# hedging of a spread option with call

We have 2 underlying $S^{1}$ and $S^{2}$ with BS dynamic under the risk-neutral measure (r constant...) I found the (big) PDE satisfied by the price function $u(t,x,y)$ of a call spread whose payoff is $(S_{T}^{1}-S_{T}^{2}-K)_{+}]$ Then I have deduced the number of shares of $S^{1}$ and $S^{1}$ and the amount of cash to hedge this option (to hold $du/dx$ shares $S^{1}$ ...) My question is: how to hedge this spread option with only a call written on $S^{1}$, a call written on $S^{2}$ and cash ?

• Am I understanding correctly that you want to do delta hedging of the multi asset option but only trading the single stock calls and cash? i.e. not trading the underlying stocks? What happens if you just match the partial deltas of the multi-option with the deltas of the single stock calls? – mbison Dec 6 '15 at 15:25
• Yes it is. At first I try to decompose the spread option in terms of calls of $S^{1}$ and $S^{2}$ but it doesn't work. But what justify to write : $\delta_{x} Spread=\Delta C^{S^{1}}$ and $\delta_{y} Spread=\Delta C^{S^{2}}$ ? – glork Dec 7 '15 at 9:22
• I don't understand your question: "what justify to write $\delta_x Spread = \Delta C^{S1}$? Is it not the case that that $Spread(S1, S2)$? Why are you taking partials with respect to x instead of S1 and S2? – mbison Dec 13 '15 at 11:38
• just a notation x for S1 and y for S2. But I don't understand your answer/question of Dec 6. Why the fact to match the partial deltas will make it work. – glork Dec 13 '15 at 18:22
• Assume the price of your derivative is given by $f(S1, S2)$. Expanding this gives $df = f_{S1}dS + f_{S2}dS2 + H$ where H stands for all the higher order ito terms. The regular call on S1 and S2 have $\Delta_1$ and $\Delta_2$. For a move in S1 the regular call will change approximately $\Delta_1 dS1$. Your special option f will change $f_{S1}$. Suppose you hold 1 special option long versus $N = \frac{f_{S1}}{\Delta_1}$ short in call option1 then you are more or less hedged. similar for S2 and call 2. you will miss out on all the gamma terms as well as the correlation term. – mbison Dec 15 '15 at 0:36