CFA Level I Derivatives - Binomial Model for Pricing Options

[Music] recall from quantitative methods that a binomial model is based on the idea that over the next period a value will change to one of two possible values one for the upmove and the other for the down move to construct a binomial model we need to know the beginning asset value the size of the up move and down move and the probabilities of the up move and down move occurring to illustrate consider a share of stock currently priced at $50 an analyst wants to estimate the expected price of the stock after one year she estimates that the size of an up move is 25% denoting the up factor as you-you is therefore 1.25 the down factor D has to be the reciprocal of U so in this case D is 0.8 she also estimated that the probability of an up move is 0.6 since they aren't mutually exclusive the probability of a down move is therefore 0.4 by multiplying the current price with the up move factor we get the expected price in the event of an up move to be $62.50 in the event of a down move the expected price is $40 by multiplying each of these values with their respective probabilities and summing them up we get an expected price for $53.50 after one year now you may be wondering how does an analyst estimate the size of an up move and its corresponding probability it'll suffice to say that the size of an up move depends much on the volatility of the stock so an analyst estimates the size of the up move based on that the probability of an up move is calculated based on the size of the moves and the risk-free rate using this formula this probability is not the actual probability of an up move it's known as the risk-neutral pseudo probability of an up move and is commonly denoted with the symbol pi the pseudo probability of a down move PI D is simply 1 minus 4 pi u the derivation of the formula is not required for the level 1 exam so you just need to know how to apply it it'll suffice for you to know that their calculation is based on an arbitrary relationship based on risk-neutral probabilities at the risk-free rate let's quickly determine if the probabilities quoted by the analyst are pseudo probabilities assuming a 7% risk-free rate let's calculate the risk-neutral pseudo probability of an up move plugging in the figures into the formula we get PI U as zero point six and PI D as zero point four so indeed the probabilities from the analyst are pseudo probabilities so how does the binomial tree help us to decide on the value of an option we know that options are derivatives in which it's performance is based solely on its underlying so by estimating the future price of the underlying we can effectively estimate the price of the option now to do that we follow a three-step process using the binomial model the first step is to calculate the payoff of the option at maturity in both the up move and down move States this is followed by calculating the expected value of the option in one year this is the probability weighted average of the payoffs in each of the up-and-down moves and finally discounting this expected value back to today at the risk-free rate let's illustrate this by going back to our earlier example given that the risk-free rate is 7% calculate the value of a one-year call option with an exercise price of $45 we've earlier calculated the expected prices one year from now in the event of an up and down move so our first step is to calculate the payoff of the call option for both the up and down moves using the formula of the payoff at expiration for a call option the payoff in the event of an up move is $17.50 the payoff in the event of a down move is 0 as the option expires out of the money the second step is to calculate the expected option value one year from now using the pseudo probabilities by multiplying the payoffs with their respective probabilities we get an expected value of $10.50 the last step is to simply discount back this value to its present value using the risk-free rate the call option is valued at nine dollars and 81 cents using this binomial pricing approach you're watching an excerpt from our comprehensive and animation library for more videos like these head on down to prep Nuggets comm let's prep Nuggets let us do the hard work for you

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Keywords: CFA Level 1, CFA Level I, Derivatives, CFA Prep Course, CFA Prep Provider, CFA Lesson, CFA Tutorial, CFA Video

Id: JeY4Bq1F-8Q

Length: 5min 31sec (331 seconds)

Published: Thu Jan 16 2020