# exact solution of B-S for price of Euro (also American) call option
CSt_formula := S*N(d[1]) - K*exp(-r*(T-t))*N(d[2]); NiM+SSxDU3RfZm9ybXVsYUc2IiwmKiZJIlNHRiUiIiItSSJOR0YlNiMmSSJkR0YlNiNGKUYpRikqKEkiS0dGJUYpLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRjVJKF9zeXNsaWJHRiU2IywkKiZJInJHRiVGKSwmSSJUR0YlRilJInRHRiUhIiJGKUY+RiktRis2IyZGLjYjIiIjRilGPg== N_function := x->(1/2)*(1+erf(x/sqrt(2)));
plot(N_function(x),x=-5..5); NiM+SStOX2Z1bmN0aW9uRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYjIiIiIiIjRi4tSSRlcmZHRiU2IyomOSRGLi1JJXNxcnRHRiU2I0YvISIiRi1GJUYlRiU= 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 d1_function := t -> (log(S/K) + (r + (1/2)*sigma^2)*(T-t))/(sigma*sqrt(T-t));
d2_function := t -> (log(S/K) + (r - (1/2)*sigma^2)*(T-t))/(sigma*sqrt(T-t)); NiM+SSxkMV9mdW5jdGlvbkc2ImYqNiNJInRHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSooLCYtSSRsb2dHRiU2IyomSSJTR0YlIiIiSSJLR0YlISIiRjMqJiwmSSJyR0YlRjMqJEkmc2lnbWFHRiUiIiMjRjNGO0YzLCZJIlRHRiVGMzkkRjVGM0YzRjNGOkY1LUklc3FydEc2JEkqcHJvdGVjdGVkR0ZDSShfc3lzbGliR0YlNiNGPUY1RiVGJUYl NiM+SSxkMl9mdW5jdGlvbkc2ImYqNiNJInRHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSooLCYtSSRsb2dHRiU2IyomSSJTR0YlIiIiSSJLR0YlISIiRjMqJiwmSSJyR0YlRjMqJEkmc2lnbWFHRiUiIiMjRjVGO0YzLCZJIlRHRiVGMzkkRjVGM0YzRjNGOkY1LUklc3FydEc2JEkqcHJvdGVjdGVkR0ZDSShfc3lzbGliR0YlNiNGPUY1RiVGJUYl CSt := subs(d[1]=d1_function(t),d[2]=d2_function(t),CSt_formula);
CSt := eval(subs(N=N_function,%)); NiM+SSRDU3RHNiIsJiomSSJTR0YlIiIiLUkiTkdGJTYjKigsJi1JI2xuRzYkSSpwcm90ZWN0ZWRHRjJJKF9zeXNsaWJHRiU2IyomRihGKUkiS0dGJSEiIkYpKiYsJkkickdGJUYpKiRJJnNpZ21hR0YlIiIjI0YpRj1GKSwmSSJUR0YlRilJInRHRiVGN0YpRilGKUY8RjdGPyNGN0Y9RilGKSooRjZGKS1JJGV4cEdGMTYjLCQqJkY6RilGP0YpRjdGKS1GKzYjKigsJkYvRikqJiwmRjpGKUY7RkJGKUY/RilGKUYpRjxGN0Y/RkJGKUY3 NiM+SSRDU3RHNiIsJiomSSJTR0YlIiIiLCYjRikiIiNGKS1JJGVyZkc2JEkqcHJvdGVjdGVkR0YwSShfc3lzbGliR0YlNiMsJCoqLCYtSSNsbkdGLzYjKiZGKEYpSSJLR0YlISIiRikqJiwmSSJyR0YlRikqJEkmc2lnbWFHRiVGLEYrRiksJkkiVEdGJUYpSSJ0R0YlRjtGKUYpRilGQEY7RkEjRjtGLEYsRitGK0YrRilGKSooRjpGKS1JJGV4cEdGLzYjLCQqJkY+RilGQUYpRjtGKSwmRitGKS1GLjYjLCQqKiwmRjZGKSomLCZGPkYpRj9GREYpRkFGKUYpRilGQEY7RkFGREYsRitGK0YrRilGOw== # Exercise L10: 6 month call, r = 0.04
# L.10 Use a binomial lattice to find the value of a value of a six-month call option with strike price # # $52, Current price $55, interest rate 4%, standard deviation 0.25, and Delta t= 1 month.
