Black–Scholes–Merton: Assumptions & Where They're Used
A quick mapping from each assumption to the exact step in the derivation and why it matters.
| Assumption | Where it's used in derivation | Why it's needed |
|---|---|---|
| Stock price follows GBM \(dS_t = \mu S_t\,dt + \sigma S_t\,dW_t\) with constant \(\mu,\sigma\) | Starting point; plug into Itô's Lemma for \(V(S,t)\) | Continuous paths, proportional volatility, lognormal prices → tractable and realistic enough for the model |
| No transaction costs, taxes, bid–ask spreads | Forming and continuously rebalancing \(\Pi = V - V_S S\) | Prevents hedging frictions; exact term-by-term cancellation relies on costless trading |
| Continuous trading | Maintain \(\Delta = V_S\) at every instant | Without continuous rebalancing, the random \(dW_t\) exposure reappears |
| Short selling allowed | Setting \(\Delta = V_S\) (which can be negative) | You must be able to short the stock to hedge options with negative delta |
| Constant risk-free rate \(r\) | Riskless-return step: \(d\Pi = r\,\Pi\,dt\) | Simplifies discounting/replication; yields constant-coefficient PDE |
| No dividends (in basic form) | In the PDE drift term \(r S V_S\) | If dividends with yield \(q\) exist, replace \(r\) with \(r - q\) in the PDE |
| No arbitrage | Equating riskless growth: \(d\Pi = r\,\Pi\,dt\) | Forces any riskless portfolio to earn exactly the risk-free rate; pins down the PDE |
| Complete market (every payoff can be replicated) | Implicit in replication/hedging argument | Guarantees the hedged portfolio matches the option's cash flows → unique price |
Resulting PDE:
\[
V_t + \frac12\sigma^2 S^2 V_{SS} + r S V_S - r V = 0
\]
with terminal condition \(V(S,T)=\text{payoff}(S_T)\) (e.g., \(\max(S-K,0)\) for a call).