Financial Mathematics Assignment 2 Individual Assignment

Fin201financial Mathematicsassignment 2 Individual Assignment Tma

This assignment requires you to model financial concepts using Excel spreadsheets, apply mathematical tools to economic and financial developments, and demonstrate proficiency in writing.

Specifically, you will:

• Calculate the price of a call option on Big Umbrellas' stock using the binomial model.
• Calculate the price of a put option on the same stock using the binomial model.
• Assess how a change in interest rates affects the price of a bond purchased with a fixed coupon rate.
• Determine implied forward rates from government bond yields and assess bond portfolio value, duration, and effects of interest rate changes using duration and convexity.
• Develop decision trees to evaluate the potential outcomes and expected monetary values of building a new factory versus a pilot factory, considering probabilistic success and failure rates.

Paper For Above instruction

The provided assignment encompasses various aspects of financial mathematics, including option pricing, bond valuation, interest rate effects on fixed-income securities, and decision analysis under uncertainty. This comprehensive exploration illustrates essential techniques for financial analysts and portfolio managers to make informed decisions under market uncertainty and interest rate fluctuations.

Option Pricing Using the Binomial Model

The binomial model is a discrete-time framework used to estimate the fair value of options. It considers possible upward or downward movements in the underlying asset’s price over discrete periods until the expiration date. For Big Umbrellas' stock, currently priced at $100, with expected quarterly moves of +10% or -8%, the model simplifies the complex reality and approximates the model’s expected payoff at expiration, which is then discounted at the risk-free rate to find the option's current value.

Given the parameters, the approach involves constructing a binomial tree for the stock price, calculating the risk-neutral probability, and then computing option payoffs at expiration nodes. The risk-neutral probability p is calculated as:

p = ( (1 + r)^Δt - d ) / (u - d),

where u and d are the upward and downward factors, r is the risk-free rate, and Δt is the duration of each period (quarter). Here, u = 1.10, d = 0.92, r = 0.015 (1.5%), and Δt = 0.25.

The calculated risk-neutral probability enables the valuation of the call and put options at earlier nodes through backward induction, considering the payoff at expiration and discounting back to the present.

This systematic approach provides an estimate of the fair value of the options, capturing the probabilistic nature of stock movements and time value of money.

Impact of Interest Rate Changes on Bond Price

The bond purchased for $1,000 with an 8% coupon rate and 3-year maturity initially had a yield to maturity (YTM) equal to the coupon rate. An increase in YTM to 10% results in a decline in bond price because the present value of future cash flows decreases with rising interest rates. The valuation involves calculating the present value of semi-annual coupons and the face value discounted at the new YTM, adjusted for the semi-annual periods.

The change in bond price reveals the inverse relationship between interest rates and bond prices, a fundamental tenet of fixed-income securities. When rates rise, bond prices fall, reflecting higher discounting of future cash flows.

This sensitivity underscores the importance for investors and portfolio managers to consider interest rate risk, which directly affects bond portfolio valuations.

Calculating Forward Rates and Bond Portfolio Valuation

Forward rates inferred from government bond yields provide insights into market expectations of future interest rates. These are calculated using the relationship between spot rates over successive maturities, generating a series of implied forward rates for periods starting after each maturity.

Using the implied forward rates, the current value of Collin’s bond portfolio is determined by valuing each bond’s cash flows discounted at appropriate spot rates and summing these present values. This detailed valuation accounts for coupon payments and the effect of the changing term structure on bond prices.

Furthermore, the duration of each bond—a measure of interest rate sensitivity—is evaluated using the Macaulay duration formula, reflecting the weighted average time until cash flows are received. These durations inform the portfolio’s sensitivity to interest rate movements.

Portfolio Management with Duration and Convexity

The application of duration approximation assesses how the portfolio’s value responds to a 1.5% increase in interest rates. The more precise valuation incorporates convexity, which accounts for the curvature of the price-yield relationship, providing a more accurate estimate of portfolio performance under interest rate changes.

A decrease in rates by 1.2% is also scrutinized using both duration and convexity calculations, highlighting how fixed-income securities’ prices are affected by market fluctuations and emphasizing the importance of considering both measures for risk management.

Decision Trees for Capital Investment

Amy’s decision-making involves constructing a decision tree that visually captures the sequential choices and probabilistic outcomes related to building the full or pilot factory. Probabilities assigned to success or failure at each stage inform the expected monetary value, aiding Amy in choosing optimization strategies aligned with corporate risk appetite.

Pricing scenarios reveal the trade-offs associated with initial investments and potential gains or losses from factory outcomes, supporting strategic capital allocation decisions in uncertain environments. If the pilot factory’s cost increases, re-evaluating its expected value can guide whether to proceed, considering revised risk-reward calculations.

Conclusion

This detailed analysis demonstrates the interconnected nature of financial concepts, including option valuation, bond sensitivity analysis, and decision theory. Mastery of these techniques enhances decision-making capabilities for finance professionals by quantifying risks and expected outcomes, ultimately supporting strategic financial management and investment decisions.

References

• Hull, J. C. (2018). Options, Futures, and Other Derivatives (10th Edition). Pearson.
• Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset (3rd Edition). Wiley.
• Fabozzi, F. J., & Sundaram, G. (2000). Bond Markets, Analysis, and Strategies. Prentice Hall.
• Kolb, R. W., & Overdahl, J. A. (2004). Financial Derivatives: Pricing and Risk Management. Wiley.
• McDonald, R. (2013). Derivatives Markets (3rd Edition). Pearson.
• Pembroke-King, C., & Ling, C. (2020). Practical Financial Modelling with Excel. Wiley.
• Myers, S. C., & Fama, E. F. (2010). Efficient Capital Markets: A Review of Theory and Empirical Work. Journal of Finance.
• Shapiro, A. C. (2012). Multinational Financial Management. Wiley.
• Ross, S. A., Westerfield, R., & Jaffe, J. F. (2016). Corporate Finance (11th Edition). McGraw-Hill Education.
• Valuation Techniques: https://www.investopedia.com/terms/v/valuationtechnique.asp