Implied volatility approximation: Brenner and Subrahmanyan method 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 the Bell notification button below so that you never miss fresh videos and tutorials you might be interested in many things stuck around patreon supporters and YouTube members for making this video possible it was also greatly appreciate if you consider supporters as well so let's check the link in description click the join button below for more details my name is saba and today we're investigating a simple way to calculate the implied volatility of an underlying using option markets that has been proposed by Brenna and sabrachmanyan in 1988 in their famous article in financial analyst journal the idea behind it is quite simple first of all it is very uh conventional to use at the money options to extract implied volatility first of all because those markets at the money markets are generally the most liquid they have got the lowest bidesk spreads and therefore you can estimate implied volatility with greater Precision less noise and second as can be shown using option Vega the sensitivity of option value to volatility of the underlying near at the money options are those that are the most sensitive to changes in implied volatility and so it makes the most sense to use add the money or at least near at the money strikes to estimate the implied volatility however um as Brenna and subrachmanyan argue the procedure that involves the newton-raphson iterative method or numeric optimization that we showed in the previous video is unnecessarily complicated if the underlying price is close to the strike price because the formula of the black trolls a model can be simplified to provide a nice close form approximation of implied volatility and if we base our calculation just on one option say one call option then we can easily calculate the implied volatility based on the premium of the call the spot price and the maturity of the option T and this constant 0.398 emerges from the numerical approximations of the standard normal distribution function that is involved in the black trolls formula so here let's apply this particular procedure to the options in Tesla as of the 24th of April 2023 with 19th of May expiry keeping in mind that the current spot price is 162.83 dollars and the Premier for respective options at near-end money strikes again as advised are as follows so first for the maturity of the option we just need to subtract the current date from the expiry date and divide by 365 as we want maturity in yes always giving us roughly 0.07 and that's all we need to know in terms of calculations to figure out the implied volatility based on the brand and subrachmanian method so for the call option let's say we want 160 strike so we just divide the call Premium by 0.398 times our spot price that's given and the square root of maturity that gives us an implied volatility of a whopping 70 again Tesla is expected to be quite volatile over the next month as it turns out given the option prices however this is not the most precise approximation that we can do because the strike of 165 is closer to our spot price than the strike of 160. so the motivation would be to go for the strike that is the closest to the Sport price and that gives us a value of 54 we can see that the gap between the two estimates is quite large both quantitatively and qualitatively however this poses an issue um as we need them to acknowledge that spot prices can be quite far from Strike prices that are available again generally strike prices are just nice round numbers and we choose an appropriate number of strikes as an exchange for example that facilitates option trading to enable enough liquidity in each of those markets but provide enough variety uh and um unavoidably sometimes the spot price is quite far away from either strikes that are available and uh for this particular reason Brandon sabrachmanyan also provided a calculation based not on an individual option but on a straddle with both a call and a port at a nearer at the money strike and their argument is that this allows you to estimate the implied volatility um even in a situation where your spot price is slightly far away from strikes that are available and it also filters out a lot of the noise associated with us spreads and maybe potential mispricings associated with call input markets so for that we need to figure out the price of a straddle again it's just the sum of call and put premiere for a particular strike and then we can calculate the implied volatility based on the half of the price of the straddle divided by the same expression 0.398 times the Sport price n times the square root of maturity that we need to lock as well and that generates very close implied volatility figures of 54 of 52 we can see that filters out a lot of noise that was associated with using just the call option say uh so this first of all Narrows down the range of implied volatility estimations and it also allows you to use this approach which is again an approximation so there is some estimation error involved but this particular method based on the straddle allows you to use this approach for a wider range of strikes and for a wider range of markets and that's the approach of Brenner and subrachmanyan to estimating the implied volatility without having to rely on numerical optimization or newton-raphson iterations please leave a like on this video from helpful in the comments below I mean to see any further suggestions for videos and business finance economics would like me to record and please don't forget to subscribe to our Channel consider the sports on patreon thank you very much and stay tuned