Q1. What is Stochastic Process?
A stochastic process, also known as a random process, is a collection of random variables that are indexed by some mathematical set. Each probability and random process are uniquely associated with an element in the set. The index set is the set used to index the random variables. The term random function is also used to refer to a stochastic or random process, because a stochastic process can also be interpreted as a random element in a function space.
Q2. What is continuous and discrete stochastic process?
Continuous and discrete stochastic processes refer to two different types of mathematical models used to describe the evolution of systems over time in a probabilistic manner. The main difference lies in the nature of time intervals at which the processes are defined.
1. Continuous Stochastic Process:
– In a continuous stochastic process, the system evolves continuously over time. The process is defined and analyzed over a continuous time domain, such as an interval of real numbers.
– The state of the process can change at any point in time, and the changes occur smoothly without any jumps or discontinuities.
– The process is often described using differential equations, such as stochastic differential equations, which involve infinitesimal changes over time.
– Examples of continuous stochastic processes include Brownian motion (Wiener process), diffusion processes, and continuous-time Markov chains.
2. Discrete Stochastic Process:
– In a discrete stochastic process, the system evolves in discrete time steps. The process is defined and analyzed at specific, distinct points in time.
– The state of the process is updated only at these discrete time points, and there are no changes between them.
– The process is often described using difference equations or recursive equations, which involve updating the state based on previous states and random fluctuations.
– Examples of discrete stochastic processes include discrete-time Markov chains, random walks, and Poisson processes.
Q3. What is Markov Process?
A Markov process, also known as a Markov chain, is a type of stochastic process that exhibits the Markov property. The Markov property states that the future behavior of the process depends only on its current state and is independent of its past history, given the current state.
Q4. What is Weiner Process?
The Wiener process, also known as Brownian motion, is one of the fundamental continuous-time stochastic processes. The Wiener process is a mathematical model that describes the random motion of particles or variables over time.
A Wiener process is a continuous-time stochastic process that has the following key properties:
1. Continuity: The Wiener process is continuous, meaning it does not have any abrupt jumps. It exhibits continuous paths, which means that it can take on any real value at any given point in time.
2. Markov Property: Like the Markov process I mentioned earlier, the Wiener process has the Markov property. This means that the future behavior of the process depends only on its current state and is independent of its past behavior.
3. Independence of Increments: The increments of the Wiener process over disjoint time intervals are independent of each other. This property makes it a memoryless process.
4. Gaussian Distribution: The increments of the Wiener process over a fixed time interval (e.g., from time t1 to time t2) are normally distributed with a mean of 0 and a variance proportional to the length of the interval (t2 – t1).A generalized Wiener process for a variable x can be defined in terms of dz as
dx = adt + bdz
The a dt term implies that x has an expected drift rate of a per unit of time. The bdz term can be regarded as adding noise or variability to the path followed by x.
The standard Wiener process is a specific case where μ = 0 and σ = 1.
Q5. Define Black Scholes Model?
The Black-Scholes model, aka the Black-Scholes-Merton (BSM) model, is a differential equation widely used to price options contracts. The Black-Scholes model requires five input variables: the strike price of an option, the current stock price, the time to expiration, the risk-free rate, and the volatility.
Though usually accurate, the Black-Scholes model makes certain assumptions that can lead to predictions that deviate from the real-world results. The standard BSM model is only used to price European options, as it does not take into account that American options could be exercised before the expiration date.
The variables c and p are the European call and European put price, S0 is the stock price at time zero, K is the strike price, r is the continuously compounded risk-free rate, is the stock price volatility, and T is the time to maturity of the option.
For another way of looking at the Black–Scholes–Merton equation for the value of a European call option, note that it can be written as:
The terms here have the following interpretation:
Q6. What are the assumptions of Black Scholes Model?
The Black-Scholes model is a mathematical model used to calculate the theoretical price of options. It makes several key assumptions:
1. Efficient markets: The model assumes that financial markets are perfectly efficient, meaning that there are no transaction costs, no restrictions on short selling, and no market frictions. Additionally, it assumes that there are no arbitrage opportunities available.
2. Constant volatility: The model assumes that the volatility of the underlying asset’s returns is constant over the life of the option. This assumption is known as the constant volatility assumption and is often considered a limitation of the Black-Scholes model since volatility in real markets tends to fluctuate.
