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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 Black-Scholes 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 higher-yielding 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
- Math, Trigraph, Black–Scholes, V1, basic options, Itˆ’s lemma