Black-Scholes model computations

[pearu]

The Black–Scholes formula calculates the price of European put and call options.

The example of the BS formula application can be formulated both as a UDF as well as UDTF.

Goal: show a simple self-contained compute example that uses numpy functions under the hood

References:

• http://janroman.dhis.org/stud/I2014/BS2/BS_Daniel.pdf
• https://towardsdatascience.com/systematically-pricing-financial-options-with-black-scholes-4998c591ccbd - use this for downloading the data
• Black-Scholes example in numba - unvectorize to get the UDF form

Notes:

• The BS formula requires CFN that can be expressed in terms of erf function. RBC supports calling erf from UDFs, however, this will not work when executing the SQL queries on CUDA-enabled omniscidb server (https://github.com/xnd-project/rbc/issues/159). As a workaround, see the numba example where the rational approximation of CFN is used.
• OmnisciDB 5.4 is required to run the tests and example notebooks

Tasks:

• [ ] get the data from example, implement a Python function that retrieves the data and make it available via rbc.examples.retrieve_black_scholes_data(...)
• [x] implement BS formula as UDF in rbc test-suite
• [x] implement BS formula as a UDTF in rbc test-suite
  • [ ] implement a minimal numpy functions support for Omnisci Columns in rbc
• [x] create a Jupyter notebook demonstrating the above

Here is an initial implementation of BS as a UDF:

``` table_name = os.path.splitext(os.path.basename(__file__))[0] @pytest.fixture(scope='module') def omnisci(): config = rbc_omnisci.get_client_config(debug=not True) omnisci = rbc_omnisci.RemoteOmnisci(**config) omnisci.sql_execute(f'DROP TABLE IF EXISTS {table_name}') omnisci.sql_execute(f'CREATE TABLE IF NOT EXISTS {table_name} (S DOUBLE);') for i in range(10): S = 10 + (random.random() - 0.5) * 5 omnisci.sql_execute(f'INSERT INTO {table_name} VALUES ({S})') yield omnisci omnisci.sql_execute(f'DROP TABLE IF EXISTS {table_name}') def test_blackscholes_udf(omnisci): @numba.njit def cnd(d): # element-wise cnd A1 = 0.31938153 A2 = -0.356563782 A3 = 1.781477937 A4 = -1.821255978 A5 = 1.330274429 RSQRT2PI = 0.39894228040143267793994605993438 K = 1.0 / (1.0 + 0.2316419 * np.abs(d)) ret_val = (RSQRT2PI * np.exp(-0.5 * d * d) * (K * (A1 + K * (A2 + K * (A3 + K * (A4 + K * A5)))))) if d > 0: return 1.0 - ret_val return ret_val signature = 'double(double, double, double, double, double, bool)' def blackscholes(stockPrice, optionStrike, optionYears, Riskfree, Volatility, getcall): S = stockPrice X = optionStrike T = optionYears R = Riskfree V = Volatility sqrtT = np.sqrt(T) d1 = (np.log(S / X) + (R + 0.5 * V * V) * T) / (V * sqrtT) d2 = d1 - V * sqrtT cndd1 = cnd(d1) cndd2 = cnd(d2) expRT = np.exp(- R * T) if getcall: callResult = S * cndd1 - X * expRT * cndd2 return callResult else: putResult = X * expRT * (1.0 - cndd2) - S * (1.0 - cndd1) return putResult omnisci(signature)(blackscholes) X = 10 T = 5 R = 0.1 V = 0.2 # Compute test output in the server descr, result = omnisci.sql_execute( f'''select S, blackscholes(S, CAST({X} as DOUBLE), CAST({T} as DOUBLE), CAST({R} as DOUBLE), CAST({V} as DOUBLE), TRUE), blackscholes(S, CAST({X} as DOUBLE), CAST({T} as DOUBLE), CAST({R} as DOUBLE), CAST({V} as DOUBLE), FALSE) from {table_name};''' ) for S, C, P in result: print(S, C, P) ```

[guilhermeleobas]

For future reference, I am using data from this website rather than live data from Yahoo Finance!. One of the columns from this site contains the approximated value for the black-scholes model.

[guilhermeleobas]

Pull Request has been submitted with implementing both the UDF and UDTF black-scholes model. https://github.com/xnd-project/rbc/pull/166

[pearu]

https://github.com/xnd-project/rbc/blob/master/notebooks/black_scholes_UDF.ipynb