An¬†option is a¬†paper, that gives you a¬†right (not¬†duty) to¬†buy stocks (or¬†anything else) on¬†a¬†specific day. For¬†example, you bought an¬†option for¬†5$ which gives you the¬†right to¬†buy 1 share of¬†Apple in¬†one year for¬†today‚Äôs price of¬†150$. You expect Apple stocks to¬†rise by¬†more than 5$. Let‚Äôs see 3 scenarios of¬†Apple price in¬†one year:
170$, we realize the¬†option and¬†get a¬†profit of¬†170-150-5 = 15$. This is a¬†200% return. If we invested in¬†a¬†share, we would have got 170-150 = 20$ profit, 13%.
130$, we don‚Äôt realize an¬†option, because now the¬†shares cost less, than in¬†the¬†contract (150$). We lose 5$ (the¬†cost of¬†an¬†option) instead of¬†130-150=-20$ if we bought a¬†share.
150$-155$, we may realize an¬†option and¬†get a¬†small loss. 153-150-5=-2$. We lost 40$ instead of¬†2% profit.
If we had bought the¬†shares, we would have risked losing all the¬†share costs. Here we risk losing only the¬†option price of¬†5$.
Options are the¬†profit guarantee for¬†a¬†seller
A¬†seller wants to¬†have a¬†profit from a¬†share. Instead of¬†thinking about whether the¬†stock price will be as¬†high as¬†he wants or¬†not, he can just say: ‚ÄúI want to¬†get 20$ profit in¬†one year from Apple shares that cost 150$, everything that is higher is yours‚ÄĚ. To¬†do so, he sells the¬†option for¬†20$. A¬†seller immediately gets a¬†profit of¬†20$ and¬†can invest then in¬†something else. Even if the¬†share price drops by¬†20$ to¬†130$, he does not¬†lose anything, as¬†well as¬†a¬†buyer.
Companies also issue options¬†‚Äď Warrants. Companies give premiums to¬†employees as¬†an¬†option for¬†buying a¬†share of¬†a¬†company. For¬†example, Elon Musk gives Tesla employees options for¬†buying tesla at¬†a¬†huge discount, that they can realize at¬†any time. So that when employees save enough, they can invest money in¬†a¬†company. It is an¬†alternative to¬†convertible bonds.
European and¬†American Options
European call (put)option is the¬†right to¬†buy(sell) a¬†unit of¬†an¬†underlying asset at¬†a¬†strike (=pre-specified) price at¬†a¬†specific point in¬†time.
American call (put)option is the¬†right to¬†buy(sell) a¬†unit of¬†an¬†underlying asset at¬†any time on¬†or¬†before an¬†expiration date of¬†an¬†option. European options can also be without an¬†expiration date.
American options are also traded in¬†Europe, in¬†the¬†Netherlands stock exchange, Euronext.liffe, for¬†example. These options are traded primarily on¬†equities: FTSE-100, CAC40, Bel-20.
Price and¬†problems with options
It is hard to¬†estimate the¬†price of¬†an¬†option. There are parameters, such as¬†risk-free rate, time to¬†expiration, and¬†volatility (variance, or¬†price fluctuation). For¬†example, low-volatility options cost less over time, while high-volatility options cost a¬†lot.
Formulas of¬†option pricing
Binomial formula for¬†pricing European options.
- K = strike price (end price, agreed before signing contract)
- N = periods of¬†expiration
- S0 = current price of¬†an¬†asset
- rf = risk-free return per¬†period
- ŌÄ= risk-neutral probability of¬†an¬†up move
- u = ratio of¬†the¬†share price to¬†the¬†prior share price, given that the¬†upstate has occurred over a¬†binomial step
- d = ratio of¬†the¬†share price to¬†the¬†prior share price, given that the¬†downstate has occurred over a¬†binomial step.
