Derivatives Analysis and Valuation (Quick Revision)
1. Master Formula Sheet
A. Forward and Futures Pricing (Cost of Carry Model)
1. Non-Income Paying Asset (Simple Compounding):
$$F_0 = S_0 \cdot \left(1 + r \cdot \frac{t}{12}\right)$$2. Continuous Compounding:
$$F_0 = S_0 \cdot e^{rt}$$3. Asset with Known Cash Dividend ($D$):
$$F_0 = (S_0 - D \cdot e^{-rt_d}) \cdot e^{rt} \quad \text{or} \quad F_0 = S_0 \cdot (1 + rt) - D$$4. Asset with Continuous Dividend Yield ($y$):
$$F_0 = S_0 \cdot e^{(r - y)t}$$5. Commodity with Storage ($u$) & Convenience Yield ($c$):
$$F_0 = S_0 \cdot e^{(r + u - c)t}$$| Arbitrage Condition | Market Status | Arbitrage Strategy Execution |
|---|---|---|
| $F_{\text{actual}} > F_{\text{fair}}$ | Futures Overvalued | Cash-and-Carry: Borrow money, buy stock in spot, sell (short) futures contract. |
| $F_{\text{actual}} < F_{\text{fair}}$ | Futures Undervalued | Reverse Cash-and-Carry: Short stock in spot, invest proceeds at risk-free rate, buy (long) futures. |
2. Stock Index Futures & Beta Hedging
Number of Contracts to Trade ($N$):
$$N = \frac{V_p \cdot (\beta^* - \beta_p)}{V_f}$$Where $V_f = \text{Futures Price} \times \text{Lot Size / Multiplier}$.
- To reduce beta ($\beta^* < \beta_p$): Result is negative $\implies$ Sell (Short) $N$ Contracts.
- To increase beta ($\beta^* > \beta_p$): Result is positive $\implies$ Buy (Long) $N$ Contracts.
- For complete hedge ($\beta^* = 0$): $N = \frac{V_p \cdot \beta_p}{V_f}$ (Sell).
Daily Marking-to-Market (MTM) Margin Rules
$$\text{Initial Margin} = \mu + 3\sigma$$
$$\text{Maintenance Margin} = 75\% \text{ of Initial Margin (or specified %)}$$
If account drops below maintenance margin $\implies$ Margin Call for full deficit back to Initial Margin.
3. Option Strategies & Payoff Matrix
| Strategy | Constituent Legs | Market Outlook | Max Profit | Max Loss | Break-Even Point(s) |
|---|---|---|---|---|---|
| Long Call | Buy Call ($X$) | Strongly Bullish | Unlimited | Premium Paid ($c$) | $X + c$ |
| Long Put | Buy Put ($X$) | Strongly Bearish | $X - p$ | Premium Paid ($p$) | $X - p$ |
| Long Straddle | Buy Call ($X$) + Buy Put ($X$) | High Volatility (Direction Neutral) | Unlimited | $c + p$ (at $S_T = X$) | $X - (c+p)$ and $X + (c+p)$ |
| Long Strangle | Buy OTM Put ($X_1$) + Buy OTM Call ($X_2$) | Extreme Volatility | Unlimited | $c + p$ (between $X_1$ and $X_2$) | $X_1 - (c+p)$ and $X_2 + (c+p)$ |
| Bull Call Spread | Buy Call ($X_1$) + Sell Call ($X_2$) ($X_2 > X_1$) | Moderately Bullish | $(X_2 - X_1) - \text{Net Debit}$ | Net Debit ($c_1 - c_2$) | $X_1 + \text{Net Debit}$ |
| Bear Put Spread | Buy Put ($X_2$) + Sell Put ($X_1$) ($X_2 > X_1$) | Moderately Bearish | $(X_2 - X_1) - \text{Net Debit}$ | Net Debit ($p_2 - p_1$) | $X_2 - \text{Net Debit}$ |
| Collar Strategy | Long Stock + Buy OTM Put ($X_1$) + Sell OTM Call ($X_2$) | Downside Protected with Capped Upside | $X_2 - S_0 - \text{Net Premium}$ | $S_0 - X_1 + \text{Net Premium}$ | $S_0 + \text{Net Premium}$ |
