In the first series, we wrapped up our lectures on options pricing. Now in the continuation we will be discussing model calibration. We will be having seven lectures on that and I'm going to start by just do a quick recap of what we've already covered on option pricing modules. What we did was we came up with a generic options pricing model. The way it worked was we said we pick a model. We discussed few models like Geometric Brownian motion, Histone, variance gamma. Then pick a model, pick your parameter set for that model, Theta for example for Geometric Brownian Motion, that would be just Sigma. If you have Histon, then for Histon it would be a five-parameter model, Kappa, Theta, Lambda, Rho, and V nought and if you have Vg, it would be Sigma Mu and Theta. Of course we can have more than that, but we just decided to discuss GBM, Histon and variance gamma VG. For your homework assignments, I might actually decide to come up with some more models for you just to get the use other models as well. Then for a fixed maturity, what that module that we develop would give us the call options prices for a range of strikes. When I'm seeing a call, of course we can do it for a put. We need to make some changes to it that I'm going to assign it actually for your assignments. Notation for model price would be for typically for model price we're showing it as v-hat. The reason I'm calling it a-hat just showing that the model prices, not the market price. Then definitely we are having the spot, a strike risk-free rate, dividend rate, and maturity and the parameter set. I put the semicolon here just to show that the first set of parameters that they're known, the known parameters are contractual parameters. This one is a parameter said that actually, depending on the model, you may have various different parameters for that. I'm emphasizing again here that the Theta is a parameter set. Now for market prices, we discussed already the modal process for market prices. Any price for options prices there are bid and ask prices as well. Now, if you are buying, you get the ask price. If you're selling, you get the bid price. Do not assume a buyer would get the bid price. Note that their bid and ask prices the market. If you're buying, you get the ask price, if you're selling a get the bid price. Definitely it makes sense that when there are more buyers than sellers market would go up and vice versa. That means when the more sellers and buyers, you will see the prices would go down. Now, when it comes to bid and ask, it depends on exactly the type of the options we're working on. For example, if you are working with large trading volumes or the so-called liquid up option contracts. But then ask are typically very tight. On the other hand, when we don't see that much volume or the so-called illiquid options. Then we'll see that the bid-ask or wider. To see this, the best thing I decided to do was I decided to go to Bloomberg. I pick for example, the Apple company. I decided to show you the options prices for Apple stock. I'm having actually 40 slides on that. That was the snapshot that I'm getting on July 23rd of 2018. I'm going from the shortest maturity to the longest maturities. There were nine of them, but I'm not showing you all, I'm just giving you a snapshot of those prices, and I decided to put all of those into excel sheet that I'm providing it to you. For the next seven modules, actually, I will be just working on these prices. A couple of things to notice here is, for example, we see the calls and the puts on this side, on the left-hand side, we're seeing the calls, on the right-hand side, we are seeing the puts. For maturity, the shortest maturity here, which was on August 17th, 2018. We are seeing various different choice going from 170 coming to 210. As I mentioned earlier, we see bid and ask, you'll see the bids are sitting on this side, the ask are sitting on the other side, and is relatively tight. For example, for a strike of 182, we see the bid and ask just being about 10 cents, and the volume are sitting here, some volumes are larger than the others. Now that's a snapshot, at the time that there was taking the options on Apple stock. Now, I'm going a bit to a longer maturity, slightly longer from August. I'm moving to September 21st, 2018, exact same scenario. Bids and ask, and you see the volume this has, for example, the highest volume and very, very low volume here. On a lower volume, as you see, the bid and ask is a bit wider than the higher volume. No question about this. This is exactly what I mentioned earlier, depending on the volume, typically, the bid and ask gets tighter or gets wider. Now, one thing to notice is for equity options, they always expire on Friday. If you go and look at these days, for example, September 21st, 2018, you will see this actually is a Friday. We go to longer maturities, the longer maturity. For example, here I'm having January 17th, 2020, which is with respect to July 23rd, 2018, is relatively long, it's more than a year, year and a half roughly. For that one you see very low volume and the bid and ask are much wider. The longest maturity on Apple that I cannot just find on Bloomberg, it goes to June 19th, 2020, almost two-year maturity. For that one, again, very, very low volume, some volume doesn't actually exist as much, and it's much wider, like $2.85. Then is just for you to give you some insight into exactly what you would see in the market when it comes to bid and ask on options prices. Typically, when we working with options, typically, we work with the so-called mid-price, which is the average between bid and ask, divided by 2. Some people like to make this fun, volume adjusted. At this point we're not getting into details of that. For the assignments, I will ask you actually to play with some of these, at this point I'm not actually getting into it. What I'm showing here to you is I'm actually plot the entire surface for all maturities and all the strikes, and just simply showing you the so-called Call Price Surface. I'm going to switch to the Python code, that what I'm doing is, I first reading the excel sheet that I have all the prices I'm grabbing from Bloomberg. I'm simply reading it. You need to do a bit of formatting, but you have the code, you can go through the code and see how it works and then simply plotting it. When I'm plotting, I want to make sure that they're put it in a matrix form, that means simply, if I'm writing it as C(K, T), C for the call, then the K would be the strike or strikes that would become so-called maturity. Having the surface, that means if anything is missing here, I'm doing interpolation to make sure that I have the surface for every a strike and every maturity because they are not necessarily they would align than in category if you look at the code as I'm going to go through it, you'll see that to make hit for every a strike and maturity, I do interpolation just to set it up to be able to plot the surface. For the surface as you see, I'm showing it in days for maturity goes from very short day to much longer day, roughly two years, a strikes goes from 172-210, and then on this side, vertically you will see the car prices goes from very short maturity to high matures. As you see, fixed maturity, as a strike increases, the prices would go down and this is consistent through every maturity. Then I'm just going to switch to the Python code and actually show you the process I've explained. Reading the Excel sheet and then simply plotting it. Here is the sample code that I'm having for plotting the surface for the option on Apple stock. That Excel sheet, I'm calling it data_ apple.xlsx, and what I'm doing is I'm simply reading it. Python already has the so-called amuse, we are using the pandas library to read the Excel is actually in that regard Python is pretty powerful. We simply reading it and as I said, you read, we have the bid and the ask. What I'm doing for the MIT, I'm simply doing the average. I'm using the mean. You can show some of the data is up to you how you want to show it. But as I mentioned earlier, I'm making a mesh for the strike range and maturity range for the strike by eyeballing you see this goes from a 170-210 for maturity. I'm just going through all and I'm picking the unique days for maturity. That's simply, that's what I'm doing. I'm putting everything in a matrix df calls, and simply setting up the grid from a vector. I'm going to a grid for both the strikes and maturity, and simply what I'm doing as I'm reading, I'm just simply populating that matrix. Nothing more, nothing less and for plotting the surface, I'm using a 3D subplot in Python and simply plotting the surface. Let me just run this one, Shift, Enter, Shift, Enter and here Shift Enter. That's exactly how I'm doing this. One more thing just to remember, as I mentioned earlier, for cases that I do not have the strike as I mentioned earlier, I do interpolation. That means for every maturity, if this strike does not exist, I do interpolation to make sure that I can populate the matrix for every maturity I have the corresponding strikes. That's as far as [inaudible]. In the next module, what we doing is we will be using the so-called disk called prices as the so-called market prices and then we show exactly what we're trying to do with these market prices. Finding a model that we can't do a nice fit to those market prices.