Let‚Äôs understand each part of¬†the¬†equation:
The¬†U and¬†D are the¬†possible outcomes (in¬†$) of¬†events. We assume, that there are always two scenarios in¬†one period of¬†time: positive and¬†negative. Some traders use 1 second as¬†a¬†period, but¬†then in¬†1 minute, there are 2^60, more than 1 billion scenarios. We assume 1 period as¬†a¬†quarter, half, or¬†a¬†full year.
We put all possible outcomes in¬†a¬†binomial tree¬†‚Äď a¬†graph, with 2 inherited sub-elements on¬†each node. Let‚Äôs see the¬†binomial tree example of¬†Apple stock, which can either double or¬†halve, and¬†we also assume, that the¬†risk-free rate is equal to¬†0 for¬†simplicity:
now we count the¬†path to¬†each final node:
It is 1 to¬†UU, 2 to¬†UD, and¬†1 to¬†DD. Now we find the¬†probability of¬†each of¬†the¬†nodes.
The¬†probability of¬†the¬†last one is 1¬†‚Äď 0.22¬†‚Äď 0.44 = 0.34. Yes, the¬†probability of¬†the¬†nodes is neither 25% nor 33%, it is not¬†similarly distributed. It can only be similarly distributed if ŌÄ=0.5
Finally, we find the¬†option price. We will assume that we need the¬†strike price at¬†the¬†moment (ATM), that is equal to¬†the¬†spot value (S0):
The¬†option price is equal to¬†99$.
Black Sholes Formula for¬†American and¬†European Options, with no-arbitrage:
- N = CDF of¬†the¬†normal distribution
- St = spot price of¬†an¬†asset (current price at¬†time t)
- K = strike price (end price)
- rf = risk-free interest rate
- t = time to¬†maturity
- ŌÉ = volatility of¬†the¬†asset (variance)
arbitrage is a¬†risk-free profit, that happens when tracking a¬†portfolio (or¬†a¬†share) costs more (or¬†less) than the¬†derivative (forward or¬†an¬†option). If the¬†share price is higher than the¬†forward strike price (end-price) + the¬†cost of¬†a¬†forward, then we short the¬†share and¬†buy a¬†forward or¬†an¬†option.
That means if an¬†Apple share costs 160$ and¬†the¬†forward cost is 0$, and¬†the¬†strike price is 150$, then we short apple stock and¬†buy forwards. We get 160-150=10$ profit. Moreover, when we short a¬†stock, we get the¬†cost of¬†a¬†share to¬†our account, that we can put in¬†a¬†bank. That brings a¬†lot of¬†profit.
Let‚Äôs use an¬†option in¬†the¬†example above. Let‚Äôs assume that an¬†option is free. We have the¬†same profit if the¬†cost of¬†apple stock is higher than 150$. If apple costs less than 150$, we just don‚Äôt exercise the¬†option and¬†get profit from shorting a¬†stock. If the¬†cost of¬†an¬†option was 10$, then we just have a¬†profit of¬†160-150-5 + 160*(risk-free rate, such as¬†a¬†bank deposit).
As¬†a¬†result, we get risk-free profit from such operations. In¬†a¬†world with unlimited buyers and¬†sellers, we could do the¬†operation over and¬†over again and¬†become the¬†richest people immediately, or¬†in¬†the¬†end of¬†a¬†period.
Put-Call parity Price of¬†a¬†call (buy) and¬†put (sell) options is different. that is because of¬†a¬†risk-free rate. We use an¬†equation:
Call price - Put price = Current price - Present value of a Strike price
Call price - Put price = Current price - Strike price / (1 + risk-free rate)^periods
# usually, a period is equal to 1 year and is named "t"
C0 - P0 = S0 - K/(1 + rf)^t
Due to¬†the¬†always non-negative risk-free rate (because otherwise, you don‚Äôt put money in¬†the¬†bank account), the¬†price of¬†a¬†call is higher than the¬†put.