It is the beginning of trade on June 6 th, 2023, and you observe the following list of options written on the Alphabet Inc. stock, all of which are due to expire in approximately two months.

The spot price of Alphabet stock was recorded as $125 at the end of the previous trading day. Answer questions 1 and 2 using this data.

option	type	strike	bid	ask	option	type	strike	bid	ask
c1	call	120	10.5	11.5	p1	put	115	2.25	3.0
c2	call	125	8.00	8.75	p2	put	125	5.0	6.25
c3	call	135	3.00	3.50	p3	put	130	7.75	9.0

1. Quantitatively illustrate how a butterfly spread can be constructed using a proper selection from among the options given above. Clearly indicate what options you use, draw the profit diagram of the position you create, and numerically identify the axis intersection points and maximum/minimum payoffs.

2. Quantitatively illustrate how a box spread can be constructed using a proper selection from among the options given above. Clearly indicate what options you use, draw the profit diagram of the position you create, and numerically identify the axis intersection points and maximum/minimum payoffs.

3. The price of a stock trading on the NYSE is currently $80 and you project that this price is equally likely to go up or down by $5 per quarter over the next six months. The continuously compounded annual riskfree rate is 5% per year across all maturities. Use a two-step binomial tree with quarterly steps to compute the theoretical price of a European call option with a strike price of $85 and six months left to expiration.

4. The price of a stock trading on Borsa Istanbul is currently 100. You project that this price is likely to go up by 11% or down by 10% per month over the next three months. The continuously compounded annual risk-free rate is 20% across all maturities. Use a three-step binomial tree with monthly steps to compute the theoretical price of an American put option with a strike price of 100 and three months left to expiration.