FinalCheatSheet_2015 - Final Cheat Sheet for IEOR E4706...

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Final “Cheat Sheet” for IEOR E4706: Foundations of FE (Fall 2015) One-Period Definitions (1) An elementary security is a security that has date t = 1 payoff of the form e j = (0 , . . . , 0 , 1 , 0 , . . . , 0), where the payoff of 1 occurs in state j . (2) A contingent claim, X , is attainable if there exists a trading strategy, θ = [ θ 0 θ 1 . . . θ N +1 ] > , such that X = S 1 θ where S 1 is the m × ( N + 1) matrix of date 1 security payoffs. (3) A type A arbitrage is a trading strategy, θ , such that S > 0 θ < 0 and S 1 θ = 0. A type B arbitrage is a trading strategy, θ , such that S > 0 θ 0, S 1 θ 0 and S 1 θ 6 = 0. (4) We say that a vector π = ( π 1 , . . . , π m ) > 0 is a vector of state prices if the date t = 0 price, P , of any attainable security, X , satisfies P = m k =1 π k X ( ω k ) . We call π k the k th state price. (5) A numeraire security is a security with a strictly positive price at all times, t . (In a multi- period world we can use a security as a numeraire if it has a strictly positive price process and does not pay dividends. If it does pay dividends then the cumulative gains process could be used as long as it is strictly positive. ) Multi-Period Definitions (6) A trading strategy is a vector, θ t = ( θ (0) t ( ω ) , . . . , θ ( N ) t ( ω )), of predictable stochastic processes that describes the number of units of each security held just before trading at time t , as a function of t and ω . (7) A self-financing trading strategy is a strategy, θ t , where changes in V t are due entirely to trading gains or losses, rather than the addition or withdrawal of cash funds. In particular, a self-financing strategy satisfies V t = N i =0 θ ( i ) t +1 S ( i ) t for t = 1 , . . . , T - 1 . (8) We say that the contingent claim C is attainable if there exists a self-financing trading strat- egy, θ t , whose value process, V T , satisfies V T = C . (9) We say that the market is complete if every contingent claim is attainable. Otherwise the market is said to be incomplete . (10) An equivalent martingale measure (EMM), Q = ( q 1 , . . . , q m ), is a set of probabilities such that q i > 0 for all i = 1 , . . . , m , and the deflated security prices are Q -martingales. (11) The cumulative gain process, G t , of a security at time t is equal to value of the security at time t plus accumulated cash payments that result from holding the security. Some Discrete-Time Results (12) (Put-Call Parity) The call and put options prices, C t and P t respectively, satisfy S t = C t - P t + d ( t, T ) K where d ( t, T ) is the discount factor for lending between dates t and the expiration date of the options, T . K is the strike of both the call option and the put option, and the security is assumed to pay no intermediate dividends. (13)

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