costs of selling and repurchasing the portfolio.Stack and Roll ? In its simplest form, a stack‐and‐roll hedge involves repeatedly buying a bundle, or “stack,” of short dated futures or forward contracts to hedge a longer‐term exposure.? Initially we enter into futures contracts to hedge exposures up to a time horizon? Just before maturity we close them out an replace them with new contract reflecting the new exposure Liquidity Issues? In any hedging situation there is a danger that losses will be realized on the hedge while the gains on the underlying exposure are unrealized ? This can create liquidity problems ? One example is Metallgesellschaft which sold long term fixed‐price contracts on heating oil and gasoline and hedged using stack and roll.Index Arbitrage? When F0 > S0e(r‐q)T an arbitrageur buys the stocks underlying the index and sells futures. ? When F0 < S0e(r‐q)T an arbitrageur buys futures and shorts or sells underlying stocks.? Index arbitrage involves simultaneous trades in futures and many different stocks. ? Very often a computer is used. ? Occasionally simultaneous trades are not possible and the theoretical no‐arbitrage relationship between F0 and S0 does not hold. Futures and Forward on Currencies: F0=S0e(r?r f)T, Consumption Assets: storage is –ve income: F0≤S0e(r?u)T where u is the storage cost per unit time as a % of the asset value, Alternatively: F0≤(S0+U)e(rT where U is the present vaulue of the storage costs. Cost of Carry:The cost of carry, c, is the storage cost plus the interest costs less the income earned ? For an investment asset F0=S0e cT, for a consumption asset:F0≤S0e cT, convenience yield on the consumption asset y: F0=

S0e(c?y)T Fu tures Prices & Expected Future Spot Prices? Suppose k is the expected return required by investors in an asset? We can invest F0e–r T at the risk‐free rate and enter into a long futures contract to create a cash inflow of ST at maturity, This shows: F0=ES t e r?k T,Futures Prices & Future Spot Prices No Systematic Risk k = r F0 = E(ST), +ve Systematic Risk k > r

F0 < E(ST), -ve Systematic Risk k < r F0 > E(ST) ?Positive systematic risk: stock indices ? Negative systematic risk: gold (at least for some periods)Duration Matching? Duration is a measure by which a bank?s asset or liability portfolio increases or decreases as the result of a 1% change in interest rate. ? Duration ma tching involves hedging against interest rate risk by

matching the durations of assets and liabilities ? It provides protection against small parallel shifts in the zero curve.

Duration‐Based Hedge Ratio V F:Contract price for interest rate futures, D F: Duration of asset underlying futures at maturity,P: Value of portfolio being hedged,D p Duration of portfolio at hedge maturity, P D p/V F D F, Limitations of Duration Based

Hedging? Assumes that only parallel shift in yield curve take place? Assumes that yield curve changes are small? When

T‐Bond futures is used assumes there will be no change in the cheapest‐to‐deliver bond.

Swaps:Definition: A swap is an agreement to exchange cash flows at specified future times according to certain specified rules. The use of an interest rate swap: ? Converting a liability from fixed rate to floating rate or floating rate to fixed rate? Converting

an investment from fixed rate to floating rate or floating rate to fixed rate.Day Count? A day count convention is specified for fixed and floating payment. Confirmations: ? it is specify the terms of a transaction? The Internat ional Swaps and Derivatives has developed Master Agreements that can be used to cover all agreements between two counterparties ? Governments now

require central clearing to be used for most standardised derivatives. e.g on borrowing/ loan: Aand B offered $20mill 5yr loan: at following: fixed: 5% (A), 6.4% (B), floating, LIBOR+0.1%(A), LIBOR=0.6%(B) and A requires, floating, B requires fix, F.I charges 0.1%

benefits: 1.4%-0.5%-0.1%=0.8%, each receive 0.4% benefits for A, the floating: LIBOR-0.3% or reciving extra 0.3% on fix, B would pay

5.4%=

6.4%-0.6%-0.4% e.g2: if XY< offer 5million on 10-investment: Fixed: 8%(X), 8.8% (Y), both floating is LIBOR:

F.I charge 0.2%Credit Risk: A swap is worth zero to a company initially, at future time its value is liable to be either +ve or –ve, the company has CR exposure only when its value is +ve. Credit Risk vs. Market Risk: CR arises from the % of a default by the counterparty, MR arises from movements in the market variables such as interest rates and exchange rates, A complication is that the CR in a swap is contingent on the values of market variable. , A company?s position in a swap has CR only when the value of the swap to the company is +ve. expected loss from a default on swap vs. the expected loss from a default on a loan. On swap it loss less than a loan, in an interst rate swap, financial institiutiion?s exposure depends on the difference b/w a fixed-rate of interest and a floating –rate of interest, it has no exposure to the notional principal, where in Loan the whole principal can be lost. Bank finds that its assets are not matched with its liablilites, it?s taking floating-rate deposits and making fixed-rate loans, how can swaps be used to offset bank. The bank is paying a floating-rate on the deposits and receiving a fixed-rate on the loans, it can offset its risk by entering into interest rate swaps in which it contracts to pay fixed and receiving floating. How to value a swap that is the exchange of a floating rate in one currency for a fixed rate in another currency? The floating payments can be valued in currency A by(i) assuming that the forward rates are realized, and (ii) discounting the resulting C.F at appropriate currency A discount rates, suppose that the value is Va, the fixed payments can be valued in currency B by discounting them at the appropriate currency B discount rates, suppose that the value is Vb, IF Q is the current exchange rate, the value of the swap in currency A is Va-QVb, or otherwise. Similarity on Currency Swaps and Interst rate swaps: both can be valued either as the difference b/w 2 bonds or as a portfolio of forward contracts.Valuation of an Interest Rate Swap? Initially interest rate swaps worth close to zero? At later times they can be valued as the difference between the value of a fixed‐rate bond and the value of a floating‐rate bond.? The fixed rate bond is valued in the usual way? Floating rate bond is valued by noting that it is at par immediately after the next payment date? Alternatively, they can be valued as a portfolio of forward rate agreements (FRAs) Mechanics of Options Markets:Option Types: 1) A call is an option to buy, 2)A put is an option to sell, 3)A European option can be exercised only at the end of its life, 4)An American option can be exercised at any time,Option Positions: 1)Long call 2)Long put 3)Short call 4)Short put. Notation: c: European call option price, p, European put price. C: American call option price, P : American put price, K strike price, T; life of option, s: volatility of stock price, s0: stock price today, s t: stock price at option maturity, D: PV of dividends paid during life of option, r: Risk-free rate for maturity T with cont. comp. Dividends & Stock Splits: N option with strike price of K,,: no adjustments are made to the option terms for cash dividends, where there is n for m stock split, the strike price is reduce to mK/n, no. of option is increase to nN/m, and stock dividends are handled similarly to stock splits. Margins: they are reuired when options are sold, when a naked option is written the margin is greater of: a total of 100% of the proceeds of the sale+ 20% of the underlying share price-the amount (if any) by which the option is out of the money, or A total of 100% of the proceeds of the sale + 10% of the underlying share price (call) or exercise price( put). Market Makers: Most exchanges use market makers to facilitate options trading, a MM quotes both bid and ask prices when requested, the MM d oesn?t know whether the individual requesting the quotes wants to buy or sell, the trading system of many exchanges can be characterised as a mixture of an order driven and quote driven markets. Warrants: it?s options that are issued by a corp or F.I, the No. of warrants outstanding is determined by the size of the original issue and changes only when they are exercised or when they expire. The issuer settles up with the holder when a warrant is exercised, when call warrants are issued by a corp on its own stock, exercise will usually lead to new treasury stock being issued. Employee Stock Option: they are a form of

remuneration issued by a company to its executives, they are usually at the money when issued, when options are exercised the company issues more stock and sells it to the option holder for the strike price, expensed on the income statement. Convertible Bonds, they are regular bonds that can be exchanged for equity at certain times in the future according to a predetermined

exchange ratio, usually a convertible is callable,the call provision is way in which the issuer can force conversion at a time earlier than the holder might otherwise choose. Lower Bound for Eup Call& Put option price: No: dividend: c≥S0?

Ke?rt,p≥Ke?rt?S0 with dividend: c≥S0?D?Ke?rt,p≥D+Ke?rt?S0

Early Exercise: Calls on a Non-dividend Paying stock: Since c0+Ke?rT=p0+S0→c≥S0?Ke?rt, Also, c0≥C0, therefore: C≥S0?Ke?rt, it means American call must worth more than its instrinsic value, a call options market value consists of C= instrinsic value+ time value. C=S?K+K?Ke?rT+P Puts on Non-Divd: P=K?S+K?Ke?rT?K+ P, when a put option is sufficiently deep in the money, Ke?rT?K will be –vely large relative to the value of call, and time

value of European put option will be –ve, In tht case, the European put will sell for less than its intrinsic value, however, its American couterpart can?t sell for less than its intrinsic value, which implies that an Ame rican put option can be worth more than an otherwise identical european option.