3. Log-normal distribution: The model assumes that the returns of the underlying asset follow a log-normal distribution. This implies that the asset prices can only be positive and that extreme price movements are highly unlikely.
4. No dividends: The model assumes that the underlying asset does not pay any dividends during the life of the option. If the underlying asset does pay dividends, the model can be adjusted to account for them.
5. Risk-free interest rate: The model assumes that there is a risk-free interest rate that is known and constant over the life of the option. This assumption allows for discounting the future payoff of the option to its present value.
6. European-style options: The Black-Scholes model is specifically designed for European-style options, which can only be exercised at expiration. It does not apply directly to American-style options, which can be exercised at any time before expiration.
Q7. Define Binomial Tree?
A binomial tree, also known as a lattice or a recombining tree, is a graphical representation of possible price movements of an underlying asset over time. It is commonly used in option pricing and other financial models.
In a binomial tree, each node represents the price of the underlying asset at a specific point in time. The tree starts at an initial price and branches out at each time step, representing the possible price movements of the asset. Typically, the tree grows in a binary fashion, meaning that at each time step, the asset price can either go up or down.
The tree is constructed based on certain assumptions, such as discrete time intervals, constant volatility, and risk-neutral probabilities. By applying these assumptions, the tree allows for the calculation of option prices at different nodes in the tree, from the final time step back to the initial time step.
The binomial tree model is particularly useful for pricing options because it provides a step-by-step representation of the underlying asset’s price movements and allows for the calculation of the option’s value at each node. This information can then be used to determine the fair value of the option or to create hedging strategies.
The binomial tree model is a simple and intuitive approach to option pricing, although it has limitations and assumptions that may not always hold in real-world markets. More complex models, such as the Black-Scholes model, build upon the principles of the binomial tree to incorporate additional factors and improve accuracy.
Q8. Important formulas to remember for Binomial Tree?
When using the binomial tree model for option pricing, there are several key formulas to remember:
1. Stock Price Calculation:
– Upward Movement: S_up = S * u
– Downward Movement: S_down = S * d
Here, S is the current stock price, u is the factor by which the stock price increases in each period, and d is the factor by which it decreases.
2. Risk-Neutral Probability Calculation:
– Probability of Upward Movement: p = (e^(r * Δt) – d) / (u – d)
– Probability of Downward Movement: q = 1 – p
Here, r is the risk-free interest rate and Δt is the time interval between periods.
3. Option Value Calculation:
– Option Value at each node: V = e^(-r * Δt) * (p * V_up + q * V_down)
Here, V_up and V_down are the option values at the next time step in the upward and downward scenarios, respectively.
4. Option Payoff Calculation:
– Call Option Payoff at Expiration: C = max(S – K, 0)
– Put Option Payoff at Expiration: P = max(K – S, 0)
Here, K is the strike price of the option.
5. Option Price Backward Induction:
Starting from the final time step (expiration), calculate the option value at each node by using the option value formula and working backward through the tree until reaching the initial time step.
These formulas form the foundation of the binomial tree model for option pricing. By using them in conjunction with the assumptions of the model, you can estimate the fair value of options at different points in time.
Q9. How does valuation of American Option differ from European
Option while using Binomial Tree?
The key concept in valuing American options with the binomial tree is the idea of early exercise. At each node, the option holder compares the option’s intrinsic value (the immediate payoff if exercised) with the option’s time value (the value of holding the option and potentially benefiting from further price movements). The holder will exercise the option early if the intrinsic value exceeds the time value.
To account for potential early exercise, the binomial tree for American options is constructed by backward induction, similar to valuing European options. However, the values at each node need to be adjusted based on the early exercise decision. This adjustment involves comparing the option’s intrinsic value with the continuation value (i.e., the value of holding the option and moving to the next node).
In summary, the key difference when using a binomial tree for European options versus American options lies in the handling of the exercise feature. European options can only be exercised at expiration, while American options have the added flexibility of early exercise. This distinction affects the valuation methodology and the calculations at each node of the binomial tree.
Q10. Explain the principle of risk-neutral valuation?
The price of an option or other derivative when expressed in terms of the price of the underlying stock is independent of risk preferences. Options therefore have the same value in a risk-neutral world as they do in the real world. We may therefore assume that the world is risk neutral for the purposes of valuing options. This simplifies the analysis. In a risk-neutral world all securities have an expected return equal to risk-free interest rate. Also, in a riskneutral world, the appropriate discount rate to use for expected future cash flows is the riskfree interest rate.
