Dynamic Hedging with Stochastic Differential Utility
Dynamic Hedging with Stochastic Differential Utility
Dynamic Hedging with Stochastic Differential Utility
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0 > h 00 (J) J w J x + h 0 (J) J wx = ¡ e r(T −t) − 1 ¢ ×<br />
·<br />
= E h ¡ 00 U ¡ ¢¢ h ¡ ¢ i<br />
WT θt<br />
−t U 0 2 ¡<br />
W<br />
θt<br />
T −t + h<br />
0<br />
U ¡ W θt<br />
=⇒ k (J) Jw 2 + h0 (J) J wx<br />
(e r(T −t) − 1) = J ww + k(J)Jw 2 =⇒<br />
=⇒ J xw = ¡ e r(T −t) − 1 ¢ J ww .<br />
T −t<br />
¢¢ ¡ ¢¸<br />
U<br />
00<br />
WT θt<br />
−t =⇒<br />
Appendix E - Risk Aversion Increases <strong>with</strong><br />
Power <strong>Utility</strong><br />
Proposition 15 With Power utility and exponential risk adjustment, risk<br />
aversion increases.<br />
Proof. Clearly cov(e −ρU , ¡ ¢ h<br />
WT θt −a−γ<br />
−t ),a=0, 1, is positive, since d exp −ρ W 1−γ<br />
dW 1−γ<br />
0, and d W −a−γ < 0. Now, to simplify the notation, let q ≡ e −ρU , p ≡<br />
dW<br />
¡ ¢<br />
W<br />
θt −1−γ<br />
T −t ,andg ≡ ¡ WT −t¢ θt −γ<br />
. We want to prove that<br />
Since<br />
E (pq)<br />
E (gq) ≥ E (p)<br />
E (g) .<br />
i<br />
<<br />
Cov (p, q) ≥ 0=⇒ E (pq) ≥ E (p) E (q) .<br />
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