5.1 Fundamentals of Derivatives: Forwards, Futures, and Swaps

Conceptual Foundation of Financial Derivatives

A derivative is a financial instrument whose economic value is derived from the price, value, or performance of an underlying asset, index, rate, or reference item. In contemporary capital markets, derivatives serve as essential tools for risk management, hedging, price discovery, and portfolio rebalancing. The underlying assets encompass equities, debt securities, foreign currencies, commodities, interest rates, and market indices.

+---------------------------------------------------------------------------------------------------+
|                                  DERIVATIVE CONTRACT CLASSIFICATION                               |
+-----------------------------------+---------------------------------------------------------------+
| Over-the-Counter (OTC) Bilateral  | Exchange-Traded Standardized Contracts                        |
+-----------------------------------+---------------------------------------------------------------+
| - Forwards (Customized terms)     | - Futures (Standardized lot size, tick, expiry)               |
| - Custom Swaps (IRS, Currency)    | - Exchange-Traded Options (Puts & Calls)                      |
| - Bilateral counterparty credit   | - Novation via Clearing Corporation (Zero counterparty risk)  |
| - Tailored settlement dates       | - Daily mark-to-market (MTM) margin settlement                |
+-----------------------------------+---------------------------------------------------------------+

Forward Contracts

A forward contract represents a bilateral agreement between two parties to buy or sell a specified underlying asset at an agreed-upon future delivery date at a pre-determined price (the forward price).

1. Customized Terms: Forwards are private bilateral transactions negotiated over-the-counter (OTC). The underlying quantity, quality grade, delivery location, and maturity date are customized to the exact commercial requirements of the counterparties.
2. Pricing Mechanism (Cost-of-Carry Model): Under the no-arbitrage assumption with no dividend yield, the theoretical forward price $F_0$ is expressed as: $$F_0 = S_0 \times e^{r \times T} \quad \text{or} \quad F_0 = S_0 \times (1 + r - q)^T$$ Where:
   • $S_0$ = Current spot price of the underlying asset
   • $r$ = Annualized risk-free interest rate (borrowing/financing cost)
   • $q$ = Annualized dividend yield or dividend cash flow
   • $T$ = Tenor or time to contract maturity in years
3. Payoff Profiles:
   • Long Forward Payoff: $\Pi_{\text{Long}} = S_T - K$
   • Short Forward Payoff: $\Pi_{\text{Short}} = K - S_T$ Where $S_T$ is the spot price at expiration and $K$ is the agreed delivery forward price.
4. Structural Limitations: Forwards suffer from counterparty default risk (the risk that one party fails to perform at settlement), illiquidity (lack of an active secondary trading market), and asymmetric search costs.

Futures Contracts

A futures contract is an exchange-traded, legally binding standardized agreement to buy or sell an underlying security, index, or commodity at a pre-determined price on a specified future expiration date.

Mechanism of Futures Trading

• Contract Standardization: The exchange specifies standardized parameters including lot size, contract expiry cycle (near-month, next-month, far-month maturing on the last Thursday of the expiry month in Indian equity markets), tick size (minimum price movement, e.g., ₹0.05), and price quotation units.
• Role of the Clearing Corporation: Through legal novation, the Clearing Corporation (such as NSE Clearing Limited - NCL) interposes itself between the buyer and the seller, acting as the buyer to every seller and seller to every buyer. This eliminates bilateral counterparty credit default risk.
• Margining Framework:
  1. Initial Margin (SPAN): Standard Portfolio Analysis of Risk (SPAN) computes the worst-case portfolio loss over a single-day holding period across multiple market stress scenarios (underlying price movements and implied volatility shifts).
  2. Extreme Loss Margin (ELM): A regulatory buffer margin mandated by SEBI (typically 3.5% for equity index and individual equity derivatives) to protect the clearing house against tail-risk events.
  3. Mark-to-Market (MTM) Settlement: At the close of each trading day, outstanding positions are revalued at the official daily settlement price. Daily gains are credited, and daily losses are debited directly to the member's clearing account in cash on a $T+1$ morning basis.

Day T Trading Session ---> Closing Daily Settlement Price Computed ---> MTM Loss Debited / Gain Credited (T+1 morning)
                          |
                          +---> Margin Deficit triggers Margin Call (Must be replenished before trade resumption)

Swaps: Interest Rate and Currency Swaps

A swap is an OTC derivative contract through which two counterparties exchange financial cash flows over a predetermined series of future dates based on a specified notional principal amount.

• Plain Vanilla Interest Rate Swap (IRS): Counterparty A agrees to pay a fixed interest rate on notional principal $N$ for $T$ years, while receiving a floating interest rate (e.g., overnight MIBOR or SOFR) from Counterparty B. $$\text{Net Cash Flow}_t = N \times (\text{Floating Rate}_t - \text{Fixed Rate}) \times \left(\frac{d_t}{365}\right)$$ Institutional asset managers utilize IRS to convert fixed-rate bond liabilities into floating-rate exposures, or conversely, to immunize bond portfolio durations without liquidating underlying physical debt securities.
• Currency Swaps: Exchange of principal and interest payment streams denominated in two different currencies at inception, with re-exchange of principal amounts at contract termination at the original spot exchange rate.

