Fundamentals of Financial Derivatives

Evaluation & Justification:

Statement is TRUE.

Technical Rationale:

By definition, a financial derivative is a contractual financial instrument whose economic value, payoffs, and cash flows are completely derived from or contingent upon the performance of a primary underlying asset, benchmark reference rate, or index:

$$f_t = g(S_t, t)$$

where $S_t$ denotes the price of the underlying asset (such as an equity share, sovereign bond, currency, physical commodity, or short-term interest rate index) and $t$ denotes time.

Without an underlying asset, a derivative has no independent intrinsic value or payoff mechanism.

Evaluation & Justification:

Statement is TRUE.

Technical Rationale:

The intrinsic value of a put option represents the immediate payoff that would be realized if the option were exercised right now:

$$\text{Intrinsic Value (Put)} = \max(X - S_t, 0)$$

where $X$ is the strike (exercise) price and $S_t$ is the current spot price of the underlying asset.

A put option is defined as in-the-money (ITM) when the current market price of the underlying asset is strictly below the strike price ($S_t < X$). Consequently:

$$\text{Intrinsic Value} = X - S_t > 0$$

Therefore, an in-the-money put option always carries a strictly positive intrinsic value.

Evaluation & Justification:

Statement is TRUE (for American calls and standard European calls on non-dividend paying assets).

Technical Rationale:

The value of an option consists of intrinsic value plus time value:

$$C = \text{Intrinsic Value} + \text{Time Value}$$

As time to expiration ($T$) increases:

1. There is a wider distribution of possible future terminal stock prices ($S_T$), increasing the probability of favorable, large upside movements.
2. Downside risk to the call buyer is truncated at zero (loss is strictly limited to the premium paid).
3. Mathematically, the option’s sensitivity to expiration time is positive:
   $$\Theta_T = \frac{\partial C}{\partial T} > 0$$
   Thus, longer-dated call options command higher premiums due to greater volatility optionality and extended holding rights.

Evaluation & Justification:

Statement is FALSE.

Technical Rationale:

In the Cox-Ross-Rubinstein (CRR) Binomial Option Pricing Model, the risk-neutral probabilities of up and down movements are determined solely by the risk-free rate ($r$) and the asset’s volatility parameters ($u$ and $d$):

$$p = \frac{(1 + r) - d}{u - d} \quad \text{(or } p = \frac{e^{r\Delta t} - d}{u - d}\text{)}, \quad 1 - p = \frac{u - (1 + r)}{u - d}$$

These risk-neutral probabilities ($p$ and $1 - p$) are identical whether pricing a call option or a put option. The only difference between pricing a call and a put lies in the terminal payoff functions at the expiration nodes:

• Call Payoff: $C_T = \max(S_T - X, 0)$
• Put Payoff: $P_T = \max(X - S_T, 0)$

The up and down probabilities are never reversed when pricing put options.

Evaluation & Justification:

Statement is FALSE.

Technical Rationale:

Neither the buyer nor the seller of a put option faces the potential for unlimited losses:

1. Put Option Buyer:
   • Maximum loss is strictly capped at the initial option premium paid ($P_0$):
     $$\text{Max Loss (Buyer)} = -P_0$$
   • Maximum profit occurs if the stock price collapses to zero ($S_T = 0$), yielding $X - P_0$.
2. Put Option Seller (Writer):
   • Maximum gain is capped at the premium received ($+P_0$).
   • Maximum loss is reached when the stock price falls to zero ($S_T = 0$):
     $$\text{Max Loss (Seller)} = -(X - P_0)$$

Because stock prices are bounded below by zero ($S_T \ge 0$), the maximum possible downside on a put option is strictly finite ($X - P_0$). Unlimited loss potential applies only to the uncovered (naked) call seller.

Evaluation & Justification:

Statement is FALSE.

