Dynamic Hedging with Stochastic Differential Utility

5.1 EXPONENTIAL RISK-ADJUSTMENT: CRRA ANDCARAIn this section, we give two simple examples for concreteness, where we specifyh (u) =−e −ρu ,withρ > 0. In both, we show that the risk aversionincreases, causing a decreasing on the pure speculative demand.First, suppose that u(w) =−e −γw ,whereγ > 0. Then 23µ JwwR = − + k(J)J w =Jà w£ ¡exp −ρU − 2γWθt= γ1+ρ E tT −tE t£exp¡−ρU − γWθtT −t¢¤ !We may interpret R as the global risk aversion parameter, as we havedone in Section 3. From this expression, we can see clearly that, with certaintyequivalent, we will need numerical methods to find the optimal hedgeratio. Also, we may realize that the risk aversion parameter increases bythe proportion ρ E t[exp(−ρU−2γWT θt−t)]> 0. Consequently the pure speculativedemand decreases 24 . Moreover, if ρ = 0, we are back to theE t[exp(−ρU−γWT θt−t)]TerminalValue-type utility example, given in the last section.¢¤23 We suppress the TWSDU superscript of θ for simplicity.24 We are not considering equilibrium effects.26