Option valuation - the Black-Scholes model (Excel) transcript

hello everyone and welcome again to nettle the best platform around distance learning in business finance economics and much much more my name is Sarah and today we're discussing option valuation using the black-scholes model we want to figure out how to assign fair values to options on set of all the particular characteristics obviously we have already discussed options in great detail they are how you can come up with imatinib strategies to capitalize on your forecasts of underlying share price but we always considered the premiere of food and call options that we are buying or selling as a given and we have never actually discussed in great detail where do these premium originate from how do they correspond to the underlying characteristics of particular options one of the simplest and one of the most famous models that basically impact wine the option premier and their characteristics of the underlying share price and the details of the option contract is the blood calls model and today we are analyzing what this model is all about on the case of HSBC stock and existing options for the ages position so our case is very simple we are starting to trade options for HSBC on the 24th of December 2019 so right before Christmas and we take options with the expiry date at the 17th of April 2020 which is again the third Friday of April typical option expiry dates for any option exchanges the share price at the end of the trading day on the 24th of December has been slightly below six pounds the chair so five hundred ninety nine point for the chair and the poop trying to value calls and puts for the strike price of 600 I may remind you generally strike prices are round numbers that go in particular increments what we need to figure out to perform the valuation first is what is the maturity of the option in years so how much time will pass from today when we want to buy or sell a boot or a call and the expiry date and it will be exercised yeah here we assume that it's a European option so it's exercisable just on the expiry date on the maturity date not at any day from today until expiry date that would be an American option so we're just considering what would happen at the very end of our trading period and the expiry date at the 17th of April so the maturity in years would just the difference we need to pirate date and the start date give us the number of days that pass from today until the expiry but to get it in years we need to divide it by 365 well because there are 365 days in a year and obviously is not as precise as we could about because we are concerned only with trading days so ideally we might have wanted to figure out how many trading pages are between l28 and then divide by 2050 to for example the number of 30 days in here but estimation is also very accurate so you can stick with that this formula and we see that the maturity is roughly a third of the year so 0.3 yes why do we need maturity well remember options are beneficially exercisable if the share price moves either out of the money or in the money so options allow you to capitalize on upward eternal volatility and the longer the time period that will pass from today until the expiry date be more relevant events happen with regards to the underlying company and the higher the magnitude of share price movements might be time is essentially volatility defined in other terms so if the maturity is higher then the expected volatility will be expected magnitude of share price movements that might happen within the breeding period is higher so the options would be more valuable if they are for higher maturity if the expiry date is further the way into the future and we need B with flow rate of this 0.77 percent roughly that is the yield to maturity of UK short-term government bonds and the current date so 24th of December 2019 there is create is required for discounting and to accommodate for the fact that generally we expect all share prices on average to appreciate over time because of the present value considerations and time value of money then we need to estimate the volatility of the underlying share price and we need to estimate it on an annual frequency because obviously we have the maturity in years and have the annual risk-free rate so how to estimate any old volatility of a particular instrument well to do that you can just get daily stock price data for the underlying share price here I've got it from 24th of December 2018 all the way until 24th of December 2019 so the very last pay but be aware of when we start trading obviously we can't use the data that's unavailable to us at the moment when we start trading and how far back do we need to go into the past to estimate storica volatility is an open question you don't want to go too far back because well the events that happened back then might not be relevant for the information the company is in today but you don't want to be too short famous as well as the larger the sample size the more precise is the estimation of volatility so here we use one year worth of Haley beta daily volatility we just need to estimate ideally returns again chair price today divided by the chair price yesterday minus 1 and we correctly get all the way down and get all the returns and then we can calculate any volatility by just applying B standard deviation for the sample photo and we select the whole area of returns and get one one zero seven percent daily volatility then we need to analyze this volatility well we can use the random walk assumption so that the stock returns are independent of both stock returns and that the share price was around the walk and just multiply the a level ability whether it's square root of the number of trading days with Mia and we all know by that point in time that there are beverage hardened 52 trading days within a year so when we multiplied by the square root of the quantity 2 but because variance of independent events scales linearly so standard deviation skills we can just multiply by the square root of the number of training days and it will give us roughly at 17 percent annualized volatility well now let's discuss the heuristics we want to be fulfilled and manifested in our option valuation model in any option valuation model we want to accommodate for three main stillest facts how do straight prices of options maturity and volatility and tribute to option premier have already discussed at stripe price effects call inputs differently if strike price is higher then calls would be less valuable and what would be more valuable because it's more valuable inherently to buy something people and to sell something event but regarding maturity and volatility as we have already discussed the presently both old and put should become more valuable when volatility or maturity increases because with increase in both volatility and maturity the share price at the expiry date is more likely to move either in or out of the money so it will positively contribute to the value of optionality regarding both calls and puts so for any option valuation model that we can consider theoretically those three stylized facts must be manifested in its inherent logic for the model to be basically consistent with the but Cole's model is one of the simplest models that have ever been suggested for option valuation and it has many other simple models in violence applies the random walk concept and the formality assumption to derive inferences about future potential share price movements what it suggests is that stock returns are normally distributed and that they have constant volatility so every single day standard deviation is the same the mean is the same and the returns tomorrow are independent of the returns today utilizing this assumption we can account for potential nightmare upsides and downsides of the share price and derive the value of optionality at the particular strike price without further adue need to estimate two key parameters that we would lock into actual model to estimate likely upsides and downsides and those are D 1 and D 2 you see here those two formulas are very similar with the exception with the sole exception of this side here the squared volatility times time is added and here it is subtracted well the logic is the following this risk-free rate comes for the time value of money consideration that one average stock prices are likely to appreciate in