Technical Rationale:

In organized futures trading:

• Initial Margin: $\text{Rs } 5,000$
• Maintenance Margin: $\text{Rs } 3,500$
• Current Margin Balance: $\text{Rs } 4,000$

1. Trigger Condition: A margin call is issued if and only if the margin balance falls strictly below the maintenance margin level ($< \text{Rs } 3,500$). Since the account balance is $\text{Rs } 4,000 > \text{Rs } 3,500$, no margin call is triggered.
2. Variation Margin Rule: If a margin call were triggered (i.e., balance $< \text{Rs } 3,500$), the investor must deposit sufficient funds to bring the balance back to the Initial Margin level ($\text{Rs } 5,000$), not merely to the maintenance margin level.

Therefore, the statement is completely false on both accounts.

Evaluation & Justification:

Statement is TRUE (from an intra-day / theoretical contract valuation perspective).

Technical Rationale:

1. Intra-day / Economic Value: Between mark-to-market settlement times, the economic value of an existing futures position fluctuates with prevailing prices:
   $$V_t = (F_t - F_0) \times \text{Contract Size} \quad (\text{for a long position})$$
   If market prices decline ($F_t < F_0$), the value of the long position becomes negative ($V_t < 0$), representing an accrued unrealized loss.
2. Post Mark-to-Market Reset: At the end of each trading day, the clearinghouse marks the contract to market, cash-settles the daily gain or loss through the margin account, and resets the daily replacement value of the futures contract to zero ($V = 0$).

Therefore, intra-day or from a cum-loss perspective, a futures contract position can indeed have negative value.

Evaluation & Justification:

Statement is TRUE.

Technical Rationale:

In interest rate swap markets:

• A Plain Vanilla Interest Rate Swap involves exchanging a fixed interest rate for a floating benchmark rate (e.g., Fixed vs. SOFR).
• A Basis Swap is formally defined as an interest rate swap where both counterparty legs are floating interest rates, referenced to two different money market indices or different tenors (for example, 1-month SOFR vs. 3-month SOFR, or SOFR vs. US Treasury Bill rate).

Since both legs are floating rate streams, it is correctly designated as a basis swap.

Evaluation & Justification:

Statement is FALSE.

Technical Rationale:

In futures markets, a futures contract is a bilateral commitment where both the long counterparty (buyer) and the short counterparty (seller) are exposed to potential price movements and subsequent default/performance risk.

To safeguard the clearinghouse against counterparty default risk, the exchange mandates that both the buyer (long position) and the seller (short position) must deposit the initial margin upon opening their positions.

(In contrast, in the options market, only the option writer/seller posts margin because option buyers pay the full premium upfront and face no default liability).

Evaluation & Justification:

Statement is TRUE.

Technical Rationale:

Given:

• Market Put Premium ($P$) = $\text{Rs } 20$
• Strike Price ($X$) = $\text{Rs } 250$
• Current Stock Price ($S_0$) = $\text{Rs } 235$

Step 1: Calculate Intrinsic Value of the Put Option

$$\text{Intrinsic Value} = \max(X - S_0, 0) = \max(250 - 235, 0) = \text{Rs } 15$$

Step 2: Calculate Time Value

$$\text{Total Put Premium} = \text{Intrinsic Value} + \text{Time Value}$$

$$\text{Time Value} = \text{Put Premium} - \text{Intrinsic Value} = 20 - 15 = \text{Rs } 5$$

The time value is precisely $\text{Rs } 5$. Hence, the statement is True.

Step-by-Step Comprehensive Solution:

1. Definition of Derivative Markets

A derivative market is a specialized financial marketplace where financial contracts (derivatives) whose values are derived from underlying assets, reference rates, commodities, or indices are traded. These markets operate via two primary organizational structures:

• Exchange-Traded Markets: Standardized contracts (e.g., futures and exchange-traded options) cleared through a central counterparty (clearinghouse) with daily marking-to-market.
• Over-the-Counter (OTC) Markets: Privately negotiated, customized bilateral contracts (e.g., forwards, swaps, and exotic options) between institutional participants.

