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Chapter 18 Option Pricing Without Perfect Replication

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2 Framework On the Edges of Arbitrage One-period Good-deal Bounds Multiple Periods and Continuous Time Extensions, Other Approaches, and Bibliography Asset Pricing

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On the Edges of Arbitrage No Arbitrage Principle is Obvious, But… Not Trade Continuously, and Transactions Costs Stochastic Interest Rate or Stochastic Stock Volatility Underlying Asset Not Traded, Real Option for Example, or Short Selling Forbidden Internal Inconsistency of Arbitrage Portfolio So Unavoidable Basis Risk Arises. Many Authors simply add market price of risk assumptions. But this leaves the questions, How Sensitive are the Results to MPR Assumption? What are the reasonable Values for MPR Asset Pricing

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On the Edges of Arbitrage We are still willing to take as given the prices of lots of assets in determining the price of an option, and in particular assets that will be used to hedge the option. We Form an “Approximate” hedge or portfolio of basis asset closest to the focus payoff, hedge most of the option’s risk. The uncertainty about the option value is reduced only to figuring out the price of the residual. Good-Deal Option Price Bounds: Systematically Searching over all Possible Assignments of the MPR to a reasonable Value, and Impose No Arbitrage Opportunities and Find Upper and Lower Bounds on the Option Price Asset Pricing

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One-period Good-deal Bounds We want to price the payoff, where for a call option. Given an N-dimensional vector of basis payoffs x, whose prices p we can observe. The good- deal bound finds the minimum and maximum value of by search over all positive discount factors that price the basis assets and have limited volatility Asset Pricing

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One-period Good-deal Bounds In order of calculation, all the combinations of binding and nonbinding constrains: - Volatility Constraint Binds, Positivity Constrain Slack - Positivity Constraint Binds, Volatility Constraint Slack - Volatility and Positivity Constraints both Bind Volatility Constraint Binds, Positivity Constrain Slack - Lagrange Multipliers - Orthogonal Decomposition Asset Pricing

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Volatility Constraint Binds, Positivity Constrain Slack Asset Pricing

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So the call price bound: Interpretation: - The first term: approximate hedge portfolio - The second term: Residual Consistent with Volatility Bound Explicit Option Pricing Formula: 8 Volatility Constraint Binds, Positivity Constrain Slack Asset Pricing

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Any discount factor m that satisfies can be decomposed as: where So minimization problem is then: 9 Algebraic Expressions Volatility Constraint Binds, Positivity Constrain Slack Asset Pricing

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Introducing Lagrange multipliers, the problem is: Introducing Kuhn-Tucker multiplier on, and taking partial derivatives wrt. in each state 10 Volatility and Positivity Constraints both Bind Direct Approach Asset Pricing

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Kuhn Multiplier Discussion - If the positivity constraint is slack, : - If the positivity constraint binds: So the solution: And Maximize: 11 Volatility and Positivity Constraints both Bind Direct Approach Asset Pricing

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Hansen, Heaton and Luttmer (1995) Interchanging Min and Max, We may obtain: Search numerically over to find the solution to this problem. 12 Volatility and Positivity Constraints both Bind Dual Approach Asset Pricing

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Positivity Constraint Binds, Volatility Constraint Slack The problem: These are the Arbitrage Bounds, Denote the lower arbitrage bound by. The minimum-variance discount factor that generates the arbitrage bound solves: Asset Pricing

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Introducing the Lagrange Multiplier: Using the same conjugate method, the optimal : This problem is equivalent to: We search numerically for to solve this problem. 14 Positivity Constraint Binds, Volatility Constraint Slack Asset Pricing

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Good-Deal Comparisons with BS and Arbitrage Bounds We can obtain closer bounds on prices with more information about the discount factor. In particular, if we know the correlation of discount factor with, we could price the option better Asset Pricing

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Multiple Periods and Continuous Time The central fact that makes good-deal bounds tractable in dynamic environments that bounds are recursive. Consider two periods version: Be equivalent to a series of one-period problems: So the two-period problem must optimize in each state of nature at 1 The recursive property only holds if we impose Asset Pricing

