Binomial methods are versatile in pricing options for it is suitable for American, European, and Asian options. With an European call option with maturity t, strike price k, spot price S, volatility σ, risk-free rate r:
For put option, the last effective term shall be in the max function shall be:
Stock’s increment, decrements, and probability to move up are given by the below respectively:
One of the CUDA samples from Nvidia is to implement the binomial model on GPU.
Finally my new laptop that sports the new GeForce Nvidia RTX 2060 arrived. It is time to check out the muscle of this little beast with the toolset I’m familiar with.
On the hardware, the laptop is a i7-8750 and 16G RAM with a Turing architecture based GeForce RTX 2060.
The laptop came with full drivers installed. Nevertheless I downloaded the latest drivers and CUDA for the most up-to-date experience. The software include Nvidia GeForce drivers, Visual Studio Express 2017, CUDA Toolkit, and TensorFlow.
Be careful when trying all these bleeding edge technologies, not only because TensorFlow 2.0 is currently in Alpha, compatibility issues may haunt like with previous 1.x TensorFlow on CUDA 10.1. I have to fallback to 10.0 to have TF happy with it (although one can always choose the compile from source approach).
And here are my favorite nbody and Mandelbrot simulation, and also the Black Scholes sample in CUDA. The diagnostic tool in VS gives a nice real time profiling interface with graphs.
Finally for this test drive – TensorFlow with GPU. The installation is smooth until I tried to verify TF on GPU. After several failed attempts I realized it could be that CUDA 10.1 may not be compatible with the TF version installed. There are couples of suggested solutions out there, including downgrading to CUDA 9, but since my GPU is the Turing series this is not an option. Actually TF supports CUDA 10 since v.13. So I finally decided to fall back CUDA to 10.0 from 10.1 and it worked!
The TI Nspire calculator is a great platform for visualizing data via interactive graphs. The built-in facility like input slider for variable value adjustment allowed dynamic visualization to complex equations, like the volatility sensitivity in delta-hedged gains used financial investment. Since this strategy involved a single call option, the volatility exposure equals the vega value of the option.
The following setup on the Nspire provided the functions to calculate the vega values.
This spreadsheet input screen stores the spot prices and the calculated Black Scholes vega values.
Finally, with the data plotting screen the graph of Delta hedged gains of volatility sensitivity is completed. An additional slider control can easily be added on it to adjust an offset variable so as to visualize scenarios under different spot price.
The R package RND computes the implied volatility for a Call option. A sample usage is given as below.
The implied volatility based on the Black-Scholes model differs from realized volatility in that the latter is a retrospective estimate of price, while the former provides insight into the future.
Realized volatility can be derived from more traditional approach like standard deviation and GARCH models. Implied volatility, on the other hand, must be found numerically because the Black-Scholes formula cannot be solved for phi in terms of other parameters. A previous installment provides more mathematical details in TI Nspire.
A much more practical approach than calculating GARCH parameters on a calculator is to do it in R. Not only is there is available packages, retrieving financial data for experimenting is also a piece of cake as the facilities built-in offered convenient access to historical data.
To use GARCH in R the library must be installed first.
To test the library, data are imported using the tSeries package.
A plot of the log return.
Before running the GARCH model, a QQ plot is reviewed.
Finally, the GARCH model is created using the command below.
With trace=off a clean model can be printed after running the model.
Came across a ten-year old article from TI on working with the Black-Scholes pricing model in TI-84. In it, a couple of examples are given to utilize various features of the TI-84 to work with the equation to derive an European call option in the Black-Scholes model. One of these method being invoking the Solver.
Entering the equation again is considered quite cumbersome, and it is not quite sure how to archive the Solver equation for later use. After a couple of tries, it become obvious that the Solver wouldn’t work with a function stored in String at all. Fortunately alternative method is found to somehow persist the equation for later use.
The trick is to make use of the following build-in functions available in the TI-84:
By making use of these two functions, the Solver will be able to handle the Black-Scholes equation which is stored in String properly. Firstly the equation must be stored in a String, and then by making use of String>Equ() function, the equation will be able to persist in one of the equation variables. In this form, the Solver will be happy to work with it in its entirety, which means all variables are considered. The equation stored in Str0 is converted to function Y0, and is then processed properly by the Solver as shown in the below screens. For persisting, this can be done in a program including the definition of the Black-Scholes formula itself.
From the “HP 12C Platinum Solutions Handbook”, an example is given on calculation of refinancing an existing mortgage (on page 7). Since the HP 12C is a special breed specializing in financial calculations, much of the steps are optimized and is different from using financial functions available on other higher end calculators like the TI Nspire. In the following re-work of the same example, the Finance Solver is called from within the Calculator Page and the Vars are recalled for calculations.
Monthly payment on existing mortgage received by lender calculation.
Monthly payment on new payment calculation.
Net monthly payment to lender, and Present value of net monthly payment calculation.