# L.11 Use a binomial lattice to find the value of a value of a six-month American put and a six-month # # European put option with the same parameters as in exercise L. 10. Fri 11/13
# Read: Sec 13.1, 13.2, 13.3, 13.5.
# Homework: 13.2, 13.4
# L.12 Use the Black-Scholes formulas to compute the values of a European call option and European put # # option with the same parameters as in Exercise L.10. (Show your work.)
T_value := evalf(6/12);
sigma_value := 0.25;
r_value := 0.04;
params := {K = 52, sigma = sigma_value, r = r_value, T = T_value}; NiM+SShUX3ZhbHVlRzYiJCIrKysrK10hIzU= NiM+SSxzaWdtYV92YWx1ZUc2IiQiI0QhIiM= NiM+SShyX3ZhbHVlRzYiJCIiJSEiIw== NiM+SSdwYXJhbXNHNiI8Ji9JIktHRiUiI18vSSZzaWdtYUdGJSQiI0QhIiMvSSJyR0YlJCIiJUYuL0kiVEdGJSQiKysrKytdISM1 CSt_params := subs(params,N=N_function,CSt); NiM+SStDU3RfcGFyYW1zRzYiLCYqJkkiU0dGJSIiIiwmI0YpIiIjRiktSSRlcmZHNiRJKnByb3RlY3RlZEdGMEkoX3N5c2xpYkdGJTYjLCQqKCwoLUkjbG5HRi82IywkRigjRikiI19GKSQiKysrXWlOISM2RilJInRHRiUkISsrKytEckY+RiksJiQiKysrKytdISM1RilGPyEiIiNGRkYsRixGKyQiKysrKys/ISIqRitGKUYpKiYtSSRleHBHRi82IywmJCErKysrKz9GPkYpRj8kIiIlISIjRiksJkYrRiktRi42IywkKigsKEY2RikkIisrKyt2ViEjN0YpRj8kISorKyt2KUY+RilGQkZHRixGK0ZIRitGKSEjXw== plot0 := plot(subs(t=0,CSt_params),S=40..60):
plot1 := plot(subs(t=1/12,CSt_params),S=40..60):
plot2 := plot(subs(t=2/12,CSt_params),S=40..60):
plot3 := plot(subs(t=3/12,CSt_params),S=40..60):
plot4 := plot(subs(t=4/12,CSt_params),S=40..60):
plot5 := plot(subs(t=5/12,CSt_params),S=40..60):
plot6 := plot(subs(t=6/12,CSt_params),S=40..60):
with(plots):
display([plot0,plot1,plot2,plot3,plot4,plot5,plot6]);
subs(S=55,t=0,CSt_params);
evalf(%); 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NiMsKCMiI2IiIiMiIiItSSRlcmZHNiRJKnByb3RlY3RlZEdGK0koX3N5c2xpYkc2IjYjLCQqJiwmLUkjbG5HRio2IyNGJSIjX0YnJCIrKytdaU4hIzZGJ0YnRiYjRidGJiQiK0NyVUdHISIqRiQqJi1JJGV4cEdGKjYjJCErKysrKz9GOUYnLCZGOkYnLUYpNiMsJComLCZGMkYnJCIrKysrdlYhIzdGJ0YnRiZGOkY7RjpGJyEjXw== NiMkIitOKFxnMychIio= # Check by substitution that CSt(S,t) actually satisfies Black-Sholes equation
BS_pde := diff(f(S,t),t)+r*S*diff(f(S,t),S) + (1/2)*sigma^2*S^2*diff(diff(f(S,t),S),S) - r*f(S,t) = 0;
