Answer to Question #175724 in Financial Math for Andrea

Question

Stock ABC is currently priced at $50. Consider two 6-month options on stock ABC:

Option 1:

$60-strike call (priced at 30% implied vol). Price works out to $1.27. Delta = +0.23  

Option 2:

$40-strike put (priced at 40% implied vol). Price works out to $1.56. Delta = -0.18

 

(Prices and deltas have been calculated via the Black-Scholes model, assuming q=0% and r=0%.)

 

Suppose a trader holds:

100 units of Option 1 and 200 units of Option 2.

 

[a] Explain the statement: “Both options are $10 out-of-the-money.”

 

[b] What’s the market value of the trader’s total position?

 

[c] How many shares of ABC should the trader buy (or sell) in order to be delta-neutral?

 

[d] What’s the total gamma of her position?

Answer 1

a) For such a stock, call options with strike prices above $10 would be OTM calls, while put options with strike prices below $10 would be OTM puts

b)Market value for call option = market price+delta

= 1.27+0.23

= 1.5

Market value for put option = 1.56-0.18

= 1.33

c)For ABC to be delta neutral, the shares for both option 1 and option 2 should be equal

Hence, they should trade 400 shares

d) Total gamma = 0.30+0.40

= 0.70