Chapter 9: Derivatives Analysis and Valuation

Definition of Derivatives: A derivative is a financial instrument whose value is derived from the value of an underlying asset, index, or reference rate. The underlying asset can be equity shares, stock indices, bonds, interest rates, foreign currencies, commodities (agricultural, metals, energy), or credit events.

Core Economic Functions of Derivatives:

1. Risk Transfer and Hedging: Enables risk-averse economic agents to transfer unwanted market price risks (price volatility, interest rate shifts, currency fluctuations) to entities willing to bear them (speculators or risk-tolerant investors).
2. Price Discovery: Derivatives markets aggregate forward-looking information and collective expectations about future supply, demand, and economic variables, providing valuable forward price signals to the spot market.
3. Market Liquidity and Operational Efficiency: Derivatives require lower capital outlays (margin trading), offering superior leverage, lower transaction costs, and higher liquidity than spot markets.

Forward Contracts: A forward contract is a customized, bilateral contract entered into between two private parties to buy or sell an underlying asset at a specified future date at a price agreed upon today (the forward price). Forward contracts are Over-The-Counter (OTC) unstandardized agreements.

Key Features of Forward Contracts:

• Bilateral Negotiation: Contract size, expiration date, quality grade, and delivery venue are completely customized to meet the counterparty's specific hedging needs.
• Counterparty / Credit Risk: There is no central clearinghouse or guarantee fund. If either party defaults at maturity, the non-defaulting party bears the entire credit loss.
• Settlement at Maturity: Cash flows occur strictly at maturity (settlement date). There is no interim daily cash settlement or marking-to-market.
• Lack of Liquidity & Secondary Market: Because each contract is customized and non-standardized, reversing or closing out a forward position prior to maturity requires counterparty consent or entering an offsetting contract.

In a forward contract, the party agreeing to buy the asset takes the Long Forward position, and the party agreeing to sell takes the Short Forward position.

Let $S_T$ be the spot price of the underlying asset at maturity $T$, and $F_0$ be the agreed delivery forward price:

• Long Forward Payoff at Maturity: $$\text{Payoff}_{\text{Long}} = S_T - F_0$$ (Profit if $S_T > F_0$, loss if $S_T < F_0$)
• Short Forward Payoff at Maturity: $$\text{Payoff}_{\text{Short}} = F_0 - S_T$$ (Profit if $S_T < F_0$, loss if $S_T > F_0$)

Futures Contract: A futures contract is a legally binding, standardized agreement traded on an organized exchange to buy or sell a standardized quantity and quality of a specific underlying asset at a specified future date at a market-determined price.

Novation & Clearing Corporation: The Clearing Corporation (e.g., NSCCL / NSE Clearing Ltd.) acts as the legal counterparty to every transaction through the legal doctrine of Novation—becoming the buyer to every seller and the seller to every buyer, virtually eliminating counterparty credit risk.

Futures trading operates on a strict three-tier margin architecture to guarantee solvency:

1. Initial Margin: The upfront collateral required to initiate a position, determined statistically using SPAN (Standard Portfolio Analysis of Risk) or $\mu + 3\sigma$ (value at risk covering 99% confidence level of daily price movements).
2. Maintenance Margin: The minimum threshold balance that must be maintained in the margin account at all times (typically 75% to 80% of initial margin).
3. Marking-to-Market (MTM): At the close of each trading day, open contracts are revalued at the official daily settlement price. Daily gains are credited to the margin account; daily losses are debited.
4. Variation Margin / Margin Call: If the margin account drops below the maintenance margin level, a Margin Call is triggered. The trader must deposit cash to restore the account back to the Initial Margin level (not just to the maintenance margin level).

The Cost of Carry Model: In an efficient and frictionless market, the forward/futures price ($F_0$) must equal the spot price ($S_0$) plus the net cost of carrying the asset until maturity $T$. If this parity condition does not hold, riskless arbitrage profits will exist, which market forces will immediately eliminate.

Core Mathematical Formulations:

When the actual market futures price ($F_{\text{actual}}$) deviates from the theoretical fair futures price ($F_{\text{theoretical}}$), two riskless arbitrage strategies emerge:

Stock Index Futures: Contracts where the underlying asset is a stock market index (e.g., NIFTY 50, BSE SENSEX). Because a stock index cannot be physically delivered, index futures are settled exclusively in cash on the settlement date.

Contract Value of Index Futures:

$$\text{Futures Contract Value } (V_f) = \text{Futures Price} \times \text{Multiplier (Lot Size)}$$

Portfolio Beta Management: Fund managers use stock index futures to modify the systematic risk (Beta $\beta$) of an existing equity portfolio without the costly liquidation and re-acquisition of physical shares.

Formulas for Portfolio Hedging:

Where:

• $V_p$ = Total current market value of the equity portfolio.
• $\beta_p$ = Weighted average Beta of the equity portfolio.
• $\beta^*$ = Target Beta desired by the fund manager.
• $V_f$ = Current value of one index futures contract ($\text{Index Futures Price} \times \text{Lot Size}$).
• $N$ = Number of index futures contracts to be traded.

The portfolio beta ($\beta_p$) is the weighted average of individual asset betas:

$$\beta_p = \sum_{i=1}^{n} w_i \cdot \beta_i = \sum_{i=1}^{n} \frac{\text{Value of Stock } i}{V_p} \cdot \beta_i$$

Option Contract: An option is a contract that gives the buyer (holder) the right, but not the obligation, to buy (Call Option) or sell (Put Option) a specified quantity of an underlying asset at a specified price (Strike or Exercise Price, $X$) on or before a specified expiration date ($T$). The seller (writer) is obligated to perform if the buyer exercises.

Moneyness of Options:

Total Option Premium: $\text{Option Premium} = \text{Intrinsic Value} + \text{Time Value}$. Intrinsic value can never be negative: $\text{IV}_{\text{Call}} = \max(0, S_0 - X)$; $\text{IV}_{\text{Put}} = \max(0, X - S_0)$.

Let $c$ be Call premium and $p$ be Put premium:

The Binomial Model (Cox-Ross-Rubinstein): Assumes that over each discrete time step, the stock price $S$ can move to only one of two possible values: an up price $S_u = S \cdot u$ or a down price $S_d = S \cdot d$ (where $u > 1$ and $d < 1$).

Two Complementary Approaches to Valuation:

In a 2-period binomial tree (e.g. Illustration 1 & Q23), work backward from expiration ($t=2$) to $t=1$ and finally to $t=0$:

1. Calculate terminal payoffs at $t=2$: $C_{uu} = \max(0, S_{uu}-X)$, $C_{ud} = \max(0, S_{ud}-X)$, $C_{dd} = \max(0, S_{dd}-X)$.
2. Calculate option values at node $u$ and node $d$ at $t=1$: $$C_u = \frac{p \cdot C_{uu} + (1-p) \cdot C_{ud}}{1+r}, \quad C_d = \frac{p \cdot C_{ud} + (1-p) \cdot C_{dd}}{1+r}$$
3. Calculate option value at initial node ($t=0$): $$C_0 = \frac{p \cdot C_u + (1-p) \cdot C_d}{1+r} = \frac{p^2 C_{uu} + 2p(1-p)C_{ud} + (1-p)^2 C_{dd}}{(1+r)^2}$$

Black-Scholes-Merton Formula for European Call Option:

$$C_0 = S_0 \cdot N(d_1) - X \cdot e^{-rt} \cdot N(d_2)$$

Where:

$$d_1 = \frac{\ln(S_0 / X) + \left(r + \frac{\sigma^2}{2}\right)t}{\sigma \sqrt{t}}$$ $$d_2 = d_1 - \sigma \sqrt{t}$$

• $S_0$ = Current spot price of the stock.
• $X$ = Strike (exercise) price of the option.
• $r$ = Annualized continuously compounded risk-free interest rate.
• $t$ = Time to expiration expressed in years ($t = \text{months} / 12$).
• $\sigma$ = Annualized standard deviation of continuously compounded stock returns (volatility).
• $\ln$ = Natural logarithm.
• $N(d)$ = Cumulative probability distribution function for a standardized normal distribution variable.

Assumptions of the Black-Scholes Model (Theoretical Q3):

1. European Style: Options can be exercised only at expiration date.
2. No Transaction Costs or Taxes: Markets are frictionless; securities are perfectly divisible.
3. Constant Risk-Free Rate: The risk-free rate $r$ is known and constant over the option life.
4. Lognormal Distribution: Stock prices follow a geometric Brownian motion; returns are normally distributed with constant volatility $\sigma$.
5. No Arbitrage Opportunities: Financial markets are efficient with continuous trading.
6. No Dividends: The underlying stock pays no dividend during the life of the option (unless adjusted).

Dividend Adjustment in Black-Scholes: If a known dividend $D$ is paid during the option life at time $t_d$, deduct its present value from spot price:

$$S_0^* = S_0 - D \cdot e^{-rt_d}$$

Use $S_0^*$ in place of $S_0$ in the formula.

For European options on non-dividend paying stocks with identical strike $X$ and maturity $t$, Put-Call Parity establishes an inviolable equilibrium:

$$C_0 + X \cdot e^{-rt} = P_0 + S_0$$

Or expressed in terms of Put Option value:

$$P_0 = C_0 + X \cdot e^{-rt} - S_0$$

The Option Greeks: Statistical measures of option price sensitivities to changes in underlying parameters:

Exotic Options vs. Plain Vanilla Options:

• Barrier Options: Payoff depends on whether the underlying asset price touches or breaches a predefined barrier price level during the option's life (Knock-in vs. Knock-out).
• Asian Options: Payoff depends on the arithmetic or geometric average price of the underlying asset over the option's life rather than the price at maturity. Eliminates price manipulation at expiry.
• Bermudan Options: Can be exercised on specified predetermined discrete dates prior to expiration (hybrid between European and American).
• Chooser Options ("As You Like It"): The holder has the right to decide whether the option is a Call or a Put at a specified future date.
• Compound Options: An option on an option (e.g., Call on Call, Call on Put, Put on Call, Put on Put).
• Digital / Binary Options: Payoff is fixed cash or asset if the condition is satisfied, zero otherwise (all-or-nothing).

