Exercise 4.19 4 point 19

[jess8]

Hello,

In exercise 4.19 part (b) and (c), would you please let me know why we need to consider delta if the call option has been delta-hedged? Call options with different strikes will have different gamma, but as a long call has positive gamma, I think the shape of the graph (change-of-value w.r.t change in stock value) will be about the same irrespective of strike value?

Thanks.
jess

[mj]

initially the value has slope 1 in the right tail.

After you delta hedge it has slope 1 - delta in the right tail. Thus the value of delta matters.

With this sort of problem the best thing is to get out EXCEL and the BS formula and plot the graphs.

[jess8]

Thank you mj. I plotted the graph using excel and here is how I looked at the problem.

(a). the change of the delta-hedged option (call) is dV = 0.5*gamma*(dS)^2
- here I assumed time didn't change hence there is no theta
- gamma and delta are both valued at the original spot (i.e. 100 in the exercise)

(b). consequently, the change of call option value is just due to gamma (when dS is small at least).

In my graph, I calculated
(i). original call value at the original spot (let's say it's V_ref)
(ii). call values due to different spot values (V_new)
(iii). the change of call value (V_new - V_ref), one for each new spot value
(iv). calculated (dV = V_new - V_ref - delta*dS)
(v). finally plotted the the dV vs dS graph.

Is this right? I also calculated 0.5*gamma*(dS)^2, and plotted its graph vs dS. As dV should be just this 0.5*gamma*(dS)^2 plus some higher order terms. It turned out the gamma graph is much steeper than the dV vs dS graph. I think this is due to the omission of higher order terms.. or I might have missed something...

thanks, jess

[mj]

the two graphs should be very similar for dS small . If they are not then something is wrong.

[jess8]

Thanks a lot. The two graphs are similar with small dS and start to differ when I increase dS.
Thanks,
Jess