Thursday, 14 May 2020

Black- Scholes Model for Non Math Users

There are two kinds of people on this planet. Mathematicians who may constitute 0.001% (or even lesser than that!) of the total population and the other lesser mortals. So if you are one of those lesser mortals (like me!) and you want to use the Black Scholes model then, this article is for you. This article is based on Prof. Damodaran's real options pricing theory.

To understand Black Scholes without delving too much into calculus, one needs to the understand two concepts - a) replication portfolio and b) arbitrage.

To value or price an option we replicate the cash flows from the option by using other financial instruments as described below.

Call = Borrowing + Buying D of the Underlying Asset

Put = Selling Short D on Underlying Asset + Lending

To replicate the cash flows on a call option without actually buying the option, we need to borrow some money (put in some personal equity) and buy D units of underlying asset. At the end of the time period if the stock price goes up you can sell the stock and repay the borrowed amount (+ interest) and the balance is your profit. Similarly with a put option, you sell short on the underlying asset at the beginning of the period and lend the amount at an interest rate i. At the end of the time period t, if the price of the underlying asset decreases and the principal + interest you received from the borrower can be used to square of your short position and you keep the difference as your profit.

The cash flows on the option and the replication portfolio are the same and in case there is a difference there is scope for arbitrage. Arbitrageurs chip in to ensure that there is an equilibrium in these prices.

In a two period binomial model, we can easily determine the price of the options by working backwards as shown in the figure below. The number of shares that you need to buy in case of a call option is given by D, B is the borrowed amount, K is the strike price, t is the number of time periods, r is the interest rate per period.

By equating the cash flows on the replication portfolio and the call, we are implicitly assuming zero arbitrage and thus we can arrive at the price of the option today. (working backwards from the cash flows at the end of the period)

In the real world, asset prices change continuously and the Black Scholes model is just an extension of the two period model in to the real world in which asset prices change continuously and are normally distributed.

Remember those pay off diagrams for calls and puts. A call option derives value only when the underlying asset is at a price above the strike price. Similarly a put option derives its value only when the price is below the strike price of the option.

Payoff on Options

Once you have an understanding of these basic ideas it is not really difficult to use the Black Scholes model which is presented below.

The Black Scholes Formula for Calls and Puts


C is the price of the call option

P is the price of the put option

S is the Stock Price

K is the strike price

y is the annual dividend yield - dividend paid/current market price of the underlying stock

t is the time period

r is the interest rate per time period

? and sigma for standard deviation

When we move from a two period model to a continuously compounding model, the value of money is assumed to compound at e units per unit of time, e is the euler's number which is equal to 2.7182818284590452353602874713527 (and more ...). So in the equations above determining call and put prices where we use K*e^(-r*t), we are just determining the present value of the strike price as we do not have to forgo or receive the strike price till the end of the period.

A call derives its value only when it is in the money, that is only when K < S, so to arrive at the price/value of the call we find the difference between present value of the stock price and strike price.

N(d1) is the option delta gives you the responsiveness in the value of the option to a change in the value of the underlying asset.

N(d2) is the risk neutral probability of the option being in the money.

In effect to arrive at the value/price of a call option we are deducting the strike price of the option multiplied by the probability of the option being in the money from the present value of the stock price multiplied by option delta. In case of the put option, we are doing the opposite we are subtracting the present value of the stock price multiplied by option delta from the strike price of the option multiplied by the probability of the option being in the money.

d1 and d2 are the mathematical derivations using calculus that you can leave for Manjul Bhargava or Brian Greene !

Template to use Black Scholes Model

Given above is the link to a spread sheet template for using the Black Scholes model. You just have to input the variables in the yellow fields to arrive at the option price.

The template has been tried on some listed stock options on CBOE and the option prices are quite close to the actual bid ask quotes. Hope you find it useful!

(Source: Prof Damodaran Real Options)

Friday, 1 May 2020

Using Regression Analysis to Screen for Stocks (Relative Valuation Approach)

1. Data Source: The data sets for this article have been sourced from (open to everyone) It looks like updates its data every day at 1930 hrs. As the earnings season progresses more numbers will get updated but most of the earnings numbers should belong to 2019. (GIGO - the quality of outputs is subject to the quality of inputs)

2. Consistency and Timing: Multiples approach to valuation is subject to whims and fancies of the analysts. For example, PE ratio which is the most widely used multiple across the global financial markets can have many variations within the rules of the game. Although price is the current market price most of the time, earnings can be previous years earnings or trailing twelve month earnings or forward earnings. An analyst who is bearish on the stock may use past earnings while an analyst who is bullish on the stock may use forward earnings. Even the price used can be the current price, weekly or monthly average or a moving average. To facilitate a fair comparison across companies consistency and timing in the definition of multiples is critical. Both the numerator and the denominator used in the multiple should match each other in consistency and timing. Example, Price to Earnings is a consistent multiple but Price to EBITDA is not a consistent multiple as it scales an equity variable to a firm level variable.

3. Descriptive Stats: Price multiples are skewed distributions as shown in the charts below and subject to selection bias. Negative multiples are not included in the sample as they are not useful in making any meaningful conclusions.

As you can see in the summary stats above, median multiple is a much better representative measure of these data sets than average multiple. These average multiples are affected by very large observations in the right tail. 

 4. Analysis and Application: There are some 5000 stocks listed on BSE and it is very difficult for ordinary unsophisticated retail investors to screen these stocks.

To be fair, it is not a great time to run this regression as many stocks may look under priced due to the impact of Covid-19. (most of the earnings numbers are of 2019 and prices are current prices) Nevertheless the results are presented below for your perusal.

Based on inputs provided by Prof. Damodaran's Relative Valuation a multiple regression has been run on Indian stock multiples and presented in the spreadsheets below.

Price to Earnings Multiple Regression:

Price to Book Value Multiple Regression:

EV to EBITDA Multiple Regression:

(It is not really difficult to run a regression on an excel sheet and predict multiples and readers may give it a shot as this can be a good starting point to make more informed investment decisions !)

A summary of regression results is presented in the table below. By substituting these variables into the equation we can find a predicted multiple. In effect, we are estimating a multiple which is the dependent variable based on its relationship with the independent variables. 

 The results obtained by using these equations can be very different from the results obtained using discounted cash flow (intrinsic value) techniques. Nevertheless, we have a starting point which we can use to investigate further into the fundamentals of under priced companies to discover undervalued companies.Similar analysis can be performed across sectors or global markets.

(Reference and Acknowledgements Prof. Damodaran)