Chapter 8— Option Pricing

Pricing model for options & factors influencing option price

Hi there. This is Sushant Reddy from Big Blind series on options trading. I’m delighted to know that you’ve come this far in my options trading course. If you love math (like I do), I’m sure you’ll enjoy this section.

For readers who aren’t too fascinated by equations, my suggestion is to skip the math part and just focus on the ‘intuitive aspects’ discussed in this tutorial. Ultimately, my experience has been that people who have an intuitive understanding of options become successful traders, even if they don’t exactly understand the math behind it.

Factors that influence option pricing

In my previous tutorials, I’ve discussed briefly about the factors influencing option pricing. Let’s understand each of these factors intuitively before we discuss the mathematical aspects of pricing.

1. Strike Price: Farther the strike price, lesser the likelihood that current stock price ends above the strike (for call option) or below the strike (for put option). Since this option is less likely to pay on expiry date, we should intuitively expect that the price we pay today to be lower.

All things equal, farther the strike, lower will be the option price.

2. Time to Expiry: Greater the time to expiry, higher the likelihood of the stock price going beyond the strike price. As time to expiry increases, chance of sudden black swan events happening (for eg., a trade war between US and China or major Brexit announcement or demonetization etc) also increases. When there are sudden large movements in stock price, an option buyer could see a huge increase in price.

All other things equal, greater the time to expiry, higher will be the option price

3. Volatility: In previous tutorials ( Time Value of Options & Volatility), I discussed the impact of volatility on option pricing. To reiterate, a higher volatility increases the likelihood of a huge swing of stock price on either direction. Potential of a huge movement away from stock price increases the likelihood of stock price going beyond strike price. High volatility increases the price of options.

All else equal, higher the volatility, greater the option price

3. Interest rates: How do interest rates impact option prices — for most option traders, I would say the impact is minimal. Most option trading in India happens for expiry dates under 1 month. For such short expiries, impact of interest rates on option pricing is minimal.

However, for the sake of discussion, high interest rate environments lead to expectation of higher returns on stocks. Reason for this is very simple: In a high interest rate scenario, if stock prices don’t generate high expected returns, everyone will stop investing in stock markets and instead, park their savings in fixed deposits/money market instruments. Higher expectation of returns leads to higher stock prices in future — all things equal, high interest rates tend to increase price of call options and reduce price of put options.

High interest rates increase call option prices and reduce put option prices. However impact of interest rates on short dated options is minimal.

For most of our discussions, I usually ignore the impact of interest rates on option prices.

4. Dividend rate: Stocks offer dividends from time to time. How does an expected dividend in future impact option price. Again, impact of dividend rate on option price is minimal — in most of our discussions, I usually ignore impact of stock dividend on options pricing.

For academic discussion, a higher dividend in future will reduce the stock price, all other things being equal ( To avoid arbitrage, markets tend to reduce stock price as soon as a dividend is announced by a company). A drop in stock price increases value of put option and reduces value of a call option.

A high dividend rate reduces price of call option and increases price of put option. For practical trading, impact of dividend on pricing is minimal.

To summarize, here is the impact of various factors on Call and Put Options:

Normal distribution of daily stock price returns

One of the biggest empirical observation about stock prices is that daily returns of any stock are normally distributed. This in essence is a foundation for all modern day option pricing models. What does this mean?

A normal distribution is a bell shaped curve that shows the probability of a particular event happening in future. Peak of bell curve represents an event that has the highest likelihood of happening. As we move away from peak, probability of such events happening reduces.

Let’s look at NIFTY index daily returns for the last 365 days — everyday, we calculate percentage return for that day and plot frequency of returns on a distribution chart. On x-axis, we have the daily return values and on y-axis we have number of instances where daily returns were within a specific range.

In the above illustration, I have plotted daily returns of NIFTY 50 for past 1 year, from 28 May 18 to 25 May 19. Over past 250 trading sessions, notice that daily returns of NIFTY have largely been between -1% to 1%. Infact, if someone asked me to bet, I’d bet that it is most likely that tomorrow’s NIFTY level will be in a range of +/- 0.25%. Notice that there are very few trading sessions when NIFTY index has moved over 2% in either direction — such scenarios constitute what we call as a ‘tail’ of a normal distribution

Option pricing — Black Scholes Model

Taking a fundamental assumption that daily returns follow a normal distribution, option markets have built a pricing model popularly known as ‘Black Scholes’ option pricing model (named after Fischer Black and Myron Scholes who went on to win Nobel Prize in Economics for his work on option pricing). As you would have guessed already, key inputs to Black Scholes option pricing model are:

1. option type (call/put)
2. strike price
3. current stock price
4. time to expiry (in years)
5. annualized volatility
6. interest rate (10 year government bond rate)
7. dividend rate (if any)

I’m skipping the mathematical rigor and actual pricing model — I’ve noticed that most readers psychologically give up when they see complex equations. I don’t want people to be intimidated by such equations — for people interested in option pricing, please email me at sushant.reddy@bigblind.in and I shall send you my option pricing and greeks spreadsheet.

To summarize, we looked at factors impacting option prices and a fundamental assumption that daily stock returns are normally distributed. I hope this tutorial was helpful — please leave your comments and feedback in the section below.