In the last video, we spoke about importance of liquidity effects in the market. Now, let's talk about how market frictions producing such effects can be modeled. There exists a few different approaches to modeling market frictions in the market. The first approach focuses on incorporating transaction costs in derivative's pricing. When the size of transaction costs at a given time step is assumed to be fixed function of meat price of a stock at this time. For example, one simple specification is provided by a proportional cost model that says that transaction costs are linear in the size of a trait. The first application of such approach to price options was done by Leland in 1985. Another direction in the literature is to model a balance of demand and supply forces in the market. In this approach, traders are not just price takers who are indifferent to trading derivatives in any quantities. Instead, the price they pay depends on the quantity they trade. The execution price over trade will therefore be determined by a current balance between buyers and sellers in the market. What this approach provides is a sort of a structural model for transaction costs. Of course, a simple proportional cost model with a single fixed transaction cost parameter Kappa is also a structural model, but such structural model with a constant rate, Kappa might be just too simplistic. So, the supply curve approach produces a richer model of transaction costs where parameters are interpreted as parameters of dynamics of demand pressure or the difference between the total demand and total supply for security. Another approach is to model feedback effects of trading stock, for example, as a part of delta-hedging of an option. There are two types of such effects that are discussed in the literature: permanent impact and temporary impact. As was noted by Rogers and Singh in 2006, models that include permanent impact effects might have some issues as models of market liquidity. Therefore as an alternative, they propose modelling liquidity effects as convex transaction costs as opposed to proportional cost models. Now, I will like to talk in a little more details about each one of these approaches. First, let's talk about transaction costs models. First, transaction costs are costs incurred in buying and selling security, that are present due to bid-ask spreads in security prices. Bid-ask spread is simply the difference between the lowest task and highest bid price in the market. Now, importance of transaction costs for pricing strongly depends on the market and security. For example, in a very a liquid market such as markets for government bonds, bid-ask spreads are very small. This makes frequently hedging possible, and it needs to earn. This makes the Black-Scholes price acceptable provided we replace a fixed volatility parameter by a market implied volatility. But for illiquid markets for example, emerging markets, the situation might be quite different. Bid-ask spreads would be quite high, so that frequent rehedging would be very costly, and this will produce a large impact on prices. The first model that included transaction costs in option prices was proposed by Leland in 1985. The model starts in discrete time and assumes that rehedging is done at every time step irrespective of whether it's optimal or not. Leland used the proportional cost model where the costs are equal to Kappa times the absolute value of the trade in the number of stocks times the stock price. The model then applies the standard Black-Scholes Delta, and assumes that a hedge portfolio has a risk-free return as in the Black-Scholes model. It turns out that for plain vanilla call and put options, this whole procedure amounts to the conventional Black-Scholes pricing formula with an adjusted volatility parameter. For example, for long call and put options, the adjusted volatility is shown in this formula on the slide. For short positions, the adjusted volatility is given by a similar formula, but with a plus sign in the bracket instead of the minus sign. Therefore, a long and short positions would have different prices, and this will generate bid-ask spreads. So, this model is very simple. However, it only works for plain vanilla call and put options, and not for other options. Some other more elaborated approaches to modeling proportional transaction costs are described in the book, Derivaties by Paul Willmott, which also discuss his own models in this vein. Now, let's talk a little bit about demand-based models. These models try to take a more fundamental view of market liquidity and price impact. They view an execution price for security as a result of a balance between buyers and sellers. When there are more buyers than sellers, this creates this balance that will drive prices up. This would bring more sellers to the market until prices will go down. In the opposite scenario, when they're are more sellers and buyers, prices will be driven down until more buyers come and push prices up. So, conceptually, what this approach tries to achieve is to produce richer models of execution price and market impact. Instead of saying that the execution price will be a fixed proportion of a mid-price, these models end up with a richer structural model of the execution price. This model is also capable of incorporating dynamic signals into a prediction of execution prices something that proportional cost models cannot do. Now, to avoid any possible confusion before we move to our next topic, I want to emphasize that there are two types of demand forces, and respectively two types of demand-based models for auction markets. The first type of demand supply balance uses the stock market dynamics. This is a popular topic in the literature that deals with stock markets, and not necessarily with auction markets. These considerations are important for auctions because they're hedged with stocks. Another balance of demand and supply applies to trading options themselves. This was discussed in the paper by Garleanu and co-authors that I already mentioned earlier. Finally, I want to briefly discuss models with convex transaction costs. This approach was proposed in 2006 by Rogers and Singh. They criticized proportional cost models for not taking into account the depths of the limit order book. In their approach, they do take into account and analyze price impact from the point of view of demand and supply balance for this stock in the market. They show that the difference between an execution price and mid-price can be determined in terms of a function that essentially inverses the demand supply difference in the market. What they then showed is that if one chooses some simple parameterizations of such function, this produces an effective transaction cost model. Unlike proportional cost models, this transaction cost model is convex or more precisely quadratic in the size of the trade. Option trading is done in this framework by solving the Hamilton Jacobi Bellman or HJB equation, which is a continuous time analog of the Bellman equation that we used extensively in the specialization. If you want to know more details of this elegant model, please consult the original paper, and while here we will move on to our next topic, which will be Models of Feedback Effects.