5.2 Options Payoffs, Moneyness, and Greeks

Option Fundamentals: Calls and Puts

An option is a derivative contract that confers upon the buyer (holder) the right, but not the obligation, to purchase or sell an underlying asset at a specified strike price ($X$ or $K$) on or before a specified expiration date. To acquire this asymmetric right, the option buyer pays a non-refundable upfront cash premium ($P$) to the option seller (writer). The option writer assumes the commensurate legal obligation to perform if the buyer elects to exercise.

• Call Option: Right to buy the underlying asset at strike price $X$.
  • Buyer Payoff at Expiry: $\Pi_{\text{Long Call}} = \max(0, S_T - X) - C_0$
  • Writer Payoff at Expiry: $\Pi_{\text{Short Call}} = -\max(0, S_T - X) + C_0$
• Put Option: Right to sell the underlying asset at strike price $X$.
  • Buyer Payoff at Expiry: $\Pi_{\text{Long Put}} = \max(0, X - S_T) - P_0$
  • Writer Payoff at Expiry: $\Pi_{\text{Short Put}} = -\max(0, X - S_T) + P_0$

+---------------------+-------------------------------+-------------------------------+
| Position            | Maximum Upside Potential      | Maximum Downside Risk         |
+---------------------+-------------------------------+-------------------------------+
| Long Call           | Theoretically Unlimited       | Limited to Premium Paid (C0)  |
| Short Call          | Limited to Premium Received   | Theoretically Unlimited       |
| Long Put            | Substantial (X - P0)          | Limited to Premium Paid (P0)  |
| Short Put           | Limited to Premium Received   | Substantial (X - P0)          |
+---------------------+-------------------------------+-------------------------------+

Moneyness of Options

Moneyness reflects the intrinsic value that would be realized if the option contract were exercised immediately at prevailing spot prices ($S$):

+--------------------+----------------------------+----------------------------+
| State of Moneyness | Call Option Condition      | Put Option Condition       |
+--------------------+----------------------------+----------------------------+
| In-The-Money (ITM) | Spot Price > Strike (S > X)| Spot Price < Strike (S < X)|
| At-The-Money (ATM) | Spot Price = Strike (S = X)| Spot Price = Strike (S = X)|
| Out-Of-The-Money   | Spot Price < Strike (S < X)| Spot Price > Strike (S > X)|
+--------------------+----------------------------+----------------------------+

An option's total market price is mathematically decomposed into two components: $$\text{Option Premium} = \text{Intrinsic Value} + \text{Time Value}$$

• Intrinsic Value: For a Call = $\max(0, S - X)$; for a Put = $\max(0, X - S)$. It can never be negative.
• Time Value (Extrinsic Value): Represents the probability premium that the underlying asset's price will move favorably prior to expiration. Time value decays non-linearly over time, collapsing to exactly zero at contract expiration.

The Option Greeks: Quantitative Risk Sensitivities

The Black-Scholes-Merton (BSM) option pricing model demonstrates that option prices are governed by five observable parameters: spot price ($S$), strike price ($X$), risk-free rate ($r$), time to maturity ($T$), dividend yield ($q$), and one unobservable parameter: implied volatility ($\sigma$). The partial derivatives measuring sensitivity to these inputs are known as the Greeks:

1. Delta ($\Delta$): First derivative of option price with respect to underlying spot price: $$\Delta_{\text{Call}} = \frac{\partial C}{\partial S} = e^{-qT} N(d_1) \in [0, +1]; \quad \Delta_{\text{Put}} = \frac{\partial P}{\partial S} = e^{-qT} [N(d_1) - 1] \in [-1, 0]$$
   • Delta measures the hedge ratio: the number of shares required to create a delta-neutral position.
   • For ATM options, Call Delta is approximately $+0.50$ and Put Delta is approximately $-0.50$.
2. Gamma ($\Gamma$): Second derivative with respect to spot price (rate of change of Delta): $$\Gamma = \frac{\partial^2 C}{\partial S^2} = \frac{e^{-qT} N'(d_1)}{S \sigma \sqrt{T}} > 0$$
   • Gamma is strictly positive for long option positions (both puts and calls). It peaks when the option is precisely At-The-Money (ATM) close to expiration, generating heightened dynamic rehedging risk.
3. Theta ($\Theta$): First derivative with respect to passage of time (time decay): $$\Theta = \frac{\partial V}{\partial t}$$
   • Theta is typically negative for long option positions, reflecting the daily erosion of time value. Time decay accelerates aggressively during the final 30 calendar days leading to expiration.
4. Vega ($ u$): Sensitivity of option price to changes in underlying implied volatility: $$ u = \frac{\partial V}{\partial \sigma} = S e^{-qT} \sqrt{T} N'(d_1) > 0$$
   • Vega is identical for both long calls and long puts. A 1% increase in implied volatility increases option premiums by the value of Vega.
5. Rho ($\rho$): Sensitivity to shifts in the risk-free benchmark interest rate: $$\rho_{\text{Call}} = \frac{\partial C}{\partial r} = X T e^{-rT} N(d_2) > 0; \quad \rho_{\text{Put}} = \frac{\partial P}{\partial r} = -X T e^{-rT} N(-d_2) < 0$$