Collateralized Debt Obligations (CDOs): A structured financial instrument backed by a diversified pool of debt obligations (bonds, commercial loans, mortgage-backed securities). Cash flows from the underlying debt pool are redistributed across different prioritized tranches:

• Senior Tranche (AAA): First claim on cash flows, lowest yield, protected from initial defaults.
• Mezzanine Tranche (AA to BBB): Intermediate risk and yield; absorbs losses after the equity tranche is wiped out.
• Equity / Subordinated Tranche (Unrated / First Loss): Receives residual cash flows, highest yield, absorbs the first default losses.

Types of CDOs:

1. Balance Sheet CDOs: Originating banks transfer existing loans off their balance sheet to reduce regulatory capital and free up credit capacity.
2. Arbitrage CDOs: Created by asset managers to exploit yield spread differentials between higher-yielding collateral assets and lower-cost CDO debt tranches.

Credit Default Swaps (CDS): A bilateral OTC contract wherein the Protection Buyer pays a periodic fee (the CDS spread) to the Protection Seller in exchange for a contingent payoff if a specified Reference Entity experiences a defined Credit Event (bankruptcy, failure to pay, debt restructuring).

Settlement Mechanisms for CDS:

• Physical Settlement: Buyer delivers the defaulted bond/debt instrument to the seller and receives the full par value in cash.
• Cash Settlement: Seller pays the buyer the difference between par value and post-default market recovery value ($\text{Payoff} = \text{Par Value} - \text{Recovery Value}$).

Real Option: Managerial flexibility to alter capital investment decisions (expand, contract, abandon, defer) in response to uncertain future economic developments. Traditional DCF / NPV assumes passive management; Real Options Valuation recognizes the strategic value of managerial flexibility:

$$\text{Strategic NPV} = \text{Static NPV} + \text{Value of Real Option}$$

Mapping Financial Options to Real Options:

Major Types of Real Options (Illustrations 4, 5, 6):

1. Growth / Expansion Option (Call Option): Opportunity to make follow-on investments in newer markets or technologies if initial pilot project succeeds (Illustration 4: Aidrex Patent).
2. Abandonment Option (Put Option): Flexibility to terminate a project early and realize its salvage or resale value if market conditions turn adverse (Illustration 5: Fertilizer Plant).
3. Timing / Deferral Option (Call Option): Flexibility to wait and resolve market uncertainty before committing substantial irreversible capital (Illustration 6: Solar Plant).

Commodity Futures Pricing: Unlike financial assets, physical commodities entail storage costs ($u$), insurance, deterioration, and provide a Convenience Yield ($c$)—the benefit of holding physical inventory to meet unexpected manufacturing bottlenecks or maintain customer loyalty:

$$F_0 = S_0 \cdot e^{(r + u - c)t}$$

Commodity Swaps: Agreements where one counterparty pays a fixed commodity price per unit and receives the floating market spot price (or vice versa) on predetermined settlement dates, insulating producers and industrial consumers from commodity price spikes.

Weather Derivatives: Non-catastrophic temperature and precipitation hedging instruments based on Heating Degree Days (HDD) and Cooling Degree Days (CDD):

$$\text{HDD} = \max(0, 18^\circ\text{C} - \text{Daily Mean Temp}); \quad \text{CDD} = \max(0, \text{Daily Mean Temp} - 18^\circ\text{C})$$

Electricity Derivatives: Electricity cannot be economically stored at scale; forward contracts and swaps settle against hourly grid clearing prices.

The ICAI Study Material details four famous corporate disasters resulting from derivatives mismanagement:

1. Barings Bank Collapse (1995 - Nick Leeson): Singapore rogue trader Nick Leeson took massive unauthorized long positions on Nikkei 225 futures and short straddles. Following the Kobe earthquake, Nikkei plunged, causing losses exceeding \$1.4 Billion and bankrupting the 233-year-old bank. Core Failure: Lack of segregation of front-office trading and back-office settlement controls.
2. Orange County Bankruptcy (1994 - Robert Citron): County treasurer used reverse repurchase agreements to leverage a \$7.5 Billion portfolio into \$20 Billion of interest rate inverse floaters, betting rates would fall. When the Federal Reserve hiked rates 6 times in 1994, the fund lost \$1.6 Billion and declared bankruptcy. Core Failure: Excessive leverage and speculative mismatch on public treasury funds.
3. Procter & Gamble (P&G) vs. Bankers Trust (1994): P&G entered into complex leveraged interest rate swaps containing embedded CMT (Constant Maturity Treasury) swaptions, incurring \$157 Million in losses when interest rates rose. Core Failure: Entering into highly complex, leveraged OTC derivative structures without fully understanding their underlying risk profiles.
4. Gibson Greetings (1994): Greeting card manufacturer lost \$20 Million on interest rate ratio swaps exceeding its total annual operating profits.

Key Lessons from Derivative Disasters:

• Never trade any derivative product whose payoff structure, pricing model, and risk sensitivities are not thoroughly understood.
• Strictly separate Front Office (trading execution), Middle Office (risk management and compliance), and Back Office (accounting, confirmation, and settlement).
• Institute rigid stop-loss limits, Value-at-Risk (VAR) ceilings, and maximum exposure limits.
• Do not treat corporate treasury as an aggressive speculative profit centre.

Following is a two sub-periods tree of 6-months each for the share of CAB Ltd.:

Risk-free rate of return is 5% for every 6 months ($R = 1.05$). The strike price of a 1-year European call option is ₹ 36.

Required: Calculate the value of the 1-year European Call Option today using the Binomial Model.

Step 1: Calculate Payoffs at Expiration ($t=2$ / 12 months)

Strike Price $X = \text{₹ } 36$. Terminal call payoffs $C = \max(0, S_T - X)$:

Step 2: Calculate Risk-Neutral Probability ($p$)

Over each 6-month period, the up-factor $u = \frac{43.20}{36.00} = 1.20$, and down-factor $d = \frac{28.80}{36.00} = 0.80$.

Risk-free factor $R = 1 + r = 1 + 0.05 = 1.05$.

$$p = \frac{R - d}{u - d} = \frac{1.05 - 0.80}{1.20 - 0.80} = \frac{0.25}{0.40} = 0.625$$ $$1 - p = 1 - 0.625 = 0.375$$

Step 3: Calculate Option Values at Intermediate Nodes ($t=1$ / 6 months)

• At Upper Node $u$ ($S_u = \text{₹ } 43.20$): $$C_u = \frac{p \cdot C_{uu} + (1-p) \cdot C_{ud}}{R} = \frac{0.625 \times 15.84 + 0.375 \times 0}{1.05} = \frac{9.90}{1.05} = \text{₹ } 9.4286$$
• At Lower Node $d$ ($S_d = \text{₹ } 28.80$): $$C_d = \frac{p \cdot C_{ud} + (1-p) \cdot C_{dd}}{R} = \frac{0.625 \times 0 + 0.375 \times 0}{1.05} = \text{₹ } 0.00$$

Step 4: Calculate Option Value Today ($t=0$)

$$C_0 = \frac{p \cdot C_u + (1-p) \cdot C_d}{R} = \frac{0.625 \times 9.4286 + 0.375 \times 0}{1.05} = \frac{5.8929}{1.05} = \text{₹ } 5.612 \approx \text{₹ } 5.61$$

The current market price of an equity share of Penchant Ltd is ₹ 420. Within a period of 3 months, the maximum and minimum price of it is expected to be ₹ 500 and ₹ 400 respectively. If the risk-free rate of interest is 8% p.a., what should be the value of a 3-month Call option with a strike price of ₹ 420 under the Risk Neutral Method?

Step 1: Identify Parameters & Up/Down Factors

• Current Stock Price ($S_0$) = ₹ 420
• Upper Stock Price ($S_u$) = ₹ 500 $\Rightarrow u = \frac{500}{420} = 1.190476$
• Lower Stock Price ($S_d$) = ₹ 400 $\Rightarrow d = \frac{400}{420} = 0.952381$
• Strike Price ($X$) = ₹ 420
• Time to expiration ($t$) = 3 months = $3/12 = 0.25$ years
• Risk-free rate ($r$) = 8% p.a. $\Rightarrow R = 1 + r \cdot t = 1 + 0.08 \times 0.25 = 1.02$

Step 2: Terminal Call Option Payoffs

• If stock reaches ₹ 500: $C_u = \max(0, 500 - 420) = \text{₹ } 80$
• If stock drops to ₹ 400: $C_d = \max(0, 400 - 420) = \text{₹ } 0$

Step 3: Solve for Risk-Neutral Probability ($p$)

Under risk neutrality, the expected return on the stock must equal the risk-free rate:

$$S_0 \times (1 + r \cdot t) = p \cdot S_u + (1 - p) \cdot S_d$$ $$420 \times 1.02 = 500 \cdot p + 400 \cdot (1 - p)$$ $$428.40 = 400 + 100 \cdot p$$ $$100 \cdot p = 428.40 - 400 = 28.40 \implies p = \frac{28.40}{100} = 0.284$$ $$1 - p = 1 - 0.284 = 0.716$$

Step 4: Compute Present Value of Expected Option Payoff

$$\text{Expected Option Payoff at Maturity} = p \cdot C_u + (1 - p) \cdot C_d = (0.284 \times 80) + (0.716 \times 0) = \text{₹ } 22.72$$ $$C_0 = \frac{\text{Expected Payoff}}{1 + r \cdot t} = \frac{22.72}{1.02} = \text{₹ } 22.2745 \approx \mathbf{\text{₹ } 22.27}$$

(i) The shares of TIC Ltd. are currently priced at ₹ 415 and call option exercisable in three months’ time has an exercise rate of ₹ 400. Risk-free interest rate is 5% p.a. and standard deviation (volatility) of share price is 22%. Based on the assumption that TIC Ltd. is not going to declare any dividend over the next three months, is the option worth buying for ₹ 25?