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Q11. What is the difference between entering into a long forward
contract when the forward price is $50 and taking a long position in a call
option with a strike price of $50?
In the first case the trader is obligated to buy the asset for $50. (The trader does not have a choice.) In the second case the trader has an option to buy the asset for $50. (The trader does not have to exercise the option.)
Q12. An investor enters into a short forward contract to sell 100,000 British pounds for US dollars at an exchange rate of 1.5000 US dollars per pound. How much does the investor gain or lose if the exchange rate at the end of the contract is (a) 1.4900 and (b) 1.5200?
(a) The investor is obligated to sell pounds for 1.5000 when they are worth 1.4900. The gain is (1.5000−1.4900) ×100,000 = $1,000.
(b) The investor is obligated to sell pounds for 1.5000 when they are worth 1.5200. The loss is (1.5200−1.5000)×100,000 = $2,000
Q13. Suppose that you write a put contract with a strike price of $40 and an expiration date in three months. The current stock price is $41 and the contract is on 100 shares. What have you committed yourself to? How much could you gain or lose?
You have sold a put option. You have agreed to buy 100 shares for $40 per share if the party on the other side of the contract chooses to exercise the right to sell for this price.
The option will be exercised only when the price of stock is below $40. Suppose, for example, that the option is exercised when the price is $30. You have to buy at $40 shares that are worth $30; you lose $10 per share, or $1,000 in total.
If the option is exercised when the price is $20, you lose $20 per share, or $2,000 in total. The worst that can happen is that the price of the stock declines to almost zero during the three-month period.
This highly unlikely event would cost you $4,000. In return for the possible future losses, you receive the price of the option from the purchaser
Q14. Suppose you own 5,000 shares that are worth $25 each. How can put options be used to provide you with insurance against a decline in the value of your holding over the next four months?
You could buy 50 put option contracts (each on 100 shares) with a strike price of $25 and an expiration date in four months. If at the end of four months the stock price proves to be less than $25, you can exercise the options and sell the shares for $25 each.
Q15. Suppose that a March call option to buy a share for $50 costs $2.50 and is held until March. Under what circumstances will the holder of the option make a profit? Under what circumstances will the option be exercised? Draw a diagram showing how the profit on a long position in the option depends on the stock price at the maturity of the option.
The holder of the option will gain if the price of the stock is above $52.50 in March. (This ignores the time value of money.) The option will be exercised if the price of the stock is above $50.00 in March.
Q16. Suppose that a June put option to sell a share for $60 costs $4 and is held until June. Under what circumstances will the seller of the option (i.e., the party with a short position) make a profit? Under what circumstances will the option be exercised? Draw a diagram showing how the profit from a short position in the option depends on the stock price at the maturity of the option.
The seller of the option will lose money if the price of the stock is below $56.00 in June. (This ignores the time value of money.) The option will be exercised if the price of the stock is below $60.00 in June.
Q17. A company knows that it is due to receive a certain amount of a foreign currency in four months. What type of option contract is appropriate for hedging?
A long position in a four-month put option can provide insurance against the exchange rate falling below the strike price. It ensures that the foreign currency can be sold for at least the strike price.
Q18. A US company expects to have to pay 1 million Canadian dollars in six months. Explain how the exchange rate risk can be hedged using (a) a forward contract and (b) an option.
The company could enter into a long forward contract to buy 1 million Canadian dollars in six months. This would have the effect of locking in an exchange rate equal to the current forward exchange rate.
Alternatively, the company could buy a call option giving it the right (but not the obligation) to purchase 1 million Canadian dollars at a certain exchange rate in six months. This would provide insurance against a strong Canadian dollar in six months while still allowing the company to benefit from a weak Canadian dollar at that time.
Q19. A trader enters into a short forward contract on 100 million yen. The forward exchange rate is $0.0090 per yen. How much does the trader gain or lose if the exchange rate at the end of the contract is (a) $0.0084 per yen; (b) $0.0101 per yen?
a) The trader sells 100 million yen for $0.0090 per yen when the exchange rate is $0.0084 per yen. The gain is 100 0 0006 ´ . millions of dollars or $60,000.
b) The trader sells 100 million yen for $0.0090 per yen when the exchange rate is $0.0101 per yen. The loss is 100 0 0011 ´ . millions of dollars or $110,000.