5.3 PMS Derivative Regulations, Hedging, Rebalancing, and Leverage Prohibition

Regulatory Mandates under SEBI (Portfolio Managers) Regulations, 2020

The Securities and Exchange Board of India (SEBI) establishes clear and rigorous statutory boundaries governing how registered Portfolio Managers may employ derivative instruments within client accounts. Unlike hedge funds or Alternative Investment Funds (AIF Category III), Portfolio Management Services (PMS) operate under a client fiduciary framework designed to manage segregated client wealth without exposing clients to speculative liquidation hazards.

+---------------------------------------------------------------------------------------------------+
|                        SEBI PMS REGULATORY DERIVATIVE BOUNDARIES MATRIX                           |
+------------------------------------+--------------------------------------------------------------+
| Permitted Activities               | Strictly Prohibited Activities                               |
+------------------------------------+--------------------------------------------------------------+
| 1. Hedging underlying equity risks | 1. Speculative trading or unhedged directional derivative bets|
| 2. Portfolio rebalancing           | 2. Net synthetic or leveraged positions (Gross exposure > 100%)|
| 3. Covered Call writing            | 3. Naked Call or Put writing without underlying asset backing|
| 4. Cash-secured Put writing        | 4. Borrowing funds or leveraging portfolio assets            |
+------------------------------------+--------------------------------------------------------------+

Absolute Prohibition of Leverage

Regulation 22(4) of the SEBI (Portfolio Managers) Regulations, 2020 explicitly stipulates:

• No Borrowing: The portfolio manager shall not borrow funds or pledge client securities to create leverage on behalf of the client.
• Cap on Derivative Exposure: The total gross exposure of the client's portfolio, taking into account underlying cash equity positions, debt holdings, and derivative notionals, must not exceed 100% of the net asset value (NAV) of the client's portfolio.
• Cash Settled Backing: When a portfolio manager executes a covered call option strategy, the client's demat account must hold the identical quantity of underlying shares. Naked option writing is strictly illegal in PMS accounts.

Practical Portfolio Applications of Derivatives in PMS

1. Portfolio Beta Hedging via Index Futures

When an equity portfolio manager anticipates a near-term macroeconomic correction or systemic market decline but wishes to avoid liquidating high-conviction fundamental equity stocks (which would trigger capital gains tax liabilities and transaction friction), the manager hedges systematic risk using equity index futures (e.g., Nifty 50 Futures):

$$\text{Number of Index Futures Contracts} = \beta_p \times \left( \frac{V_p}{V_f} \right)$$ Where:

• $\beta_p$ = Beta of the client's equity portfolio relative to the market index
• $V_p$ = Total market value of the equity portfolio being hedged
• $V_f$ = Notional contract value of one index futures contract ($V_f = \text{Futures Index Price} \times \text{Lot Size}$)

Practical Hedging Case Study

Assume a PMS discretionary client has an equity portfolio valued at ₹20,00,00,000 (₹20 Crore) with an estimated portfolio beta of $\beta_p = 1.25$. The Nifty 50 Futures contract trades at 24,000 with a lot size of 50 units.

1. Value of one Nifty futures contract: $$V_f = 24,000 \times 50 = ₹12,00,000$$
2. Number of contracts required for a complete delta hedge: $$N = 1.25 \times \left( \frac{₹20,00,00,000}{₹12,00,000} \right) = 1.25 \times 166.67 = 208.33 \approx 208 \text{ contracts}$$ By selling (shorting) 208 Nifty 50 futures contracts, the portfolio manager reduces the systematic market beta of the client's portfolio to approximately 0, protecting the portfolio against systemic drawdowns while preserving alpha in the underlying equity holdings.

2. Protective Put Strategy (Portfolio Insurance)

A portfolio manager purchases Out-of-the-Money (OTM) or At-the-Money (ATM) index put options to establish a definitive floor beneath the portfolio value while retaining 100% of the upside participation. The cost of the put option acts as an insurance premium.

Portfolio Payoff with Protective Put:
Value
  ^
  |                     / (Unhedged Portfolio Upside)
  |                    /
  |      +------------/   (Insured Portfolio Payoff: Floor established)
  |      |           /
  |      |          /
  +------+---------+-------------------> Underlying Market Index
         Floor = Strike - Premium

3. Covered Call Writing for Yield Enhancement

In a range-bound, stagnant, or moderately bullish market, the portfolio manager writes OTM call options against existing long equity holdings in the client's demat account. If the stock remains below the strike price at expiration, the option expires worthless, and the client retains the premium received, boosting overall portfolio yield.