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Consider pricing a European call option on asset that is not a traded asset, but correlated with a traded asset that can be used as an approximate hedge Terminal Payoff: Basis Asset: Underling Asset: The risk cannot be hedged by the basis asset, so the market price of risk will matter to the option price Goal: Discount factor to price and, has instantaneous volatility 17 Basis Risk and Real Options (One Dimension) Asset Pricing

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SDF solution form: Interpretation:, So the Good-deal bound is given by: And the Result: 18 Basis Risk and Real Options (One Dimension) Asset Pricing

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Market Prices of Risk: if an asset has a price process that loads on a shock, then its expected return: With Sharpe ratio: In the above Real Option Example: 19 Basis Risk and Real Options (One Dimension) Asset Pricing

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Basis Assets: -dimension, Dividends rate Underlying Assets: -dimension Goal: Value an asset that pays continuous dividends at rate, and terminal payment 20 Continuous time (High Dimensions) Asset Pricing

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For small time intervals, discretize: It can also be expressed: Discount Factor Form: 21 Continuous time (High Dimensions) Asset Pricing

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Volatility Constraint: Market price of risk: is the vector of market prices of risks of the shocks Solutions: Guess the lower bound follows: 22 Continuous time (High Dimensions) Asset Pricing

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Continuous time (High Dimensions) The Optimal SDF Theorem: The lower bound discount factor follows: And satisfy the restriction: Where: Asset Pricing

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Proof: And Guess: So Problem: 24 Continuous time (High Dimensions) Asset Pricing

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This is a linear objective in with a quadratic constraint. Therefore, as long as, the constraint binds and the optimal : So plugging the optimal value: For clarity, and exploiting the fact that does not load on, write the term as 25 Continuous time (High Dimensions) Asset Pricing

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The Optimal SDF: Plug in the definition of, we may obtain: The Good-Deal Option Price Bounds Theorem: The lower bound is the solution to the partial differential equation 26 Continuous time (High Dimensions) Asset Pricing

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Subject to the boundary conditions provided by the focus asset payoff. Replacing + with – before the square root gives the partial differential equation satisfied by the upper bound 27 Continuous time (High Dimensions) Asset Pricing

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In general, the process depends on the parameters, so without solving the PDE we don’t not know how to spread the loading of across the multiple sources of risk whose risk prices we do not observe. Thus, we cannot use an integration approach to find the bound; we cannot characterize enough to calculate: 28 Continuous time (High Dimensions) Asset Pricing

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However, if there is only shock, then we don ’ t have to worry about how loading of spreads across multiple sources of risk. can be determined simply by the volatility constraint. in this special case and are scalars. Theorem: In the special case that there is only one extra noise driving the process, we can find the lower bound discount factor directly from 29 Continuous time (High Dimensions) Asset Pricing

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The one-period good-deal bound is the dual to the HJ bound with positivity --- Hansen and Jagannathan (1991) study the minimum variance of positive discount factors that correctly price a given set of assets. The Good-deal bound interchanges the position of the option pricing equation and the variance of the discount factor. The techniques for solving the bound, are exactly those of the HJ bound in this one-period setup. This kind of problem needs weak but credible discount factor restrictions that lead to tractable and usefully tight bounds 30 Extensions, Other Approaches, and Bibliography Asset Pricing

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Extensions, Other Approaches, and Bibliography Several other similar restrictions: Monotonicity Restrictions: Levy (1985), Constantinide (1998), the discount factor declines monotonically with a state variable; marginal utility should decline with wealth. The good-deal bounds allow the worst case that marginal utility growth is perfectly correlated with a portfolio of basis and focus assets. So more credible correlation setup obtains tighter bounds Gain-Loss Restriction: Bernardo and Ledoit (2000) use the restriction to sharpen the no-arbitrage restriction. They show that this restriction has a beautiful portfolio inter-pretation -- corresponds to limited gain-loss Asset Pricing

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Define and as the gains and losses of an excess return, then: Bernardo and Ledoit also suggest, where is an explicit discount factor, such as the consumption based model or CAPM There alternatives are really not competitors The continuous-time treatment has not considered jumps and if with jumps, both positivity and volatility constraints will bind 32 Extensions, Other Approaches, and Bibliography Asset Pricing

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33 Thanks Your suggestion is welcome! Asset Pricing

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