ArbitrageFree Pricing Models


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1 ArbitrageFree Pricing Models Leonid Kogan MIT, Sloan , Fall 2010 c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
2 Outline 1 Introduction 2 Arbitrage and SPD 3 Factor Pricing Models 4 RiskNeutral Pricing 5 Option Pricing 6 Futures c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
3 Option Pricing by Replication The original approach to option pricing, going back to Black, Scholes, and Merton, is to use a replication argument together with the Law of One Price. Consider a binomial model for the stock price Payoff of any option on the stock can be replicated by dynamic trading in the stock and the bond, thus there is a unique arbitragefree option valuation. Problem solved? c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
4 Drawbacks of the Binomial Model The binomial model (and its variants) has a few issues. If the binomial depiction of market dynamics was accurate, all options would be redundant instruments. Is that realistic? Empirically, the model has problems: one should be able to replicate option payoffs perfectly in theory, that does not happen in reality. Why build models like the binomial model? Convenience. Unique option price by replication is a very appealing feature. How can one make the model more realistic, taking into account lack of perfect replication? c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
5 Arbitrage and Option Pricing PRICING REDUNDANT OPTIONS STOCK BOND ARBITRAGE ARGUMENT OPTION PRICE PRICING NON REDUNDANT OPTIONS ARBITRAGE FREE MODEL: STOCK BOND OPTION EMPIRICAL ESTIMATION OPTION PRICE c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
6 Arbitrage and Option Pricing Take an alternative approach to option pricing. Even when options cannot be replicated (options are not redundant), there should be no arbitrage in the market. The problem with nonredundant options is that there may be more than one value of the option price today consistent with no arbitrage. Change the objective: construct a tractable joint model of the primitive assets (stock, bond, etc.) and the options, which is 1 2 Free of arbitrage; Conforms to empirical observations. When options are redundant, no need to look at option price data: there is a unique option price consistent with no arbitrage. When options are nonredundant, there may be many arbitragefree option prices at each point in time, so we need to rely on historical option price data to help select among them. We know how to estimate dynamic models (MLE, QMLE, etc.). Need to learn how to build tractable arbitragefree models. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
7 Absence of Arbitrage Consider a finitehorizon discretetime economy, time = {0,..., T }. Assume a finite number of possible states of nature, s = 1,..., N t = 0 t = 1 t = 2 s 1 s2 s 3 s 4 Definition (Arbitrage) Arbitrage is a feasible cash flow (generated by a trading strategy) which is nonnegative in every state and positive with nonzero probability. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
8 Absence of Arbitrage Definition (Arbitrage) Arbitrage is a feasible cash flow (generated by a trading strategy) which is nonnegative in every state and positive with nonzero probability. We often describe arbitrage as a strategy with no initial investment, no risk of a loss, and positive expected profit. It s a special case of the above definition. Absence of arbitrage implies the Law of One Price: two assets with the same payoff must have the same market price. Absence of arbitrage may be a very weak requirement in some settings and quite strong in others: Equities: few securities, many states. Easy to avoid arbitrage. Fixed income: many securities, few states. Not easy to avoid arbitrage. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
9 Absence of Arbitrage Fundamental Theorem of Asset Pricing (FTAP) Proposition (FTAP) Absence of arbitrage is equivalent to existence of a positive stochastic process {π t (s) > 0} such that for any asset with price P t (P T = 0) and cash flow D t, P t (s) = E t or, in return form, πt+1 (s) E t π t (s) R t+1(s) T u=t+1 π u (s) π t (s) D u(s) = 1, R t+1 (s) = P t+1(s) + D t+1 (s) P t (s) Stochastic process π t (s) is also called the stateprice density (SPD). FTAP implies the Law of One Price. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
