Implied volatility explained: Solver and Newton-Raphson (Excel) transcript

hi everyone and welcome again to nettle the go-to place to learn about business finance economics and much much more please don't forget to subscribe to our Channel and click that Bell notification button below so that you never miss fresh videos and tutorials you might be interested in many thanks our current patreon supporters and YouTube members for making this video possible it would also greatly appreciate if you consider supporting this as well so picture the link in description click the join button below for more details my name is sava and today we're investigating implied volatility which is one of the most important Concepts in option valuation as it highlights one of the most fundamental sources of option value which is volatility of the underlying and second it presents the informational value of option markets because we can use option prices to form Market implied predictions of the volatility of the underline itself and today we'll investigate the example of Tesla and it is currently as of the 24th of April 2023 trading at 162.83 dollars per share we have also got some data on option prices uh based on the strikes that are quite close to the price of the underlying so strikes that are close to being at the money and we have got data on the most recent premiere on both call and Port options those call and put options are all for 19th of May expiring so around the month until expiring and we'll use those option prices to try to figure out what does the market believe is going to be the level of volatility of Tesla returns in the nearest month uh for that we need to First choose the option that will base our implied volatility calculations on and let's say we are starting at 160 strike let's input 160 and we'll compare the implied volatility that's given by the call option and the put option and discuss some potential ramifications of that but first of all the conceptual procedure uh behind calculating the implied volatility we all know that the black Shields model can be used to extract the fair value of either a call or a put option based on the current price of the underlying the strike price the risk-free rates the maturity of the option and the volatility of the underlying most of the time everything except the volatility of the underlying are directly observable therefore we can plug all of the parameters that are known apart from the volatility of the underlying assume that the blackshells model correctly prices the option and figure out the only unknown variable which is the volatility basically implied volatility is the level of the volatility of the underline that makes black shows valuations Fair so first of all we need to figure out the maturity in years for that we can subtract the current date from the expiry date and divide by 265 so here we've got the measure difference 0.07 years slightly less than a month uh the level of volatility we just guess and then allow it to converge to the value that makes black shows valuation fat so let's say thirty percent started at 30 risk-free rate can be retrieved directly from let's say government bond yields so let's assume it to be around four percent uh then we can calculate D1 and D2 as in usual black holes using uh all of the data that we know and the guess of our volatility level so for D1 we need to figure out the natural logarithm of the current spot price for the underlying subtract the discounted strike price so strike price times the exponent of minus the risk-free rate times maturity in years that's the numerator in the denominator will have the volatility times the square root of maturity for D1 we add half of volatility times square root of maturity and for D2 We need to subtract it so we can either copy this formula and change this plus sign into a minus sign or we could simply subtract volatility times the square root of maturity from D1 which will give us D2 and then we can plug in the fair value formulas for both colonput so for the call we'll multiply the current spot price by the standard normal distribution cumulative of D1 one for cumulative and subtract the discounted strike price so strike price times the exponent of negative risk-free rate times maturity and that we multiply by the standard normal distribution of D2 comma 1 for cumulative we see that the fair value of the call if we assume volatility to be 30 is 6.85 whereas uh in the real world it's 11.9 so if we believe that all other assumptions of black trolls are correct and all other inputs are indeed correct then the only way how this can deviate from The Real World Market price is that our volatility assumption underestimated but drastically so the level of implied volatility of the Tesla stock so Tesla implied volatility would be much higher than 30 percent for this price to be justified however we can also perform the same calculation based on puts for that we need just to change the plus signs to minus signs in the fair value of the call formula that would be quite a bit quicker so we change this plus into minus this plus into a minus this one essential plus and this plus into a minus the reflect the boot called parity condition and that produces a very similar picture the fair value of the port is much lower than the market price for the port at 160 strike if we assume a volatility level of 30 again meaning that the implied volatility level is much higher than 30 percent but how much higher for that we could either use a numerical optimization algorithm like solver that's built into Excel or just as in the good old days use the newton-raphson iterative procedure I'll show you both ways so let's start with the solver algorithm first in solver we need to specify our task let's say we are calculating the implied volatility based on the call option at the strike price of 360. so we want our fair value of the call that is just calculated in cell B12 to be equal to 11.9 which is exactly the market price that we observe and we can only change the volatility that we have guessed initially and the only thing that's left to do is to press solve and then we see that first of all the fair value of the call converges directly to 11.9 which is the market price and volatility uh goes up to 60.57 which is a spectacularly high level of volatility Tesla being a notoriously volatile stock if we would like to do it based on the put option we can click solve once again change our objective cell which will be now the fair value of the put and input the value of the market price of the put which is 6.25 we press solve and based on the uh fair value of the put option implied volatility is quite a bit lower at 46 still dramatically higher