Homework #1 for MAFS 5220

"""Created on Sun Nov  6 16:26:56 2016@author: Stan WangAssignment_1 Delta hedging of an option"""import scipy as spimport mathimport numpy as npimport scipy.stats as ssdef d1(S0, K, r, sigma, T):    return (np.log(S0/K) + (r + sigma**2 / 2) * T)/(sigma * np.sqrt(T))def d2(S0, K, r, sigma, T):    return (np.log(S0/K) + (r - sigma**2 / 2) * T)/(sigma * np.sqrt(T))# the Black-Shoes option pricedef BS_Call(S0, K, r, sigma, T):    return S0 * ss.norm.cdf(d1(S0, K, r, sigma, T)) - K * np.exp(-r * T) * ss.norm.cdf(d2(S0, K, r, sigma, T))def MC_deltaHedging_Call(S               # stock  price                        ,X               # strike price                        ,T               # maturity                        ,sigma           # volatility                        ,expectedReturn  # the stock's expected Return                        ,r               # riskless return rate                        ,numMC           # number of Monte Carlo                        ,numsim          # number of days to the maturity                        ):    dt = T/float(numsim)    drift=(r-0.5*sigma*sigma)*dt    sigmasqrtdt = sigma * math.sqrt(dt)    portfolio = sp.zeros([numMC],dtype=float)    interest = np.exp(r*dt)    # Assume that we adjust our portfolio at the beginning of the business day;    for j in range(0,numMC):        # At the first day        #print("In the first day: ")        stockPrice = S        callValue01 = BS_Call(stockPrice,X,r,sigma,T)        # according to BS, delta=N(d1)        delta01 = ss.norm.cdf(d1(stockPrice,X,r,sigma,T))        # we long one call option, and short delta stock to hedge it.        # and we put the extra money(positive or negative)        # into the money account        moneyAccount =  delta01 * stockPrice - callValue01        print("the stock price is {}".format(stockPrice))        print("to hedge one call option, we sell {} shares of stock".format(delta01))        # From  the second day to the last day        for i in range(1,numsim):            print("in the {} day: ".format(i+1))            # assume that the stock price is GBM            e = sp.random.normal(0,1)            stockPrice *= np.exp(drift + sigmasqrtdt * e)            print("the stock price is {}".format(stockPrice) )            # every day the money in the money account will earn or pay            # interest at the rate of r;            moneyAccount *= interest            # the call value and corresponding delta;            callValue02 = BS_Call(stockPrice,X,r,sigma,T-i*dt)            delta02 = ss.norm.cdf(d1(stockPrice,X,r,sigma,T-i*dt))            # P&L of this call option;            PnL_call = callValue02 - callValue01            print("the P&L of the option is {}".format(PnL_call))            print("the new delta is {}".format(delta01))            # due to new delta, we need to buy or sell stocks at the value of            # delta01 - delta02 which means buying if positive, or selling if            # negative;            print("the amount of stock we changed is {}".format(delta01-delta02))            # adjust the value of money account;            moneyAccount +=(delta02-delta01)*stockPrice            delta01 = delta02            callValue01=callValue02        # At maturity, the call value is given by terminal condition;        e = sp.random.normal(0,1);        stockPrice *= math.exp(drift+sigmasqrtdt * e)        print("the stock price at maturity is {}".format(stockPrice))        callValue = max(stockPrice-X,0)        print("the call value at maturity is {}".format(callValue))        # we can caculated the portfolio value at maturity;        moneyAccount *= interest        portfolio[j] = moneyAccount+callValue-delta01*stockPrice        print("the portfolio at maturity is {}".format(portfolio[j]))    # After numMC of Monte Cal    portfolioValue =sp.mean(portfolio)    return portfolioValueprint("the result from Monte Carlo for delta hedging strategy is {}".format(MC_deltaHedging_Call(100,100,1,0.1,0.1,0.05,1,360)))# Remark: we can NOT make or lose money by using the delta hedging strategy under the assumptions of Black-Sholes model.