10 Absence of Arbitrage From SPD to No Arbitrage Prove one direction (the easier one): if SPD exists, there can be no arbitrage. Let W t denote the portfolio value, (θ 1,..., θ N ) are holdings of risky assets 1,..., N. Manage the portfolio between t = 0 and t = T. An arbitrage is a strategy such that W 0 0 while W T 0, W T = 0. The trading strategy is selffinancing if it does not generate any cash in or outflows except for time 0 and T. Formal selffinancing condition i i i i i W t+1 = θ t+1 P t+1 = θ t (P t+1 + D t+1 ) i i c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
11 Introduction Arbitrage and SPD Factor Pricing Models RiskNeutral Pricing Option Pricing Futures Absence of Arbitrage From SPD to No Arbitrage Show that selffinancing implies that π t W t = E t [π t+1 W t+1 ]: i i self financing i i i E t [π t+1 W t+1 ] = E t [π t+1 θ t+1 P t+1 ] = E t π t+1 θ t (P t+1 + D t+1 ) FTAP i i = θ t π t P t = π t W t i i Start at 0 and iterate forward π 0 W 0 = E 0 [π 1 W 1 ] = E 0 [E 1 [π 2 W 2 ]] = E 0 [π 2 W 2 ] =... = E 0 [π T W T ] Given that the SPD is positive, it is impossible to have W 0 0 while W T 0, W T = 0. Thus, there can be no arbitrage. i c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
12 Absence of Arbitrage Fundamental Theorem of Asset Pricing (FTAP) When there exists a full set of statecontingent claims (markets are complete), there is a unique SPD consistent with absence of arbitrage: π t (s) is the price of a statecontingent claim paying $1 in state s at time t, normalized by the probability of that state, p t. The reverse is also true: if there exists only one SPD, all options are redundant. When there are fewer assets than states of nature, there can be many SPDs consistent with no arbitrage. FTAP says that if there is no arbitrage, there must be at least one way to introduce a consistent system of positive state prices. We drop explicit statedependence and write π t instead of π t (s). c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
13 Example: Binomial Tree SPD Options are redundant: any payoff can be replicated by dynamic trading. FTAP implies that p π t+1(u) π t+1 (d) S (1 + r) + (1 p) (1 + r) = 1 t+1 = us, π t+1 (u) π t π t p π t+1 (u) π t+1 (d) p u + (1 p) d = 1 S t, π t πt π t SPD is unique up to normalization π 0 = 1: (1 p) S t+1 = ds, π t+1 (d) π t+1 (u) r d = π t p(1 + r) u d π t+1 (d) 1 u (1 + r) = π t (1 p)(1 + r) u d c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
14 ArbitrageFree Models using SPD Discounted Cash Flow Model (DCF) Algorithm: A DCF Model 1 Specify the process for cash flows, D t. 2 Specify the SPD, π t. 3 Derive the asset price process as P t = E t T u=t+1 π u π t D u (DCF) To make this practical, need to learn how to parameterize SPDs in step (2), so that step (3) can be performed efficiently. Can use discretetime conditionally Gaussian processes. SPDs are closely related to riskneutral pricing measures. Useful for building intuition and for computations. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
15 SPD and the Risk Premium Let R t+1 be the gross return on a risky asset between t and t + 1: π t+1 E t R t+1 = 1 π t Using the definition of covariance, E t π t+1 (1 + r t ) = 1 π t Conditional Risk Premium and SPD Beta Risk Premium t E t [R t+1 ] (1 + r t ) = (1 + r t )Cov t R t+1, π t+1 π t c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
16 SPD and CAPM CAPM says that risk premia on all stocks must be proportional to their market betas. CAPM can be reinterpreted as a statement about the SPD pricing all assets. Assume that π t+1 /π t = a br t M +1 where R M is the return on the market portfolio. (The above formula may be viewed as approximation, if it implies negative values of π). Using the general formula for the risk premium, for any stock j, j E t R t +1 (1 + r j t ) = const Cov t (R t +1, R t M +1) The above formula works for any asset, including the market return. Use this to find the constant: E t j j Cov t (R t +1, R t M R t +1 (1 + r M t ) = E t R t +1 (1 + r t ) M Var t (R t +1 ) +1 ) c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
17 SPD and MultiFactor Models Alternative theories (e.g., APT), imply that there are multiple priced factors in returns, not just the market factor. Multifactor models are commonly used to describe the crosssection of stock returns (e.g., the FamaFrench 3factor model). Assume that π t+1 /π t = a + b 1 F t b K F t K +1 where F k, k = 1,..., K are K factors. Factors may be portfolio returns, or nonreturn variables (e.g., macro shocks). Then risk premia on all stocks have factor structure E t K j R t +1 (1 + r j t ) = λ k Cov t R t +1, F t k +1 k=1 Factor models are simply statements about the factor structure of the SPD. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
