# Derivation and examples single and multi period call pricing e1 := u*a + R*b = C[u]; e2 := d*a + R*b = C[d];NiM+SSNlMUc2Ii8sJiomSSJ1R0YlIiIiSSJhR0YlRipGKiomSSJSR0YlRipJImJHRiVGKkYqJkkiQ0dGJTYjRik=NiM+SSNlMkc2Ii8sJiomSSJkR0YlIiIiSSJhR0YlRipGKiomSSJSR0YlRipJImJHRiVGKkYqJkkiQ0dGJTYjRik=subs(solve({e1,e2},{a,b}));NiM8JC9JImJHNiIqKCwmKiZJInVHRiYiIiImSSJDR0YmNiNJImRHRiZGK0YrKiZGL0YrJkYtNiNGKkYrISIiRitJIlJHRiZGMywmRipGK0YvRjNGMy9JImFHRiYqJiwmRixGM0YxRitGK0Y1RjM=C = subs(%,a+b); C = simplify(rhs(%)); NiMvSSJDRzYiLCYqJiwmJkYkNiNJImRHRiUhIiImRiQ2I0kidUdGJSIiIkYwLCZGL0YwRitGLEYsRjAqKCwmKiZGL0YwRilGMEYwKiZGK0YwRi1GMEYsRjBJIlJHRiVGLEYxRixGMA==NiMvSSJDRzYiKigsKiomSSJSR0YlIiIiJkYkNiNJImRHRiVGKiEiIiomRilGKiZGJDYjSSJ1R0YlRipGKiomRjJGKkYrRipGKiomRi1GKkYwRipGLkYqRilGLiwmRjJGKkYtRi5GLg==qformula := (R-d)/(u-d); Cformula := (1/R)*(q*C[u] + (1-q)*C[d]); C = Cformula; subs(q=qformula,%); simplify(%);NiM+SSlxZm9ybXVsYUc2IiomLCZJIlJHRiUiIiJJImRHRiUhIiJGKSwmSSJ1R0YlRilGKkYrRis=NiM+SSlDZm9ybXVsYUc2IiomSSJSR0YlISIiLCYqJkkicUdGJSIiIiZJIkNHRiU2I0kidUdGJUYsRiwqJiwmRixGLEYrRihGLCZGLjYjSSJkR0YlRixGLEYsNiMvSSJDRzYiKiZJIlJHRiUhIiIsJiomSSJxR0YlIiIiJkYkNiNJInVHRiVGLEYsKiYsJkYsRixGK0YoRiwmRiQ2I0kiZEdGJUYsRixGLA==NiMvSSJDRzYiKiZJIlJHRiUhIiIsJiooLCZGJyIiIkkiZEdGJUYoRiwsJkkidUdGJUYsRi1GKEYoJkYkNiNGL0YsRiwqJiwmRixGLComRitGLEYuRihGKEYsJkYkNiNGLUYsRixGLA==NiMvSSJDRzYiKigsKiomSSJSR0YlIiIiJkYkNiNJImRHRiVGKiEiIiomRilGKiZGJDYjSSJ1R0YlRipGKiomRjJGKkYrRipGKiomRi1GKkYwRipGLkYqRilGLiwmRjJGKkYtRi5GLg==params := {S=40,u=1.125,d=0.9,R=1.05, K=42};NiM+SSdwYXJhbXNHNiI8Jy9JIlNHRiUiI1MvSSJ1R0YlJCIlRDYhIiQvSSJkR0YlJCIiKiEiIi9JIlJHRiUkIiQwIiEiIy9JIktHRiUiI1U=qvalue := subs(params,qformula);NiM+SSdxdmFsdWVHNiIkIitubW1tbSEjNQ==Cvalue := subs(q=qvalue,Cformula);NiM+SSdDdmFsdWVHNiIqJkkiUkdGJSEiIiwmJkkiQ0dGJTYjSSJ1R0YlJCIrbm1tbW0hIzUmRis2I0kiZEdGJSQiK0xMTExMRjAiIiI=C_u_value := eval(subs(params,max(u*S-K,0))); C_d_value := eval(subs(params,max(d*S-K,0)));NiM+SSpDX3VfdmFsdWVHNiIkIiUrSSEiJA==NiM+SSpDX2RfdmFsdWVHNiIiIiE=Cvalue := subs(C[u]=C_u_value,C[d]=C_d_value,Cvalue);NiM+SSdDdmFsdWVHNiIsJCokSSJSR0YlISIiJCIrKysrKz8hIio=Cvalue := subs(params,Cvalue);NiM+SSdDdmFsdWVHNiIkIiswPncvPiEiKg==# Exercise L10: 6 month call, delta t = 1 month r = 0.04 deltat_value := evalf(1/12); sigma_value := 0.25; u_value := evalf(exp(sigma_value*sqrt(deltat_value))); d_value := evalf(1/u_value); r_value := 0.04; R_value := 1.0 + deltat_value*r_value; params := {S=55, u = u_value, d=d_value, R = R_value, K = 52}; qvalue := subs(params,qformula); NiM+SS1kZWx0YXRfdmFsdWVHNiIkIitMTExMJCkhIzY=NiM+SSxzaWdtYV92YWx1ZUc2IiQiI0QhIiM=NiM+SSh1X3ZhbHVlRzYiJCIrV24kWzIiISIqNiM+SShkX3ZhbHVlRzYiJCIrLiZRUEkqISM1NiM+SShyX3ZhbHVlRzYiJCIiJSEiIw==NiM+SShSX3ZhbHVlRzYiJCIrTExMLjUhIio=NiM+SSdwYXJhbXNHNiI8Jy9JIlNHRiUiI2IvSSJLR0YlIiNfL0kidUdGJSQiK1duJFsyIiEiKi9JIlJHRiUkIitMTEwuNUYxL0kiZEdGJSQiKy4mUVBJKiEjNQ==NiM+SSdxdmFsdWVHNiIkIitYZ1JdXSEjNQ==C_uuuuuu_value := eval(subs(params,max(((u^6)*S-K),0))); C_uuuuud_value := eval(subs(params,max(((u^5)*(d)*S-K),0))); C_uuuudd_value := eval(subs(params,max(((u^4)*(d^2)*S-K),0))); C_uuuddd_value := eval(subs(params,max(((u^3)*(d^3)*S-K),0))); C_uudddd_value := eval(subs(params,max(((u^2)*(d^4)*S-K),0))); C_uddddd_value := eval(subs(params,max(((u)*(d^5)*S-K),0))); C_dddddd_value := eval(subs(params,max(((d^6)*S-K),0)));NiM+SS9DX3V1dXV1dV92YWx1ZUc2IiQiK11wVSFHJCEiKQ==NiM+SS9DX3V1dXV1ZF92YWx1ZUc2IiQiKzclPjE5IyEiKQ==NiM+SS9DX3V1dXVkZF92YWx1ZUc2IiQiK1ZyK2E2ISIpNiM+SS9DX3V1dWRkZF92YWx1ZUc2IiQiKisrKyskISIpNiM+SS9DX3V1ZGRkZF92YWx1ZUc2IiIiIQ==NiM+SS9DX3VkZGRkZF92YWx1ZUc2IiIiIQ==NiM+SS9DX2RkZGRkZF92YWx1ZUc2IiIiIQ==C_uuuuu_value := subs(params,subs(q=qvalue,C[u]=C_uuuuuu_value,C[d]=C_uuuuud_value,Cformula)); C_uuuud_value := subs(params,subs(q=qvalue,C[u]=C_uuuuud_value,C[d]=C_uuuudd_value,Cformula)); C_uuudd_value := subs(params,subs(q=qvalue,C[u]=C_uuuudd_value,C[d]=C_uuuddd_value,Cformula)); C_uuddd_value := subs(params,subs(q=qvalue,C[u]=C_uuuddd_value,C[d]=C_uudddd_value,Cformula)); C_udddd_value := subs(params,subs(q=qvalue,C[u]=C_uudddd_value,C[d]=C_uddddd_value,Cformula)); C_ddddd_value := subs(params,subs(q=qvalue,C[u]=C_uddddd_value,C[d]=C_dddddd_value,Cformula));NiM+SS5DX3V1dXV1X3ZhbHVlRzYiJCIrPEtDMkYhIik=NiM+SS5DX3V1dXVkX3ZhbHVlRzYiJCIrJDQnelk7ISIpNiM+SS5DX3V1dWRkX3ZhbHVlRzYiJCIrcyR5KClHKCEiKg==NiM+SS5DX3V1ZGRkX3ZhbHVlRzYiJCIrKD4mMzU6ISIqNiM+SS5DX3VkZGRkX3ZhbHVlRzYiJCIiIUYnNiM+SS5DX2RkZGRkX3ZhbHVlRzYiJCIiIUYnC_uuuu_value := subs(params,subs(q=qvalue,C[u]=C_uuuuu_value,C[d]=C_uuuud_value,Cformula)); C_uuud_value := subs(params,subs(q=qvalue,C[u]=C_uuuud_value,C[d]=C_uuudd_value,Cformula)); C_uudd_value := subs(params,subs(q=qvalue,C[u]=C_uuudd_value,C[d]=C_uuddd_value,Cformula)); C_uddd_value := subs(params,subs(q=qvalue,C[u]=C_uuddd_value,C[d]=C_udddd_value,Cformula)); C_dddd_value := subs(params,subs(q=qvalue,C[u]=C_udddd_value,C[d]=C_ddddd_value,Cformula));NiM+SS1DX3V1dXVfdmFsdWVHNiIkIis3TjZ2QCEiKQ==NiM+SS1DX3V1dWRfdmFsdWVHNiIkIitTN10pPSIhIik=NiM+SS1DX3V1ZGRfdmFsdWVHNiIkIis0OCVRVCUhIio=NiM+SS1DX3VkZGRfdmFsdWVHNiIkIisxNT4sdyEjNQ==NiM+SS1DX2RkZGRfdmFsdWVHNiIkIiIhRic=C_uuu_value := subs(params,subs(q=qvalue,C[u]=C_uuuu_value,C[d]=C_uuud_value,Cformula)); C_uud_value := subs(params,subs(q=qvalue,C[u]=C_uuud_value,C[d]=C_uudd_value,Cformula)); C_udd_value := subs(params,subs(q=qvalue,C[u]=C_uudd_value,C[d]=C_uddd_value,Cformula)); C_ddd_value := subs(params,subs(q=qvalue,C[u]=C_uddd_value,C[d]=C_dddd_value,Cformula));NiM+SSxDX3V1dV92YWx1ZUc2IiQiKyVmdjZvIiEiKQ==NiM+SSxDX3V1ZF92YWx1ZUc2IiQiKzMheilmIikhIio=NiM+SSxDX3VkZF92YWx1ZUc2IiQiK0d4dCdmIyEiKg==NiM+SSxDX2RkZF92YWx1ZUc2IiQiK3InW2gjUSEjNQ==C_uu_value := subs(params,subs(q=qvalue,C[u]=C_uuu_value,C[d]=C_uud_value,Cformula)); C_ud_value := subs(params,subs(q=qvalue,C[u]=C_uud_value,C[d]=C_udd_value,Cformula)); C_dd_value := subs(params,subs(q=qvalue,C[u]=C_udd_value,C[d]=C_ddd_value,Cformula));NiM+SStDX3V1X3ZhbHVlRzYiJCIrYSR6KFs3ISIpNiM+SStDX3VkX3ZhbHVlRzYiJCIrREpRKVEmISIqNiM+SStDX2RkX3ZhbHVlRzYiJCIrUyVbZVwiISIqC_u_value := subs(params,subs(q=qvalue,C[u]=C_uu_value,C[d]=C_ud_value,Cformula)); C_d_value := subs(params,subs(q=qvalue,C[u]=C_ud_value,C[d]=C_dd_value,Cformula));NiM+SSpDX3VfdmFsdWVHNiIkIitaSTBXKikhIio=NiM+SSpDX2RfdmFsdWVHNiIkIitZPUJdTSEiKg==C_value := subs(params,subs(q=qvalue,C[u]=C_u_value,C[d]=C_d_value,Cformula));NiM+SShDX3ZhbHVlRzYiJCIrUidbVD8nISIq