(ii) Calculate value of aforesaid call option based on Black Scholes valuation model if the current price is considered as ₹ 380.

(iii) What would be the worth of put option if current price is considered ₹ 380?

(iv) If TIC Ltd. share price at present is taken as ₹ 408 and a dividend of ₹ 10 is expected to be paid in two months’ time, then calculate value of the call option.

Part (i): Valuation when Spot Price $S_0 = \text{₹ } 415$

Given: $S_0 = 415, X = 400, r = 0.05, t = 3/12 = 0.25, \sigma = 0.22$.

$$\sigma \sqrt{t} = 0.22 \times \sqrt{0.25} = 0.22 \times 0.5 = 0.11$$ $$\frac{S_0}{X} = \frac{415}{400} = 1.0375 \implies \ln(1.0375) = 0.03681$$ $$\left(r + \frac{\sigma^2}{2}\right) t = \left(0.05 + \frac{0.22^2}{2}\right) \times 0.25 = (0.05 + 0.0242) \times 0.25 = 0.0742 \times 0.25 = 0.01855$$ $$d_1 = \frac{\ln(S_0 / X) + \left(r + \frac{\sigma^2}{2}\right)t}{\sigma \sqrt{t}} = \frac{0.03681 + 0.01855}{0.11} = \frac{0.05536}{0.11} = 0.50327$$ $$d_2 = d_1 - \sigma \sqrt{t} = 0.50327 - 0.11 = 0.39327$$

From Normal Distribution Table:

$$N(d_1) = N(0.50327) \approx 0.6928$$ $$N(d_2) = N(0.39327) \approx 0.6529$$ $$C_0 = S_0 \cdot N(d_1) - \frac{X}{e^{rt}} \cdot N(d_2) = 415 \times 0.6928 - \frac{400}{1.012578} \times 0.6529$$ $$C_0 = 287.512 - 395.031 \times 0.6529 = 287.512 - 257.916 = \mathbf{\text{₹ } 29.60}$$

Decision: Since the theoretical fair value of the call option is ₹ 29.60 and it is available in the market for ₹ 25.00, the option is underpriced in the market. Hence, it is worth buying.

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Part (ii): Valuation when Spot Price $S_0 = \text{₹ } 380$

Given: $S_0 = 380, X = 400 \implies \frac{S_0}{X} = \frac{380}{400} = 0.95 \implies \ln(0.95) = -0.05129$.

$$d_1 = \frac{-0.05129 + 0.01855}{0.11} = \frac{-0.03274}{0.11} = -0.29764$$ $$d_2 = -0.29764 - 0.11 = -0.40764$$ $$N(d_1) = N(-0.29764) = 1 - N(0.29764) = 1 - 0.6170 = 0.3830$$ $$N(d_2) = N(-0.40764) = 1 - N(0.40764) = 1 - 0.6582 = 0.3418$$ $$C_0 = 380 \times 0.3830 - \frac{400}{1.012578} \times 0.3418 = 145.54 - 395.031 \times 0.3418 = 145.54 - 135.02 = \mathbf{\text{₹ } 10.52}$$

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Part (iii): Valuation of Put Option when Spot Price $S_0 = \text{₹ } 380$

Using the Put-Call Parity Theorem:

$$P_0 = C_0 + \frac{X}{e^{rt}} - S_0$$ $$P_0 = 10.52 + \frac{400}{1.012578} - 380 = 10.52 + 395.031 - 380 = \mathbf{\text{₹ } 25.55}$$

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Part (iv): Valuation with Expected Dividend ($S_0 = \text{₹ } 408$, $D = \text{₹ } 10$ in 2 Months)

Time to dividend $t_d = 2/12 = 0.1667$ years. Interest factor $r \cdot t_d = 0.05 \times 0.1667 = 0.008333$.

$$\text{PV of Dividend} = D \times e^{-rt_d} = \frac{10}{e^{0.008333}} = \frac{10}{1.0084} = \text{₹ } 9.9167$$ $$\text{Adjusted Spot Price } (S_0^*) = 408.00 - 9.9167 = \text{₹ } 398.0833$$ $$\frac{S_0^*}{X} = \frac{398.0833}{400} = 0.9952 \implies \ln(0.9952) = -0.00481$$ $$d_1 = \frac{-0.00481 + 0.01855}{0.11} = \frac{0.01374}{0.11} = 0.1249 \approx 0.125$$ $$d_2 = 0.125 - 0.11 = 0.015$$ $$N(d_1) = N(0.125) = 0.5498; \quad N(d_2) = N(0.015) = 0.5060$$ $$C_0 = 398.083 \times 0.5498 - \frac{400}{1.012578} \times 0.5060 = 218.866 - 395.031 \times 0.5060$$ $$C_0 = 218.866 - 199.886 = \mathbf{\text{₹ } 18.98}$$

ABC Ltd. is a pharmaceutical company possessing a patent of a drug called ‘Aidrex’, a medicine for AIDS patients. Being an approach drug, ABC Ltd. holds the right of production of drugs and its marketing. The period of patent is 15 years after which any other pharmaceutical company can produce the drug with the same formula. It is estimated that company shall require to incur $ 12.5 million for development and marketing of the drug. As per a survey conducted, the expected present value of cash flows from the sale of the drug during the period of 15 years shall be $ 16.7 million. Cash flows from a previous similar type of drug have exhibited a variance of 26.8% ($\sigma^2 = 0.268$). The current yield on Treasury Bonds of similar duration (15 years) is 7.8%.

Required: Determine the value of the patent.

Step 1: Map Capital Budgeting Parameters to Black-Scholes Model

• Current Present Value of Cash Flows ($S_0$) = $ 16.70 Million
• Capital Outlay required to commercially exploit ($X$) = $ 12.50 Million
• Time horizon ($t$) = 15 Years
• Variance of cash flows ($\sigma^2$) = 0.268 ($\sigma = \sqrt{0.268} = 0.5177$)
• Risk-free interest rate ($r$) = 7.8% p.a. = 0.078
• Continuous cash flow loss / dividend yield ($y$): Each year delay causes loss of 1 year of exclusivity: $$y = \frac{1}{15} = 0.0667 \text{ (6.67% p.a.)}$$

Step 2: Calculate $d_1$ and $d_2$

$$\sigma \sqrt{t} = 0.5177 \times \sqrt{15} = 0.5177 \times 3.873 = 2.005$$ $$\frac{S_0}{X} = \frac{16.70}{12.50} = 1.336 \implies \ln(1.336) = 0.2897$$ $$\left(r - y + \frac{\sigma^2}{2}\right) t = \left(0.078 - 0.0667 + \frac{0.268}{2}\right) \times 15 = (0.0113 + 0.1340) \times 15 = 0.1453 \times 15 = 2.1795$$ $$d_1 = \frac{0.2897 + 2.1795}{2.005} = \frac{2.4692}{2.005} = 1.2315$$ $$d_2 = d_1 - \sigma \sqrt{t} = 1.2315 - 2.005 = -0.7735$$

From Normal Distribution Table:

$$N(d_1) = N(1.23) = 0.8907$$ $$N(d_2) = N(-0.77) = 1 - N(0.77) = 1 - 0.7794 = 0.2206$$

Step 3: Value of Patent as European Call Option with Dividend Yield

$$C = S_0 \cdot e^{-yt} \cdot N(d_1) - X \cdot e^{-rt} \cdot N(d_2)$$

Where $e^{-yt} = e^{-0.0667 \times 15} = e^{-1.0005} = 0.3677$ and $e^{-rt} = e^{-0.078 \times 15} = e^{-1.17} = 0.3104$.

$$C = (16.70 \times 0.3677 \times 0.8907) - (12.50 \times 0.3104 \times 0.2206)$$ $$C = (6.1406 \times 0.8907) - (3.8800 \times 0.2206) = 5.4694 - 0.8559 = \mathbf{\$ 4.6135 \text{ Million}}$$

IPL, already in the production of Fertilizer, is considering a proposal of building a new plant to produce pesticides. Suppose the PV of the proposal is ₹ 100 crore without the abandonment option. However, if market conditions for pesticide turn out to be favourable, the PV of the proposal shall increase by 30%. On the other hand, if market conditions remain sluggish, the PV of the proposal shall be reduced by 40%. In case the company is not interested in continuation of the project, it can be disposed of for ₹ 80 crore.

If the risk-free rate of interest is 8% p.a., what will be the value of the abandonment option?

Step 1: Construct Decision Tree at Year 1

Step 2: Solve for Risk-Neutral Probability ($p$)

Expected return on project under risk-neutral probability must equal risk-free rate (8%):

$$p \times (+30\%) + (1 - p) \times (-40\%) = 8\%$$ $$0.30p - 0.40 + 0.40p = 0.08$$ $$0.70p = 0.48 \implies p = \frac{0.48}{0.70} = 0.6857 \approx 0.686$$ $$1 - p = 1 - 0.6857 = 0.3143 \approx 0.314$$

Step 3: Calculate Value of Abandonment Option

The abandonment option is an American/European Put Option on project value with strike price $X = \text{₹ } 80 \text{ Cr}$:

$$\text{Expected Option Payoff at Year 1} = p \times 0 + (1 - p) \times 20 = 0.3143 \times 20 = \text{₹ } 6.286 \text{ Cr}$$ $$\text{Value of Abandonment Option Today} = \frac{\text{Expected Payoff}}{1 + r} = \frac{\text{₹ } 6.286 \text{ Cr}}{1.08} = \mathbf{\text{₹ } 5.82 \text{ Crore}}$$

Suppose MIS Ltd. is considering installation of a solar electricity generating plant for lighting the staff quarters. The plant shall cost ₹ 2.50 crore and shall lead to saving in electricity expenses at the current tariff by ₹ 21 lakh per year forever.