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2. Major Economic Functions of Derivative Markets

1. Risk Shifting and Hedging (Risk Management):

   • Enables commercial businesses, banks, and institutional investors to transfer unwanted price, currency, commodity, or interest rate risks to speculators who are willing to bear them.
   • For example, an agricultural producer locks in future crop prices via futures contracts, eliminating catastrophic downside price volatility.

2. Price Discovery:

   • Derivative markets aggregate global supply and demand information, reflecting market participants’ forward expectations faster and more accurately than spot markets.
   • Futures and forward prices serve as authoritative benchmarks for future spot market equilibrium:
     $$F_0 = S_0 e^{(r + u - y)T}$$

3. Enhancing Market Liquidity and Lowering Transaction Costs:

   • Because derivatives require only a fraction of total contract value as initial margin, capital efficiency is substantially higher than in spot cash markets.
   • Trading transaction costs and bid-ask spreads in index futures are typically far lower than trading all component stocks in the cash market.

4. Promoting Market Efficiency and Arbitrage:

   • Derivative pricing models (such as put-call parity and cost of carry) create tight arbitrage links between spot and derivative prices.
   • Any temporary mispricing is swiftly eliminated by arbitrageurs, ensuring that asset prices reflect intrinsic fair value across all market segments.

5. Facilitating Speculation and Investment Strategies:

   • Derivatives permit investors to execute directional views and synthetic asset allocations (e.g., short-selling an overvalued index without physical borrowing) safely and cost-effectively.

Step-by-Step Comprehensive Solution:

Given:

• Current stock price ($S_0$) = $\text{Rs } 140$
• Strike price of put option ($X$) = $\text{Rs } 130$
• Expiration period ($T$) = $6$ months
• Put option purchase premium ($P_0$) = $\text{Rs } 12$

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a. Determine Gain or Loss at Expiration for $S_T \in \{150, 130, 120, 110\}$:

The payoff to a long put option is:

$$\text{Payoff} = \max(X - S_T, 0)$$

The net profit (gain or loss) is:

$$\text{Net Profit} = \text{Payoff} - P_0 = \max(130 - S_T, 0) - 12$$

1. At $S_T = \text{Rs } 150$ (Out-of-the-Money):

   $$\text{Payoff} = \max(130 - 150, 0) = 0$$
   $$\text{Net Profit} = 0 - 12 = \mathbf{-\text{Rs } 12 \text{ (Loss of Rs 12)}}$$

2. At $S_T = \text{Rs } 130$ (At-the-Money):

   $$\text{Payoff} = \max(130 - 130, 0) = 0$$
   $$\text{Net Profit} = 0 - 12 = \mathbf{-\text{Rs } 12 \text{ (Loss of Rs 12)}}$$

3. At $S_T = \text{Rs } 120$ (In-the-Money):

   $$\text{Payoff} = \max(130 - 120, 0) = \text{Rs } 10$$
   $$\text{Net Profit} = 10 - 12 = \mathbf{-\text{Rs } 2 \text{ (Loss of Rs 2)}}$$

4. At $S_T = \text{Rs } 110$ (Deep In-the-Money):

   $$\text{Payoff} = \max(130 - 110, 0) = \text{Rs } 20$$
   $$\text{Net Profit} = 20 - 12 = \mathbf{+\text{Rs } 8 \text{ (Gain of Rs 8)}}$$

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b. Breakeven Price of the Underlying at Expiration:

The breakeven price occurs when Net Profit is zero:

$$\text{Net Profit} = (X - S_T^*) - P_0 = 0$$

$$S_T^* = X - P_0 = 130 - 12 = \mathbf{\text{Rs } 118}$$

Consistency Check with Part (a):

• At $S_T = 118$: $\text{Profit} = (130 - 118) - 12 = 12 - 12 = \text{Rs } 0$ (Exact breakeven).
• When $S_T = 120 > 118$: The price has not dropped enough to cover the premium, resulting in a loss of $\text{Rs } 2$.
• When $S_T = 110 < 118$: The price drop exceeds breakeven, resulting in a positive gain of $\text{Rs } 8$. The results are completely consistent.

