Let's go to an example on the so-called hedgers. Let's assume you own Apple stock in your portfolio. Let's say in your 401K or in your portfolio you have bunch of Apple stocks. How would you hedge your position? By hedging means, let's say you have the Apple which is at 190 and you want to protect yourself, what happens because it's sitting on for your 401K or is sitting somewhere that in six months you may need money and you want to cash out. What happens in the case that the stock goes lower, then maybe face value today if you have 100 of this, it looks as if you have $19,000 in your portfolio, but if it goes to 140, for example, then from 19,000, you go down to 14,000, then what you would do is, you would buy a put option because you want to now protect your so-called downside risk, and I'm going to give you an example exactly how that works. One more time again, I'm asking this question, what's the benefit of buying the put option, the hedger, She protects the value of her portfolio against big drops in the value of this sort, as I said, that's simply means protecting your downside risk is very important. Remember, in the case of a speculator, a speculator did not own the stock, he or she was just speculating. Here the hedger owns the stock and wants to protect the downside risk. Let's go over the example, the hedger example. Again, we go with two scenarios. In the top one, the hedger did not buy a put option, her today's portfolio is $19,000, I said it earlier. Why? Because today's price is $190 and my assumption is she owns 100 apple stocks in her portfolio. Now, let's again go through the same scenarios. I mean, the scenarios here I'm exaggerating and actually even a bit more. Let's say future price, that means three months or six months from now, it could go down to 130, the worst-case scenario will go up to 250 bucks. Assuming that you don't have any protection, that means you're not hedging your position, what happens is future portfolio value. That means what happens three or six months from now, in this case would be $13,000 if it goes down to $130, $16,000 or the best case scenario would be $25,000. Remember, 13,000 with respect to 19,000 you're down $6,000 already. Now let's look at the case that this speculator bought a six months maturity put as a strike 180, and you paid eight dollars for that. Now, one more time, you save wired paying eight dollar, that's exactly what this options pricing module's trying to do, trying to help you to value the option. Why eight? Why not nine? Why not seven? We'll discuss this later. Let's assume that you bought a six months maturity, something that six months from now would expire, a put option with a strike 180 and you paid today eight dollars for it. That means today your immediate loss is minus $800, immediate loss because you already paid that. That's out of your pocket. But let's look at the scenarios. Now, this is again today's price, five different scenarios. The worst-case scenario going down to $130, the best-case scenario going up to 250. But I'm going to go through each and then what I want you to see is exactly how the hedging your position with a put option would help you with your downside risk. Let's go through it. The payoff. I'm saying the payoff here is $5,000 if he goes down to 130, why? Because that would be different between 130, please do not say 190 because the strike is 180. That means what you would gain is 180 minus 130, which is 50, multiplied by 100 becomes $5,000. Remember this is a put. That means when it goes below the strike you make money. Is a put versus the call, which in the next slide, I'm actually go through a pictorial to explain to how that works. Now for this case, then it goes to 160 would be the difference between 160 and 180, which is your strike. You make $20 multiplied by 100 because it's on 100 stocks is $2,000. Now for the case of 190, because 190 is now above 180, the pay off is zero because you have the right, you're not going to exercise you in a way, it wouldn't makes sense to exercise because is a put option and you never exercised anything which is below the strike price, which is 180, zero and zero for all of this. Now let's look at the future portfolio value. This is very important. Here is today's portfolio exactly as above. That's us for us to. This Is your payoff, which you got it from here. This is the $800 that you paid that I say over here. Then if you sum all of these up, that would become $17,200, much better than this $13,000. The next case, again, the value of the portfolio was 16,000 exactly like here, $2,000 a payoff from here, and $800 that you paid day one. If you do that again, 17,200, do not assume that's by accident is 17,200. That's your chap on your downside, that means the value of the portfolio, no matter what, never goes below 17,200, which is not far from this, no. The worst-case scenario below 19,000 by paying $800 today, you're protecting your downside risk. Let's look at the upside. On the upside here, no payoff from 19,000 which is the value of the portfolio minus 800 is 182,002,124. You may say, as you see, this is lower than this by 800, that's the premium you paid, but at least you're protecting your downside risk. Let's just summarize the observations. What was the effect of when they put option for the hedger? By buying the put option, she hedged her position as a portfolio value would never go below certain level. For us that certain level was $17,200 no matter what, no matter how low the apple stock goes. That's actually how people, and you could imagine a portfolio manager. When it comes to her portfolio, she actively hedging all the positions is having on behalf of her clients. She has tones of apples, Microsoft, Facebook, all those stocks. Actually what she would do is she would go out there and actually would change the positions by buying various different put options on those stocks that she has in her portfolio. Of course, if you have just 100 of them, you buy one option or depends, maybe a fractional, we'll discuss this later, but if we have thousands of them then definitely you would go to buying 10 call options. Thank you.