Risk Management- Final Revision PDF

Risk Management – Revision Chapter 4 Definition of the Binomial Option Pricing Model: The Binomial Option Pricing Model is a financial model used to determine the theoretical value of options using a simplified approach. This model represents the possible price paths that an underlying asset might t...

Risk Management – Revision Chapter 4 Definition of the Binomial Option Pricing Model: The Binomial Option Pricing Model is a financial model used to determine the theoretical value of options using a simplified approach. This model represents the possible price paths that an underlying asset might take over time. The key concept behind the binomial model is that it divides the time to expiration into potentially numerous steps or intervals. At each interval, the price of the underlying asset can move up or down by a specific factor. This leads to a binomial tree of possible asset prices, and the option value is calculated at the nodes of this tree based on the probabilities of moving up or down, the risk-free rate, and the option's payoff at expiration. Steps of Usage: 1. 2. 3. 4. Step 1: Create the binomial tree Step 2: Expand the binomial tree Step 3: Calculate the option payoff at expiration Step 4: Work backwards to calculate the option price The binomial model is particularly useful because it can be applied to options that have various features and exercise rules, such as American options, which can be exercised at any time before expiration. The flexibility to adjust for different scenarios and parameters makes it a vital tool in financial decision-making One-Period Binomial Model In a one-period model, the time to expiration of the option is divided into just one interval. Two-Period Binomial Model A two-period model splits the time to expiration into two intervals. A hedge portfolio offers several benefits, primarily centered around risk management and return stability. Here are the key advantages of maintaining a hedge portfolio: 1. Risk Reduction The fundamental purpose of a hedge portfolio is to mitigate risk. By holding investments that are expected to perform well under varying market conditions, the overall risk of the portfolio is reduced. For instance, if a portfolio contains both stocks and bonds, the bonds can provide stability and income even if the stock market declines, thus reducing the portfolio's volatility. 2. Diversification Hedging involves diversification, which is the strategy of spreading investments across different financial instruments, industries, and other categories to reduce exposure to any single asset or risk. A well-diversified portfolio can handle market fluctuations better because the performance of different assets can offset losses in others. 3. Protection Against Market Volatility During periods of high market volatility, a hedge portfolio can protect against significant losses. By including assets that are inversely correlated or less correlated with the market, such as gold or certain types of derivatives like options and futures, the portfolio can maintain more stable values. Problems Q1) Consider an option pricing scenario using a one-period binomial model. The current stock price (S) is $80, and the strike price (X) of the option is also $80. The stock price can either move up by 25% (u = 1.25) or down by 20% (d = 0.80) by the end of the period. The risk-free rate (r) for the period is 7%. Step 1: Calculate the possible future values of the stock (S_u and S_d) The current stock price (S) is $80. The upward price movement factor (u) is 1.25, and the downward movement factor (d) is 0.80. Calculate the stock price if it moves up: 𝑆𝑢=𝑆×𝑢=80×1.25=100 Calculate the stock price if it moves down: 𝑆𝑑=𝑆×𝑑=80×0.80=64 Step 2: Compute the option values at the end of the period (C_u and C_d) The strike price (X) is $80. Calculate the option value if the stock price goes up max(100−80,0)=20 Calculate the option value if the stock price goes down max(64−80,0)=0 Step 3: Calculate the risk-neutral probability (p) of the stock moving up The risk-free rate (r) is 0.07. Step 4: Calculate the present value of the expected option payoff (C) Q2) Consider an option pricing scenario using a one-period binomial model. The current stock price (S) is $80, and the strike price (X) of the option is also $80. The stock price can either move up by 35%. The risk-free rate (r) for the period is 8%. Step 1: Calculate the possible future values of the stock (S_u and S_d) The current stock price (S) is $80. The upward price movement percentage is 35%, so the upward movement factor (u) is 1.35. Assuming a downward movement factor (d) is not given, let’s use a typical down factor of 0.80 (if you have a different value for d, let me know): Calculate the stock price if it moves up: 𝑆𝑢=𝑆×𝑢=80×1.35= 108 Calculate the stock price if it moves down: 𝑆𝑑=𝑆×𝑑=80×0.80= 64 Step 2: Compute the option values at the end of the period (C_u and C_d) The strike price (X) is $80. Calculate the option value if the stock price goes up Step 3: Calculate the risk-neutral probability (p) of the stock moving up The risk-free rate (r) is 0.08. Step 4: Calculate the present value of the expected option payoff (C) Q3 Consider the scenario where you are evaluating the value of a put option using the one-period binomial model. The current stock price is $100, and the strike price of the put option is also $100. Over the next period, the stock price could either increase by 20% or decrease by 25%. The risk-free interest rate over the period is 7%. Chapter 5 Black, Scholes, Merton and the 1997 Nobel Prize Historical Volatility is a statistical measure of the dispersion of returns for a given security or market index, calculated by determining the standard deviation of price changes over a specific past period. Chapter 6 Different Holding Periods the context of financial analysis or investment strategies, these differing holding periods may be used to analyze how different lengths of investment impact returns, risks, or other financial metrics Buy a Call: Purchasing a call option grants the buyer the right, but not the obligation, to buy the underlying asset at a specified price (strike price) before the option expires. Write a Call: Writing a call option involves the seller granting the buyer the right to purchase the underlying asset at a specified price, potentially obligating the seller to sell the asset if the option is exercised. Buy a Put: Buying a put option gives the buyer the right, but not the obligation, to sell the underlying asset at a predetermined price before the option expires. Write a Put: Writing a put option means the seller grants the buyer the right to sell the underlying asset at a specified price, potentially obligating the seller to buy the asset if the option is exercised. Covered Calls: A covered call strategy involves holding a long position in an asset while simultaneously selling call options on the same asset to generate income from the option premiums. Protective Put: A protective put strategy involves buying put options for an asset that one already owns to hedge against potential losses in the asset's value. Chapter 12 "Interest Rate Forwards and Options" refer to financial derivatives where forwards contractually lock in future interest rates for borrowing or lending, while options provide the right, but not the obligation, to enter into such transactions at predetermined rates. Forward Rate Agreements Definition A forward contract in which the underlying is A forward contract in which the underlying is an interest rate Q1) A company enters into a forward rate agreement (FRA) to hedge against interest rate fluctuations. The FRA is based on a 120 day contract. The notional amount of the agreement is $10 million, and the agreed upon fixed rate is 5 percent. If the LIBOR rate at the time of settlement is 4 percent, calculate the payoff of the FRA. Q2) XYZ enters into a forward rate agreement (FRA) to hedge against interest rate fluctuations. The FRA is based on a 30 day contract. The notional amount of the agreement is $18 million, and the agreed upon fixed rate is 3 percent. If the LIBOR rate at the time of settlement is 6 percent, calculate the payoff of the FRA Examples of LIBOR Corporate Loans Interest Rate Swaps Savings Accounts Investment Funds Commercial Paper Good Luck! Dr. Mohammed Saharti