Connecting the dots: Black Scholes, Volatility and Implied Volatility

Question

I am a first year Management & Finance undergrad preparing for my second year Finance courses, given that term 3 and exams have pretty much been cancelled for all British first years.

During that preparation, I am a little bit lost in connecting the dots in option pricing. For option pricing models (binomial, trinomial, Black Scholes) you would need

• Underlying's price
• Strike price
• Current date
• Expiry date
• Dividend yield
• Volatility

Now, what I am stuck with is the volatility and the role of the implied volatility. My problem is that the Black Scholes equation can be used to both calculate the price of an option using the volatility, but also calculate the implied volatility given the option price. How can that work? If I am given the price of an option, it already incorporates the volatility used in the formula. But how would I arrive at the correct option price in the first place when I don't know the implied volatility?

Secondly, I was wondering how exactly options exchanges, say CBOE, arrive at their option prices. If the only information that I am given information is a stock's dividend yield, its entire price history, and the relevant information of an American option like strike price and date of expiry, how would I go about calculating its price? Which volatility would I use?

Thank you very much for your help.

Answer 1

Your question makes perfect sense; one has to define volatility. Volatility can be used interchangeably for a number of different metrics. Realized volatility - the observed volatility of the underlying asset (and btw, there are many quite different ways of measuring it). Implied volatility - the number you get when you run your option pricer in reverse. Volatility - the number that you put into your option pricer.

First question: your difficulty is only one of definition. When using an option model, you put in the volatility parameter that you think is correct. On the other hand, the implied volatility is what someone else put into their model to get their price that is currently being shown in the market. If you don't know the implied volatility, that is another way of saying that you don't know the market price. If you need to price an option without knowing where the market is, you have to come up with your own estimate of volatility to put into it.

Second question: the CBOE or any other exchange does not come up with prices. Those prices are posted at the exchange by traders, usually market makers. They send prices to the exchange where they wish to buy or sell options.

While one could write books about how to price an option in the absence of market pricing, a good start is to measure realized volatility and use that as an input. A good modifier to that would be some sort of a modifier based on upcoming events that impact the volatility and/or pricing of a stock. For instance, a significant economic data release, an earnings report, or an FDA approval process for a biotech.

Answer 2

1. The basic difference is that for calculating the option's price within the classic BS-framework, you mostly use the historical vol (which is extracted from time series with a model). But this is only a theoretical (arbitrage free) price. At an option's exchange, you will see supply and demand meeting each other. Assuming perfect and efficient capital markets, the price has to incorporate all information, so while in the BS-framework all other parameters are known, vol is just an estimation from a model and therefore the only unknown parameter which could explain any differences in prices.

2. Also from the 1st question, exchanges generally build prices from meeting supply and demand, not by calculating theoretical prices.