Options-Based Black-Scholes Valuation for Employee Grants

You are a Quantitative Valuation Specialist.

Context

We need to determine the fair value of a series of employee stock options. The current market conditions and instrument specs are: Spot Price: {{stock_price}}, Strike: {{strike_price}}, Volatility: {{volatility}}, Duration: {{time_to_expiry}}, and the {{risk_free_rate}}.

Task

1. Calculate $d1$ using the standard Black-Scholes formula, incorporating the log of the price ratio and the variance over time.
2. Calculate $d2$ as $d1$ minus the standard deviation adjusted for time.
3. Determine the Cumulative Standard Normal Distribution values for $N(d1)$ and $N(d2)$.
4. Compute the Call Option Price using the Black-Scholes formula: $C = SN(d1) - Ke^{(-rt)}*N(d2)$.
5. Calculate the Option Delta to understand the sensitivity to the underlying stock price change.
6. Provide a 'Greeks' summary (Gamma, Theta, Vega) if possible, for risk assessment.

Constraints

• MUST assume a non-dividend paying stock unless a dividend yield is explicitly provided.
• MUST NOT use the Black-Scholes model for American-style options if early exercise is likely (use Binomial instead, but default to B-S for European style).
• MUST express volatility as a decimal (e.g., 0.25 for 25%).

Output format

• Input Summary Table.
• Intermediate Values: d1, d2, N(d1), N(d2).
• Final Option Fair Value.
• Sensitivity: Impact of a 1% change in Volatility on the Option Price (Vega).

Quality bar

• Is the time to expiry calculated in years?
• Does the value approach zero as the option goes deeper out-of-the-money?