scipy.optimize.fmin_cg — SciPy v1.6.0 Reference Guide

Here is the implementation in python of the algorithm in this article:

#! /usbin/python #---------- # This unusual and intriguing algorithm was originally invented # by Michael V. Klibanov, Professor, Department of Mathematics and Statistics, # University of North Carolina at Charlotte. It is published in the following # paper: # M.V. Klibanov, A.V. Kuzhuget and K.V. Golubnichiy, # "An ill-posed problem for the Black-Scholes equation # for a profitable forecast of prices of stock options on real market data", # Inverse Problems, 32 (2016) 015010. #---------- # Script assumes it's called by crontab, at the opening of the market #----- import numpy as np import pause, datetime from bs4 import BeautifulSoup import requests # Quadratic interpolation of the bid and ask option prices, and linear interpolation in between (https://people.math.sc.edu/kellerlv/Quadratic_Interpolation.pdf) def funcQuadraticInterpolationCoef(values): # There is 'scipy.interpolate.interp1d' too y = np.array(values) A = np.array([[1,0,0],[1,-1,1],[1,-2,4]]) return np.linalg.solve(A,y) # https://en.wikipedia.org/wiki/Polynomial_regression def funcUab(t,coef): return coef[2]*t**2 + coef[1]*t + coef[0] def funcF(s, sa, sb, ua, ub): return (s-sb)*(ua-ub)/(sa-sb) + ub # Initialize the volatility and option lists of 3 values optionBid = [0] # dummy value to pop in the loop optionAsk = [0] # dummy value to pop in the loop volatility = [0] # dummy value to pop in the loop # Initalization for the loop Nt = 4 # even number greater than 2: 4, 6, ... Ns = 2 # even number greater than 0: 2, 4, ... twotau = 2 # not a parameter... alpha = 0.01 # not a parameter... dt = twotau / Nt # time grid step dimA = ( (Nt+1)*(Ns+1), (Nt+1)*(Ns+1) ) # Matrix A dimensions dimb = ( (Nt+1)*(Ns+1), 1 ) # Vector b dimensions A = np.zeros( dimA ) # Matrix A b = np.zeros( dimb ) # Vector b portfolio = 1000000 # Money 'available' securityMargin = 0.00083 # EMPIRICAL: needs to be adjusted when taking into account the transaction fees (should rise, see the article p.8) # Wait 10mn after the opening of the market datet = datetime.datetime.now() datet = datetime.datetime(datet.year, datet.month, datet.day, datet.hour, datet.minute + 10) pause.until(datet) # Record the stock and option values and wait 10mn more def funcRetrieveStockOptionVolatility(): # Stock stock_data_url = "https://finance.yahoo.com/quote/MSFT?p=MSFT" stock_data_html = requests.get(data_url).content stock_content = BeautifulSoup(stock_data_html, "html.parser") stock_bid = content.find("td", {'class': 'Ta(end) Fw(600) Lh(14px)', 'data-test': "BID-value"}) print(stock_bid) stock_ask = content.find("td", {'class': 'Ta(end) Fw(600) Lh(14px)', 'data-test': "ASK-value"}) print(stock_ask) stockOptVol[0] = stock_bid.text.split()[0] stockOptVol[1] = stock_ask.text.split()[0] # Option option_data_url = "https://finance.yahoo.com/quote/MSFT/options?p=MSFT&date=1631836800" option_data_html = requests.get(option_data_url).content option_content = BeautifulSoup(option_data_html, "html.parser") call_option_table = content.find("table", {'class': 'calls W(100%) Pos(r) Bd(0) Pt(0) list-options'}) calls = call_option_table.find_all("tr")[1:] it = 0 for call_option in calls: it+=1 print("it = ", it) if "in-the-money " in str(call_option): itm_calls.append(call_option) print("in