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c. Maximum Profit and Maximum Loss:

• Maximum Loss: Occurs when the put option expires unexercised ($S_T \ge X = 130$). The loss is strictly limited to the premium paid:
  $$\text{Maximum Loss} = -P_0 = \mathbf{-\text{Rs } 12}$$
• Maximum Profit: Occurs when the stock price falls to its absolute theoretical floor of zero ($S_T = 0$):
  $$\text{Maximum Profit} = (X - 0) - P_0 = 130 - 12 = \mathbf{+\text{Rs } 118}$$

Final Answer:

• a. Net Gain/Loss: $S_{150}: -\text{Rs } 12$, $S_{130}: -\text{Rs } 12$, $S_{120}: -\text{Rs } 2$, $S_{110}: +\text{Rs } 8$
• b. Breakeven Price: Rs 118
• c. Maximum Loss: Rs 12; Maximum Profit: Rs 118

Step-by-Step Comprehensive Solution:

Given:

• Current stock price ($S_0$) = $\text{Rs } 260$
• Strike price ($X$) = $\text{Rs } 250$
• Time to expiration ($T$) = $6 \text{ months} = \frac{6}{12} = 0.5 \text{ years}$
• Volatility / Standard deviation ($\sigma$) = $0.30$
• Annual risk-free rate ($r$) = $6\% = 0.06$

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Step 1: Black-Scholes-Merton (BSM) Call Option Formula

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

where:

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

$$d_2 = d_1 - \sigma \sqrt{T}$$

and $N(\cdot)$ represents the cumulative standard normal distribution function.

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Step 2: Calculate $d_1$ and $d_2$

1. Calculate $\ln(S_0 / X)$:

   $$\frac{S_0}{X} = \frac{260}{250} = 1.04$$
   $$\ln(1.04) \approx 0.039221$$

2. Calculate $\left( r + \frac{\sigma^2}{2} \right) T$:

   $$\frac{\sigma^2}{2} = \frac{0.30^2}{2} = \frac{0.09}{2} = 0.045$$
   $$r + \frac{\sigma^2}{2} = 0.06 + 0.045 = 0.105$$
   $$\left( r + \frac{\sigma^2}{2} \right) T = 0.105 \times 0.5 = 0.0525$$

3. Calculate denominator $\sigma \sqrt{T}$:

   $$\sigma \sqrt{T} = 0.30 \times \sqrt{0.5} = 0.30 \times 0.707107 \approx 0.212132$$

4. Evaluate $d_1$:

   $$d_1 = \frac{0.039221 + 0.0525}{0.212132} = \frac{0.091721}{0.212132} \approx 0.432378 \approx \mathbf{0.43}$$

5. Evaluate $d_2$:

   $$d_2 = d_1 - \sigma \sqrt{T} = 0.432378 - 0.212132 = 0.220246 \approx \mathbf{0.22}$$

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

Using standard normal distribution tables:

$$N(0.43) \approx 0.6664 \quad (\text{exact } N(0.4324) = 0.66727)$$

$$N(0.22) \approx 0.5871 \quad (\text{exact } N(0.2202) = 0.58715)$$

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Step 4: Calculate Present Value of Strike Price

$$X e^{-rT} = 250 \times e^{-(0.06 \times 0.5)} = 250 \times e^{-0.03}$$

Since $e^{-0.03} \approx 0.9704456$:

$$X e^{-rT} = 250 \times 0.9704456 \approx \text{Rs } 242.6114$$

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Step 5: Compute Call Option Price ($C$)

$$C = 260 \times 0.66727 - 242.6114 \times 0.58715$$

$$C = 173.49 - 142.45 = \mathbf{\text{Rs } 31.04}$$

(Using 2-decimal rounded table values: $C = 260(0.6664) - 242.61(0.5871) = 173.26 - 142.44 = \text{Rs } 30.82$).