18 SPDs and RiskNeutral Pricing One can build models by specifying the SPD and computing all asset prices. It is typically more convenient to use a related construction, called riskneutral pricing. Riskneutral pricing is a mathematical construction. It is often convenient and adds something to our intuition. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
19 RiskNeutral Measures The DCF formulation with an SPD is mathematically equivalent to a change of measure from the physical probability measure to the riskneutral measure. The riskneutral formulation offers a useful and tractable alternative to the DCF model. Let P denote the physical probability measure (the one behind empirical observations), and Q denote the riskneutral measure. Q is a mathematical construction used for pricing and only indirectly connected to empirical data. Let B t denote the value of the riskfree account: t 1 B t = (1 + r u ) u=0 where r u is the riskfree rate during the period [u, u + 1). c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
20 RiskNeutral Measures Q is a probability measure under which T T = E P π u = E Q B t P t t D u t D u, for any asset with cash flow D π t u=t+1 B u u=t+1 Under Q, the standard DCF formula holds. Under Q, expected returns on all assets are equal to the riskfree rate: E Q [R t+1 ] = 1 + r t t If Q has positive density with respect to P, there is no arbitrage. There may exist multiple riskneutral measures. Q is also called an equivalent martingale measure (EMM). c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
21 SPD and Change of Measure Construct riskneutral probabilities from the SPD. Consider our treemodel of the market and let C(ν t ) denote the set of time(t + 1) nodes which are children of node ν t. Define numbers q(ν t+1 ) for all nodes ν t+1 C(ν t ) by the formula q(ν t+1 ) = (1 + r t )q(ν t ) π(ν t+1)p(ν t+1 ) π(ν t )p(ν t ) Recall that the ratio p(ν t+1 )/p(ν t ) is the nodeν t conditional probability of ν t+1. q(ν t+1 ) > 0 and π(ν t+1 )p(ν t+1 ) q(ν t+1 ) = q(ν t ) (1 + r t ) π(ν t )p(ν t ) ν t+1 C(ν t ) ν t+1 C(ν t ) = q(ν t )E P t π(ν t+1 ) (1 + rt ) = q(ν t ) π(ν t ) c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
22 SPD and Change of Measure q(ν t ) define probabilities. Are these riskneutral probabilities? For any asset i, 1 Q q(ν t+1 ) 1 E t Rt+1 i = Rt+1 i 1 + r t q(ν t ) 1 + r t ν t+1 C(ν t ) (1 + r t ) π(ν t+1)p(ν t+1 ) 1 = π(ν t )p(ν t ) 1 + r t ν t+1 C(ν t ) p(νt+ 1) π(ν = p(νt ) π(ν ν t+1 C(ν t ) = E P t = 1 π t+1 i Rt π +1 t Conclude that q(ν t ) define riskneutral probabilities. t+ 1) i Rt t ) +1 R t i +1 c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
23 Example: Binomial Tree RiskNeutral Measure FTAP implies that S t+1 = us, π t+1 (u) q 1 + r d qu+(1 q)d = 1+r t q = S t, π t u d Alternatively, compute transition (1 q) probability under Q using the SPD S t+1 = ds, π t+1 (d) π: q = p(1 + r t ) π t+1(u) π t r t d 1 + r t d = p(1 + r t ) = p(1 + r t ) u d u d c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
24 NormalityPreserving Change of Measure Results Under P, ε P N(0, 1). Define a new measure Q, such that under Q, ε P N( η, 1). Let ξ = d Q dp. Then, ξ(ε P ) = exp (εp + η) 2 + (εp ) 2 = exp ηε P η The change of measure is given by a lognormal random variable ξ(ε P ) serving as the density of the new measure. NormalityPreserving Change of Measure dq dp = e ηεp η 2 /2 ε Q = ε P + η N(0, 1) under Q c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
25 Price of Risk When P and Q are both Gaussian, we can define the price of risk (key notion in continuoustime models). Consider an asset with gross return R t+1 = exp µ t σ 2 t /2 + σ t ε P t+1, E t [R t+1 ] = exp(µ t ) where ε P t+1 N(0, 1), IID, under the physical Pmeasure. Let the shortterm riskfree interest rate be exp(r t ) 1 Let the SPD be π t+1 = π t exp r t η 2 t /2 η t ε P t+1 Recall ε Q = ε P t t + η t Under Q, the return distribution becomes R t+1 = exp µ t σ 2 t /2 σ t η t + σ t ε Q t+1 where ε Q t+1 N(0, 1), IID, under the Qmeasure. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
