Now something else I can do is, I can actually go back to this expression here, and just do some simple algebraic manipulations to get the following. I can also say that the P$L is equal to delta S times Delta S over S, plus Gamma S squared over 2 times Delta S over S all to be squared, plus Vega times Delta Sigma. Now, I can write this. So this is my return, Delta S over S is the return on the stock price. I'm going to call Delta. This Delta here is Delta C Delta S. So Delta times S is often called the ESP, standing for equivalent stock position or the dollar Delta. Delta S over S all to be squared, well, this is my return squared. So this quantity here, Gamma S squared over 2 is sometimes called dollar Gamma. So dollar Gamma is this expression here. What I should emphasize is that in practice, market participants option traders are just investors who happen to invest in options as well as underlying securities stocks and futures and so on. What they will do is, they will often know what the ESP is of their option. So they will typically know the ESP, the equivalent stock position, they will know their dollar Gamma, and they will know their Vega. Knowing these quantities will help them understand how their portfolio behaves as the underlying stock moves and as the volatility parameter Sigma changes. So for example, let's consider the following situation. Suppose the ESP, the equivalent stock position, is equal to one million dollars. Now, this might come about because maybe S is $100, Delta is a half, so a half times 100 is 50. But maybe I've got thousands of these options and altogether, they combine to give me an ESP of one million dollars. Maybe my dollar Gamma is equal to, let's say 500 K, or $500,000. Suppose my Vega is equal to $100,000 per one percent change in Sigma. Now, suppose Delta S over S is equal to 10 percent. So suppose the underlying stock has increased by 10 percent. Maybe this is over the next day. Suppose Sigma goes to whatever the previous value was plus two percentage points? Well, then I can use this expression and these quantities here to approximate my P$L. So in this case, my P$L or profit and loss on my option position will be approximately equal to my ESP, which is one million dollar times 10 percent. So it's going to be one million times 10 percent, plus my dollar Gamma which is 500K, times return squared. My return is 10 percent, so 10 percent squared is one percent, plus my Vega, which is 100K per one percent. So it's plus 100K, and volatility, my Delta Sigma is changed by two percent, so that's times 0.02. This is equal to, let's see, it's a 100K, plus 5K plus 2K, so that's equal to 107K. Of course, I should mention that one can get very different options. So one can get options with very different ESPs, Gammas, and Vegas. What's also interesting is that people use this not just for a single option, but for an entire portfolio of options. One can have an entire portfolio and can compute the ESP for the entire portfolio, and the dollar Gamma for the entire portfolio, and indeed the Vega for the entire portfolio. So one will then understand or maybe represent the exposure of the portfolio in terms of the ESP, the dollar Gamma, and Vega. Typically, of course, they will also know the Theta for the portfolio and maybe some of the other Greeks as well that we haven't had time to go into. So the Greeks are very important, people understand their risk sensitivities in terms of these Greeks. Now, as I've shown you here, understanding your Greeks and typically writing them in terms of quantities like ESP, equivalent stock position, or dollar Gamma, and so on is a very good idea, and market participants do use these approximations all the time. But it is also worth pointing out that for very large moves, an S or Sigma,10 can break down, it will no longer work. The reason it will no longer work is because Taylor's theorem isn't valid for very large moves in S are Sigma. Now, I won't go into any further details on this, but you have to understand, the Taylor's theorem gives you a good approximation for relatively small changes in S and relatively small changes in Sigma. If you get very large moves or extreme moves, then these approximations break down, and they aren't very accurate. In that case, what people often use is scenario analysis. So here's one slide giving you an idea of what is going on with scenario analysis. So what I've shown here is the following. It's an example of a pivot table that I've constructed in Excel. If you don't know what a pivot table is, well then you can use the help facilities in Excel to figure out what they are and how to use them. They can be very useful in many situations. We're not going to say anything more about them here. What we've assumed here is that we've got an options portfolio. The options portfolio is written on the S$P 500. We've got lots of options and maybe we've also got futures in our portfolio. What we've done is we've considered two stresses. We've stressed the underlying security price. In this case, the underlying security price as I said is the S$P 500. Down on this axis, we're considering stresses where the S$P 500 falls by one percent, or it falls by two percent, or five percent, up as much as 20 percent. We're also considering situations where the S$P 500 increases by 1 percent, 2 percent, 5 percent, 10 percent, and 20 percent. Across the x axis up here, we're considering stresses on volatility. So this vol here refers to the Sigma parameter we've been discussing. So this is the segment that enters into the Black-Scholes formula. What we're doing is we're considering Sigma, going into Sigma plus one percentage points, Sigma plus two percentage points, up to Sigma plus 10 percentage points, and down to minus 10 percentage points as well. Then in any one cell, we can see what the profit or loss is on the portfolio at that particular scenario. So for example, down here, this corresponds to the S$P increasing five percent and implied volatility's increasing by five percentage points. Well, in that case, I'm going to see a loss of 4,322. Now, if this is in units of dollars, then its represents $4,322, but maybe it's in units of 1,000, in which case, it represents a loss of $4.3 million. So this is an example of a scenario analysis. People do this all the time in finance with derivatives portfolios. They stress their risk factors. In this case, the risk factors are the prices of the underlying security, and the volatility of the options. Then they re-evaluate their portfolios and these new scenarios and compute the profit or loss in these scenarios. So this is an alternative approach to risk management. It gives you a more global approach than the approach given to us by the Greeks that we saw on the previous slide, where we just use Delta, Vega, and Gamma, and Theta, and so on, to analyze the risk of small changes in the underlying parameters. Here, we're looking at much larger changes in the underlying parameters, be the volatility or the underlying security, and then we figure out what the P$L is in these scenarios. It is important to choose the risk factors and stress levels carefully. It's pretty straightforward to do this with a vanilla options portfolio. By vanilla, I mean where the options are pretty standard or straightforward like European call options. But if you're trying to do scenario analysis with very complex portfolios, portfolios containing complex derivative securities. Understanding what these risk factors are can be a challenge in and of itself. Moreover, figuring out what the appropriate stress levels are, and by stress level I mean a two percent, or 10 percent, or five volatility points. They're examples of stress levels. So with very complex derivatives portfolios, figuring out what appropriate stress levels are can also be very challenging. If you don't believe me, you can just think of what went down during the financial crisis when people were working with CDOs, and asset-backed securities, and ABS-CDOs, and so on. In these situations, the portfolios are very complex with many, many risk factors and understanding how to do scenario analysis with these portfolios was more or less impossible. Indeed, this is one of the many reasons that explain what went down during the financial crisis, and the difficulties with these exotic structured products.