MA3269: Mathematical Finance I
AY2013/2014, Semester 1, Lecturer: Ng Wee Seng
Course Coverage:
1. Theory of Interest
2. Bonds
3. Expected Utility Theory
4. Mean-Variance Analysis
5. Portfolio Theory & CAPM
6. Basic Option Theory
This is the first course in mathematical finance and it overlaps significantly with EC3333. This module provides a broad overview of finance and portfolio management with emphasis on mathematical rigour.
The theory of interest is fundamental to all financial topics. Without interest rate, there is no incentive for lenders to channel their excess funds to borrowers, resulting in allocation of funds that is dynamically inefficient. In this chapter, the different methods of compounding (annual, monthly, daily, continuous), the concept of time value, and their applications to loan structures are covered in detail.
Bonds are one of the most common security in the financial market. It is a debt obligation between two counterparty that specifies a face value, redemption value, maturity date and coupon rate. Essentially, the bond issuer (debtor) is taking a loan from the buyer (creditor) and pays only the coupon rate (interest rate) computed based on the face value until the maturity date, when a fixed sum (redemption value) is repaid to the creditor. Usually, the price of this bond is the present value of the face value and the redemption value is equal to the face value. The Macaulay duration and bond convexity are the first and second order measure of the sensitivity of the bond price to changes in interest rates, respectively. In this chapter, the term structure of interest rate is also covered extensively.
The Expected Utility Theory studies the investor's behaviour under uncertainty. In particular, the theory studies the risk attitude of investors when the payoff is uncertain. The most important concept of this topic is perhaps the certainty equivalent, that is, the certain payoff that gives a utility level equivalent to the expected utility derived from the lottery. The Arrow-Pratt measures of risk aversion include absolute and relative risk aversion, and these measures can be used compare the degree of risk aversion between two investors.
Mean-variance analysis examines the trade-offs between return and risk in a portfolio of assets. This forms the basis for analysis using Markowitz's porfolio theory. In summary, an efficient portfolio must consist only assets that have the highest return given the level of risk. The set of such assets constitute the efficient frontier in the mean-variance graph.
Markowitz's portfolio theory asserts that the ideal portfolio of risky assets for all investors is any relative proportion of the market portfolio. Thus, the portfolios of investors only differ in the proportion of wealth invested into risk-free assets (as determined by their risk aversion). The CAPM asserts that the risk premium of any assets or efficient portfolio is equals to its beta multiplying the risk premium of the market portfolio. The beta measures the volatility of the asset/portfolio with respect to the market, with the market portfolio having a beta of 1.
Basic Option Theory introduces the vanilla options (put and call) and the various trading strategies involving these options. An important equation in the theory is the put-call parity which should always hold in the absence of arbitrage. The last part introduces the Binomial Option Pricing Model (BOPM) that prices options based on binomial probabilities in discrete time.
In this module, the famous Black-Scholes Option Pricing Model (BSOPM), and other derivatives (exotic options, forward, futures and swaps) are not covered. This module gives a simple overview of the mathematics involve in investment and trading strategies. However, it is worthy to note that such models are highly stylized and are generally not applicable directly in the real world. Nevertheless, they are useful because these models have important features that modern portfolio management and derivative trading strategies still retain, but yet simple enough as an introduction with academic rigour.
Workload: Heavy
Difficulty: Moderate
Grade: B
Course Coverage:
1. Theory of Interest
2. Bonds
3. Expected Utility Theory
4. Mean-Variance Analysis
5. Portfolio Theory & CAPM
6. Basic Option Theory
This is the first course in mathematical finance and it overlaps significantly with EC3333. This module provides a broad overview of finance and portfolio management with emphasis on mathematical rigour.
The theory of interest is fundamental to all financial topics. Without interest rate, there is no incentive for lenders to channel their excess funds to borrowers, resulting in allocation of funds that is dynamically inefficient. In this chapter, the different methods of compounding (annual, monthly, daily, continuous), the concept of time value, and their applications to loan structures are covered in detail.
Bonds are one of the most common security in the financial market. It is a debt obligation between two counterparty that specifies a face value, redemption value, maturity date and coupon rate. Essentially, the bond issuer (debtor) is taking a loan from the buyer (creditor) and pays only the coupon rate (interest rate) computed based on the face value until the maturity date, when a fixed sum (redemption value) is repaid to the creditor. Usually, the price of this bond is the present value of the face value and the redemption value is equal to the face value. The Macaulay duration and bond convexity are the first and second order measure of the sensitivity of the bond price to changes in interest rates, respectively. In this chapter, the term structure of interest rate is also covered extensively.
The Expected Utility Theory studies the investor's behaviour under uncertainty. In particular, the theory studies the risk attitude of investors when the payoff is uncertain. The most important concept of this topic is perhaps the certainty equivalent, that is, the certain payoff that gives a utility level equivalent to the expected utility derived from the lottery. The Arrow-Pratt measures of risk aversion include absolute and relative risk aversion, and these measures can be used compare the degree of risk aversion between two investors.
Mean-variance analysis examines the trade-offs between return and risk in a portfolio of assets. This forms the basis for analysis using Markowitz's porfolio theory. In summary, an efficient portfolio must consist only assets that have the highest return given the level of risk. The set of such assets constitute the efficient frontier in the mean-variance graph.
Markowitz's portfolio theory asserts that the ideal portfolio of risky assets for all investors is any relative proportion of the market portfolio. Thus, the portfolios of investors only differ in the proportion of wealth invested into risk-free assets (as determined by their risk aversion). The CAPM asserts that the risk premium of any assets or efficient portfolio is equals to its beta multiplying the risk premium of the market portfolio. The beta measures the volatility of the asset/portfolio with respect to the market, with the market portfolio having a beta of 1.
Basic Option Theory introduces the vanilla options (put and call) and the various trading strategies involving these options. An important equation in the theory is the put-call parity which should always hold in the absence of arbitrage. The last part introduces the Binomial Option Pricing Model (BOPM) that prices options based on binomial probabilities in discrete time.
In this module, the famous Black-Scholes Option Pricing Model (BSOPM), and other derivatives (exotic options, forward, futures and swaps) are not covered. This module gives a simple overview of the mathematics involve in investment and trading strategies. However, it is worthy to note that such models are highly stylized and are generally not applicable directly in the real world. Nevertheless, they are useful because these models have important features that modern portfolio management and derivative trading strategies still retain, but yet simple enough as an introduction with academic rigour.
Workload: Heavy
Difficulty: Moderate
Grade: B