Dean's Office Official Model Question Paper

Financial Derivative

A financial derivative is a financial contract whose value is derived from the value, price, or performance of an underlying asset (stocks, bonds, currencies, interest rates, commodities).

Four Primary Contracts: Forwards, Futures, Options, and Swaps.

Forward vs. Futures Contract

• Forward Contract: A private, bilateral Over-The-Counter (OTC) agreement, fully customized, bearing counterparty default risk, settled at maturity.
• Futures Contract: A standardized contract traded on an organized exchange, guaranteed by an exchange clearinghouse, with daily mark-to-market margin settlements.

Call Option vs. Put Option

• Call Option: Grants the buyer the right, but not the obligation, to buy an underlying asset at a specified strike price on or before a specified expiration date.
• Put Option: Grants the buyer the right, but not the obligation, to sell an underlying asset at a specified strike price on or before a specified expiration date.

Put-Call Parity Formula

$$C + K e^{-rT} = P + S_0$$

Where:

• $C$ = Price of European call option
• $K e^{-rT}$ = Present value of strike price $K$
• $P$ = Price of European put option
• $S_0$ = Current spot price of underlying asset

Moneyness of Call Options

For a call option with spot price $S$ and strike price $K$:

• In-The-Money (ITM): $S > K$ (Positive intrinsic value $S - K$).
• At-The-Money (ATM): $S = K$ (Zero intrinsic value).
• Out-Of-The-Money (OTM): $S < K$ (Zero intrinsic value; would expire worthless if exercised today).

Binomial Option Pricing Model

a. Risk-Neutral Probability Calculation

Parameters:

• $S_0 = 100$
• $u = 1.20$ (Up factor)
• $d = 0.90$ (Down factor)
• $r = 0.05 \implies e^{rT} = 1 + r = 1.05$ (annual compounding)

$$\text{Risk-Neutral Probability } q = \frac{e^{rT} - d}{u - d} = \frac{1.05 - 0.90}{1.20 - 0.90} = \frac{0.15}{0.30} = \mathbf{0.50}$$

$$1 - q = 1 - 0.50 = \mathbf{0.50}$$

b. Call Option Valuation ($K = 105$)

Option payoffs at maturity ($T = 1$):

• Up State ($S_u = 120$):
  $$C_u = \max(0, S_u - K) = \max(0, 120 - 105) = 15$$
• Down State ($S_d = 90$):
  $$C_d = \max(0, S_d - K) = \max(0, 90 - 105) = 0$$

Discounted Expected Payoff:

$$C_0 = \frac{q C_u + (1 - q) C_d}{1 + r} = \frac{0.50(15) + 0.50(0)}{1.05} = \frac{7.50}{1.05} \approx \mathbf{7.14}$$

The fair value of the European call option today is Rs 7.14.

Black-Scholes-Merton (BSM) Model and Option Greeks

1. BSM Model Formula for European Call

$$C = S_0 N(d_1) - K e^{-rT} N(d_2)$$

$$d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma \sqrt{T}}, \quad d_2 = d_1 - \sigma \sqrt{T}$$

• Five Inputs: Current spot price ($S_0$), Strike price ($K$), Risk-free rate ($r$), Time to expiration ($T$), and Volatility of underlying asset ($\sigma$).

2. The Option Greeks

• Delta ($\Delta$): Rate of change of option price with respect to changes in underlying asset price ($\frac{\partial C}{\partial S}$). Used for delta-hedging.
• Gamma ($\Gamma$): Rate of change of Delta with respect to underlying asset price ($\frac{\partial^2 C}{\partial S^2}$).
• Vega ($\nu$): Sensitivity of option price to changes in underlying asset volatility ($\frac{\partial C}{\partial \sigma}$).
• Theta ($\Theta$): Time decay; rate of change of option price as time to maturity passes ($\frac{\partial C}{\partial t}$).
• Rho ($\rho$): Sensitivity of option price to changes in the risk-free interest rate ($\frac{\partial C}{\partial r}$).

Option Trading Strategies

1. Protective Put (Synthetic Call)

• Long Stock + Long Put.
• Objective: Downside insurance: protects stock holdings against price collapse while retaining unlimited upside potential.

