## WHAT I WANT TO DO

In Sympy, **A** is defined as (-K * erf(sqrt(2) * (2 * T * r - T * sigma2 + 2 * log(S/K)) / (4 * sqrt(T) * sigma)) - K + S * (erf(sqrt(2) * (2 * T * r + T * sigma2 + 2 * log(S/K)) / (4 * sqrt(T) * sigma)) + 1) * exp(T * r)) * exp(-T * r) / 2; **B** is defined as (-K * erf(sqrt(2) * (2 * T * r - T * sigma2 + 2 * log(S/K)) / (4 * sqrt(T) * sigma)) + K + S * (erf(sqrt(2) * (2 * T * r + T * sigma2 + 2 * log(S/K)) / (4 * sqrt(T) * sigma)) - 1) * exp(T * r)) * exp(-T * r) / 2

Please write a Sympy code to represent B in terms of A (note that the formulas for A and B contain some common intermediate variables and expressions, which can be eliminated, so please eliminate these common intermediate variables and expressions in the Sympy code), and simplify it to obtain the expression for B.

## GPT4 answer

```
# Re-defining the symbols and common terms as per the new prompt
A = (-K*erf(sqrt(2)*(2*T*r - T*sigma**2 + 2*log(S/K))/(4*sqrt(T)*sigma)) - K + S*(erf(sqrt(2)*(2*T*r + T*sigma**2 + 2*log(S/K))/(4*sqrt(T)*sigma)) + 1)*exp(T*r))*exp(-T*r)/2
B = (-K*erf(sqrt(2)*(2*T*r - T*sigma**2 + 2*log(S/K))/(4*sqrt(T)*sigma)) + K + S*(erf(sqrt(2)*(2*T*r + T*sigma**2 + 2*log(S/K))/(4*sqrt(T)*sigma)) - 1)*exp(T*r))*exp(-T*r)/2
# Express B in terms of A
B_expr = B.subs(A, A)
# Simplify the expression
B_simplified = simplify(B_expr)
B_simplified
Result
(-K*erf(sqrt(2)*(2*T*r - T*sigma**2 + 2*log(S/K))/(4*sqrt(T)*sigma)) + K + S*(erf(sqrt(2)*(2*T*r + T*sigma**2 + 2*log(S/K))/(4*sqrt(T)*sigma)) - 1)*exp(T*r))*exp(-T*r)/2
```

## PROBLEM

From the description above, it’s evident that A and B share many common intermediate variables and expressions, which can actually be eliminated and simplified.

#### Question 1:

We want to manually simplify and solve the above problem using Sympy. We’ve tried using collect, subs, simplify, and cancel, but none of these worked. Please help us.

#### Question 2:

We want to design a prompt to instruct ChatGPT to help us generate Sympy code to solve the above problem, but we can’t seem to get it right. We would appreciate your assistance.