Can I derive the BSM put price from the call price using put-call parity instead of memorizing two formulas?

Yes, this is a perfectly legitimate strategy used by experienced candidates and traders alike. Memorise one formula deeply, derive the other on the fly. Here is the exact mechanics with the gotchas you need to know.

The European Put-Call Parity Formula (continuous dividends):

Rearranging:

Step-by-step exam application:

Suppose the exam gives you $S_0 = 100$, $K = 100$, $r = 0.05$, $q = 0$, $\sigma = 0.20$, $T = 1$ and asks for both call and put.

1. Compute $d_1 = 0.35$, $d_2 = 0.15$
2. Look up $N(0.35) \approx 0.6368$, $N(0.15) \approx 0.5596$
3. Compute call: $c = 100 \cdot 1 \cdot 0.6368 - 100 \cdot 0.9512 \cdot 0.5596 = 63.68 - 53.23 = \$10.45$
4. Apply parity: $p = 10.45 - 100 \cdot 1 + 100 \cdot 0.9512 = 10.45 - 100 + 95.12 = \$5.57$

You wrote one set of $N(\cdot)$ lookups and got both prices. That saves about 3 minutes on the exam.

The four gotchas:

Gotcha 1 — European only. Put-call parity holds only for European options. American puts can be optimal to exercise early, so the parity is replaced by an inequality.

Gotcha 2 — Matching discount factors. Both PV terms use the same $r$ and $T$. If the exam gives you a forward price $F$ directly instead of $S_0$, the parity becomes $c - p = (F - K) \cdot e^{-rT}$.

Gotcha 3 — Discrete dividends. If the stock pays a known cash dividend $D$ before expiry, replace $S_0$ with $S_0 - D \cdot e^{-rT}$. The continuous-$q$ form assumes a dividend rate, not lump sums.

Gotcha 4 — Cash settlement vs. physical. For cash-settled index options, parity works as stated. For physically settled equity options where the holder must take delivery, parity can break slightly because of dividend timing.

On the exam:

Item-set questions on options frequently give you $S_0$, $K$, $r$, $q$, $\sigma$, $T$ and ask for several derived numbers (call price, put price, delta of call, delta of put, breakeven). Memorising one formula plus put-call parity covers 80% of those. Adding the parity-based delta relationship ($\Delta_{\text{put}} = \Delta_{\text{call}} - e^{-qT}$) covers the rest.

The reverse direction:

If the exam gives you a put price and asks for the call, just rearrange: $c = p + S_0 \cdot e^{-qT} - K \cdot e^{-rT}$. Same formula, both directions.