26 Price of Risk Under Q, R t+1 = exp µ t σ 2 t /2 σ t η t + σ t ε Q By definition of the riskneutral probability measure, ε Q t+1, t+1 N(0, 1) E Q t [R t+1 ] = exp(r t ) µ t σ t η t = r t The risk premium (measured using log expected gross returns) equals σ t is the quantity of risk, η t is the price of risk. µ t r t = σ t η t Models with timevarying price of risk, η t, exhibit return predictability. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
27 Overview As an example of how riskneutral pricing is used, we consider the problem of option pricing when the underlying asset exhibits stochastic volatility. The benchmark model is the BlackScholes model. Stock (underlying) volatility in the BS model is constant. Empirically, the BS model is rejected: the implied volatility is not the same for options with different strikes. There are many popular generalizations of the BS model. We explore the model with an EGARCH volatility process. The EGARCH model addresses some of the empirical limitations of the BS model. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
28 The BlackScholes Model Consider a stock with price S t, no dividends. Assume that S t+1 = exp µ σ 2 /2 + σε P S t t+1, where ε P t+1 N(0, 1), IID, under the physical Pmeasure. Assume that the shortterm interest rate is constant. Assume that under the Qmeasure, S t+1 = exp r σ2 /2 + σε Q S t t+1, ε Q t+1 N(0, 1), IID The timet price of any European option on the stock, with a payoff C T = H(S T ), is given by = E Q e r(t t) H(S T ) C t t This is an arbitragefree model. Prices of European call and put options are given by the BlackScholes formula. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
29 Implied Volatility The BlackScholes model expresses the price of the option as a function of the parameters and the current stock price. European Call option price C(S t, K, r, σ, T ) Implied volatility σ i of a Call option with strike K i and time to maturity T i is defined by C i = C(S t, K i, r, σ i, T i ) Implied volatility reconciles the observed option price with the BS formula. Option prices are typically quoted in terms of implied volatilities. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
30 How consistent are market option prices with the BS formula? Figure 2(a) shows the decrease of σ imp with the strike level of options on the S&P 500 index with a fixed expiration of 44 days, as observed on May 5, This asymmetry is commonly called the volatility skew. Figure 2(b) shows the increase of σ imp with the time to expiration of If option atthemoney prices satisfied options. the This BS variation assumptions, is generally all implied called volatilities the volatility would be the same, term equal structure. to σ, the In volatility this paper of we the will underlying refer to them price collectively process. as Empirically, the volatility implied smile. volatilities depend on the strike and time to maturity. Implied Volatility Smile (Smirk) FIGURE 2. Implied Volatilities of S&P 500 Options on May 5, 1993 (a) (b) option implied volatility (%) O O O O O O O O O O option strike (% of spot) option implied volatility (%) O O O O option expiration (days) O Source: E. Derman, I. Kani, 1994, The Volatility Smile and Its Implied Tree, Quantitative Strategies Research Notes, Goldman Sachs In Figure 2(a) the data for strikes above (below) spot comes from call (put) prices. In Figure 2(b) the average of atthemoney call and put implied volatilities are used. You can see that σ imp falls as the strike c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
31 EGARCH Model Consider a model of stock returns with EGARCH volatility process. Assume the stock pays no dividends, and shortterm interest rate is constant. Stock returns are conditionally lognormally distributed under the Qmeasure St ln = r σt2 1 + σ t 1 ε Q S t, 1 2 t ε Q t N(0, 1), IID Conditional expected gross return on the underlying asset equals exp(r). Conditional volatility σ t follows an EGARCH process under Q σ 2 σ 2 ε Q 2 ln t = a 0 + b 1 ln t 1 + θε Q t 1 + γ t 1 π Option prices can be computed using the riskneutral valuation formula. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
32 Option Valuation by Monte Carlo We use Monte Carlo simulation to compute option prices. Using the valuation formula = E Q e r(t t) H(S T ) C t t Call option price can be estimated by simulating N trajectories of the n underlying asset S u, n = 1,..., N, under Q, and averaging the discounted payoff 1 C t N N e r (T t) n max(s T K, 0) n=1 Resulting option prices are arbitragefree because they satisfy the riskneutral pricing relationship. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
