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Unformatted text preview: 16 2 Basic Options Var [ ξ ] = E £ ξ 2 / E [ ξ ] 2 = 1 a 2 e 3 b 2 1 a 2 e 2 b 2 = e 2 b 2 a 2 ( e b 2 1) . 11. a) Show that if an investment is risk free, then theoretically its return rate must be the spot interest rate. b) Using this fact and Itˆ o’s lemma, derive the BlackScholes equation. Solution : a) Let r r , r be the return and spot interest rates respectively. If r r > r , then an arbitrageur will borrow money from bank and invest in the higheryielding opportunity. In response to the pressure of demand, we would expect that the bank will raise its interest rate to attract money. Therefore r r will not be greater than r . Similarly, r r is also not less than r . Thus theoretically r r = r . b) Let V ( S, t ) be the value of an option. By Itˆ o’s lemma we have: dV = ∂V ∂S dS + µ ∂V ∂t + 1 2 σ 2 S 2 ∂ 2 V ∂S 2 ¶ dt. Now construct a portfolio which consists of one option and Δ units of the underlying asset. Its value is: Π = V ΔS. During a time step dt the holder of the portfolio earns dΠ = dV ΔdS ΔSD dt, where we assume that the underlying asset pays out a dividend D Sdt during that period, D being a constant. Using the expression for dV , we have dΠ = µ ∂V ∂S Δ ¶ dS + • ∂V ∂t + 1 2 σ 2 S 2 ∂ 2 V ∂S 2 ΔSD ‚ dt. Choose Δ = ∂V ∂S . Then we get: dΠ = µ ∂V ∂t + 1 2 σ 2 S 2 ∂ 2 V ∂S 2 ∂V ∂S SD ¶ dt. Because this investment has no risk, we arrive at rΠdt = r µ V ∂V ∂S S ¶ dt = µ ∂V ∂t + 1 2 σ 2 S 2 ∂ 2 V ∂S 2 ∂V ∂S SD ¶ dt, i.e., ∂V ∂t + 1 2 σ 2 S 2 ∂ 2 V ∂S 2 + ( r D ) S ∂V ∂S rV = 0 ....
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 Spring '10
 Zhu
 Math, Trigraph, Black–Scholes, V1, basic options, Itˆ’s lemma

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