Finance & Options

Lesson 1 of 12

Options: Calls, Puts and Payoffs

Options work like insurance on a stock: what calls and puts pay at expiry, how to draw and combine their payoffs, and what in, at and out of the money mean.

Insurance on a stock

Say you own a share at $100 and you're worried about a crash. You can pay someone $2 today for the right to sell them your share at $95 in three months' time. That contract is a put option. The fixed price K=95K = 95 is the strike, the date TT is the expiry, and the $2 you pay is the premium.

A call option is the mirror image: the right, with no obligation, to buy at KK. Write STS_T for the stock price at expiry. You never have to use either option, so you exercise only when it helps you, and at expiry they pay

call: (ST−K)+=max⁡(ST−K,0),put: (K−ST)+=max⁡(K−ST,0).\text{call: } (S_T - K)^+ = \max(S_T - K, 0), \qquad \text{put: } (K - S_T)^+ = \max(K - S_T, 0).

Now check the insurance. If the stock crashes to 60, the put pays 95−60=3595 - 60 = 35, so share plus put is worth 95. You paid 102 in total, so the loss is 7, and that is the worst case at any price. At ST=130S_T = 130 the put expires worthless, and you keep the gain of 30 minus the premium, a profit of 28.

A European option can be exercised only at TT. An American option can be exercised at any time up to TT. Most interview questions and all the formulas in this track use European options unless they say otherwise.