Q20. The current price of a stock is $94, and three-month call options with a strike price of $95 currently sell for $4.70. An investor who feels that the price of the stock will increase is trying to decide between buying 100 shares and buying 2,000 call options (20 contracts). Both strategies involve an investment of $9,400. What advice would you give? How high does the stock price have to rise for the option strategy to be more profitable?
The investment in call options entails higher risks but can lead to higher returns.
If the stock price stays at $94, an investor who buys call options loses $9,400 whereas an investor who buys shares neither gains nor loses anything.
If the stock price rises to $120, the investor who buys call options gains is
Option Gain = 2000 X (120 – 95) – 9400 = $40,600
An investor who buys shares gains = 100 X (120 – 94) = $2,600
The strategies are equally profitable if the stock price rises to a level, S, where
100 X (S – 94) = 2000(S – 95) – 9400 or S =100
The option strategy is therefore more profitable if the stock price rises above $100.
Q21. Suppose that you enter into a short futures contract to sell July silver for $17.20 per ounce. The size of the contract is 5,000 ounces. The initial margin is $4,000, and the maintenance margin is $3,000. What change in the futures price will lead to a margin call? What happens if you do not meet the margin call?
There will be a margin call when $1,000 has been lost from the margin account. This will occur when the price of silver increases by 1,000/5,000 = $0.20. The price of silver must therefore rise to $17.40 per ounce for there to be a margin call. If the margin call is not met, your broker closes out your position.
Q22. Distinguish between the terms open interest and trading volume?
The open interest of a futures contract at a particular time is the total number of long positions outstanding. (Equivalently, it is the total number of short positions outstanding.) The trading volume during a certain period of time is the number of contracts traded during this period.
Q23. What is the difference between the operation of the margin accounts administered by a clearing house and those administered by a broker?
The margin account administered by the clearing house is marked to market daily, and the clearing house member is required to bring the account back up to the prescribed level daily. The margin account administered by the broker is also marked to market daily. However, the account does not have to be brought up to the initial margin level on a daily basis. It has to be brought up to the initial margin level when the balance in the account falls below the maintenance margin level. The maintenance margin is usually about 75% of the initial margin.
Clearing houses act as a central counterparty between buyers and sellers. They assume the role of the buyer to every seller and the seller to every buyer. This means they are exposed to the risk of potential defaults by traders. To mitigate this risk, clearing houses have stringent risk management procedures. Clearing houses set and enforce margin requirements for trades processed through their systems. These margin requirements are typically higher than those set by brokers. Clearing houses aim to minimize the risk of default and protect the financial stability of the markets they serve.
Unlike clearing houses, brokers are not central counterparties to trades. They don’t assume the risk of default. Instead, they facilitate the trading activity of their clients. If a trader defaults on their obligations, the broker may face some risk, but it’s limited to that particular client’s account. Brokers also set and enforce margin requirements, but these are often lower than those set by clearing houses. The requirements may vary based on the broker’s risk management policies.
Q24. What are the most important aspects of the design of a new futures contract?
The most important aspects of the design of a new futures contract are the specification of the underlying asset, the size of the contract, the delivery arrangements, and the delivery months.
Q25. Explain the difference between bilateral and central clearing for OTC derivatives.
In bilateral clearing the two sides enter into an agreement governing the circumstances under which transactions can be closed out by one side, how transactions will be valued if there is a close out, how the collateral posted by each side is calculated, and so on. In central clearing a CCP stands between the two sides in the same way that an exchange clearing house stands between two sides for transactions entered into on an exchange
Q26. A cattle farmer expects to have 120,000 pounds of live cattle to sell in three months. The livecattle futures contract traded by the CME Group is for the delivery of 40,000 pounds of cattle. How can the farmer use the contract for hedging? From the farmer’s viewpoint, what are the pros and cons of hedging?
The farmer can short 3 contracts that have 3 months to maturity. If the price of cattle falls, the gain on the futures contract will offset the loss on the sale of the cattle. If the price of cattle rises, the gain on the sale of the cattle will be offset by the loss on the futures contract. Using futures contracts to hedge has the advantage that the farmer can greatly reduce the uncertainty about the price that will be received. Its disadvantage is that the farmer no longer gains from favorable movements in cattle prices.