4. Binomial Model & Black-Scholes-Merton Formula
A. Binomial Model & Risk-Neutral Pricing
1. Hedge Ratio (Delta $\Delta$):
$$\Delta = \frac{C_u - C_d}{S_u - S_d}$$2. Risk-Neutral Probability ($p$):
$$p = \frac{R - d}{u - d} \quad \text{where } R = 1 + r \cdot t \text{ (or } e^{rt}\text{)}$$3. Single-Period Call Value:
$$C_0 = \frac{p \cdot C_u + (1-p) \cdot C_d}{R}$$B. Black-Scholes-Merton Model
$$C_0 = S_0 \cdot N(d_1) - X \cdot e^{-rt} \cdot N(d_2)$$
$$d_1 = \frac{\ln(S_0 / X) + \left(r + \frac{\sigma^2}{2}\right)t}{\sigma \sqrt{t}}; \quad d_2 = d_1 - \sigma \sqrt{t}$$
C. Put-Call Parity Theorem
$$C_0 + X \cdot e^{-rt} = P_0 + S_0 \implies P_0 = C_0 + X \cdot e^{-rt} - S_0$$
5. The Option Greeks at a Glance
| Greek | Formula / Definition | Call Value | Put Value | Peak Value |
|---|---|---|---|---|
| Delta ($\Delta$) | $\frac{\partial C}{\partial S}$ | $0 < \Delta < 1$ ($N(d_1)$) | $-1 < \Delta < 0$ ($N(d_1)-1$) | Deep ITM ($\to 1$) |
| Gamma ($\Gamma$) | $\frac{\partial^2 C}{\partial S^2} = \frac{\partial \Delta}{\partial S}$ | Always $> 0$ | Always $> 0$ | At-The-Money (ATM) |
| Theta ($\Theta$) | $\frac{\partial C}{\partial t}$ | Typically $< 0$ (Decay) | Typically $< 0$ (Decay) | ATM near Expiration |
| Vega ($\nu$) | $\frac{\partial C}{\partial \sigma}$ | Always $> 0$ | Always $> 0$ | At-The-Money (ATM) |
| Rho ($\rho$) | $\frac{\partial C}{\partial r}$ | Always $> 0$ | Always $< 0$ | Deep ITM Call |
6. Credit Derivatives & Real Options Mapping
| Real Option Type | Option Equivalent | Underlying Asset ($S$) | Strike Price ($X$) | Strategic Value Added |
|---|---|---|---|---|
| Growth / Expansion | Call Option | PV of Future Cash Inflows | Follow-on Capex Required | Captures upside if initial market test succeeds. |
| Abandonment | Put Option | Project Continuation Value | Salvage / Liquidation Value | Floors project downside if conditions turn sluggish. |
| Timing / Deferral | Call Option | PV of Future Project Inflows | Initial Investment Outlay | Value of waiting to resolve tariff/market uncertainty. |
7. High-Frequency Exam Traps & Shortcuts
- Continuous vs Simple Discounting: If exponential values ($e^{rt}$) are provided in question footnotes, use continuous compounding ($S e^{rt}$). Otherwise, use simple interest $S(1+rt)$.
- Portfolio Beta with Cash: Cash always has a beta of zero ($\beta_{\text{cash}} = 0$). Remember to include cash in total portfolio value $V_p$ when weighting!
- Sensex Multiplier vs Nifty Lot: Sensex multiplier is typically 50. Nifty lots in past exams are 25, 50, 100, or 200. Always multiply index price by lot size to obtain $V_f$!
- Margin Call Top-up: The amount of money to deposit on margin call brings the balance back to Initial Margin, NOT just to the maintenance margin.
- Option Payoff Tables: When constructing payoff tables for combinations (Straddle, Collar), always tabulate columns: $S_T$, Call Payoff, Put Payoff, Total Gross Payoff, Total Premium Paid, Net Profit/(Loss).