Final Answer: The fair theoretical value of the call option is Rs 31.04 (approx. Rs 31.00).

Step-by-Step Comprehensive Solution:

Given:

• Current spot price of underlying asset ($S_0$) = $\text{Rs } 900$
• Maturity of futures contract ($T$) = $30 \text{ days} = \frac{30}{365} \text{ years} \approx 0.08219 \text{ years}$
• Annual risk-free interest rate ($r$) = $5\% = 0.05$

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a. Calculate Futures Price:

Under the Cost of Carry Model without storage costs:

• Continuous Compounding:

  $$F_0 = S_0 e^{rT} = 900 \times e^{0.05 \times \frac{30}{365}} = 900 \times e^{0.0041096}$$
  Since $e^{0.0041096} \approx 1.004118$:
  $$F_0 = 900 \times 1.004118 = \mathbf{\text{Rs } 903.71}$$

• Discrete Compounding:

  $$F_0 = S_0 \left(1 + r \frac{T}{365}\right) = 900 \left(1 + 0.05 \times \frac{30}{365}\right) = 900(1 + 0.00411) = \mathbf{\text{Rs } 903.70}$$

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b. Calculate Futures Price with Storage Costs of Rs 30 at Expiration:

When storage costs ($U = \text{Rs } 30$) are incurred and paid at contract expiration ($t = T$):

$$F_0 = S_0 e^{rT} + U$$

$$F_0 = 903.71 + 30 = \mathbf{\text{Rs } 933.71}$$

(Alternatively, if the storage cost is quoted in present value terms $U_0 = 30 e^{-rT}$, then $F_0 = (S_0 + U_0)e^{rT} = 900 e^{rT} + 30 = \text{Rs } 933.71$).

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c. Similarities Between Futures and Forward Contracts:

1. Underlying Objective & Purpose: Both are derivative commitments to purchase or sell a specified quantity of an underlying asset at a pre-agreed price on a set future date.
2. Payoff Structure: Both contracts possess linear, symmetric payoff profiles:
   $$\text{Payoff (Long)} = S_T - K, \quad \text{Payoff (Short)} = K - S_T$$
3. Core Pricing Principles: Both are priced using the no-arbitrage Cost of Carry Model ($F = S_0 e^{(r+u-y)T}$).
4. Economic Utility: Both serve identical core functions: hedging against adverse price movements, speculative positioning, and arbitrage trading.
5. Zero Value at Inception: In the absence of transaction costs, both contracts have an initial theoretical market value of zero at inception ($V_0 = 0$).

Final Answer:

• a. Futures Price: Rs 903.71
• b. Futures Price with Storage Cost: Rs 933.71
• c. Key Similarities: Same linear payoff, same no-arbitrage cost of carry pricing, zero initial value, and shared hedging/speculative functionality.

Step-by-Step Comprehensive Solution:

Given Parameters:

• Position: Long 1 Gold Futures Contract
• Contract size: $1 \text{ kg}$
• Initial contract price ($P_0$): $\text{Rs } 8,400,000$ per kg
• Initial Margin deposit: $\text{Rs } 200,000$
• Maintenance Margin: $\text{Rs } 150,000$

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1. Daily Mark-to-Market Ledger Table:

Day	Settlement Price (Rs)	Daily Price Change (Rs)	Daily Gain / Loss (Rs)	Cumulative Gain / Loss (Rs)	Margin Balance (Before Call)	Margin Call (Deposit)	Ending Margin Balance
0	8,400,000	—	—	—	—	—	200,000
1	8,380,000	-20,000	-20,000	-20,000	180,000	0	180,000
2	8,304,000	-76,000	-76,000	-96,000	104,000	96,000	200,000
3	8,300,000	-4,000	-4,000	-100,000	196,000	0	196,000
4	8,650,000	+350,000	+350,000	+250,000	546,000	0	546,000
5	8,350,000	-300,000	-300,000	-50,000	246,000	0	246,000

Notes on Day 2:

• On Day 2, the margin balance fell to $\text{Rs } 104,000$, which is strictly below the maintenance margin of $\text{Rs } 150,000$.
• A Margin Call of $\text{Rs } 96,000$ was issued to restore the account back to the Initial Margin level of $\text{Rs } 200,000$ ($200,000 - 104,000 = 96,000$).

