In this video, I would like to talk about liquidity and by this I mean the market liquidity of course and not liquidity of fluids in physics. So what is liquidity? Surprisingly, there is no single commonly accepted definition of market liquidity. In general, liquidity can be defined as the ease of trading securities. This notion can include several different phenomena in the market. One source of market liquidity is provided by exogenous transaction costs such as brokerage fees, processing costs, and transaction taxes. The other source is endogenous and involves considerations related to a balance of demand and supply and forces in the market described in terms of imbalance of demand and supply called demand pressure. Another related endogenous liquidity risk factor is called the inventories, and simply put inventory risk is the risk of not being able to fully convert your assets into their expected cash values. A simple example would be owning a car or a house. If you have a car and want to sell it for its book value is actually far from obvious that you will be able to do it. It all depends on the demand side, that is on the number of potential buyers to your car. If you have only one bidder and he or she bids below you ask price, you either have to sell lower than you planned or to walk away from the deal. A third origin of liquidity effects in the market is related to the role of a private information in trading. If a market player wants to make a large trade, this may be based on some private information that only this player possesses and therefore, trading with uninformed counter-party may end up in a lose. Yet another source of private information in the market is information about order flow. For example, a trading desk may know that a hedge fund needs to liquidate a large position and this rate will likely move the market. And in this case, the desk can sell early at a higher price and buy later at a lower price to make a profit. Yet one more source of liquidity is a difficulty to find a counter-party that is willing to trade a given security in a given quantity, and this is called source friction. This effect is particularly relevant in over-the-counter or OTC markets, in which there is no central marketplace. Now, if investors require compensation for bank liquidity risk then liquidity costs should affect security prices, and more than that, liquidity varies in time. Therefore, liquidity premium requested by investors will also vary in time. All these makes security evaluation much harder than under more conventional and less realistic assumptions of frictionless markets. Let's therefore first talk about the standard approach of standard asset pricing. This approach rests on the assumption of perfectly liquid market, that is a market without any frictions. This assumption is usually combined with one of the three further assumptions that are also related to one another. These are the paradigms of competitive market equilibrium, agent optimality, and the no arbitrage principle. In the last video, we spoke about competitive market equilibrium and how no arbitrage principle is a weaker form of market equilibrium paradigm. No arbitrage is essentially a statement that you cannot be sovereignty that is with probability one make money in one state of nature without paying money in at least one other state. It means that you cannot have a guaranteed profit by trading options. Now, no arbitrage so many holes in the market on average because if it were not true someone would already get infinitely rich, but applying this principle to every single option trait is clearly only an approximation of the classical finance. While the accuracy of this approximation depends on the market, it brings along many simplifications and this is the main reason it's used so widely in both the industry and academia. If strict no arbitrage holds. it's equivalent to the existence of the so-called stochastic discount factor, mt. If we have a security with a dividend process dt, then the price, pt, of the security is given by the expectation of its next period value, plus the next dividend and multiplied by the ratio of this stochastic discount factors at time, t plus one mt, as shown in this formula. We can also iterate this formula over all time steps and obtain an equivalent form of this formula shown in the right hand side. It says that the security price is equal to a total expected dividend discounted by the stochastic discount factor. This is the main equation of the standard asset pricing theory, which can also be obtained using arguments based on optimality of representative agent. It turns out that if investor preferences are expressed by an additive utility function, which we can call u of c sub t over consumption process ct, then the stochastic discount factor can be simply computed as a derivative of one step utility with respect to consumption, ct, which is also called the marginal utility. Now, in real markets frictions are always present and they do impact prices. And a simple way to see it is to notice that market-makers who provide liquidity would simply not exist otherwise. If frictions did not import prices they simply would have no way to make any money and this means that frictionless markets do not exist. This point was explained very clearly in a review of liquidity in asset pricing by Amihud, Mendelson and Pedersen written in 2005. There are also examples in a marketplace where securities with identical or very similar cash flows can have quite different prices. For example, newly issued treasury bills called on-the-run bills, often traded at the lower yield which means a higher price than similar off-the-run bills that you're issued earlier. A very similar effect exist also for on-the-run and off-the-run credit indexes such as CDX index in the US market, as well as for some other instruments. The only possible explanation for such differences is an impact of liquidity, but this means that in real markets single unique stochastic discount factor, mt, that would apply to all securities simply does not exist. In a frictionless market described by the standard asset pricing theory, the price of a security depends only on its cash flows, while the pricing kernel mt is universal for all securities and depends only on the utility of a representative investor, but now effects of illiquidity make such a construction impossible. Still in some liquidity models, liquidity adjustments amount to some kind of modifications over pricing kernel, but in other models, for example in transaction cost based models, there is no pricing kernel. Pricing now becomes dependent not only on future cash flows over security as in the standard approach but also on utilities of individual investors and liquidity of the security. Also, individually you choose may depend on private information. Therefore, models that address liquidity in the market should relax the assumption that all investors have the same information, and it's of course needless to say that all these brings even more complexity to the world of financial modelling for options. We will talk about some of these approaches in our next video.