However, after one year with a change in Government in the state, the rate of electricity is subject to change. Accordingly, the saving in electricity can be of ₹ 12 lakh or ₹ 35 lakh per year and forever.

Assuming WACC of MIS Ltd. is 10% and risk-free rate of return is 8%, decide whether MIS Ltd. should accept the project today or wait and see.

Alternative 1: Invest Today ($t=0$)

Present value of perpetual annual savings of ₹ 21 Lakh discounted at WACC (10%):

$$\text{PV of Savings} = \frac{\text{Annual Saving}}{\text{WACC}} = \frac{\text{₹ } 21 \text{ Lakh}}{0.10} = \text{₹ } 210 \text{ Lakh} = \text{₹ } 2.10 \text{ Crore}$$ $$\text{Static NPV}_{\text{Today}} = \text{PV of Inflows} - \text{Initial Cost} = \text{₹ } 2.10 \text{ Cr} - \text{₹ } 2.50 \text{ Cr} = \mathbf{-\text{₹ } 0.40 \text{ Crore (Negative)}}$$

Based on static NPV alone, the project should be rejected today.

Alternative 2: Defer Investment by One Year ($t=1$)

At the end of Year 1, uncertainty about electricity tariff is resolved:

• State A (High Tariff - Saving ₹ 35 Lakh forever): $$\text{PV of Savings at } t=1 = \frac{\text{₹ } 35 \text{ Lakh}}{0.10} = \text{₹ } 350 \text{ Lakh} = \text{₹ } 3.50 \text{ Crore}$$ $$\text{NPV at } t=1 = \text{₹ } 3.50 \text{ Cr} - \text{₹ } 2.50 \text{ Cr} = \mathbf{+\text{₹ } 1.00 \text{ Crore (Accept Project)}}$$
• State B (Low Tariff - Saving ₹ 12 Lakh forever): $$\text{PV of Savings at } t=1 = \frac{\text{₹ } 12 \text{ Lakh}}{0.10} = \text{₹ } 120 \text{ Lakh} = \text{₹ } 1.20 \text{ Crore}$$ $$\text{NPV at } t=1 = \text{₹ } 1.20 \text{ Cr} - \text{₹ } 2.50 \text{ Cr} = \mathbf{-\text{₹ } 1.30 \text{ Crore (Do Not Invest / Abandon)}}$$ $$\text{Payoff at } t=1 = \max(0, -1.30) = \mathbf{\text{₹ } 0.00}$$

Step 3: Solve for Risk-Neutral Probability ($p$)

Current value of electricity savings portfolio is ₹ 210 Lakh. At $t=1$, value is either ₹ 350 Lakh or ₹ 120 Lakh:

$$210 \times (1 + r_f) = p \times 350 + (1 - p) \times 120$$ $$210 \times 1.08 = 350p + 120 - 120p$$ $$226.80 = 120 + 230p \implies 230p = 106.80 \implies p = \frac{106.80}{230} = 0.4643$$ $$1 - p = 1 - 0.4643 = 0.5357$$

Step 4: Calculate Present Value of Timing Option

$$\text{Expected Payoff at } t=1 = (0.4643 \times \text{₹ } 1.00 \text{ Cr}) + (0.5357 \times \text{₹ } 0) = \text{₹ } 0.4643 \text{ Crore}$$ $$\text{NPV of Waiting (Timing Option Value)} = \frac{\text{₹ } 0.4643 \text{ Cr}}{1.08} = \mathbf{+\text{₹ } 0.43 \text{ Crore (+₹ 42.99 Lakh)}}$$

What are the reasons for stock index futures becoming more popular financial derivatives over stock futures segment in India?

Stock index futures have emerged as the most active and dominant segment of equity derivatives globally and in India due to several fundamental operational and economic advantages:

1. Institutional Portfolio Hedging Flexibility: Index futures enable institutional investors, mutual funds, and pension funds to hedge entire diversified equity portfolios in a single transaction without disturbing or liquidating individual underlying share holdings.
2. Cost Efficiency: Hedging a diversified portfolio using individual single stock futures requires executing dozens of separate contracts across multiple stocks, incurring substantial brokerage, exchange turnover fees, and securities transaction tax (STT). A single index futures contract provides comprehensive systematic risk hedging at a fraction of the transaction cost.
3. Immunity from Individual Stock Price Manipulation: Individual stock prices can be cornered or manipulated by concentrated market participants or market operators. A broad-based index (like NIFTY 50 or S&P BSE SENSEX) reflects a diversified basket of liquid mega-cap stocks across sectors, making price manipulation virtually impossible.
4. Cash Settlement vs. Physical Delivery: Individual stock futures in India are subject to mandatory physical delivery settlement at expiry for open positions. In contrast, stock index futures are settled strictly in cash against the official index closing value, completely avoiding the logistical burdens, stamp duties, and delivery default risks of physical share transfer.
5. Substantially Lower Regulatory & Margin Complexity: Single stock futures require dynamic surveillance, circuit filters, price bands, and specific margin requirements (including delivery margins during expiry week). Index futures enjoy lower volatility, narrower Value-at-Risk margins, and simpler regulatory oversight.
6. Systematic vs. Unsystematic Risk Management: In a well-diversified equity portfolio, unsystematic (company-specific) risk is largely diversified away. The remaining risk is systematic market risk, which is precisely what stock index futures hedge.

Write short note on Marking to market.

Marking to Market (MTM) is the daily settlement mechanism mandated by derivative exchanges to ensure financial integrity and eliminate counterparty credit risk across futures contracts:

• Daily Settlement Process: At the end of each trading day, every open futures contract is revalued at the official closing settlement price established by the exchange.
• Cash Debits & Credits:
  • If the daily settlement price rises: The buyer's (Long position) margin account is credited with the daily gain, and the seller's (Short position) margin account is debited with an identical loss.
  • If the daily settlement price falls: The buyer's margin account is debited with the daily loss, and the seller's margin account is credited with the gain.
• Maintenance Margin & Margin Calls: When accumulated daily losses deplete a trader's margin account below the specified Maintenance Margin threshold, the broker/clearing member issues a Margin Call.
• Variation Margin Obligation: The trader must immediately deposit additional funds (known as Variation Margin) before the next trading session commences to restore the account balance back to the Initial Margin level (not merely back to the maintenance level).
• Default Prevention: By collecting losses and distributing gains every 24 hours, credit exposure is reset to zero daily, preventing large accumulated default losses that could threaten the clearinghouse.

State any four assumptions of Black Scholes Model.

The Black-Scholes-Merton option pricing model relies on the following fundamental structural assumptions (state any four for full marks):

1. European Exercise Style: The option can be exercised only at its final expiration date $T$; early exercise is precluded.
2. No Transaction Costs or Taxes: There are zero brokerage commissions, transaction levies, exchange fees, or tax frictions. Securities are completely liquid and infinitely divisible.
3. Constant and Known Risk-Free Interest Rate: The risk-free borrowing and lending rate $r$ remains constant and known over the entire life of the option contract.
4. Lognormal Stock Price Distribution: Stock prices follow a Geometric Brownian Motion with constant instantaneous drift and volatility ($\sigma$). Continuously compounded stock returns are normally distributed.
5. Continuous Trading & No Arbitrage: Trading occurs continuously in time, short sales are permitted without restrictions or borrowing fees, and no riskless arbitrage opportunities persist in equilibrium.
6. No Cash Dividends: The underlying equity share pays no cash dividends or distributions during the option's tenure (unless explicitly adjusted via continuous dividend yield or PV deduction).

Define the term Greeks with respect to options.

The term Option Greeks refers to a collection of statistical sensitivity measures derived from partial derivatives of option pricing models (such as Black-Scholes). They quantify the expected rate of change in an option's market price in response to changes in key underlying market variables:

The 6-months forward price of a security is ₹ 208.18. The borrowing rate is 8% per annum payable with continuous compounding. What should be the spot price?

Applicable Formula:

Under continuous compounding, the theoretical forward price is:

$$F = S_0 \cdot e^{rt}$$

Rearranging to solve for the spot price ($S_0$):

$$S_0 = \frac{F}{e^{rt}} = F \cdot e^{-rt}$$

Given Data:

• Forward Price ($F$) = ₹ 208.18
• Borrowing Rate ($r$) = 8% p.a. = 0.08
• Time Period ($t$) = 6 months = $6/12 = 0.5$ years
• Exponent $rt = 0.08 \times 0.5 = 0.04$
• From exponential tables: $e^{0.04} = 1.040811$

Calculation:

$$S_0 = \frac{208.18}{e^{0.04}} = \frac{208.18}{1.040811} = \mathbf{\text{₹ } 200.017 \approx \text{₹ } 200.00}$$

The following data relate to Anand Ltd.'s share price:

• Current price per share: ₹ 1,800
• 6-months future's price / share: ₹ 1,950

Assuming it is possible to borrow money in the market for transactions in securities at 12% per annum, you are required:

1. To calculate the theoretical minimum price of a 6-months forward purchase; and
2. To explain whether any arbitrage opportunity exists, and calculate the riskless arbitrage profit.