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2. Price Level that Triggers a Margin Call:

A margin call is triggered whenever cumulative losses cause the margin balance to drop below the maintenance margin:

$$\text{Maximum Tolerable Price Drop} = \text{Initial Margin} - \text{Maintenance Margin} = 200,000 - 150,000 = \text{Rs } 50,000$$

From the initial entry price of $\text{Rs } 8,400,000$:

$$\text{Trigger Price} = 8,400,000 - 50,000 = \mathbf{\text{Rs } 8,350,000}$$

Any settlement price strictly below Rs 8,350,000 triggers a margin call.

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3. Consequence If Investor Does Not Deposit Margin Call:

If the investor fails to deposit the required variation margin within the designated timeframe (typically before next day’s market open):

1. Immediate Liquidation: The broker has the legal right and contractual duty to liquidate (close out) the investor’s open position by selling 1 gold futures contract at the prevailing market price.
2. Settlement of Deficit: Any realized losses and execution fees are deducted from the remaining margin deposit.
3. Client Liability: If market slippage causes the account to go into negative equity, the investor remains legally liable to repay the deficiency to the brokerage firm.

Final Answer:

• Daily & Cumulative Gains/Losses: See ledger table above.
• Margin Call: Triggered on Day 2 for Rs 96,000.
• Trigger Price: Rs 8,350,000.
• Default Action: Mandatory broker liquidation of the position.

Step-by-Step Comprehensive Solution:

Given:

• Notional Principal ($N$) = $\text{Rs } 50,000,000$
• Fixed interest rate paid by Dealer to Asset Manager = $5.5\%$ per annum
• Floating return paid by Asset Manager to Dealer = Small-cap index return ($R_{\text{index}}$)
• Swap settlement frequency: Semiannual ($180 / 360 = 0.5$ years)
• Base index value at $t = 0$: $I_0 = 234.10$

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Step 1: Calculate the Fixed Inflow from Dealer to Manager

The semiannual fixed payment owed by the dealer is constant across both scenarios:

$$\text{Fixed Payment} = N \times r_{\text{fixed}} \times \left(\frac{180}{360}\right)$$

$$\text{Fixed Payment} = 50,000,000 \times 0.055 \times 0.5 = 50,000,000 \times 0.0275 = \mathbf{\text{Rs } 1,375,000}$$

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a. Scenario A: Small-Cap Index rises to $I_1 = 238.41$:

1. Calculate Index Return:

   $$R_{\text{index}} = \frac{I_1 - I_0}{I_0} = \frac{238.41 - 234.10}{234.10} = \frac{4.31}{234.10} \approx 0.01841094 \quad (1.84109\%)$$

2. Calculate Floating Payment Owed by Manager to Dealer:

   $$\text{Equity Payment} = N \times R_{\text{index}} = 50,000,000 \times 0.01841094 \approx \text{Rs } 920,546.78$$

3. Net Payment Calculation:

   $$\text{Net Cash Flow to Manager} = \text{Fixed Payment Received} - \text{Equity Payment Paid}$$
   $$\text{Net Cash Flow} = 1,375,000 - 920,546.78 = +\mathbf{\text{Rs } 454,453.22}$$

Conclusion for (a): The Dealer pays the Asset Manager Rs 454,453.22.