Q27. Explain how CCPs work. What are the advantages to the financial system of requiring all standardized derivatives transactions to be cleared through CCPs?
A CCP stands between the two parties in an OTC derivative transaction in much the same way that a clearing house does for exchange-traded contracts. It absorbs the credit risk but requires initial and variation margin from each side. In addition, CCP members are required to contribute to a default fund. The advantage to the financial system is that there is a lot more collateral (i.e., margin) available and it is therefore much less likely that a default by one major participant in the derivatives market will lead to losses by other market participants. There is also more transparency in that the trades of different financial institutions are more readily known.
Q28. Trader A enters into futures contracts to buy 1 million euros for 1.3 million dollars in three months. Trader B enters in a forward contract to do the same thing. The exchange rate (dollars per euro) declines sharply during the first two months and then increases for the third month to close at 1.3300. Ignoring daily settlement, what is the total profit of each trader? When the impact of daily settlement is taken into account, which trader does better?
The total profit of each trader in dollars is 0.03×1,000,000 = 30,000. Trader B’s profit is realized at the end of the three months. Trader A’s profit is realized day-by-day during the three months. Substantial losses are made during the first two months and profits are made during the final month. It is likely that Trader B has done better because Trader A had to finance its losses during the first two months.
Remember, a future contract is settled daily while a forward contract is settled at the maturity.
Q29. What position is equivalent to a long forward contract to buy an asset at K on a certain date and a put option to sell it for K on that date?
The long forward contract provides a payoff of ST − K where ST is the asset price on the date and K is the delivery price. The put option provides a payoff of max (K−ST, 0). If ST > K the sum of the two payoffs is ST – K. If ST < K the sum of the two payoffs is 0. The combined payoff is therefore max (ST – K, 0). This is the payoff from a call option. The equivalent position is therefore a call option.
Q30. A company has derivatives transactions with Banks A, B, and C which are worth +$20 million, −$15 million, and −$25 million, respectively to the company. How much margin or collateral does the company have to provide in each of the following two situations?
a) The transactions are cleared bilaterally and are subject to one-way collateral agreements where the company posts variation margin, but no initial margin. The banks do not have to post collateral.
b) The transactions are cleared centrally through the same CCP and the CCP requires a total initial margin of $10 million.
a) If the transactions are cleared bilaterally, the company has to provide collateral to Banks A, B, and C of (in millions of dollars) 0, 15, and 25, respectively. The total collateral required is $40 million.
b) If the transactions are cleared centrally they are netted against each other and the company’s total variation margin (in millions of dollars) is –20 + 15 + 25 or $20 million in total. The total margin required (including the initial margin) is therefore $30 million.
Q31. Explain what is meant by basis risk when futures contracts are used for hedging.
Basis risk arises from the hedger’s uncertainty as to the difference between the spot price and futures price at the expiration of the hedge. Basis risk arises because the futures contract may not perfectly mirror the price movements of the underlying asset. There are several reasons for this basis risk:
a) Delivery and Settlement Dates: Futures contracts have specific delivery and settlement dates, and the prices are determined at those future dates. The underlying asset, on the other hand, may have a different price movement during that period, leading to a basis.
b) Differences in Assets: Sometimes, the futures contract might not be based precisely on the same asset that the hedger owns. For example, a farmer may use corn futures to hedge their wheat crop. While both are related commodities, their prices can diverge, creating basis risk.
c) Market Factors: The basis risk can also be influenced by changes in supply and demand dynamics, interest rates, storage costs, or changes in the overall market sentiment, which affect the futures contract and underlying asset differently.
d) Liquidity and Counterparty Risk: In some cases, the futures market might have lower liquidity compared to the spot market for the underlying asset, leading to price discrepancies and basis risk.
Q32. Explain what is meant by a perfect hedge. Does a perfect hedge always lead to a better outcome than an imperfect hedge? Explain your answer.
A perfect hedge is one that completely eliminates the hedger’s risk. A perfect hedge does not always lead to a better outcome than an imperfect hedge. It just leads to a more certain outcome. Consider a company that hedges its exposure to the price of an asset. Suppose the asset’s price movements prove to be favorable to the company. A perfect hedge totally neutralizes the company’s gain from these favorable price movements. An imperfect hedge, which only partially neutralizes the gains, might well give a better outcome.
