{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

hw8-1 - 3 Exotic Options 137 1 rT A i.e(1 D0 T)S(1 rT)A < 0...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
3 Exotic Options 137 implies S > 1 + rT 1 + D 0 αT A , i.e., - (1 + D 0 αT ) S + (1 + rT ) A < 0. Thus if 1 + rT 1 + D 0 αT < 1 α , then when S > A α , the PDE cannot be used. If 1 + rT 1 + D 0 αT > 1 α , then when S > 1 + rT 1 + D 0 αT A , the PDE cannot be used; while A α < S < 1 + rT 1 + D 0 αT A , we have - (1 + D 0 αT ) S + (1 + rT ) A > 0 and the PDE can be used; Putting those cases together, we have that when S < max A α , 1 + rT 1 + D 0 αT A , the PDE can be used, and otherwise the PDE cannot be used. Consequently S f ( A, T ) = A max 1 α , 1 + rT 1 + D 0 αT . b) When A < E , max( A - E, 0) = 0, so the PDE always can be used. When A > E , max( A - E ) = A - E . Thus ( A - E ) ∂t + L SAT ( A - E ) = S - A T · 1 - r ( A - E ) = 1 T [ S + rTE - (1 + rT ) A ] . The last expression is greater than zero if S + rTE 1 + rT > A , and less than zero if S + rTE 1 + rT < A . If S + rTE 1 + rT < E , then A > E implies A > S + rTE 1 + rT , i.e., S + rTE - (1+ rT ) A < 0. Thus if S + rTE 1 + rT < E , then when A > E , the PDE cannot be used. If S + rTE 1 + rT > E , then when A > S + rTE 1 + rT , we have S + rTE - (1+ rT ) A < 0 and the PDE cannot be used; while E < A < S + rTE 1 + rT , we have S + rTE - (1 + rT ) A > 0 and the PDE can be used. Putting those cases together, we have that when A < max E, S + rTE 1 + rT , the PDE can be used, and otherwise the PDE cannot be used. Consequently A f ( S, T ) = max E, S + rTE 1 + rT . 9. Suppose that sampling is done discretely at t = t 1 , t 2 , · · · , t K , where 0 t 1 < t 2 < · · · < t K T . Let H ( t ) = max ( S ( t 1 ) , · · · , S ( t i * ( t ) )) , where i * ( t ) is the number of samplings before time t. Assume dS = μSdt + σSdX and the dividends are paid continuously with dividend yield D 0 .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
138 3 Exotic Options Let V ( S, H, t ) be the value of a lookback option with discrete sampling. Derive the PDE and the jump condition for such a lookback option by using a portfolio Π = V ( S, H, t ) - ΔS (without using the general PDE for derivative securities). Solution : Because dH ( t ) = ( 0, if t 6 = t i , i = 1 , 2 , · · · , K, max ( S ( t i ) , H ( t - i )) - H ( t - i ) , if t = t i , i = 1 , 2 , · · · , or K and dH ( t ) dt = K X i =1 £ max ( S ( t ) , H ( t - )) - H ( t - )/ δ ( t - t i ) , where H ( t - ) = lim ε 0 H ( t - ε ) with ε > 0, using Itˆ o’s lemma, we have dV = ∂V ∂S dS + ∂V ∂H dH + ∂V ∂t + 1 2 σ 2 S 2 2 V ∂S 2 dt. Thus if we set Π = V ( S, H, t ) - ΔS, then we have = ∂V ∂S dS + ∂V ∂H dH + ∂V ∂t + 1 2 σ 2 S 2 2 V ∂S 2 dt - Δ ( dS + D 0 Sdt ) = ∂V ∂S - Δ dS + ∂V ∂H dH + ∂V ∂t + 1 2 σ 2 S 2 2 V ∂S 2 - ΔD 0 S dt. Let us choose ∂V ∂S = Δ, then this portfolio is risk-free and the return rate is r : ∂V ∂H dH + ∂V ∂t + 1 2 σ 2 S 2 2 V ∂S 2 - ∂V ∂S D 0 S dt = r V ( S, H, t ) - ∂V ∂S S dt. Noticing the expression for dH ( t ) dt , this relation can be rewritten as ∂V ∂t + 1 2 σ 2 S 2 2 V ∂S 2 + ( r - D 0 ) S ∂V ∂S + K X i =1 £ max ( S ( t ) , H ( t - )) - H ( t - )/ δ ( t - t i ) ∂V ∂H - rV = 0 .
Background image of page 2
3 Exotic Options 139 This means that at t 6 = t i , i = 1 , 2 , · · · , K , V fulfills ∂V ∂t + 1 2 σ 2 S 2 2 V ∂S 2 + ( r - D 0 ) S ∂V ∂S - rV = 0 , 0 S, 0 H and at t = t i , i = 1 , 2 , · · · , or K , the equation ∂V ∂t + K X i =1 £ max ( S ( t ) , H ( t - )) - H ( t - )/ δ ( t - t i ) ∂V ∂H = 0 , 0 S, 0 H holds. It is a hyperbolic equation, and the characteristic relation is dH dt = K X i =1 £ max ( S ( t ) , H ( t - )) - H ( t - )/ δ ( t - t i ) .
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}