490_lecture15_2011 - Lecture 15 No Arbitrage Pricing...

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Lecture 15 No Arbitrage Pricing Pricing of Forward Contracts Forward and Future Prices The Binomial Price Process State Prices For (v 1 ,...,v n ) choose the Arrow-Debreu Security for state i, i.e, v i =1, and v j =0 for j ! i. Let x i be the resulting portfolio. The cost of the portfolio is q i =px i =p 1 x i 1 +...+p n x i n . We say that q i is the state price for state i . v 1 v 2 ! v n ! " # # # # $ % & & & & = v 1,1 v 2,1 " v n ,1 v 1,2 v 2,2 " v n ,2 ! ! # ! v 1, n v 2, n " v n , n ! " # # # # # $ % & & & & & x 1 x 2 ! x n ! " # # # # $ % & & & & State Prices For (v 1 ,...,v n ) choose the Arrow-Debreu Security for state i, i.e, v i =1, and v j =0 for j ! i. Let x i be the resulting portfolio. The cost of the portfolio is q i =px i =p 1 x i 1 +...+p n x i n . We say that q i is the state price for state i . By construction the state prices have the property that p i = q 1 ,..., q n ( ) v i ,1 v i ,2 ! v i , n ! " # # # # # $ % & & & & & = q i v i , s s = 1 n ' . State Prices In matrix notation, state prices q 1 ,...,q n must satisfy: p 1 p 2 ! p n ! " # # # # $ % & & & & = q 1 ,..., q n ( ) v 1,1 v 2,1 " v n ,1 v 1,2 v 2,2 " v n ,2 ! ! # ! v 1, n v 2, n " v n , n ! " # # # # # $ % & & & & & Risk Neutral Valuation Let q 1 ,...,q l be strictly positive state prices. And suppose that an asset generates payoffs v 1 , .... ,v s . Then we already know that the price p of the asset must satisfy p = q s v s . s = 1 l ! Risk Neutral Valuation Let q 1 ,...,q l be strictly positive state prices. And suppose that an asset generates payoffs v 1 , .... ,v s . Then we already know that the price p of the asset must satisfy p = q s v s . s = 1 l ! Payoff of riskless asset with price 1 (if exists)? v=(1+r 0 ,1+r 0 ,..., 1+r 0 ) . Thus, (1 + r 0 ) q s = 1 s = 1 l ! q s = 1 1 + r 0 s = 1 l !
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Risk Neutral Valuation Let q 1 ,...,q l be strictly positive state prices. Asset with payoffs v 1 , .... ,v s . p = q s v s . s = 1 l ! Define ! i = q i q s s = 1 l " i = q i 1 + r 0 ( ) Then ! i " 0 and ! i i = 1 l " = 1 i.e., they are probabilities. Thus, we can write p = E v s 1 + r 0 ! " # $ % & q s = 1 1 + r 0 s = 1 l ! Risk Neutral Valuation Dividing by p yields: Thus, we can write p = E v s 1 + r 0 ! " # $ % & 1 = E v s p 1 + r 0 ! " # # # # $ % & & & & = E r s + 1 1 + r 0 ! " # $ % & Thus, 1 + r = E [ r s + 1] Risk Neutral Valuation Thus, 1 + r = E [ r s + 1] This yields r = E [ r s ] With risk neutral valuation, the expected return of any asset must be equal to the riskless rate. Risk Neutral Valuation If the arbitrage principle holds then there exist a probability distribution over the states such that for any asset p = E v s 1 + r 0 ! " # $ % & where p is the price of the asset, v the value in the next period, and r 0 the riskless rate. This probability distribution is referred to as the equivalent martingale measure . r = E [ r s ] Forward Contract on Stocks p = E v s 1 + r 0 ! " # $ % & In a forward contract, the long side of the market has the promises to buy an asset at a price K, T time periods from now. Forward Contract on Stocks p = E v s 1 + r 0 ! " # $ % & In a forward contract, the long side of the market has the promises to buy an asset at a price K at T time. T t current date delivery date
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Forward Contract on Stocks In a forward contract, the long side of the market has the promises to buy an asset at a price K at T time.
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