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Black-Scholes-Merton Model | NSE Option Pricing Strategy | Quantra by QuantInsti - YouTube
Channel: Quantra
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In the previous video lecture,
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we looked at an analogy to
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pricing options through a dice game.
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We computed the expected payoff
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of the game and concluded that
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the fair price of playing the game
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should be equal to the expected payoff.
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In this video lecture, we will extend
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this analogy to intuitively understand
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the most popular options pricing
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model, the Black Scholes Model,
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also known as Black-Scholes-Merton Model
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or BSM model.
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The BSM formula estimates the
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prices of call and put options.
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Before looking at the formula,
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let us look at the payoff of a call option.
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Suppose, the strike price is INR 100 and
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the stock price is INR 105
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therefore, the profit by exercising
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the option is INR 5, gained by
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buying the stock at INR 100 from
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the option seller and selling it
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at INR 105 in the market.
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In other words, when we buy
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a call option, we pay the strike price and
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receive the stock price.
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Next, we need to understand that
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the option will only be exercised
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when the stock expires in the money.
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We will now compute
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the expected values of the two components.
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The first component
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is payment of strike price given
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the stock ends in the money.
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The expected value will be equal
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to negative of the Strike Price (X)
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multiplied by the probability of
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Stock Price (St) being
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greater than Strike Price (X).
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A negative sign indicates money is
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going out of our account.
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The second component is the receipt
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of stock price given that the stock
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ends in the money.
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The expected value of this term will be
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the stock price at the time of execution,
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multiplied by the probability of
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Stock Price (St) being greater than Strike
Price (X).
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Kindly note that the stock price
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at execution is given by
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the expected value of St,
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given that St is greater than X.
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On combining these two terms
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we get the equation Ct,
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price of call option, is equal to
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expected value of St given that St
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is greater than X, multiplied by
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the probability of St greater than X,
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minus, X into the probability of St
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greater than X.
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This formula intuitively explains
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the Black Scholes Formula,
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the equation for which is,
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Ct is equal to
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St into N of d1 minus X multiplied by
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e raised to the power of
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minus rt multiplied by N of d2.
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Where N is to denote the
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standard normal cumulative
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distribution function.
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d1 and d2 are variables that are defined as shown.
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r is the risk free interest rate,
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sigma is the volatility of returns of
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the underlying asset,
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St is the price of the underlying,
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X is the strike price of the option and
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t is the time to expiration.
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You can now compare the
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intuitive formula with the mathematical
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formula to see that intuitively
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we have understood the BSM model.
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It can be seen that the term N(d2) can
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be compared to the probability of St
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being greater than X.
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There will be a slight difference
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in the N(d1) term, which in addition
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to the probability of St being greater
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than X, will also have
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certain components from the
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expected value of St given that
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option expires in the money,
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from our intuitive formula.
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Let us look at an example to understand
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intuitively how the BSM formula works.
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Assume that stock for company ABC is
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trading at INR 100, and that we are
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the option writers interested in selling
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a call option on stock ABC at a
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strike price of INR 105, expiring one month
from now.
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Assuming that probability of
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the option being in the money
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at the time of expiry is 0.6,
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and the expected value of the stock
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price given that the stock is
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in the money is INR 110.
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Then, by the intuitive formula,
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the option premium should be equal to;
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110 multiplied by 0.6, minus, 105
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multiplied by 0.6, that is, 66 minus 63,
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which is equal to INR 3.
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110 into 0.6 is the component relating to
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selling the stock in the market,
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and 105 into 0.6 is the component for
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buying the stock by exercising the option.
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In the upcoming units,
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a couple of multiple choice questions
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will test your understanding of
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the BSM model, following which
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you will learn how to implement
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this model in Python.
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