Solution to Homework Two

Solution to Homework Two - MAFS 5030 Quantitative Modeling...

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Unformatted text preview: MAFS 5030 Quantitative Modeling of Derivatives Securities Solution to Homework Two Course Instructor: Prof. Y.K. Kwok 1. ⇐ part: The trading strategy H with V < 0 and V 1 ( ω ) ≥ , ∀ ω ∈ Ω, domi- nates the zero-holding trading strategy hatwide H = (0 0 ··· 0) T . The zero- holding strategy gives hatwide V 1 ( ω ) = V < 0, so V 1 ( ω ) > hatwide V 1 ( ω ) for all ω ∈ Ω. Thus, H dominates hatwide H . ⇒ part: Existence of a dominant trading strategy means there exists a trading strategy H = ( h 1 ··· h M ) T such that V = 0 and V 1 ( ω ) > , ∀ ω ∈ Ω. Let G * min = min ω G * ( ω ) = min ω M summationdisplay m =1 h m Δ S * m . Since G * ( ω ) = V * 1- V * > 0, we have G * min > 0. Consider the new trading strategy with hatwide h m = h m for m = 1 , ··· ,M, hatwide h =- G * min- M summationdisplay m =1 h m S * m (0) . Now, hatwide V * = hatwide h + M summationdisplay m =1 hatwide h m S * m (0) =- G * min < 0; while hatwide V * 1 ( ω ) = hatwide h + M summationdisplay m =1 hatwide h m S * m (1; ω ) =- G * min + M summationdisplay m =1 h m Δ S * m ( ω ) ≥ , by virtue of the definition of G * min . Thus, hatwide H = ( hatwide h 1 ··· hatwide h M ) T is a trading strategy that gives hatwide V < , hatwide V 1 ( ω ) ≥ , ∀ ω ∈ Ω. 2. For the given securities model, the discounted terminal payoff matrix: S (1; Ω) =   1 . 1 1 . 1 1 . 1 2 . 2 1 . 1 3 . 3   and initial price vector S (0) = (1 4). 1 (a) With h = 4 ,h 1 =- 1, V = (1 4) parenleftbigg 4- 1 parenrightbigg = 0 V 1 ( ω ) = S (1; Ω) parenleftbigg 4- 1 parenrightbigg =   3 . 3 2 . 2 1 . 1   > , V * 1 ( ω ) =   3 2 1   . Thus parenleftbigg 4- 1 parenrightbigg is a dominant trading strategy. (b) G * = V * 1- V * =   3 2 1   . (c) We shall use the result in Question 1. Now, G * min = min ω G * ( ω ) = 1 so that hatwide h =- 1- (- 1)(4) = 3. Take hatwide H = parenleftbigg 3- 1 parenrightbigg , then hatwide V = (1 4) parenleftbigg 3- 1 parenrightbigg =- 1 < hatwide V 1 = S (1; Ω)...
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This note was uploaded on 02/01/2012 for the course MATH A taught by Professor A during the Spring '11 term at HKU.

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Solution to Homework Two - MAFS 5030 Quantitative Modeling...

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