#! /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 closureDon'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
X*Y = 0Where X* is the transposed conjugate of X.
max{ singular values of A } < bI've tried to understand the min-max theorem but have not got far - if this theorem is relevant to me then please say and if possible give an example.
max{width} = max{height} = (2^16)-1 = 65535I'm not looking just for the best upper bound (i.e. the lowest upper bound). Any upper bound might be enough.
max{ singular values of A } < bI've tried to understand the min-max theorem but have not got far - if this theorem is relevant to me then please say and if possible give an example.
max{width} = max{height} = (2^16)-1 = 65535I'm not looking just for the best upper bound (i.e. the lowest upper bound). Any upper bound might be enough.
glm::mat4 dual_quat_to_mat4(const glm::fdualquat &dq) { glm::fdualquat norm_dq = glm::normalize(dq); // get an identity matrix glm::mat4 mat; // extract rotational information float scalar = norm_dq.real.w; float x = norm_dq.real.x; float y = norm_dq.real.y; float z = norm_dq.real.z; mat[0][0] = (scalar * scalar) + (x * x) - (y * y) - (z * z); mat[0][1] = (2 * x * y) + (2 * scalar * z); mat[0][2] = (2 * x * z) - (2 * scalar * y); mat[1][0] = (2 * x * y) - (2 * scalar * z); mat[1][1] = (scalar * scalar) + (y * y) - (x * x) - (z * z); mat[1][2] = (2 * y * z) + (2 * scalar * x); mat[2][0] = (2 * x * z) + (2 * scalar * y); mat[2][1] = (2 * y * z) - (2 * scalar * x); mat[2][2] = (scalar * scalar) + (z * z) - (x * x) - (y * y); // extract translational information glm::fquat trans = (norm_dq.dual * 2.0f) * glm::conjugate(norm_dq.real); mat[3][0] = trans.x; mat[3][1] = trans.y; mat[3][2] = trans.z; // the last row remains untouched; that row is the realm of clip space and perspective division return mat; }Again, my problem is that all my self-defined standard vectors (defined to establish a consistent set of unit vectors) flipped around. My world up vector of <0.0f, +1.0f, 0.0f> became <0.0f, -1.0f, 0.0f>, my world left vector of <-1.0f, 0.0f, 0.0f> became <+1.0f, 0.0f, 0.0f>, and my world forward vector of <0.0f, 0.0f, -1.0f> became <0.0f, 0.0f, +1.0f>.
The complex conjugate transpose matrix is also called Hermitian transpose. In addition, this type of matrix is usually denoted as A H or A*. Example of the conjugate transpose of a matrix. Here is an example of how to find the conjugate transpose of a matrix: First, we transpose matrix A: A conjugate transpose "A *" is the matrix taking the transpose and then taking the complex conjugate of each element of "A". A A The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix. If A = [a ij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A would be the transpose of A. of It is denoted by A′or (A T).In other words, if A = [a ij] mxn,thenA′ = [a ji] nxm.For example, How to Transpose a Matrix. Matrix transposes are a neat tool for understanding the structure of matrices. Features you might already know about matrices, such as squareness and symmetry, affect the transposition results in obvious ways.... It is very convenient in numpy to use the .T attribute to get a transposed version of an ndarray.However, there is no similar way to get the conjugate transpose. Numpy's matrix class has the .H operator, but not ndarray. Because I like readable code, and because I'm too lazy to always write .conj().T, I would like the .H property to always be available to me. The complex conjugate transpose of a matrix interchanges the row and column index for each element, reflecting the elements across the main diagonal. The operation also negates the imaginary part of any complex numbers. For example, if B = A' and A (1,2) is 1+1i , then the element B (2,1) is 1-1i. The Conjugate Transpose of a Matrix. We are about to look at an important theorem which will give us a relationship between a matrix that represents the linear A unitary matrix is a matrix whose inverse equals it conjugate transpose.Unitary matrices are the complex analog of real orthogonal matrices. If U is a square, complex matrix, then the following conditions are equivalent :. U is unitary.. The conjugate transpose U* of U is unitary.. U is invertible and U − 1 = U*.. The columns of U form an orthonormal basis with respect to the inner product Conjugate transpose. In mathematics, the conjugate transpose (or Hermitian transpose) of an m-by-n matrix A {\displaystyle {\boldsymbol {A}}} with complex entries, is the n-by-m matrix obtained from A {\displaystyle {\boldsymbol {A}}} by taking the transpose and then taking the complex conjugate of each entry (the complex conjugate of a + i b {\displaystyle a+ib} being a − i b {\displaystyle A matrix that is equal to its conjugate transpose is called Hermitian (or self-adjoint). In other words, is Hermitian if and only if. Example Consider the matrix Then its conjugate transpose is As a consequence is Hermitian. Denote by the -th entry of and by the -th entry of .
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In this video we discuss: 1) What is Conjugate of a Matrix? 2) Properties of Conjugate of a Matrix. 3) Transposed Conjugate of a Matrix. 4) Properties of Transposed Conjugate of a Matrix. 5 ... Transpose of a matrix and it's properties. To ask your doubts on this topic and much more, click here:http://www.techtud.com/video-lecture/lecture-transpose Conjugate of Matrix and it's Properties. To ask your doubts on this topic and much more, click here: http://www.techtud.com/video-lecture/lecture-conjugate-m... #Types of Matrices#Types of matrices and example#conjugate matrixHermitian&Skew-HMatrixCBSE-IIT #Types of matrices like orthogonal,conjugate ,transpose conjugate of a matrix, #Hermitian ... In this video we'll learn linear algebra matrices topic named TRANSPOSED CONJUGATE OF A MATRIX. This will be helpful for solving difficult questions asked in... In this video you are going to learn the basic understanding of the chapter matrices. You can watch the previous Chapter videos. Link of playlist is given below- Matrix- https://www.youtube.com ... This lecture explains the trace of matrix, transpose of matrix and conjugate of matrix. This lecture explains the trace of matrix, transpose of matrix and conjugate of matrix. Transpose of a Matrix Example 1 - Duration: ... Mathematics: Conjugate of Matrix - Duration: 6:35. ... Transpose of a matrix product ... In this video we'll learn about conjugate transpose matrix then we'll learn about HERMITION and SKEW HERMITION matrix in English with the help of examples. In mathematics, the conjugate transpose or Hermitian transpose of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking t...
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