than 30 percent we initially assumed we can see a massive deviation UH 60 46 percent or depending on whether we select the call option or the put option that might signal that either bidask spreads are quite large on this Market or that black shows model does not really work for Tesla and there are other considerations apart from volatility that can impact the pricing of those options primarily let's say heavy tail considerations that would lead investors to price heat not only volatility but higher moments of the return distribution But continuing to assume black trolls works we can now change the story to 165 and reevaluate our implied volatility for those two options let's say we click solve now we want to vary the fair value of the 165 call and we want it to be equal to 9.1 which is the market price this gives us a value of 58 very close to 60 we had from the 160 call and finally if we want to evaluate it based on a 165 strike but we can allow cell B13 to converge to the value of 8.39 which is the market price of 165 but click solve and get a volatility figure of 44 percent so we see that the range of implied volatile destinations for Tesla is quite wide however what about the old school newton-raphson algorithm let's say that we want to estimate our implied volatility based on 165 strike put and we start with the same initial yes volatility of 30 what we need to do then is to see how much does the fair value of the port or call depending on what you use deviate from the market price then we need to figure out how sensitive the fair value of the option is the changes in volatility and use the newton-raphson procedure to converge to the true level of implied volatility using a set of iterations we need to calculate D1 and D2 We need to calculate Phi value and we need to calculate the sensitivity of the fair value of the option to volatility and this in the option trading tradition is denoted as Vega which is a pseudo-greek letter there is no such grouplet as Vega but it denotes quite famously the sensitivity of fair values of options to volatility quite uh handle it it is the same for both calls and puts both call options and put options are sensitive to change in volatility in a very similar way so we can use the same formula for Vega regardless of whether you're using a call or a put in this formula uses the spot price P it uses the probability density function of the standard normal distribution applied to D1 and it uses maturity to figure out the sensitivity of the option price to volatility we'll use it in our newton-raphson procedure first of all we need do you want a D2 however this is nothing new natural logarithm of the underlying price blocked minus the national logarithm of discounted strike price strike price clocked times the exponent of negative risk for rate locked times maturity lot that wraps up the numerator in the denominator we input our assumed level of volatility times the square root of maturity locked and for the one we add half of the assumed level of volatility which is over here we do not lock it as we wanted to converge with iterations and multiply by the square root of maturity lot that gives us D1 for D2 We just subtract our assumed level of volatility times the square root of maturity locked for fair value again we are basing it on the put so we need to translate this formula in the language of excel so minus the spot price locked times the standard normal distribution of negative D1 one for cumulative plus the appropriate discounted strike price so strike price locked so the expert of negative risk for rate locked times maturity locked and times the normal standard distribution of negative V2 one for cumulative we see that at 30 percent the fair value of the 165 strike put is quite below 8.39 we need to figure out how sensitive this variable value is to a change in implied volatility and figure out the how much do we need to increase this volatility in the next iteration for that we need the Vega calculation and that is quite easy to do spot price locked times the standard normal distribution of D1 and here we need to input 0 for probability density so not cumulative and Times by the square root of maturity we see that Vega is equal to 16.93 which basically is a an indicator of by how much does the fair value or the price of the option in this case uh put option at 165 strike increase or decrease subject to increases or decreases in volatility so if volatility increases by one percentage point we can assume here that the fair value of the option would increase by around 0.17 dollars and we can check that if we increase our volatility by one percentage point we can see that it the fair value jumps from 6.03 to 6.2 so this particular estimation is quite um useful for this sort of analysis but we want to figure out at which level of volatility the fair value of the option matches the market price exactly and for that we need to perform the iteration we need to add to our volatility the difference between our fair value which we love minus the estimated fair value divided by estimated beta and that gives us a volatility of 43.94 in the first iteration in this formula we can see that if our fair value is below the option price we increase our implied volatility and if it's above we reduce our implied volatility quite consistent with the logic of option pricing and also given the fact that Vega is positive and now we just need to enforce the same calculations again and here we see that even in the first iteration our third value is strikingly close to the market price at 8.39 which means that 43.94 implied volatility is close enough to the true value of implied volatility we can actually see it over here we estimated using solver is 43.91 but what happens if we do another iteration in this case we converge to 8.59 almost exactly and we converge to the true value of 43.91 which we estimated previously using solver using just two iterations via the newton-raphson method and this is the method that used to be quite dominant uh when um numerical optimization tools were not readily available and option Traders had to do with what they had this is how you estimate implied volatility of the underlying directly using option markets using a black shoals model and either solver on Youth and rapson however there are simplified and much more straightforward approaches that estimate implied volatility that will address in our future videos please leave a like on this video if you found it helpful in the comments below I'm able to see any further suggestions for videos and Business Financial economics so let me record and please don't forget subscribe to our Channel consider supports on patreon thank you very much and stay tuned