DeepSeek-R1 Quantitative Derivatives & Black-Scholes Greeks

You are a Principal Quantitative Strategist and Financial Mathematics Researcher.

Formulate, prove, and compute the option pricing and risk sensitivity profile under the following market conditions:
- Underlying Asset & Price (S): {SPY (S&P 500 ETF) at Spot Price S = $580.00}
- Strike Price (K) & Tenor (T): {Strike K = $590.00, Time to Expiry T = 45 days (0.1233 years)}
- Volatility (sigma) & Risk-Free Rate (r): {Implied Volatility sigma = 16.5% (0.165), 10-Yr US Treasury r = 4.25% (0.0425), Dividend Yield q = 1.30% (0.013)}
- Option Style & Payoff: {European Call & Put (Black-Scholes-Merton with Continuous Dividend)}

### MANDATORY DEEPSEEK-R1 CHAIN OF THOUGHT:
1. <think>
   - State the Black-Scholes-Merton Partial Differential Equation (PDE):
     $$\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - r V = 0$$
   - Derive $d_1$ and $d_2$ terms showing risk-neutral drift and cumulative normal distribution $N(d_1)$ and $N(d_2)$.
   - Calculate exact analytical Greeks:
     - Delta ($\Delta = \frac{\partial V}{\partial S}$)
     - Gamma ($\Gamma = \frac{\partial^2 V}{\partial S^2}$)
     - Vega ($\nu = \frac{\partial V}{\partial \sigma}$)
     - Theta ($\Theta = \frac{\partial V}{\partial t}$)
     - Rho ($\rho = \frac{\partial V}{\partial r}$)
   - Evaluate jump-diffusion or volatility smile skew for out-of-the-money (OTM) strikes.
   - Formulate hedging bounds: simulate delta-neutral dynamic rebalancing costs under transaction friction.
</think>

### DELIVERABLES:
1. **Exact Analytical Price & Greeks Matrix**: Formatted table with numerical values rounded to 4 decimal places.
2. **Python / NumPy Vectorized Implementation**: Production-grade function calculating both Call and Put values along with the complete Greek suite.
3. **Hedging & Sensitivity Report**: Explanation of Gamma pin risk and Theta decay acceleration during the final 5 trading days prior to expiration.