Hedging Asian options (Yang et al. (2011)).

a) Compute the Asian option price $f(t,S_t,{\Lambda }_t)$ when ${\Lambda }_t/T⩾K$.

b) Compute the hedging portfolio allocation $({\xi }_t,{\eta }_t)$ when ${\Lambda }_t/T⩾K$.

c) At maturity we have $f(T,S_T,{\Lambda }_T)={({\Lambda }_T/T-K)}^{+}$, hence ${\xi }_T=0$ and $${\eta }_T{A}_T=A_T\frac{e^{-rT}}{A_0}(\frac{{\Lambda }_T}{T}-K)1_{\{{\Lambda }_T>KT\}}={(\frac{{\Lambda }_T}{T}-K)}^{+}.$$

d) Show that the Asian option with payoff ${({\Lambda }_T-K)}^{+}$can be hedged by the self-financing portfolio

$${\xi }_t=\frac{1}{S_t}(f(t,S_t,{\Lambda }_t)-e^{-(T-t)r}(\frac{{\Lambda }_t}{T}-K)h(t,\frac{1}{S_t}(\frac{{\Lambda }_t}{T}-K)))$$

in the asset $S_t$ and

$${\eta }_t=\frac{e^{-rT}}{A_0}(\frac{{\Lambda }_t}{T}-K)h(t,\frac{1}{S_t}(\frac{{\Lambda }_t}{T}-K)),0⩽t⩽T$$ in the riskless asset

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