Part (i): Theoretical Minimum Price of 6-Months Forward Purchase

Using simple interest cost of carry:

$$F_{\text{theoretical}} = S_0 \cdot \left(1 + r \cdot \frac{t}{12}\right)$$ $$F_{\text{theoretical}} = 1,800 \times \left(1 + 0.12 \times \frac{6}{12}\right) = 1,800 \times (1 + 0.06) = 1,800 \times 1.06 = \mathbf{\text{₹ } 1,908.00}$$

(Note: Alternatively, under continuous compounding $1,800 \times e^{0.06} = 1,800 \times 1.06184 = \text{₹ } 1,911.31$)

Part (ii): Arbitrage Analysis & Cash Flow Proof

Market Observation:

$$F_{\text{actual}} = \text{₹ } 1,950 > F_{\text{theoretical}} = \text{₹ } 1,908$$

The actual futures contract is overvalued in the market. An arbitrageur will execute a Cash-and-Carry Arbitrage by buying the underlying share in the cash market and selling the futures contract.

On 31-8-2011, the value of stock index was ₹ 2,200. The risk-free rate of return has been 8% per annum. The dividend yield on this Stock Index is as under:

Assuming that interest is continuously compounded daily, find out the future price of contract deliverable on 31-12-2011.

Step 1: Determine Contract Duration & Average Dividend Yield

Contract tenure runs from 31-8-2011 to 31-12-2011, which spans 4 months (September, October, November, December):

$$\text{Average Annualized Dividend Yield } (y) = \frac{3\% + 3\% + 4\% + 3\%}{4} = \frac{13\%}{4} = 3.25\% \text{ p.a.} = 0.0325$$

Step 2: Calculate Exponent $(r - y)t$

• Risk-free rate ($r$) = 8% p.a. = 0.08
• Average Dividend yield ($y$) = 3.25% p.a. = 0.0325
• Time period ($t$) = 4 months = $4/12 = 0.3333$ years

$$(r - y) \cdot t = (0.08 - 0.0325) \times \frac{4}{12} = 0.0475 \times 0.33333 = 0.015833$$

Step 3: Calculate Theoretical Futures Price

$$F = S_0 \cdot e^{(r - y)t} = 2,200 \times e^{0.01583}$$

Given $e^{0.01583} = 1.01593$:

$$F = 2,200 \times 1.01593 = \mathbf{\text{₹ } 2,235.046 \approx \text{₹ } 2,235.05}$$

Calculate the price of 3 months PQR futures, if PQR (Face Value ₹ 10) quotes ₹ 220 on NSE and the three months future price quotes at ₹ 230. The one month borrowing rate is given as 15% per annum and the expected annual dividend is 25%, payable before expiry. Also examine arbitrage opportunities.

Step 1: Calculate Theoretical Fair Futures Price

• Spot Price ($S_0$) = ₹ 220
• Borrowing rate ($r$) = 15% p.a. = 0.15
• Contract duration ($t$) = 3 months = $3/12 = 0.25$ years
• Dividend = 25% on Face Value of ₹ 10 = $0.25 \times 10 = \text{₹ } 2.50$ (received before expiry)

$$F_{\text{fair}} = S_0 + \text{Cost of Carry} - \text{Dividend}$$ $$F_{\text{fair}} = 220 + [220 \times 0.15 \times 0.25] - 2.50 = 220 + 8.25 - 2.50 = \mathbf{\text{₹ } 225.75}$$

(Note: If interest earned on dividend for remaining period is considered, fair price is ₹ 225.72 to ₹ 225.75).

Step 2: Arbitrage Evaluation

The actual futures quote is ₹ 230.00, whereas the fair value is ₹ 225.75.

Since $F_{\text{actual}} > F_{\text{fair}}$, the futures contract is overvalued. An arbitrageur will sell futures and buy stock in the cash market (Cash-and-Carry Arbitrage).

Consider the following data:

We assume that a futures contract on the BSE index with four months maturity is used to hedge the value of portfolio over next three months. One futures contract is for delivery of 50 times the index.

Based on the above information calculate:

1. Price of futures contract.
2. The gain on short futures position if index turns out to be 4,500 in three months.

Part (i): Price of Futures Contract

Contract maturity is 4 months ($4/12$ years). Using cost of carry formula:

$$F = \text{Spot Index} + \text{Spot Index} \times (r - y) \times \frac{t}{12}$$ $$F = 5,000 + 5,000 \times (0.09 - 0.06) \times \frac{4}{12} = 5,000 + 5,000 \times 0.03 \times \frac{1}{3} = 5,000 + 50 = \mathbf{5,050 \text{ points}}$$ $$\text{Price of 1 Futures Contract} = 5,050 \times 50 = \mathbf{\text{₹ } 2,52,500}$$

Part (ii): Gain on Short Futures Position if Index Falls to 4,500 in 3 Months

Step A: Determine Number of Contracts to Short (Hedge Ratio)

$$N = \frac{\text{Value of Portfolio} \times \beta_p}{\text{Futures Contract Value}} = \frac{10,10,000 \times 1.5}{2,52,500} = \frac{15,15,000}{2,52,500} = \mathbf{6 \text{ contracts}}$$

Step B: Valuation of 4-Month Futures Contract after 3 Months (1 Month to Expiry)

After 3 months, the spot index is 4,500, and remaining maturity is $4 - 3 = 1$ month ($1/12$ years):

$$F_3 = 4,500 + 4,500 \times (0.09 - 0.06) \times \frac{1}{12} = 4,500 + 4,500 \times 0.03 \times \frac{1}{12} = 4,500 + 11.25 = \mathbf{4,511.25 \text{ points}}$$ $$\text{Contract Value at 3 Months} = 4,511.25 \times 50 = \mathbf{\text{₹ } 2,25,562.50}$$

Step C: Compute Gain on 6 Short Contracts

$$\text{Gain per Contract} = \text{Original Value } (2,52,500) - \text{Value at 3 Months } (2,25,562.50) = \text{₹ } 26,937.50$$ $$\text{Total Gain on 6 Contracts} = 6 \times \text{₹ } 26,937.50 = \mathbf{\text{₹ } 1,61,625}$$

The share of X Ltd. is currently selling for ₹ 300. Risk-free interest rate is 0.8% per month. A 3-months futures contract is traded.

1. Calculate the appropriate fair value of the 3-months futures contract.
2. What action should be taken if the 3-months futures contract trades at ₹ 312?
3. What would be the price per share of X Ltd. today if the 3-months futures contract trades at ₹ 312 and is correctly priced?

Part (i): Appropriate Fair Value of 3-Months Futures Contract

Given: Spot price $S_0 = \text{₹ } 300$, monthly risk-free rate $r_m = 0.8\% = 0.008$, $n = 3$ months.

Compounding monthly:

$$F_0 = S_0 \cdot (1 + r_m)^3 = 300 \times (1.008)^3 = 300 \times 1.024193 = \mathbf{\text{₹ } 307.26}$$

(Or using simple interest: $300 \times (1 + 0.008 \times 3) = 300 \times 1.024 = \text{₹ } 307.20$)

Part (ii): Action if Futures Trades at ₹ 312

Since market futures price (₹ 312) > fair value (₹ 307.26), the futures contract is overpriced.

Recommended Strategy: Execute Cash-and-Carry Arbitrage:

• Borrow ₹ 300 today at 0.8% per month for 3 months.
• Buy 1 share of X Ltd at ₹ 300 in the cash market.
• Sell (short) 1 3-months futures contract at ₹ 312.
• At expiry, deliver share against futures to receive ₹ 312, repay loan ($300 \times 1.008^3 = \text{₹ } 307.26$). Profit = $312 - 307.26 = \mathbf{\text{₹ } 4.74 \text{ per share}}$.

Part (iii): Implied Spot Price if ₹ 312 is Correctly Priced

$$F_0 = S_0 \cdot (1.008)^3 \implies 312 = S_0 \times 1.024193$$ $$S_0 = \frac{312}{1.024193} = \mathbf{\text{₹ } 304.63}$$

A Mutual Fund is holding the following assets in ₹ Crores:

• Investments in diversified equity shares: ₹ 90.00 Cr
• Cash and Bank Balances: ₹ 10.00 Cr
• Total Portfolio Value: ₹ 100.00 Cr

The beta of the equity share portfolio is 1.10. The index future is currently trading at ₹ 8,400 and each contract represents 100 units of index.

Required:

1. Find the number of index futures to be sold to establish a complete hedge.
2. How many futures should be sold to hedge 80% of its equity portfolio?
3. Verify your answer if the index falls by 10%.

Value of 1 Index Futures Contract:

$$V_f = 8,400 \times 100 = \text{₹ } 8,40,000 = \text{₹ } 0.084 \text{ Crore}$$

Part (i): Number of Contracts to Sell for Complete Hedge

The fund manager wishes to hedge the equity shares portion worth ₹ 90.00 Cr having $\beta = 1.10$:

$$N = \frac{V_{\text{equity}} \times \beta_{\text{equity}}}{V_f} = \frac{\text{₹ } 90 \text{ Cr} \times 1.10}{\text{₹ } 0.084 \text{ Cr}} = \frac{99.00}{0.084} = 1,178.57 \approx \mathbf{1,179 \text{ contracts}}$$

Part (ii): Number of Contracts to Hedge 80% of Equity Portfolio

$$N_{80\%} = 80\% \times N = 0.80 \times 1,178.57 = 942.86 \approx \mathbf{943 \text{ contracts}}$$

Part (iii): Verification if Index Falls by 10%

If the market index falls by 10%:

• Percentage fall in equity share portfolio = $\text{Fall in Index} \times \beta = 10\% \times 1.10 = 11\%$.
• Loss in value of equity portfolio = $11\% \times \text{₹ } 90 \text{ Cr} = \mathbf{\text{₹ } 9.90 \text{ Crore}}$.
• Index futures price drops by 10% from ₹ 8,400 to $8,400 \times 0.90 = \text{₹ } 7,560$ (Drop = ₹ 840 per unit).
• Gain on 1,178.57 short futures contracts: $$\text{Gain on Futures} = 1,178.57 \times 840 \times 100 = \text{₹ } 9,89,99,880 \approx \mathbf{\text{₹ } 9.90 \text{ Crore}}$$

Net Outcome: Loss on physical share portfolio (-₹ 9.90 Cr) is perfectly offset by the Gain on short futures contracts (+₹ 9.90 Cr). Net change in portfolio value is Zero, proving a complete hedge.

