Option pricing with transaction costs (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 click that Bell notification button below so that you never miss fresh videos and tutorials you might be interested in many thanks to our current patreon supporters and YouTube members for making this video possible and I'd also greatly appreciate if you consider supporting us as well so push the link in description and click the join button below for more details my name is sava and today we're investigating an application or an extension of black shows model for option pricing when transaction costs are not zero black shows model is very famous and the quite widely applied to price options and one of the simplified assumptions it uses is that transaction costs are zero however there is a pretty straightforward extension of the model developed by Leyland in 1985 that allows for non-zero transaction costs the assumptions are as follows we have got a fixed round trip transaction cost as a percentage of Target volume so let's say we can express it in basis points or as percentages and we have got the so-called revision interval or the holding period for our option position which is notated as delta T and it might not be equal to the maturity of the option again if we are just buying and holding if our option strategy is speculative and revolves around just guessing what price is going to be at expiry well then our revision interval our holding period is exactly the same as maturity but if we are doing Dynamic hydrogen our revision interval might be smaller than that again setting revision interval greater than maturity would not be very meaningful in this regard everything else um exact same inputs as in the conventional black shells and what Legend has proven is that you can use Black Shoals as is but revise the volatility calculation by including the term that is the on your round trap trading cost K and your revision interval or your trading Horizon delta T in this adjustment so that's input our model parameters revise the volatility as per the level in 1985 procedure and study how trading costs across various revision intervals affect the values of coal and put options so let's say we have got a strike price of 100 and the underlying price of 100 let's value the at the money option as it's the simplest and the most commonly traded options let's say our annualized volatility is 30 again that's unrevised that can be estimated from historical data for example or from other techniques let's say the maturity of the option is a month again in all models like that we express maturity in years so one month would be a 12th of a year let's say a risk rate is four percent quite typical for these days and let's say our transaction cost is one percent again it's a round trip uh transaction cost so if it costs 50 basis points to sell 50 basis points to buy we include both one percent if you have got just a one-way transaction cost for the London model application you just double it again this is something that you need to keep in mind and let's say that our revision interval would be a week so let's say 1 over 52 as there are 52 weeks in the year approximately and now we can perform the revision of our volatility to plug it into the land extension of the black trolls model for that we need to multiply our starting volatility by the square root of one plus the square root of 2 divided by pi times our round trap transaction cost k divided by the initial volatility Sigma times the square root of the revision interval delta T and having closed the appropriate number of parentheses we can see that due to transaction costs the effective revised volatility is slightly inflated compared to our 30 percent initially again quite um intuitively we can see that if the transaction cost is actually zero then this volatility figure is equivalent to the starting value of the volatility and the level model is equivalent to the black trolls mode but let's see what a transaction cost of one percent actually does to Fair values of our call and put options and for that we need to calculate D1 and D2 just as in the conventional black trolls just using the revised volatility or Sigma hat in place of Sigma so we input the logarithm of the underlying price minus the logarithm of appropriately discounted strike so we input the strike price and multiply it by the exponent of negative risk-free rate times maturity of the option in years that completes the numerator calculations and then we just divide it by the revised volatility Sigma hat times the square root of our option maturity subtract half and we add half times the revived volatility or Sigma hat times the square root of maturity E1 of 0.0825 for D2 We can simply subtract Sigma hat times the square root of maturity from the D1 calculation just as in the conventional black trowels or we could copy this formula and convert this plus into a minus both ways would work so let's show you the first way we copy this formula here change this plus into a minus and that generates a D2 figure of minus 0.0120 alternatively we could have subtracted Sigma hat times the square root of maturity from our D1 figure and we would have obtained the equivalent result and now for the five values of calls and puts we need to use D1 and D2 figures as well as the cumulative distribution function of the standard normal distribution again this extension does not abandon the normality assumption or the constant volatility assumption for that matter that can be more problematic than the zero transaction cost assumption All Things Considered uh here we can multiply the underlying price by the standard normal distribution Norm as this of D1 and input 1 for cumulative and then we subtract the appropriately discounted strike so we multiply by the exponent of minus risk-free rate times maturity we multiplied by the Staten normal distribution of detail cumulative and that gives us a fair value of a call at around four now for the fair value of the put just as in standard blackshells you can copy this formula and change pluses into minuses both in front of the price of strike and in front of D1 and D2 so this minus needs to be changed into a plus and this plus needs to be changing to a minus and we get the fair value of the port of 3.6 now let's perform some comparative Statics uh first see whether the conventional relationships between uh model parameters and option values hold so for example if we increase our strike uh just as with conventional option pricing calls become cheaper and puts become more expensive given the fact that it's more valuable to sell someone at a higher price and less valuable to buy something at a higher price if the strike price is reduced the reverse is true because it's more valuable to have a call with low strike to be able to buy something for cheap rather than to be able to sell something for a cheap price this is quite understandable then we can also play around with the volatility figures if volatility goes down both Fair values of calls and pods go down as low volatility means lower potential payoff for both of our options as it's less likely that the underlying price will go deep out of the money on deep in the money and if volatility goes up the fair values of both calls and puts increase as it's more likely now that the underlying price will be deep out of the money or deep in the money at expiring and the same is true for maturity if our option is more long-term for example two months it increases the fair value and if it's more short term let's say two weeks so 2 over 52 it reduces Fair values of calls and Bots and that preserves the heuristic that a maturity and volatility are ultimately faces of the same concept that result in more substantial deviations from the strike price and that would potentially generate a greater payoff in particular scenario for either holding a portal holding a call option so back to our initial specification let's investigate how changes in transaction cost specific parameters affect valuation if we reduce our transaction cost to 0.5 for example both calls and puts become less valuable because the effective volatility in this case drops if transaction cost goes up fair values are increased quite dramatically given that effective volatility Sigma hat is also increased to Greater degree and the effect of transaction costs is more potent the more frequent the revision interval is and that can be observed both mathematically as square root of delta T here is in the denominator and both from a financial point of view as the uh more frequently you trade the more trading costs affects your ultimate payoff so if we revise not a weekly but daily one over 365 for example well that generates quite a big inflation in our revised volatility corresponding to increases and fair values of the options and if we go for a less frequent revision interval and more long-term holding period let's say one month matching our maturity that's what's appropriate if you're just buying and holding your option strategy then the impact is less pronounced what is also quite useful to observe from this particular model is that it generates quite a notable heterogeneities in option values across traders that face different levels of transaction costs and that face different investment Horizons so an option that might be overvalued to a Trader that has gotten long investment Horizon well it might as well be overvalued for a Trader that has a very short investment Horizon and therefore even if trading costs are the same across all Traders different investment Horizons or different revision intervals would lead to disagreements about the fair values of those contracts and would trigger quite a bit of trading between investors with short Horizons and investors with long Horizons and this is the fundamentals behind introducing transaction costs into black channels option evaluation and some comparative Statics in terms of how the those parameters affect Fair values of calls and Bots please leave a like on this video if you found helpful in the comments below I'll make it to see any further suggestions for videos and business finders economics would let me record and please don't forget to subscribe to our Channel consider support us on patreon thank you very much and stay tuned