2. Covered Call

• Long Stock + Short Call.
• Objective: Income generation in neutral or mildly bullish markets; writing calls generates premium income, but caps upside at strike price.

3. Long Straddle

• Long Call + Long Put at the same strike price and expiration date.
• Objective: Bet on high volatility; profits if the asset price moves sharply in either direction (e.g., ahead of major earnings announcements or court verdicts).

4. Bull Call Spread

• Long Call at lower strike $K_1$ + Short Call at higher strike $K_2$.
• Objective: Moderately bullish view; premium paid for $K_1$ is partially offset by premium received from $K_2$, reducing capital outlay while capping maximum profit.

Interest Rate Swaps and Comparative Advantage

An Interest Rate Swap is an agreement between two parties to exchange interest payment streams (fixed for floating) on an agreed notional principal.

The Comparative Advantage Argument:

Assume Company A and Company B face the following borrowing rates:

• Company A has an absolute advantage in both, but a comparative advantage in Fixed debt (spread = 2.5% vs 1.0%).
• Company B has a comparative advantage in Floating debt.
• Total Swap Gain: $2.5\% - 1.0\% = \mathbf{1.5\%}$ total potential interest savings.
• By entering an IRS, Company A issues fixed debt and swaps into floating, while Company B issues floating and swaps into fixed, splitting the 1.5% savings via an intermediary swap bank.

Case Analysis: Commodity Derivative Hedging for Himalayan Petroleum Importers Ltd.

a. Financial Payoff Comparison (500,000 Barrels)

Scenario (i): Oil Spikes to USD 110 per Barrel (Severe Geopolitical Crisis)

• 1. Futures Hedge:
  $$\text{Effective Cost} = \text{Locked Futures Price} = \text{USD } 82 / \text{bbl}$$
  $$\text{Total Cost} = 500{,}000 \times \text{USD } 82 = \mathbf{\text{USD } 41{,}000{,}000}$$
• 2. Call Option ($K = \text{USD } 82$, Premium = $\text{USD } 4$):
  • Exercise call at USD 82; Net effective price = $\text{USD } 82 + 4 = \text{USD } 86 / \text{bbl}$. $\text{Total Cost} = 500{,}000 \times \text{USD } 86 = \mathbf{\text{USD } 43{,}000{,}000}$
• 3. Zero-Cost Collar ($K_{\text{Call}} = \text{USD } 84, K_{\text{Put}} = \text{USD } 76$):
  • Call exercised at USD 84; Put expires worthless. Net premium paid = USD 0. $\text{Total Cost} = 500{,}000 \times \text{USD } 84 = \mathbf{\text{USD } 42{,}000{,}000}$

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Scenario (ii): Oil Collapses to USD 65 per Barrel (Peace Accord)

• 1. Futures Hedge:
  • Obligated to buy at futures price of USD 82; cannot benefit from market drop. $\text{Total Cost} = 500{,}000 \times \text{USD } 82 = \mathbf{\text{USD } 41{,}000{,}000}$
• 2. Call Option:
  • Let call expire worthless; buy at spot market (USD 65) + sunk premium (USD 4) = $\text{USD } 69 / \text{bbl}$. $\text{Total Cost} = 500{,}000 \times \text{USD } 69 = \mathbf{\text{USD } 34{,}500{,}000} \quad \text{(Saves USD 6.5M!)}$
• 3. Zero-Cost Collar:
  • Call expires worthless; counterparty exercises Put option forcing firm to buy at USD 76.
    $$\text{Total Cost} = 500{,}000 \times \text{USD } 76 = \mathbf{\text{USD } 38{,}000{,}000}$$

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b. Strategic Evaluation of Hedging Instruments

Recommendation: The Zero-Cost Collar is optimal: it caps maximum exposure at USD 84 without requiring USD 2 million in cash premiums from working capital.

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c. Basis Risk in Nepalese Fuel Imports

• Basis Risk: The risk that the price difference between the derivative benchmark (SGX Platt’s Singapore fuel) and the physical delivery cost in Nepal fluctuates unpredictably.
• Drivers: Overland tanker transport costs from Haldia/Barauni to Raxaul, Indian Oil Corporation (IOC) refinery processing margins, transit port demurrage, and USD/NPR currency fluctuations.