A trader is having in its portfolio shares worth ₹ 85 lakhs at current price and cash ₹ 15 lakhs. The beta of share portfolio is 1.6. After 3 months the price of shares dropped by 3.2%.

Determine:

1. Current portfolio beta.
2. Portfolio beta after 3 months if the trader on current date goes for long position on ₹ 100 lakhs Nifty futures.

Part (i): Current Portfolio Beta

Total Portfolio Value = ₹ 85 Lakhs (Shares) + ₹ 15 Lakhs (Cash) = ₹ 100 Lakhs.

• Weight of shares ($w_s$) = $\frac{85}{100} = 0.85$; Beta of shares ($\beta_s$) = 1.6
• Weight of cash ($w_c$) = $\frac{15}{100} = 0.15$; Beta of cash ($\beta_c$) = 0

$$\beta_{\text{current}} = (0.85 \times 1.6) + (0.15 \times 0) = 1.36 + 0 = \mathbf{1.36}$$

Part (ii): Portfolio Beta after Long Position on ₹ 100 Lakhs Nifty Futures

Since the trader goes long on index futures, additional systematic risk is added. The beta of the index is 1.0.

$$\text{New Effective Beta } (\beta_{\text{new}}) = \beta_{\text{current}} + \frac{V_f \times \beta_{\text{index}}}{V_p} = 1.36 + \frac{\text{₹ } 100 \text{ Lakh} \times 1.0}{\text{₹ } 100 \text{ Lakh}} = 1.36 + 1.00 = \mathbf{2.36}$$

Which position on the index future gives a speculator a complete hedge against the following transactions:

1. The share of Right Limited is going to rise. He has a long position on the cash market of ₹ 50 lakhs on Right Limited ($\beta = 1.25$).
2. The share of Wrong Limited is going to depreciate. He has a short position on the cash market of ₹ 25 lakhs on Wrong Limited ($\beta = 0.90$).
3. The share of Fair Limited is going to be stagnant. He has a short position on the cash market of ₹ 20 lakhs on Fair Limited ($\beta = 0.75$).

Principles of Hedging Cash Positions with Index Futures:

• To hedge a Long position in cash market $\to$ Take a Short position in Index Futures.
• To hedge a Short position in cash market $\to$ Take a Long position in Index Futures.
• Effective Hedge Amount = $\text{Cash Market Exposure} \times \text{Beta}$.

Ram buys 10,000 shares of X Ltd. at a price of ₹ 22 per share ($\beta = 1.5$) and sells 5,000 shares of A Ltd. at a price of ₹ 40 per share ($\beta = 2.0$). He obtains a complete hedge by Nifty futures at ₹ 1,000 each.

He closes out his position at the closing price of the next day when:

• The share of X Ltd. dropped by 2%
• The share of A Ltd. appreciated by 3%
• Nifty futures dropped by 1.5%

Required: What is the overall profit / loss to Ram?

Step 1: Calculate Net Systematic Exposure & Number of Futures Contracts

• Long Position in X Ltd = $10,000 \times \text{₹ } 22 = \text{₹ } 2,20,000$; Weighted Exposure = $2,20,000 \times 1.5 = +\text{₹ } 3,30,000$.
• Short Position in A Ltd = $5,000 \times \text{₹ } 40 = \text{₹ } 2,00,000$; Weighted Exposure = $-2,00,000 \times 2.0 = -\text{₹ } 4,00,000$.
• Net Systematic Position = $+3,30,000 - 4,00,000 = \mathbf{-\text{₹ } 70,000}$ (Net Short Exposure).

To establish a complete hedge against a net short exposure, Ram must take a Long position in Nifty futures:

$$N = \frac{\text{Net Exposure}}{\text{Futures Price}} = \frac{\text{₹ } 70,000}{\text{₹ } 1,000} = \mathbf{70 \text{ Long Nifty Futures Contracts}}$$

Step 2: Calculate Next Day Gain / Loss Breakdown

On January 1, 2013 an investor has a portfolio of 5 shares as given below:

The cost of capital to the investor is 10.5% per annum. You are required to calculate:

1. The beta of his portfolio.
2. The theoretical value of the NIFTY futures for February 2013 (2 months maturity) if spot NIFTY is 5,900.
3. The number of contracts of NIFTY the investor needs to sell to get a full hedge until February for his portfolio if lot size is 200 units.
4. The number of contracts of NIFTY to trade to reduce portfolio beta to 0.6.
5. The number of contracts of NIFTY to trade to increase portfolio beta to 1.2.

Part (i): Calculation of Portfolio Beta

$$\text{Portfolio Beta } (\beta_p) = \mathbf{0.849}$$

Part (ii): Theoretical Value of NIFTY Futures for February 2013 (2 Months)

$$F = S_0 \cdot \left(1 + r \cdot \frac{t}{12}\right) = 5,900 \times \left(1 + 0.105 \times \frac{2}{12}\right) = 5,900 \times (1 + 0.0175) = \mathbf{6,003.25 \text{ points}}$$

(Note: Using continuous compounding $5,900 \times e^{0.105 \times 2/12} = 5,900 \times 1.01765 = 6,004.16$)

$$\text{Value of 1 NIFTY Contract } (V_f) = 6,003.25 \times 200 = \mathbf{\text{₹ } 12,00,650}$$

Part (iii): Contracts to Sell for Full Hedge ($\beta^* = 0$)

$$N = \frac{V_p \cdot \beta_p}{V_f} = \frac{1,88,54,860 \times 0.849}{12,00,650} = \frac{1,60,07,776}{12,00,650} = 13.33 \approx \mathbf{13 \text{ contracts (Sell)}}$$

Part (iv): Contracts to Sell to Reduce Beta to 0.6 ($\beta^* = 0.6$)

$$N = \frac{V_p \cdot (\beta_p - \beta^*)}{V_f} = \frac{1,88,54,860 \times (0.849 - 0.600)}{12,00,650} = \frac{1,88,54,860 \times 0.249}{12,00,650} = \frac{46,94,860}{12,00,650} = 3.91 \approx \mathbf{4 \text{ contracts (Sell)}}$$

Part (v): Contracts to Buy to Increase Beta to 1.2 ($\beta^* = 1.2$)

$$N = \frac{V_p \cdot (\beta^* - \beta_p)}{V_f} = \frac{1,88,54,860 \times (1.200 - 0.849)}{12,00,650} = \frac{1,88,54,860 \times 0.351}{12,00,650} = \frac{66,18,056}{12,00,650} = 5.51 \approx \mathbf{6 \text{ contracts (Buy)}}$$

Details about portfolio of shares of an investor is as below:

The investor thinks that the risk of portfolio is very high and wants to reduce the portfolio beta to 0.91. He is considering two alternative strategies:

1. Dispose of a part of his existing portfolio to acquire risk-free securities.
2. Take appropriate position on Nifty Futures which are currently traded at 8,125 and each Nifty point is worth ₹ 200.

You are required to determine:

1. Portfolio beta.
2. The value of risk-free securities to be acquired under Strategy 1.
3. The number of shares of each company to be disposed of under Strategy 1.
4. The number of Nifty contracts to be bought / sold under Strategy 2.

Part (1): Calculate Portfolio Beta

$$\text{Current Portfolio Beta } (\beta_p) = \mathbf{1.30}$$

Part (2): Value of Risk-Free Securities to be Acquired

Let $w_e$ be the proportion of portfolio retained in equity shares, and $(1 - w_e)$ be invested in risk-free securities ($\beta_{rf} = 0$) to achieve target beta $\beta^* = 0.91$:

$$\beta^* = w_e \cdot \beta_p + (1 - w_e) \cdot 0$$ $$0.91 = w_e \times 1.30 \implies w_e = \frac{0.91}{1.30} = 0.70 \text{ (70%)}$$ $$\text{Proportion of Risk-Free Securities} = 1 - 0.70 = 0.30 \text{ (30%)}$$ $$\text{Value of Risk-Free Securities to Acquire} = 30\% \times \text{₹ } 5,000 \text{ Lakh} = \mathbf{\text{₹ } 1,500 \text{ Lakh (₹ 15.00 Crore)}}$$

Part (3): Number of Shares to Dispose of from Each Company

To maintain relative portfolio proportions, the investor disposes of 30% of holdings in each company:

Part (4): Number of Nifty Contracts to Trade under Strategy 2

Value of 1 Nifty Futures Contract = $8,125 \times 200 = \text{₹ } 16,25,000 = \text{₹ } 16.25 \text{ Lakh}$.

$$N = \frac{V_p \cdot (\beta_p - \beta^*)}{V_f} = \frac{\text{₹ } 5,000 \text{ Lakh} \times (1.30 - 0.91)}{\text{₹ } 16.25 \text{ Lakh}} = \frac{5,000 \times 0.39}{16.25} = \frac{1,950}{16.25} = \mathbf{120 \text{ contracts}}$$

Since the objective is to reduce systematic risk, the investor must Sell (Short) 120 Nifty futures contracts.

On April 1, 2015, an investor has a portfolio consisting of eight securities as shown below:

The cost of capital for the investor is 20% per annum. The current Nifty value is 8,500 and Nifty futures minimum contract size is 100 units. The investor desires to reduce the beta of his portfolio to 0.50 for a period of 3 months.

You are required to:

1. Calculate the beta of the portfolio.
2. Calculate the theoretical fair value of 3-month Nifty futures.
3. Determine the number of Nifty futures contracts to be sold.
4. Simulate the investor's gain or loss if Nifty index falls by 10% after 3 months.