33 Simulation Use oneweek time steps. Calibrate the parameters using the estimates in Day and Lewis (1992, Table 3) Calibrate the interest rate a 0 b 1 θ γ exp(52 r) = exp(0.05) Start all N trajectories with the same initial stock price and the same initial volatility, σ 0 = 15.5%/ 52. Compute implied volatilities for Call/Put options with different strikes. Plot implied volatilities against BlackScholes deltas of Put options (Δ t = P t / S t ). c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
34 Volatility Smile Weeks 24 Weeks 36 Weeks 16 Implied Volatility (%) Option Delta c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
35 Futures We want to build an arbitragefree model of futures prices. In case of deterministic interest rates and costless storage of the underlying asset, futures price is the same as the forward price. In most practical situations, storage is not costless, so simple replication arguments are not sufficient to derive futures prices. Want to model futures prices for multiple maturities in an arbitragefree framework. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
36 RiskNeutral Pricing Φ T t is the timet futures price for a contract maturing at T. Futures are continuously settled, so the holder of the long position collects Φ T t Φ T t 1 each period. Continuous settlement reduces the likelihood of default. The market value of the contract is always equal to zero. In the riskneutral pricing framework T B t Q E t Φ T u Φ T u 1 = 0 for all t B u u=t+1 We conclude (using backwards induction and iterated expectations) that the futures prices must satisfy E Q t Φ T t+1 Φ T t = 0 for all t and thus Φ T t = E Q t Φ T T = E Q t [S T ] where S T is the spot price at T. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
37 AR(1) Spot Price Suppose that the spot price follows an AR(1) process under the Pmeasure S t S = θ(s t 1 S) + σε P t, ε P t N(0, 1), IID Assume that the riskneutral Qmeasure is related to the physical Pmeasure by the stateprice density πt+ 1 = exp r t η 2 ηε P t+1 πt 2 Market price of risk η is constant. To compute futures prices, use ε Q t = ε P t + η. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
38 AR(1) Spot Price Under the riskneutral measure, spot price follows ησ ησ S t = S + θ S t 1 S + σε Q t 1 θ 1 θ Define a new constant S Q S ησ 1 θ Then S t S Q = θ S t 1 S Q + σε Q t Under Q, spot price is still AR(1), same meanreversion rate, but different longrun mean. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
39 AR(1) Spot Price To compute the futures price, iterate the AR(1) process forward Q Q Q S t+1 S = θ S t S + σε t+1 Q Q S + σε Q t+2 S = θ S t+1 S t+2 = θ 2 S t S Q Q Q + σε t+2 + θσε t+ 1. Q Q S = θ n + σε Q t+n θ n 2 σε Q t+2 + θ n 1 σε Q t+n S S t S We conclude that Φ T t = E Q t [S T ] = S Q 1 θ T t + θ T t S t Futures prices of various maturities given by the above model do not admit arbitrage. t+1 c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
40 Expected Gain on Futures Positions What is the expected gain on a long position in a futures contract? Under Q, the expected gain is zero: E Q t Φ T t+1 Φt T = 0 for all t Futures contracts provides exposure to risk, ε P. This risk is compensated, with the market price of risk η. Φ T t+1 Φ T t = σθ T t 1 ε Q t+1 = ησθt t 1 + σθ T t 1 ε P t+1 (Use ε Q t+1 = η + εp t+1 ) Under P, expected gain is nonzero, because ε Q has nonzero mean under P E P t Φ T t+1 Φt T = ησθ T t 1 for all t Can estimate model parameters, including η, from historical futures prices. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
41 Summary Existence of SPD or riskneutral probability measure guarantees absence of arbitrage. Factor pricing models, e.g., CAPM, are models of the SPD. Can build consistent models of multiple options by specifying the riskneutral dynamics of the underlying asset. BlackScholes model, volatility smiles, and stochastic volatility models. c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
42 Readings Back 2005, Chapter 1. E. Derman, I. Kani, 1994, The Volatility Smile and Its Implied Tree, Quantitative Strategies Research Notes, Goldman Sachs. T. Day, C. Lewis, 1992, Stock Market Volatility and the Information Content of Stock Index Options, Journal of Econometrics 52, c Leonid Kogan ( MIT, Sloan ) ArbitrageFree Pricing Models , Fall / 48
43 MIT OpenCourseWare Analytics of Finance Fall 2010 For information about citing these materials or our Terms of Use, visit:
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