Homework #2 for MAFS 5220

""" @author: Stan Wang assignment 2 : binomial tree for American options; """ import networkx as nx from math import * # build the binomial tree. At each node, the value stands for stock price; def bgoptiongrid(s,T,r,sigma,n): # some useful parameters; deltaT = T/n u = exp(sigma * sqrt(deltaT)) d = 1.0/u a = exp(r*deltaT) p = (a-d)/(u-d) # G stands for the binomial tree; G = nx.Graph() G.add_node((0,0),value=s,time=0) for i in range(0,n+1): for j in range(0,i+1): if i<n: currentvalue = G.node[(i,j)]['value'] G.add_node((i+1,j),value = currentvalue*d, time = (i+1)*deltaT) G.add_node((i+1,j+1), value = currentvalue*u, time = (i+1)*deltaT) G.add_edge((i,j),(i+1,j),value = 1.0 - p) G.add_edge((i,j),(i+1,j+1), value = p) return G class SimpleCall: def __init__(self,strike,maturity): self.strike = strike self.maturity = maturity def payoff(self,price): return max(price - self.strike, 0) class SimplePut(SimpleCall): def payoff(self,price): return max(self.strike - price, 0) def SetEuropeanPayoff(G,n,derivative): for i in range(0,n+1): G.node[(n,i)]['option'] = derivative.payoff(G.node[(n,i)]['value']) def EuropeanBackwardInduction(G,n,discount): for i in range(n-1,-1,-1): for j in range(0,i+1): nextdown= G.node[(i+1,j)]['option'] nextup =G.node[(i+1,j+1)]['option'] nextdownprob = G[(i,j)][(i+1,j)]['value'] nextupprob = G[(i,j)][(i+1,j+1)]['value'] undis = nextup * nextupprob + nextdown * nextdownprob G.node[(i,j)]['option'] = discount * undis return G.node[(0,0)]['option'] def AmericanBackwardInduction(G,n,discount): for i in range(n-1,-1,-1): for j in range(0,i+1): nextdown= G.node[(i+1,j)]['option'] nextup =G.node[(i+1,j+1)]['option'] nextdownprob = G[(i,j)][(i+1,j)]['value'] nextupprob = G[(i,j)][(i+1,j+1)]['value'] undis = nextup * nextupprob + nextdown * nextdownprob G.node[(i,j)]['option'] = max(discount * undis,derivative.payoff(G.node[(i,j)]['value'])) return G.node[(0,0)]['option'] s = 100 sigma = 0.1 strike = 100 T = 1 r = 0.05 n = 5 stepdiscount = exp(-r * T/n) derivative = SimpleCall(strike,T) # derivative = SimplePut(strike,T) G = bgoptiongrid(s,T,r,sigma,n) SetEuropeanPayoff(G,n,derivative) price_euro = EuropeanBackwardInduction(G,n,stepdiscount) price_amer = AmericanBackwardInduction(G,n,stepdiscount) print (price_euro) print (price_amer) #Remark: for call options, the European type and the American type have the same price if the underlying asset does not pay enough dividends; but for the put value, the American type is more valueable than the European type. # another way to code def binomialcall(s,x,T,r,sigma,n, e_a # 0 means European,1 means Ameican; ): deltaT = T/n u = exp(sigma * sqrt(deltaT)) d = 1.0/u a = exp(r*deltaT) p = (a-d)/(u-d) v = [[0 for j in range(i+1)] for i in range(n+1)] for j in range(0,n+1): v[n][j] = max(s * u **j * d ** (n-j) - x,0.0) for i in range(n-1,-1,-1): for j in range(0,i+1): v[i][j] = max(exp(- r * deltaT) * (p * v[i+1][j+1] + (1.0 - p) * v[i+1][j]), e_a*max(s*u**j*d**(i-j)-x,0.0)) return v[0][0] def binomialput(s,x,T,r,sigma,n, e_a # 0 means European,1 means Ameican; ): deltaT = T/n u = exp(sigma * sqrt(deltaT)) d = 1.0/u a = exp(r*deltaT) p = (a-d)/(u-d) v = [[0 for j in range(i+1)] for i in range(n+1)] for j in range(0,n+1): v[n][j] = max(x - s * u **j * d ** (n-j),0.0) for i in range(n-1,-1,-1): for j in range(0,i+1): v[i][j] = max(exp(- r * deltaT) * (p * v[i+1][j+1] + (1.0 - p) * v[i+1][j]), e_a*max(x - s*u**j*d**(i-j),0.0)) return v[0][0] e_a = 1 print (binomialput(s,strike,T,r,sigma,n,e_a)) print (binomialcall(s,strike,T,r,sigma,n,e_a))