Part (i): Calculation of Portfolio Beta

$$\text{Portfolio Beta } (\beta_p) = \frac{\sum (\text{Value}_i \times \beta_i)}{\sum \text{Value}_i} = \frac{10,95,832.30}{9,94,450} = \mathbf{1.102}$$

Part (ii): Theoretical Fair Value of 3-Month Nifty Futures

Spot Nifty = 8,500; $r = 20\%$ p.a.; $t = 3/12 = 0.25$ years.

$$F = S_0 \cdot \left(1 + r \cdot \frac{t}{12}\right) = 8,500 \times (1 + 0.20 \times 0.25) = 8,500 \times 1.05 = \mathbf{8,925 \text{ points}}$$ $$\text{Value of 1 Nifty Futures Contract } (V_f) = 8,925 \times 100 = \mathbf{\text{₹ } 8,92,500}$$

Part (iii): Number of Nifty Futures Contracts to Sell ($\beta^* = 0.50$)

$$N = \frac{V_p \cdot (\beta_p - \beta^*)}{V_f} = \frac{9,94,450 \times (1.102 - 0.50)}{8,92,500} = \frac{9,94,450 \times 0.602}{8,92,500} = \frac{5,98,658.90}{8,92,500} = 0.6708 \approx \mathbf{1 \text{ contract (Sell)}}$$

Part (iv): Impact if Nifty Index Falls by 10% after 3 Months

• Spot Nifty falls by 10%: $8,500 \times (1 - 0.10) = 7,650$.
• At expiration of 3-month futures, Futures Price converges to Spot Price = 7,650 points.
• Expected return on stock portfolio under CAPM: $$\% \Delta \text{ Portfolio} = \beta_p \times (-10\%) = 1.102 \times (-10\%) = -11.02\%$$ $$\text{Loss on Equity Portfolio} = -11.02\% \times \text{₹ } 9,94,450 = \mathbf{-\text{₹ } 1,09,588.39}$$
• Gain on shorting 0.6708 futures contract (sold at 8,925, settled at 7,650): $$\text{Gain on Futures} = 0.6708 \times (8,925 - 7,650) \times 100 = 0.6708 \times 1,275 \times 100 = \mathbf{+\text{₹ } 85,527.00}$$ (Or if 1 full contract is rounded: $1 \times 1,275 \times 100 = +\text{₹ } 1,27,500$)
• Net Change in Portfolio Value (with 0.6708 contract): $$\text{Net Loss} = -1,09,588.39 + 85,527.00 = \mathbf{-\text{₹ } 24,061.39}$$ $$\% \text{ Net Loss} = \frac{-24,061.39}{9,94,450} = -2.42\% \approx -5\% \text{ after adjusting for 3-month cost of capital}$$

Sensex futures are traded at a multiple of 50. Consider the following quotations of Sensex futures in 10 trading days during February 2009:

Abhishek bought one Sensex futures contract on February 04. The average daily absolute change in the value of contract is ₹ 10,000 and standard deviation of these changes is ₹ 2,000. The maintenance margin is 75% of initial margin.

Required: Determine the daily balances in the margin account and payment on margin calls, if any.

Step 1: Compute Initial Margin & Maintenance Margin Threshold

$$\text{Initial Margin} = \mu + 3\sigma = \text{₹ } 10,000 + 3 \times \text{₹ } 2,000 = \text{₹ } 10,000 + \text{₹ } 6,000 = \mathbf{\text{₹ } 16,000}$$ $$\text{Maintenance Margin} = 75\% \times \text{Initial Margin} = 0.75 \times \text{₹ } 16,000 = \mathbf{\text{₹ } 12,000}$$

Step 2: Construct Daily Marking-to-Market Schedule

Mr. A purchased a 3-month call option for 100 shares in XYZ Ltd. at a premium of ₹ 30 per share, with an exercise price of ₹ 550. He also purchased a 3-month put option for 100 shares of the same company at a premium of ₹ 5 per share with an exercise price of ₹ 450. The market price of the share on the date of purchase of options is ₹ 500.

Required: Calculate the profit / loss if the market price of the share after 3 months is:

1. ₹ 350
2. ₹ 600

Initial Premium Cash Outflow Today ($t=0$):

$$\text{Total Premium Paid} = (100 \times \text{₹ } 30) + (100 \times \text{₹ } 5) = \text{₹ } 3,000 + \text{₹ } 500 = \mathbf{\text{₹ } 3,500 \text{ (₹ 35 per share)}}$$

Case 1: Market Price falls to ₹ 350 at Maturity

• Call Option ($X = 550$): Since $S_T = 350 < X = 550$, the call option is Out-of-The-Money. Lapse (Do not exercise). Payoff = ₹ 0.
• Put Option ($X = 450$): Since $S_T = 350 < X = 450$, the put option is In-The-Money. Exercise Put. $$\text{Gross Payoff per share} = 450 - 350 = \text{₹ } 100$$ $$\text{Total Gross Payoff on 100 shares} = 100 \times 100 = \text{₹ } 10,000$$
• Net Profit / (Loss): $$\text{Net Profit} = \text{Gross Payoff } (10,000) - \text{Total Premium Paid } (3,500) = \mathbf{+\text{₹ } 6,500 \text{ (+₹ 65 per share)}}$$

Case 2: Market Price rises to ₹ 600 at Maturity

• Call Option ($X = 550$): Since $S_T = 600 > X = 550$, the call option is In-The-Money. Exercise Call. $$\text{Gross Payoff per share} = 600 - 550 = \text{₹ } 50$$ $$\text{Total Gross Payoff on 100 shares} = 100 \times 50 = \text{₹ } 5,000$$
• Put Option ($X = 450$): Since $S_T = 600 > X = 450$, the put option is Out-of-The-Money. Lapse (Do not exercise). Payoff = ₹ 0.
• Net Profit / (Loss): $$\text{Net Profit} = \text{Gross Payoff } (5,000) - \text{Total Premium Paid } (3,500) = \mathbf{+\text{₹ } 1,500 \text{ (+₹ 15 per share)}}$$

The market received a rumour about ABC Corporation’s tie-up with a multinational company. This has induced the market price to move up. If the rumour is false, the ABC Corporation stock price will probably fall dramatically. To protect from this, an investor has bought call and put options (Straddle):

• Purchased one 3-months call with strike price of ₹ 42 for ₹ 2 premium (100 shares).
• Purchased one 3-months put with strike price of ₹ 40 for ₹ 1 premium (100 shares).

Required:

1. Determine the investor’s position if the tie-up succeeds and price rises to ₹ 43, ₹ 45, or ₹ 50.
2. Determine the investor’s position if the tie-up fails and price falls to ₹ 36.

Total Initial Premium Paid:

$$\text{Call Premium} = 100 \times \text{₹ } 2 = \text{₹ } 200; \quad \text{Put Premium} = 100 \times \text{₹ } 1 = \text{₹ } 100$$ $$\text{Total Premium Outlay} = 200 + 100 = \mathbf{\text{₹ } 300 \text{ (₹ 3.00 per share)}}$$

Equity share of PQR Ltd. is presently quoted at ₹ 320. The market price of the share after 6 months has the following probability distribution:

A put option with a strike price of ₹ 300 can be written. You are required to find out the expected value of the option at maturity (i.e., 6 months).

A Put option gives the holder the right to sell at ₹ 300. The buyer exercises whenever $S_T < 300$. The payoff to the option holder is $\max(0, 300 - S_T)$:

You as an investor had purchased a 4-month call option on the equity shares of X Ltd. of Face Value ₹ 10, of which the current market price is ₹ 132 and the exercise price is ₹ 150. You paid an option premium of ₹ 10. You expect the price to increase in 4 months; your expected price distribution is:

Compute the following:

1. Expected share price at the end of 4 months.
2. Value of the call option at the end of 4 months, if expected price materializes.
3. Is the option worth buying for ₹ 10?

Part (i): Expected Share Price at End of 4 Months

$$E(S) = \sum (P_i \times S_i) = (0.05 \times 120) + (0.20 \times 140) + (0.50 \times 160) + (0.10 \times 180) + (0.15 \times 190)$$ $$E(S) = 6.00 + 28.00 + 80.00 + 18.00 + 28.50 = \mathbf{\text{₹ } 160.50}$$

Part (ii): Expected Value of Call Option at Expiry ($X = 150$)

Part (iii): Decision Evaluation

Since the expected terminal value of the call option is ₹ 14.00, while the premium paid is ₹ 10.00, the investor generates an expected net profit of $14.00 - 10.00 = \mathbf{+\text{₹ } 4.00 \text{ per share}}$.

Therefore, the option is definitely worth buying for ₹ 10.

Mr. X established the following strategy on Delta Corporation’s stock:

1. Purchased one 3-month call option with a premium of ₹ 30 and an exercise price of ₹ 550.
2. Purchased one 3-month put option with a premium of ₹ 5 and an exercise price of ₹ 450.

Delta Corporation’s stock is currently selling at ₹ 500. Determine profit or loss, if the price of Delta Corporation’s stock after 3 months:

1. Remains at ₹ 500
2. Falls to ₹ 350
3. Rises to ₹ 600

Total Initial Premium Paid:

$$\text{Total Premium} = \text{Call Premium (₹ 30)} + \text{Put Premium (₹ 5)} = \mathbf{\text{₹ } 35.00}$$

The equity share of VCC Ltd. is quoted at ₹ 210. A 3-month call option is available at a premium of ₹ 10 per share and a 3-month put option is available at a premium of ₹ 5 per share.

1. The strike price in both cases is ₹ 220; and
2. The share price on the exercise day is ₹ 200, ₹ 210, ₹ 220, ₹ 230, ₹ 240.

Required: Find the net payoff for the holder of the call option and the holder of the put option. Also indicate the price range at which each option may be gainfully exercised.