However, in practice, achieving a perfect hedge can be challenging, if not impossible, due to various factors:
a) Basis Risk: As mentioned earlier, basis risk is the discrepancy between the price movements of the hedging instrument (e.g., futures contract) and the underlying asset. It’s challenging to completely eliminate basis risk, leading to an imperfect hedge.
b) Transaction Costs: Executing a perfect hedge often involves transaction costs, such as brokerage fees, spread costs, or margin requirements. These costs can reduce the overall effectiveness of the hedge.
c) Market Liquidity: In some cases, the derivatives market might lack sufficient liquidity for the hedger to execute a perfect hedge, making it difficult to match the exact exposure of the underlying asset.
d) Time Mismatch: Futures contracts have specific expiration dates, and options have specific periods during which they are valid. This might not align perfectly with the hedger’s desired time horizon, leading to imperfect hedges.
Q35. How to hedge cross currency exposure. For example, if you want to trade in USA while you are in Germany, how would to hedge your trade from cross currency exposure?
Hedging cross-currency exposure involves mitigating the risk arising from transactions, assets, or liabilities denominated in a foreign currency. For example, if a company based in the United States has financial dealings in euros, it faces cross-currency risk due to fluctuations in the euro-to-dollar exchange rate. To hedge cross-currency exposure effectively, here are some common strategies:
1. Forward Contracts: A forward contract is a widely used hedging instrument. It allows you to lock in an exchange rate today for a future currency transaction. By entering into a forward contract, you can ensure that the amount you will receive or pay in the foreign currency remains constant, regardless of exchange rate fluctuations.
2. Currency Options: Currency options provide the holder with the right, but not the obligation, to buy or sell a specific amount of foreign currency at a predetermined exchange rate (strike price) on or before the option’s expiration date. Depending on your exposure and risk appetite, you can use either call options (to hedge foreign currency receivables) or put options (to hedge payables).
3. Currency Swaps: A currency swap is a financial contract in which two parties exchange principal and interest payments in different currencies over a specified period. It can be used to hedge both currency and interest rate risk. Currency swaps can be particularly useful for long-term cross-currency exposures.
4. Money Market Hedges: Money market instruments, such as currency futures and currency ETFs (Exchange-Traded Funds), can also be used for hedging cross-currency exposure. Currency futures are standardized contracts to buy or sell a specific amount of currency at a predetermined exchange rate on a specified future date.
5. Netting and Matching: If your company has significant cross-currency transactions with the same counterparty, you can consider netting and matching those transactions. This involves offsetting payables and receivables in the same currency, reducing the overall exposure and hedging the net amount.
Q36. Difference between Trading Book and Banking Book?
In the context of financial institutions, such as banks, the terms “Trading Book” and “Banking Book” refer to two distinct categories used for managing and accounting for different types of financial instruments. These categorizations are important for regulatory and risk management purposes. Let’s explore the differences between the two:
1. Trading Book:
The Trading Book encompasses all financial instruments that are actively traded by the bank’s trading desk for short-term profit-making purposes. These instruments are typically bought and sold in financial markets with the intention of profiting from short-term price movements or market fluctuations. The primary objective of the trading book is to generate trading income or capital gains. Instruments commonly included in the trading book are:
a. Stocks and Equities: Shares of publicly traded companies.
b. Bonds: Debt securities issued by governments or corporations.
c. Derivatives: Financial contracts whose value is derived from an underlying asset, such as futures, options, and swaps.
d. Forex (Foreign Exchange) Instruments: Currencies traded against each other in the foreign exchange market.
e. Commodities: Physical goods like gold, oil, agricultural products, etc., traded in commodity markets.
The positions in the trading book are actively managed by traders, and their values are marked-to-market regularly, meaning they are revalued at the current market prices to reflect their fair value on the balance sheet. This can lead to frequent fluctuations in the book’s value.
2. Banking Book:
The Banking Book, also known as the Banking Book of the Banking Book of Interest Rate Risk (IRRBB) or the Non-Trading Book, comprises financial assets that are not actively traded for speculative purposes. Instead, these assets are held by the bank for longer-term purposes, primarily to earn interest income or for banking services provided to customers. Instruments commonly included in the banking book are:
a. Loans and Mortgages: Loans given to individuals, businesses, or governments.
b. Fixed-Income Securities: Bonds and other debt instruments that the bank holds until maturity.
c. Deposits: Customer deposits, such as savings accounts, current accounts, and certificates of deposit.
