Question

9.18 Consider the June 165, 170, and 175 call option prices in Table 9.1 a. Does convexity hold if you buy a butterfly spread, buying at the ask price and b. Does convexity hold if you sell a butterfly spread, buying at the ask price and c. Does convexity hold if you are a market-maker either buying or selling a selling at the bid? selling at the bid? butterfly, paying the bid and receiving the ask?

Answer 1

a. Convexity means that slopes on the strike vs premium curve is increasing. so if we have a premium K3< K2<K1 then slope from K3 to K2<= slope K2 to K1

Since Strike price are symmetric lambda is equals to 0.5 therefore to buy a long butterfly spread we buy the 165 strike call,sell two 170 strike call and Buy 175 strike call.

[C(K1)-C(K2)]/(K2-K1) = (6.15-3.30)/(170-165)=0.57 ( Bid and ask price will alter in this case)

[C(K2)-C(K3)]/(K3-K2) = (3.30-1.38)/(175-170)=0.384 ( Bid and ask price will alter in this case)

It is also holding the convexity.

b. In case of short butterfly spread, we are selling the 165 strike call, Buy two 170 strike call and sell one 175 strike call.

[C(K1)-C(K2)]/(K2-K1) = (6.25-3.20)/(170-165)=0.61

[C(K2)-C(K3)]/(K3-K2) = (3.20-1.43)/(175-170)=0.354

So we are holding the convexity

C. A market maker can buy a butterfly spread at the price we sell it for. Therefore, the above convexity conditions are the ones relevant for market maker. Convexity is not violated from market makers perspective.