Part (i): Net Payoff for Call Option Buyer ($X = 220$, Premium = ₹ 10)

Part (ii): Net Payoff for Put Option Buyer ($X = 220$, Premium = ₹ 5)

Sumana wanted to buy shares of EIL which has a range of ₹ 411 to ₹ 592 a month later. The present price per share is ₹ 421. Her broker informs her that the price of this share can soar up to ₹ 522 within a month or so, so that she should buy a one-month CALL of EIL with strike price of ₹ 522. The risk-free rate is 3.6% p.a. continuously compounded ($e^{0.003} \approx 1.003$ or monthly factor). Calculate the probability of price moving up under the risk-neutral method.

Step 1: Compute Up and Down Movement Factors

• Current Stock Price ($S_0$) = ₹ 421
• Upper Price ($S_u$) = ₹ 592 $\implies u = \frac{592}{421} = 1.40618$
• Lower Price ($S_d$) = ₹ 411 $\implies d = \frac{411}{421} = 0.97625$
• Time period ($t$) = 1 month = $1/12 = 0.0833$ years
• Risk-free factor $R = e^{rt} = e^{0.036 \times (1/12)} = e^{0.003} = 1.003005$

Step 2: Solve for Risk-Neutral Probability ($p$)

$$p = \frac{R - d}{u - d} = \frac{1.0030 - 0.97625}{1.40618 - 0.97625} = \frac{0.02675}{0.42993} = \mathbf{0.0622 \text{ (6.22%)}}$$ $$1 - p = 1 - 0.0622 = \mathbf{0.9378 \text{ (93.78%)}}$$

Mr. Dayal is interested in purchasing equity shares of ABC Ltd. which are currently selling at ₹ 600 each. He expects that price of share may go up to ₹ 780 or may go down to ₹ 480 in three months. The chances of occurring such variations are 60% and 40% respectively. A call option on the shares of ABC Ltd. can be exercised at the end of three months at a strike price of ₹ 630.

Required:

1. What combination of share and option should Mr. Dayal select if he wants a perfect hedge?
2. What should be the value of option today (risk-free rate is 10% p.a.)?
3. What is the expected rate of return on the option?

Step 1: Determine Terminal Option Payoffs ($X = 630$)

• If stock rises to ₹ 780: $C_u = \max(0, 780 - 630) = \mathbf{\text{₹ } 150}$
• If stock falls to ₹ 480: $C_d = \max(0, 480 - 630) = \mathbf{\text{₹ } 0}$

Part (i): Perfect Hedge Combination (Hedge Ratio $\Delta$)

$$\Delta = \frac{C_u - C_d}{S_u - S_d} = \frac{150 - 0}{780 - 480} = \frac{150}{300} = \mathbf{0.50}$$

Hedging Strategy: Purchase 0.50 shares of ABC Ltd. for every 1 Call option written (sold), or purchase 1 share and write 2 call options.

Part (ii): Value of Option Today

Terminal payoff of the hedged portfolio (Long 0.5 shares + Short 1 Call):

• If stock rises to ₹ 780: Portfolio Value = $(0.50 \times 780) - 150 = 390 - 150 = \text{₹ } 240$.
• If stock falls to ₹ 480: Portfolio Value = $(0.50 \times 480) - 0 = 240 - 0 = \text{₹ } 240$.

Since the portfolio is completely riskless, its present value discounted at the risk-free rate ($10\%$ for 3 months = $2.5\%$):

$$\text{PV of Hedged Portfolio} = \frac{\text{₹ } 240}{1 + 0.10 \times 0.25} = \frac{240}{1.025} = \text{₹ } 234.146$$

Current cost of hedged portfolio = $\Delta \cdot S_0 - C_0 = (0.50 \times 600) - C_0 = 300 - C_0$.

$$300 - C_0 = 234.146 \implies C_0 = 300 - 234.146 = \mathbf{\text{₹ } 65.85}$$

Part (iii): Expected Rate of Return on the Option

$$\text{Expected Option Payoff at Maturity} = (0.60 \times 150) + (0.40 \times 0) = \text{₹ } 90.00$$ $$\text{Expected 3-Month Return} = \frac{90.00 - 65.85}{65.85} = \frac{24.15}{65.85} = 36.67\%$$ $$\text{Annualized Expected Rate of Return} = 36.67\% \times \frac{12}{3} = \mathbf{146.70\% \text{ p.a.}}$$

Consider a two-year call option with a strike price of ₹ 50 on a stock the current price of which is also ₹ 50. Assume that there are two time periods of one year each, and in each year the stock price can move up or down by an equal percentage of 20% ($u = 1.20, d = 0.80$). The risk-free interest rate is 6% p.a.

Required: Using the binomial option model, calculate the probability of price moving up and down. Also draw a two-step binomial tree and determine the value of the option today.

Step 1: Compute Stock Prices in 2-Period Tree

• Current Stock Price ($S_0$) = ₹ 50.00
• Period 1: $S_u = 50 \times 1.20 = \text{₹ } 60.00; \quad S_d = 50 \times 0.80 = \text{₹ } 40.00$
• Period 2:
  • $S_{uu} = 60 \times 1.20 = \text{₹ } 72.00$
  • $S_{ud} = 60 \times 0.80 = \text{₹ } 48.00$
  • $S_{dd} = 40 \times 0.80 = \text{₹ } 32.00$

Step 2: Terminal Call Payoffs at Year 2 ($X = 50$)

• $C_{uu} = \max(0, 72.00 - 50.00) = \mathbf{\text{₹ } 22.00}$
• $C_{ud} = \max(0, 48.00 - 50.00) = \mathbf{\text{₹ } 0.00}$
• $C_{dd} = \max(0, 32.00 - 50.00) = \mathbf{\text{₹ } 0.00}$

Step 3: Risk-Neutral Probabilities

$$p = \frac{R - d}{u - d} = \frac{1.06 - 0.80}{1.20 - 0.80} = \frac{0.26}{0.40} = \mathbf{0.65}$$ $$1 - p = 1 - 0.65 = \mathbf{0.35}$$

Step 4: Roll Back to Value Option Today ($t=0$)

Using the two-period risk-neutral discounting formula:

$$C_0 = \frac{p^2 \cdot C_{uu} + 2p(1-p) \cdot C_{ud} + (1-p)^2 \cdot C_{dd}}{(1 + r)^2}$$ $$C_0 = \frac{(0.65)^2 \times 22.00 + 2(0.65)(0.35) \times 0 + (0.35)^2 \times 0}{(1.06)^2}$$ $$C_0 = \frac{0.4225 \times 22.00}{1.1236} = \frac{9.295}{1.1236} = \mathbf{\text{₹ } 8.2725 \approx \text{₹ } 8.27}$$

The current market price of an equity share of Penchant Ltd is ₹ 420. Within a period of 3 months, the maximum and minimum price of it is expected to be ₹ 500 and ₹ 400 respectively. If the risk-free rate of interest is 8% p.a., what should be the value of a 3-month Call option under the Risk Neutral method at the strike rate of ₹ 450? Given $e^{0.02} = 1.0202$.

Step 1: Solve for Risk-Neutral Probability ($p$)

Using continuous compounding: $R = e^{rt} = e^{0.08 \times 0.25} = e^{0.02} = 1.0202$.

$$S_0 \cdot e^{0.02} = p \cdot S_u + (1 - p) \cdot S_d$$ $$420 \times 1.0202 = 500p + 400(1 - p)$$ $$428.484 = 400 + 100p \implies 100p = 28.484 \implies p = \mathbf{0.2848}$$ $$1 - p = 1 - 0.2848 = \mathbf{0.7152}$$

Step 2: Terminal Call Payoffs ($X = 450$)

• If stock reaches ₹ 500: $C_u = \max(0, 500 - 450) = \mathbf{\text{₹ } 50}$
• If stock drops to ₹ 400: $C_d = \max(0, 400 - 450) = \mathbf{\text{₹ } 0}$

Step 3: Calculate Fair Option Value Today

$$C_0 = \frac{p \cdot C_u + (1 - p) \cdot C_d}{e^{rt}} = \frac{0.2848 \times 50 + 0}{1.0202} = \frac{14.24}{1.0202} = \mathbf{\text{₹ } 13.958 \approx \text{₹ } 13.96}$$

From the following data for a certain stock, find the value of a call option:

Step 1: Compute Intermediate Terms

$$\frac{S_0}{X} = \frac{80}{75} = 1.06667 \implies \ln(1.0667) = 0.06456$$ $$\sigma \sqrt{t} = 0.40 \times \sqrt{0.50} = 0.40 \times 0.7071 = 0.28284$$ $$\left(r + \frac{\sigma^2}{2}\right) t = \left(0.12 + \frac{0.16}{2}\right) \times 0.50 = (0.12 + 0.08) \times 0.50 = 0.20 \times 0.50 = 0.10$$

Step 2: Calculate $d_1$ and $d_2$

$$d_1 = \frac{0.06456 + 0.10}{0.28284} = \frac{0.16456}{0.28284} = \mathbf{0.5818}$$ $$d_2 = d_1 - \sigma \sqrt{t} = 0.5818 - 0.28284 = \mathbf{0.2990}$$

Step 3: Cumulative Normal Probabilities $N(d_1)$ and $N(d_2)$

• For $d_1 = 0.5818$: $N(0.58) = 0.7190$ and $N(0.59) = 0.7224$. $$N(0.5818) = 0.7190 + 0.18 \times (0.7224 - 0.7190) = 0.7190 + 0.0006 = \mathbf{0.7196}$$
• For $d_2 = 0.2990 \approx 0.30$: $N(0.30) = \mathbf{0.6179}$

Step 4: Compute Call Option Value

$$C_0 = S_0 \cdot N(d_1) - X \cdot e^{-rt} \cdot N(d_2)$$ $$C_0 = (80 \times 0.7196) - (75 \times 0.94176 \times 0.6179)$$ $$C_0 = 57.568 - (70.632 \times 0.6179) = 57.568 - 43.643 = \mathbf{\text{₹ } 13.925 \approx \text{₹ } 13.93}$$