Unlike the trading book, positions in the banking book are generally held until maturity or until the loans are paid back, and their values are not marked-to-market frequently. Instead, they are accounted for at their historical cost, with adjustments for impairments and credit losses.
Regulatory bodies, such as the Basel Committee on Banking Supervision, impose different capital and risk management requirements for both trading book and banking book positions, reflecting their varying levels of risk and the short-term versus long-term nature of their activities. Proper management of both books is essential for maintaining the financial stability of a bank and meeting regulatory compliance.
Q37. Suppose that a European call option to buy a share for $100.00 costs $5.00 and is held until maturity. Under what circumstances will the holder of the option make a profit? Under what circumstances will the option be exercised? Draw a diagram illustrating how the profit from a long position in the option depends on the stock price at maturity of the option.
Ignoring the time value of money, the holder of the option will make a profit if the stock price at maturity of the option is greater than $105. This is because the payoff to the holder of the option is, in these circumstances, greater than the $5 paid for the option. The option will be exercised if the stock price at maturity is greater than $100. Note that if the stock price is between $100 and $105 the option is exercised, but the holder of the option takes a loss overall. The profit from a long position is as shown
Q38. An investor buys a European put on a share for $3. The stock price is $42 and the strike price is $40. Under what circumstances does the investor make a profit? Under what circumstances will the option be exercised? Draw a diagram showing the variation of the investor’s profit with the stock price at the maturity of the option.
The investor makes a profit if the price of the stock on the expiration date is less than $37. In these circumstances the gain from exercising the option is greater than $3. The option will be exercised if the stock price is less than $40 at the maturity of the option. The variation of the investor’s profit with the stock price
Q39. An investor sells a European call on a share for $4. The stock price is $47 and the strike price is $50. Under what circumstances does the investor make a profit? Under what circumstances will the option be exercised? Draw a diagram showing the variation of the investor’s profit with the stock price at the maturity of the option.
The investor makes a profit if the price of the stock is below $54 on the expiration date. If the stock price is below $50, the option will not be exercised, and the investor makes a profit of $4. If the stock price is between $50 and $54, the option is exercised and the investor makes a profit between $0 and $4. The variation of the investor’s profit with the stock price is
Q40. Suppose that a European put option to sell a share for $60 costs $8 and is held until maturity. Under what circumstances will the seller of the option (the party with the short position) make a profit? Under what circumstances will the option be exercised? Draw a diagram illustrating how the profit from a short position in the option depends on the stock price at maturity of the option.
Ignoring the time value of money, the seller of the option will make a profit if the stock price at maturity is greater than $52.00. This is because the cost to the seller of the option is in these circumstances less than the price received for the option. The option will be exercised if the stock price at maturity is less than $60.00. Note that if the stock price is between $52.00 and $60.00 the seller of the option makes a profit even though the option is exercised. The profit from the short position is as shown
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Q41. Describe the terminal value of the following portfolio: a newly entered-into long forward contract on an asset and a long position in a European put option on the asset with the same maturity as the forward contract and a strike price that is equal to the forward price of the asset at the time the portfolio is set up. Show that the European put option has the same value as a European call option with the same strike price and maturity.
The terminal value of the long forward contract is: St – F0
where St is the price of the asset at maturity and F0 is the forward price of the asset at the time the portfolio is set up. (The delivery price in the forward contract is also F0 .)
The terminal value of the put option is: max (F0 – St, 0) ,
The terminal value of the portfolio is therefore: max (0, St – F0)
This is the same as the terminal value of a European call option with the same maturity as the forward contract and an exercise price equal to F0 .
This result is illustrated
Q42. A company declares a 2-for-1 stock split. Explain how the terms change for a call option with a strike price of $60.
The strike price is reduced to $30, and the option gives the holder the right to purchase twice as many shares.
Q43. The treasurer of a corporation is trying to choose between options and forward contracts to hedge the corporation’s foreign exchange risk. Discuss the advantages and disadvantages of each.
Forward contracts lock in the exchange rate that will apply to a particular transaction in the future. Options provide insurance that the exchange rate will not be worse than some level. The advantage of a forward contract is that uncertainty is eliminated as far as possible. The disadvantage is that the outcome with hedging can be significantly worse than the outcome with no hedging. This disadvantage is not as marked with options. However, unlike forward contracts, options involve an up-front cost.