the future this Plus accounts for the fact that this is the potential outside and this fact count for the potential downside it's multiplied by time because well volatility scales linearly with time and this term over here the natural logarithm of the center price so the share price at the start of the trading period divided by the strike price notated as X here becomes for how far away from the center price the strike price is because well obviously if the strike price is very far away from the center price it's less likely that the share price at the expiry date will be either in or out of the money depending on whether you consider a corner put up with those d1 and d2 values so for d1 we hope the parentheses and first coat the numerator so natural labyrinth of the center price divided by the strike price and then we add another break e annual risk-free rate plus the volatility this is fair so standard deviation squared over 2 times maturity and then we close the brackets and that's our numerator or the denominator we need to divide by another bracket which would consist of the standard deviation that is annualised times the square root of maturity and here I can have the same very same logic that Ariens scales linearly and standard deviation scales at a square root so having applied that we can see that our d1 is positive and roughly 0.06 now to calculate D 2 we actually don't need to input the whole formula once again but we can do it you can just copy the formula from this cell exit over here and change that corresponds to this added volatility term to the minus that corresponds to the downside or potential downside given the random walk terms of detail and here we see that DT is negative and it's roughly 0.03 now given all those input we can just plug d1 and d2 into our fair value formulas that can help us to derive the fair values of both calls and puts and here we can see that build for loads again a very similar and B only difference are the signs well why is so because well we know that call options are exercised out of the money so when P share price is higher than the strike price so here the sign is positive we want the share price to be higher than the strike price for the call to be valuable and if the miners because both are exercised in the money so we would like the share price to drop below the strike price for the put option to be valuable so now we can input this formula for the fair value of the co-option so we multiplied the center price by the standard normal distribution more azbest using t1 is our ginsbot and we want to be accumulated here so we put in 1 and I would subtract the right price and the exponent of - risk-free rate times maturity that is basically exponential discounting that allows you to immediately its count by any rate for any time period and then we multiplied by the standard normal distribution of the same weekend we need it cumulative so this represents the potential of the stock price to move out of the money so the fair value of the call and increases with this term and this emphasizes the fact that the share price can also move in the money and then also would not be exercised and the logic is the same F inverse for boots so we need to apply this formula and we see that the fair value of the call option is 23 point 18 no we can again just copy this formula and modify it slightly to get the fair value of the boot as well does that we just need to convert some process into minuses so we need a minus before the first bit reflecting this we need the - before d1e to convert this - into it was and we need to convert this plus into minus and that performs our formula for the fair value of the gold into the formula for the fair value of the what was the formula and see that the fair value for a boot is slightly low and 22 point Bashar now let's assume that our underlying assumptions change and let's figure out how it affects fair values of calls and puts let's assume that our strike price increases for example it's now not 600 but still coming in 10 and we see that as expected the fair value of the call drops and the fair value of the boat increases again just as we can expect then if the strike price drops to 594 double then the reverse had the fair value of the call increases and the fair value of the boot decreases because again it's more valuable to buy something cheap the Transvaal able to tell us something well return back with a color shall we and test what would happen if the expiry date of the option is different let's assume that instead of 17th of April 20 20 our experiment is actually 17th of April 20-21 our maturity will increase roughly by one year let's see what would happen to option fair value given this scenario well the fair value of both options have increased dramatically 259 and 43 respectively so the longer is the trading period the more events can happen that would impact the share price and it's more likely that the share price will move either in or out of the money so when maturity of the option increases it increases the fair value of both calls and puts returning back to this case let's figure out what would happen if we reduce the negative so our expiry date would be closer to the present day of 24th of December so for example if it's 17th of January 2021 well the fair values of both calls and puts would drop reflecting the fact that for the term options are less valuable because unless a movement is likely to occur for the underlying shipment and naturally if we consider various changes in volatility for example if it increases from 1% to 2% per day we have and respective changes increases in option premium if it on the other hand drops to 0.5% that they then we have our respective reduction of value but now the logic that can be applied in terms of both of it should buy or sell parents options is to compare their fare values with their market prices let's see the current option premier that in place on the market on 24th of December when the 19 are the phone so those are purchase and ask prices so the fair value of the call is much higher than the ask price for the call it means that the call options are undervalued because their fair value is higher than the market price which you can hold the object well you always want to buy undervalued instruments in finance so theoretically you could profit from holding a call option for HSBC but well for the option the fair value is lower than the bid price the book it means that the option is overwhelmed it's price is than the fair value well what it would mean is that you could theoretically write the option good option for HSBC at 28.5 and given its fair value is lower than that you would theoretically profit from the transaction and that was primarily at least roughly the underlying logic of one of the most famous investment companies in the late 90s the long-term capital management budget that they applied valuation models for stocks and derivatives and tried to hold undervalued stocks and derivatives and sell over all its doctor derivatives well what is the main limitation of the black Scholes model that corresponded actually to the demise to the failure of long term management in 1998 was the fact that after old model seems the volatility to remain constant and as we all know that is not the best description of real-world financial markets what happens sometimes is that there are periods of high volatility and the periods of volatility so if you have pluck'd in some historical volatility values into your black Scholes model and see that options are overvalued I just mean that volatility the market expects in the future it's higher than what was the case in the past and that is what we can see now on the financial markets when implied volatility so the that would justify the option premiere that currently in place on the market are much higher than historical I was a volatility because I'm surprising even in how turbulent of times we are apparently in Lebanon so beware of the violently applying the petrels model for your option training but still it's very useful to know at least their inherent concept to understand where those option premium come from please leave a like under this video if you found it helpful in the comments below this video please suggest other topics in business economics or finance that you would like me to explain in the future and don't forget to subscribe to our Channel thank you very much stay tuned