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b. Scenario B: Small-Cap Index rises to $I_1 = 241.27$:

1. Calculate Index Return:

   $$R_{\text{index}} = \frac{I_1 - I_0}{I_0} = \frac{241.27 - 234.10}{234.10} = \frac{7.17}{234.10} \approx 0.03062794 \quad (3.06279\%)$$

2. Calculate Floating Payment Owed by Manager to Dealer:

   $$\text{Equity Payment} = N \times R_{\text{index}} = 50,000,000 \times 0.03062794 \approx \text{Rs } 1,531,396.84$$

3. Net Payment Calculation:

   $$\text{Net Cash Flow to Manager} = \text{Fixed Payment Received} - \text{Equity Payment Paid}$$
   $$\text{Net Cash Flow} = 1,375,000 - 1,531,396.84 = -\mathbf{\text{Rs } 156,396.84}$$

Conclusion for (b): The Asset Manager pays the Dealer Rs 156,396.84.

Final Answer:

• a. Dealer pays Asset Manager Rs 454,453.22
• b. Asset Manager pays Dealer Rs 156,396.84

Step-by-Step Comprehensive Solution:

Given Parameters:

• Option Type: American Call Option (exercise allowed at any node)
• Total Expiration: $T = 1 \text{ year}$
• Number of periods: $n = 2$ (each binomial step $\Delta t = 6 \text{ months} = 0.5 \text{ years}$)
• Current stock price ($S_0$): $\text{Rs } 400$
• Strike price ($X$): $\text{Rs } 350$
• Risk-free rate per period ($r$): $5\% = 0.05$ (or $1 + r = 1.05$)
• Up-factor per period ($u$): $1 + 0.20 = 1.20$
• Down-factor per period ($d$): $1 - 0.20 = 0.80$

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Step 1: Compute the Risk-Neutral Probabilities ($p$ and $1-p$)

In the binomial model, the risk-neutral probability of an upward movement is:

$$p = \frac{(1 + r) - d}{u - d} = \frac{1.05 - 0.80}{1.20 - 0.80} = \frac{0.25}{0.40} = 0.625$$

The risk-neutral probability of a downward movement is:

$$1 - p = 1 - 0.625 = 0.375$$

Notice that $0.80 < 1.05 < 1.20$ ($d < 1 + r < u$), ensuring absence of arbitrage in the underlying stock.

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Step 2: Construct the Stock Price Tree

• At $t = 0$ (Initial):

  $$S_0 = 400$$

• At $t = 1$ (6 Months):

  $$S_u = S_0 \times u = 400 \times 1.20 = 480$$
  $$S_d = S_0 \times d = 400 \times 0.80 = 320$$

• At $t = 2$ (12 Months / Expiration):

  $$S_{uu} = S_u \times u = 480 \times 1.20 = 576$$
  $$S_{ud} = S_u \times d = 480 \times 0.80 = 384$$
  $$S_{dd} = S_d \times d = 320 \times 0.80 = 256$$

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Step 3: Compute Terminal Call Payoffs at Expiration ($t = 2$)

At maturity, call payoff is $C = \max(S_2 - X, 0)$ with $X = 350$:

1. Node $uu$ ($S_{uu} = 576$):
   $$C_{uu} = \max(576 - 350, 0) = \max(226, 0) = \mathbf{226}$$
2. Node $ud$ ($S_{ud} = 384$):
   $$C_{ud} = \max(384 - 350, 0) = \max(34, 0) = \mathbf{34}$$
3. Node $dd$ ($S_{dd} = 256$):
   $$C_{dd} = \max(256 - 350, 0) = \max(-94, 0) = \mathbf{0}$$

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Step 4: Backward Induction to $t = 1$ (6 Months)

For an American option, the value at each intermediate node is the maximum of immediate exercise value and continuation (holding) value:

$$C_1 = \max \left( S_1 - X, \, \frac{p C_{\text{up}} + (1 - p) C_{\text{down}}}{1 + r} \right)$$

