[MUSIC] In this video we're going to discuss the use of options derivative contracts to ensure against market fault. So, we start from the perspective of an investor who has a long position. So he's holding a portfolio invested in a well-diversified portfolio hold in, for example, the equivalent of a stock index, okay? Let's assume that we have correctly described the optimal asset mix. And the part of the portfolio allocation is constituted by investments in the S&P 500. So we hold all these securities in their appropriate proportions. And, let's say that we have a relatively pessimistic view at the short term. We expect the long run expected return to be favorable, and we want to keep that position in the S&P 500, but over the short term, we have some doubt about the stability of the market, and we fear a market fall. So, the question we are asking here, is how can we actually protect our portfolio against such a detrimental variation in the market? So the first thing that comes to mind, is we could simply sell our possession. We could sell our possession, wait for the market fall that we expect to actually happen, and then buy the possession back. This of course comes with significant transaction costs, and this is particularly true for retail investors. So modify the portfolio structure by selling the entire portfolio and buying it back after the period where we expect to observe a market fall is going to be very costly in terms of transaction costs. Also, if we happen to be wrong, and the market rises. If we have sold our position, obviously we won't be able to benefit from this. So another way of protecting the portfolio against the market fall, is to buy an insurance. So how do we buy an insurance in the context of investment in financial securities? Well, we add a pay off that corresponds to an insurance against the market fall. So, we're going to take the situation of a long position in the S&P 500 that we want to protect. And to do so we're going to buy a derivative contract. The question is do we buy a put or do we buy a call? So you've discussed with in the previous video the structure of derivative contracts and what their payoff actually look like. What we know is that a put option is going to benefit the owner of this contract. When the market fall, the good option is a right to sell the underlying at a given price and at a given date, and this right becomes extremely valuable when the underlying value.. Okay, so let's see graphically how this situation would look like. The green line on this graph represents the value at, let's say, an horizon of three months, the value of the underlying security. The scale here is just representative and goes from zero to 200. Let's say we start today at 100, somewhere in the middle of the graph. This is the current level of the S&P 500 this year. If things goes well, the S&P 500 goes up, the value of our portfolio goes up along the green line. If things bad, The S&P level falls and the value of our portfolio goes down along the green line again. So we want to add a payoff to this initial situation, our holding of the S&P 500 which is going to compensate the losses that we might incur when the market falls. And this is the red dotted line that is added below the green line in this graph. And you see that these corresponds to the pay off of the put option with a strike price equal to 100. So this is a contract that is going to benefits its holder when the market falls below this strike price level of 100. And you see that it will exactly compensate any fall below the level of 100. If actually I add up these two lines to create the pay off generated by holding simultaneously, the position in S&P 500 and they put option written on the S&P 500 with a strike price of a $100. This is the combined payoff that I will obtain. And you see that this combined payoff provides a guarantee. Right? If things go well, we will move up along this blue line on the righthand side of the level of 100. If things go bad, well we stay at a level of 100 and our capital is guaranteed. So by purchasing an option, a put option in this context to protect against market fall, we simultaneously guarantee the initial level of the portfolio and participate in any increase in the market. Okay, so this almost looks too good to be true, but as we have extensively discussed in this course, there are no free lunch. Okay. So the option insurance that we purchase the put option that we purchase of course is not free. And this insurance is going to affect the distribution of the return and is going to come, in the form of cost, which will reduce the overall return of the portfolio. So I'm going to show you in the next graph two return distributions. One in red is going to be the return distribution assuming that we don't have the position. And the other one in blue which will be superimposed will describe the distribution of return if we also purchase the insurance. The two distribution are very very different. As you can see, the red distribution goes from minus 100, to a large value, so it means that if things go really really poorly, we could actually lost the entire investment made in the underline stack. The blue distribution. It's a little bit strange, it has such a very big mode, a little bit below zero and apparently there's nothing to the left of the distribution, it is bounded below. And this corresponds to the maximum loss that we can incur by holding simultaneously the stock because the option provides a guarantee there is a minimum level below which we will never fall. Hence, the difference between the red and the blue curve is also manifest in the level of their average. The average of the red distribution, okay so the expected return if we don't hedge is actually going to be higher than the expected return if we hedge. However, the compensation, the risk associated with the blue distribution, the hedged portfolio, it's going to be significantly lower and in particular it's going to remove all that downside risk to the left of the distribution that we actually don't like. And the risk overall is going to be lower for the hedge position. So in conclusion, we can temporarily insure our portfolio against market fall by buying a put option. Now this can be done only if there exists trade and securities, traded derivative securities written exactly on the underline that we want to protect. We can protect the evolution of the market, the evolution of the specific stock, but not necessarily the evolution of an entire portfolio. It might be the case that we have long position in stocks that have no derivative contract traded on it. And therefore in these cases, we cannot purchase interest contract. But in general, if we think about hedging against market variation, against market fall, these contracts are very liquid. It's very easy to trade them, and they're accessible on financial markets. And we can add them to the portfolio allocation to protect against the outsiders.