print(` If f(S,t) = CSt is a solution the following should give 0 = 0`);
simplify(subs(f(S,t)=CSt,BS_pde)); NiM+SSdCU19wZGVHNiIvLCotSSVkaWZmR0kqcHJvdGVjdGVkR0YqNiQtSSJmR0YlNiRJIlNHRiVJInRHRiVGMCIiIiooSSJyR0YlRjFGL0YxLUYpNiRGLEYvRjFGMSooSSZzaWdtYUdGJSIiI0YvRjgtRik2JEYsLUkiJEdGKjYkRi9GOEYxI0YxRjgqJkYzRjFGLEYxISIiIiIh NiNJaW5+SWZ+ZihTLHQpfj1+Q1N0fmlzfmF+c29sdXRpb25+dGhlfmZvbGxvd2luZ35zaG91bGR+Z2l2ZX4wfj1+MEc2Ig== NiMvIiIhRiQ= # Use put-call parity to determine price of Euro. put
PSt := CSt + K*exp(-r*(T-t)) - S; NiM+SSRQU3RHNiIsKiomSSJTR0YlIiIiLCYjRikiIiNGKS1JJGVyZkc2JEkqcHJvdGVjdGVkR0YwSShfc3lzbGliR0YlNiMsJCoqLCYtSSNsbkdGLzYjKiZGKEYpSSJLR0YlISIiRikqJiwmSSJyR0YlRikqJEkmc2lnbWFHRiVGLEYrRiksJkkiVEdGJUYpSSJ0R0YlRjtGKUYpRilGQEY7RkEjRjtGLEYsRitGK0YrRilGKSooRjpGKS1JJGV4cEdGLzYjLCQqJkY+RilGQUYpRjtGKSwmRitGKS1GLjYjLCQqKiwmRjZGKSomLCZGPkYpRj9GREYpRkFGKUYpRilGQEY7RkFGREYsRitGK0YrRilGOyomRjpGKUZGRilGKUYoRjs= print(` Uf PSt is a solution the following sould give 0 = 0`);
simplify(subs(f(S,t)=PSt,BS_pde)); NiNJVX5VZn5QU3R+aXN+YX5zb2x1dGlvbn50aGV+Zm9sbG93aW5nfnNvdWxkfmdpdmV+MH49fjBHNiI= NiMvIiIhRiQ= PSt_params := subs(params,PSt); NiM+SStQU3RfcGFyYW1zRzYiLCoqJkkiU0dGJSIiIiwmI0YpIiIjRiktSSRlcmZHNiRJKnByb3RlY3RlZEdGMEkoX3N5c2xpYkdGJTYjLCQqKCwoLUkjbG5HRi82IywkRigjRikiI19GKSQiKysrXWlOISM2RilJInRHRiUkISsrKytEckY+RiksJiQiKysrKytdISM1RilGPyEiIiNGRkYsRixGKyQiKysrKys/ISIqRitGKUYpKiYtSSRleHBHRi82IywmJCErKysrKz9GPkYpRj8kIiIlISIjRiksJkYrRiktRi42IywkKigsKEY2RikkIisrKyt2ViEjN0YpRj8kISorKyt2KUY+RilGQkZHRixGK0ZIRitGKSEjX0ZMRjtGKEZG plot0 := plot(subs(t=0,PSt_params),S=40..60):
plot1 := plot(subs(t=1/12,PSt_params),S=40..60):
plot2 := plot(subs(t=2/12,PSt_params),S=40..60):
plot3 := plot(subs(t=3/12,PSt_params),S=40..60):
plot4 := plot(subs(t=4/12,PSt_params),S=40..60):
plot5 := plot(subs(t=5/12,PSt_params),S=40..60):
plot6 := plot(subs(t=6/12,PSt_params),S=40..60):
with(plots):
display([plot0,plot1,plot2,plot3,plot4,plot5,plot6]);
subs(S=55,t=0,PSt_params);
evalf(%); Error, numeric exception: division by zero
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