1. At Up-Node ($S_u = 480$):

   • Continuation value:
     $$C_u^{\text{cont}} = \frac{p C_{uu} + (1 - p) C_{ud}}{1 + r} = \frac{0.625(226) + 0.375(34)}{1.05}$$
     $$C_u^{\text{cont}} = \frac{141.25 + 12.75}{1.05} = \frac{154.00}{1.05} \approx 146.6667$$
   • Immediate exercise value:
     $$S_u - X = 480 - 350 = 130$$
   • Decision: Since $146.6667 > 130$, it is optimal to hold (continue):
     $$C_u = \mathbf{146.6667}$$

2. At Down-Node ($S_d = 320$):

   • Continuation value:
     $$C_d^{\text{cont}} = \frac{p C_{ud} + (1 - p) C_{dd}}{1 + r} = \frac{0.625(34) + 0.375(0)}{1.05} = \frac{21.25}{1.05} \approx 20.2381$$
   • Immediate exercise value:
     $$\max(320 - 350, 0) = 0$$
   • Decision:
     $$C_d = \mathbf{20.2381}$$

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Step 5: Backward Induction to $t = 0$ (Today)

• Continuation value:
  $$C_0^{\text{cont}} = \frac{p C_u + (1 - p) C_d}{1 + r} = \frac{0.625(146.6667) + 0.375(20.2381)}{1.05}$$
  $$C_0^{\text{cont}} = \frac{91.6667 + 7.5893}{1.05} = \frac{99.2560}{1.05} \approx 94.5295$$
• Immediate exercise value at $t = 0$:
  $$S_0 - X = 400 - 350 = 50$$
• Decision: Since $94.53 > 50$:
  $$C_0 = \mathbf{\text{Rs } 94.53}$$

The maximum price you should be willing to pay for this American call option is Rs 94.53.

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Step 6: Investment Strategy if Market Price is Rs 30

The theoretical fair price of the call is Rs 94.53, but it is trading in the market at Rs 30. The call option is substantially underpriced (undervalued) by $\text{Rs } 64.53$.

Recommended Arbitrage / Investment Strategies:

Strategy 1: Instantaneous Risk-Free Cash Arbitrage

Because this is an American option, it can be exercised immediately at $t = 0$:

1. Buy the Call Option in the market at the mispriced market price of $\text{Rs } 30$.
2. Exercise the Call Option immediately, paying the strike price $X = \text{Rs } 350$ to receive $1$ share of stock. Total outlay $= 30 + 350 = \text{Rs } 380$.
3. Simultaneously sell the share of stock in the spot market at the current market price $S_0 = \text{Rs } 400$.
4. Immediate Risk-Free Arbitrage Profit:
   $$\text{Arbitrage Profit} = S_0 - X - C_{\text{market}} = 400 - 350 - 30 = \mathbf{+\text{Rs } 20 \text{ per share}}$$
   This is a zero-risk, instantaneous pure arbitrage profit.

Strategy 2: Dynamic Replicating Portfolio Arbitrage (Hold to Expiration)

If the investor wants to capture the entire theoretical mispricing of Rs 64.53:

1. Buy the undervalued call at $\text{Rs } 30$.
2. Delta-hedge by short-selling $\Delta$ shares of stock:
   $$\Delta = \frac{C_u - C_d}{S_u - S_d} = \frac{146.67 - 20.24}{480 - 320} = \frac{126.43}{160} \approx 0.7902 \text{ shares}$$
3. Lend the proceeds at the risk-free rate of $5\%$ per period.
4. Over time, this riskless hedged portfolio produces a present-value gain equal to the full mispricing ($94.53 - 30 = \text{Rs } 64.53$).

Final Answer:

• Maximum Willing Price: Rs 94.53
• Investment Strategy: Execute an immediate exercise arbitrage by buying the call at Rs 30, immediately exercising at Rs 350, and selling the stock at Rs 400 for a guaranteed instant profit of Rs 20 per share.