the money") itm_put_data = [] for td in BeautifulSoup(str(itm_calls[-1]), "html.parser").find_all("td"): itm_put_data.append(td.text) print(itm_put_data) if itm_put_data[0] == 'MSFT210917C00220000': # One single option stockOptVol[2] = float(itm_put_data[4]) stockOptVol[3] = float(itm_put_data[5]) stockOptVol[4] = float(itm_put_data[-1].strip('%')) else: otm_calls.append(call_option) print("out the money") print("bid = ", option_bid, "\nask = ", option_ask, "\nvol = ",option_vol) return stockOptVol # Record option and volatility stockOptVol = funcRetrieveStockOptionVolatility() optionBid.append(stockOptVol[2]) optionAsk.append(stockOptVol[3]) optionVol.append(stockOptVol[4]) # Wait another 10mn to record a second value for the quadratic interpolation datet = datetime.datetime.now() datet = datetime.datetime(datet.year, datet.month, datet.day, datet.hour, datet.minute + 10) pause.until(datet) stockOptVol = funcRetrieveStockOptionVolatility() optionBid.append(stockOptVol[2]) optionAsk.append(stockOptVol[3]) optionVol.append(stockOptVol[4]) tradeAtTimeTau = False tradeAtTimeTwoTau = False # Run the loop until 30mn before closure datet = datetime.datetime.now() datetend = datetime.datetime(datet.year, datet.month, datet.day, datet.hour + 6, datet.minute + 10) while datet <= datetend: datet = datetime.datetime(datet.year, datet.month, datet.day, datet.hour, datet.minute + 10) optionBid.pop(0) optionAsk.pop(0) optionVol.pop(0) stockOptVol = funcRetrieveStockOptionVolatility() stockBid = stockOptVol[0] stockAsk = stockOptVol[1] optionBid.append(stockOptVol[2]) optionAsk.append(stockOptVol[3]) optionVol.append(stockOptVol[5]) # Trade if required if tradeAtTimeTau == True or tradeAtTimeTwoTau == True: # sell if tradeAtTimeTau == True: portfolio += min(optionAsk[2],sellingPriceAtTimeTau) * 140 # sell 140 options bought 10mn ago tradeAtTimeTau = tradeAtTimeTwoTau sellingPriceAtTimeTau = sellingPriceAtTimeTwoTau sellingPriceAtTimeTwoTau = false else: # forecast the option when no trading # Interpolation coefa = funcQuadraticInterpolationCoef(optionAsk) # quadratic interpolation of the option ask price coefb = funcQuadraticInterpolationCoef(optionBid) # quadratic interpolation of the option bid price coefs = funcQuadraticInterpolationCoef(optionVol) # quadratic interpolation of the volatility sigma sa = stockAsk # stock ask price sb = stockBid # stock bid price ds = (sa - sb) / Ns # stock grid step for k in range (0, Ns+1): # fill the matrix and the vector for j in range (0, Nt+1): Atemp = np.zeros( dimA ) btemp = np.zeros( dimb ) print("k = {k}, j = {j}".format(k=k,j=j)) if k == 0: Atemp[ k*(Nt+1)+j, k*(Nt+1)+j ] = 1 btemp[ k*(Nt+1)+j ] = funcUab(j*dt,coefb) elif k == Ns: Atemp[ k*(Nt+1)+j, k*(Nt+1)+j ] = 1 btemp[ k*(Nt+1)+j ] = funcUab(j*dt,coefa) elif j == 0: Atemp[ k*(Nt+1)+j, k*(Nt+1)+j ] = 1 btemp[ k*(Nt+1)+j ] = funcF( k*ds+sb, sa, sb, funcUab(j*dt,coefa), funcUab(j*dt,coefb) ) elif j == Nt: # do nothing pass else: # main case akj = 0.5*(255*13*3)* funcUab(j*dt, coefs)**2 * (k*ds + sb)**2 dts = (twotau-dt)/Nt * (sa-sb-ds)/Ns #---------- #----- Integral of the generator L #---------- #----- time derivative #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j+1) ] = dts / dt**2 # k,j+1 ~ k,j+1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j-1) ] = dts / dt**2 # k,j-1 ~ k,j-1 #----- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j-1) ] = - dts / dt**2 # k,j+1 ~ k,j-1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j+1) ] = - dts / dt**2 # k,j-1 ~ k,j+1 #---------- #----- stock derivative #---------- Atemp[ (k+1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] = akj**2 * dts / ds**4 # k+1,j ~ k+1,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] = 4 * akj**2 * dts / ds**4 # k,j ~ k,j Atemp[ (k-1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] = akj**2 * dts / ds**4 # k-1,j ~ k-1,j #----- Atemp[ (k+1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] = -2 * akj**2 * dts / ds**4 # k+1,j ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] = -2 * akj**2 * dts / ds**4 # k,j ~ k+1,j #----- Atemp[ (k-1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] = -2 * akj**2 * dts / ds**4 # k-1,j ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] = -2 * akj**2 * dts / ds**4 # k,j ~ k-1,j #----- Atemp[ (k+1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] = akj**2 * dts / ds**4 # k+1,j ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] = akj**2 * dts / ds**4 # k-1,j ~ k+1,j #---------- #----- time and stock derivatives #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j+0) ] = akj * dts / (dt*ds**2) # k,j+1 ~ k+1,j Atemp[ (k+1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+1) ] = akj * dts / (dt*ds**2) # k+1,j ~ k,j+1 #----- Atemp[ (k+0)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j+0) ] = - akj * dts / (dt*ds**2) # k,j-1 ~ k+1,j Atemp[ (k+1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j-1) ] = - akj * dts / (dt*ds**2) # k+1,j ~ k,j-1 #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j+0) ] = -2 * akj * dts / (dt*ds**2) # k,j+1 ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+1) ] = -2 * akj * dts / (dt*ds**2) # k,j ~ k,j+1 #----- Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j+0) ] = 2 * akj * dts / (dt*ds**2) # k,j-1 ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j-1) ] = 2 * akj * dts / (dt*ds**2) # k,j ~ k,j-1 #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j+0) ] = akj * dts / (dt*ds**2) # k,j+1 ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+1) ] = akj * dts / (dt*ds**2) # k-1,j ~ k,j+1 #----- Atemp[ (k+0)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j+0) ] = - akj * dts / (dt*ds**2) # k,j-1 ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j-1) ] = - akj * dts / (dt*ds**2) # k-1,j ~ k,j-1 #---------- #---------- #----- Regularisation term - using alpha = 0.01 #---------- #---------- #----- H2 norm: 0 derivative #---------- Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] += alpha # k,j ~ k,j #----- coef = funcF( k*ds+sb, sa, sb, funcUab(j*dt,coefa), funcUab(j*dt,coefb) ) btemp[ (k+0)*(Nt+1)+(j+0) ] += alpha * 2 * coef #---------- #----- H2 norm: time derivative #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j+1) ] += alpha / dt**2 # k,j+1 ~ k,j+1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j-1) ] += alpha / dt**2 # k,j-1 ~ k,j-1 #----- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j-1) ] += -alpha / dt**2 # k,j+1 ~ k,j-1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j+1) ] += -alpha / dt**2 # k,j-1 ~ k,j+1 #----- coef = ( funcF( k*ds+sb, sa, sb, funcUab((j+1)*dt,coefa), funcUab((j+1)*dt,coefb) ) \ - funcF( k*ds+sb, sa, sb, funcUab((j-1)*dt,coefa), funcUab((j-1)*dt,coefb) ) ) / dt btemp[ (k+0)*(Nt+1)+(j+1) ] += alpha * 2 * coef btemp[ (k+0)*(Nt+1)+(j-1) ] += - alpha * 2 * coef #---------- #----- H2 norm: stock derivative #---------- Atemp[ (k+1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += alpha / ds**2 # k+1,j ~ k+1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] += alpha / ds**2 # k-1,j ~ k-1,j #----- Atemp[ (k+1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] += -alpha / ds**2 # k+1,j ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += -alpha / ds**2 # k-1,j ~ k+1,j #----- coef = ( funcUab(j*dt,coefa) - funcUab(j*dt,coefb) ) / (sa - sb) btemp[ (k+1)*(Nt+1)+(j+0) ] += alpha * 2 * coef btemp[ (k-1)*(Nt+1)+(j+0) ] += - alpha * 2 * coef #---------- #----- H2 norm: stock and time derivative #---------- Atemp[ (k+1)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j+1) ] += alpha / (ds*dt) # k+1,j+1 ~ k+1,j+1 Atemp[ (k-1)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j+1) ] += alpha / (ds*dt) # k-1,j+1 ~ k-1,j+1 Atemp[ (k-1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j-1) ] += alpha / (ds*dt) # k-1,j-1 ~ k-1,j-1 Atemp[ (k+1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j-1) ] += alpha / (ds*dt) # k+1,j-1 ~ k+1,j-1 #---------- Atemp[ (k+1)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j+1) ] += -alpha / (ds*dt) # k+1,j+1 ~ k-1,j+1 Atemp[ (k+1)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j-1) ] += -alpha / (ds*dt) # k+1,j+1 ~ k+1,j-1 Atemp[ (k+1)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j-1) ] += alpha / (ds*dt) # k+1,j+1 ~ k-1,j-1 #----- Atemp[ (k-1)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j+1) ] += -alpha / (ds*dt) # k-1,j+1 ~ k+1,j+1 Atemp[ (k+1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j+1) ] += -alpha / (ds*dt) # k+1,j-1 ~ k+1,j+1 Atemp[ (k-1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j+1) ] += alpha / (ds*dt) # k-1,j-1 ~ k+1,j+1 #---------- Atemp[ (k-1)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j-1) ] += alpha / (ds*dt) # k-1,j+1 ~ k+1,j-1 Atemp[ (k-1)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j-1) ] += -alpha / (ds*dt) # k-1,j+1 ~ k-1,j-1 #----- Atemp[ (k+1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j+1) ] += alpha / (ds*dt) # k+1,j-1 ~ k-1,j+1 Atemp[ (k-1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j+1) ] += -alpha / (ds*dt) # k-1,j-1 ~ k-1,j+1 #---------- Atemp[ (k+1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j-1) ] += -alpha / (ds*dt) # k+1,j-1 ~ k-1,j-1 #----- Atemp[ (k-1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j-1) ] += -alpha / (ds*dt) # k-1,j-1 ~ k+1,j-1 #---------- coef = ( funcUab((j+1)*dt,coefa) - funcUab((j+1)*dt,coefb) \ - funcUab((j-1)*dt,coefa) + funcUab((j-1)*dt,coefb) ) / (dt * (sa - sb)) btemp[ (k+1)*(Nt+1)+(j+1) ] += alpha * 2 * coef / (ds*dt) btemp[ (k-1)*(Nt+1)+(j+1) ] += - alpha * 2 * coef / (ds*dt) btemp[ (k-1)*(Nt+1)+(j-1) ] += - alpha * 2 * coef / (ds*dt) btemp[ (k+1)*(Nt+1)+(j-1) ] += alpha * 2 * coef / (ds*dt) #---------- #----- H2 norm: stock second derivative #---------- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j+1) ] += alpha / dt**4 # k,j+1 ~ k,j+1 Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] += 4 * alpha / dt**4 # k,j ~ k,j Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j-1) ] += alpha / dt**4 # k,j-1 ~ k,j-1 #----- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j+0) ] += -2 * alpha / dt**4 # k,j+1 ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+1) ] += -2 * alpha / dt**4 # k,j ~ k,j+1 #----- Atemp[ (k+0)*(Nt+1)+(j+1), (k+0)*(Nt+1)+(j-1) ] += alpha / dt**4 # k,j+1 ~ k,j-1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j+1) ] += alpha / dt**4 # k,j-1 ~ k,j+1 #----- Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j-1) ] += -2 * alpha / dt**4 # k,j ~ k,j-1 Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j+0) ] += -2 * alpha / dt**4 # k,j-1 ~ k,j #---------- #----- H2 norm: time second derivative #---------- Atemp[ (k+1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += alpha / ds**4 # k+1,j ~ k+1,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] += 4 * alpha / ds**4 # k,j ~ k,j Atemp[ (k+1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += alpha / ds**4 # k-1,j ~ k-1,j #----- Atemp[ (k+1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] += -2 * alpha / ds**4 # k+1,j ~ k,j Atemp[ (k+0)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += -2 * alpha / ds**4 # k,j ~ k+1,j #----- Atemp[ (k+1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] += alpha / ds**4 # k,j ~ k,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] += alpha / ds**4 # k,j ~ k,j #----- Atemp[ (k+0)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] += -2 * alpha / ds**4 # k,j ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+0), (k+0)*(Nt+1)+(j+0) ] += -2 * alpha / ds**4 # k-1,j ~ k,j #---------- coef = ( funcF( k*ds+sb, sa, sb, funcUab((j+1)*dt,coefa), funcUab((j+1)*dt,coefb) ) \ - 2 * funcF( k*ds+sb, sa, sb, funcUab((j+0)*dt,coefa), funcUab((j+0)*dt,coefb) ) \ + funcF( k*ds+sb, sa, sb, funcUab((j-1)*dt,coefa), funcUab((j-1)*dt,coefb) ) ) / dt**2 btemp[ (k+0)*(Nt+1)+(j+1) ] += alpha * 2 * coef / dt**2 btemp[ (k+0)*(Nt+1)+(j+0) ] += - alpha * 4 * coef / dt**2 btemp[ (k+0)*(Nt+1)+(j-1) ] += alpha * 2 * coef / dt**2 #---------- #---------- #----- Boundary de-computation #---------- if k+1 == Ns: Atemp[ (k+1)*(Nt+1)+(j+0), (k+1)*(Nt+1)+(j+0) ] = 0 # k+1,j ~ k+1,j Atemp[ (k+1)*(Nt+1)+(j+1), (k+1)*(Nt+1)+(j+1) ] = 0 # k+1,j+1 ~ k+1,j+1 Atemp[ (k+1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j-1) ] = 0 # k+1,j-1 ~ k+1,j-1 btemp[ (k+1)*(Nt+1)+(j+0) ] = 0 # k+1,j btemp[ (k+1)*(Nt+1)+(j+1) ] = 0 # k+1,j+1 btemp[ (k+1)*(Nt+1)+(j-1) ] = 0 # k+1,j-1 if k-1 == 0: Atemp[ (k-1)*(Nt+1)+(j+0), (k-1)*(Nt+1)+(j+0) ] = 0 # k-1,j ~ k-1,j Atemp[ (k-1)*(Nt+1)+(j+1), (k-1)*(Nt+1)+(j+1) ] = 0 # k-1,j+1 ~ k-1,j+1 Atemp[ (k-1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j-1) ] = 0 # k-1,j-1 ~ k-1,j-1 btemp[ (k-1)*(Nt+1)+(j+0) ] = 0 # k-1,j btemp[ (k-1)*(Nt+1)+(j+1) ] = 0 # k-1,j+1 btemp[ (k-1)*(Nt+1)+(j-1) ] = 0 # k-1,j-1 if j-1 == 0: Atemp[ (k+0)*(Nt+1)+(j-1), (k+0)*(Nt+1)+(j-1) ] = 0 # k,j-1 ~ k,j-1 Atemp[ (k+1)*(Nt+1)+(j-1), (k+1)*(Nt+1)+(j-1) ] = 0 # k+1,j-1 ~ k+1,j-1 Atemp[ (k-1)*(Nt+1)+(j-1), (k-1)*(Nt+1)+(j-1) ] = 0 # k-1,j-1 ~ k-1,j-1 btemp[ (k+0)*(Nt+1)+(j-1) ] = 0 # k,j-1 btemp[ (k+1)*(Nt+1)+(j-1) ] = 0 # k+1,j-1 btemp[ (k-1)*(Nt+1)+(j-1) ] = 0 # k-1,j-1 #---------- pass print("-----") print("Atemp = ") print(Atemp) print("-----") print("btemp = ") print(btemp) print("-----") print("-----") A = A + Atemp b = b + btemp print("-----") print("A = ") print(A) print("-----") print("b = ") print(b) print("-----") print("-----") input("Press Enter to continue...") # Conjugate gradient algorithm: https://en.wikipedia.org/wiki/Conjugate_gradient_method x = np.zeros(N).reshape(N,1) r = b - np.matmul(A,x) p = r rsold = np.dot(r.transpose(),r) for i in range(len(b)): Ap = np.matmul(A,p) alpha = rsold / np.matmul(p.transpose(),Ap) x = x + alpha * p r = r - alpha * Ap rsnew = np.dot(r.transpose(),r) if np.sqrt(rsnew) < 1e-16: break p = r + (rsnew / rsold) * p rsold = rsnew print("it = ", i) print("rsold = ", rsold) # Trading strategy sm = (sa + sb)/2 if x[Ns/2*(Nt+1)+Nt/2] >= optionAsk[0] + securityMargin: tradeAtTimeTau = True sellingPriceAtTimeTau = x[Ns/2*(Nt+1)+Nt/2] portfolio -= 140 * optionAsk # buy 140 options if x[Ns/2*(Nt+1)+Nt] >= optionAsk[0] + securityMargin: tradeAtTimeTwoTau = True sellingPriceAtTimeTwoTau = x[Ns/2*(Nt+1)+Nt] portfolio -= 140 * optionAsk # buy 140 options pause.until(datet) # Wait 10mn before the next loop pause.until(datet) datet = datetime.datetime.now() # Time should be around 20mn before closure datet = datetime.datetime(datet.year, datet.month, datet.day, datet.hour, datet.minute + 10) if tradeAtTimeTau == True: # sell stockOptVol = funcRetrieveStockOptionVolatility() optionAsk.pop(0) optionAsk.append(stockOptVol[3]) portfolio += min(optionAsk[2],sellingPriceAtTimeTau) * 140 # Wait 10mn more to sell the last options pause.until(datet) # it should be around 10mn before closure if tradeAtTimeTwoTau == True: # sell stockOptVol = funcRetrieveStockOptionVolatility() optionAsk.pop(0) optionAsk.append(stockOptVol[3]) portfolio += min(optionAsk[2],sellingPriceAtTimeTwoTau) * 140 # Market closure 

Don't put money on this as I'm still debugging (I bet you half a bitcoin I have mistaken a few indices in the H_2 norm)... Here is the discretisation formula I used, to copy-paste on latexbase:

\documentclass[12pt]{article} \usepackage{amsmath} \usepackage[latin1]{inputenc} \title{Klibanov algorithm} \author{Discretisation formula} \date{\today} \begin{document} \maketitle Let $$ a_{k,j} = \frac12\sigma(j\delta_\tau)^2\times(255\times13\times3)\times(k\delta_s+s_a)^2, $$ then \begin{alignat*}{3} J_\alpha(u) = & \sum_{k=1}^{N_s} \sum_{j=1}^{N_t} \left| \frac{u_{k,j+1} - u_{k,j-1}}{\delta_\tau} + a_{k,j} \frac{u_{k+1,j} - 2u_{k,j} + u_{k-1,j}}{\delta_s^2}\right|^2\frac{2\tau - \delta_\tau}{N_t}\frac{s_a - s_b - \delta_s}{N_s}\\ & + \alpha \sum_{k=1}^{N_s} \sum_{j=1}^{N_t} \left| u_{k,j} - F_{k,j}\right|^2 \\ & \qquad + \left| \frac{u_{k,j+1} - u_{k,j-1}}{\delta_t} - \frac{F_{k,j+1} - F_{k,j-1}}{\delta_t}\right|^2 \\ & \qquad + \left| \frac{u_{k+1,j} - u_{k-1,j}}{\delta_s} - \frac{u_{a,j} - u_{b,j}}{s_a - s_b}\right|^2 \\ & \qquad + \left| \frac{(u_{k+1,j+1} - u_{k-1,j+1}) - (u_{k+1,j-1} - u_{k-1,j-1})}{\delta_s\delta_t} \right. \\ & \qquad \qquad \left. - \frac{(u_{a,j+1} - u_{b,j+1}) - (u_{a,j-1} - u_{b,j-1})}{(s_a-s_b)\delta_t}\right|^2 \\ & \qquad + \left| \frac{u_{k,j+1} - 2u_{k,j} + u_{k,j-1}}{\delta_\tau^2} - \frac{F_{k,j+1} - 2F_{k,j} + F_{k,j-1}}{\delta_\tau^2} \right|^2 \\ & \qquad + \left| \frac{u_{k+1,j} - 2u_{k,j} + u_{k-1,j}}{\delta_s^2}\right|^2 \end{alignat*} %% \left| \right|^2 with $\tau = 1$ unit of time (for example 10mn). \end{document} 

Let me know if you see something wrong... And if you want to contribute, feel free