18.S096 | Fall 2013 | Undergraduate

# Topics in Mathematics with Applications in Finance

## Related Resources

### Useful Literature

Hull, John. Options, Futures, and Other Derivatives. Prentice Hall, 1997. ISBN: 9780138874988.

Baxter, Martin, and Andrew Rennie. Financial Calculus: An Introduction to Derivative Pricing. Cambridge University Press, 1996. ISBN: 9780521552899. [Preview with Google Books]

Wilmott, Paul, Sam Howison, and Jeff Dewynne. The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press, 1995. ISBN: 9780521497893. [Preview with Google Books]

Grinold, Richard C., and Ronald N. Kahn. Active Portfolio Management: Quantitative Theory and Applications. Probus Pubulication Company, 1995. ISBN: 9781557388247.

Fabozzi, Frank J., Sergio M. Focardi, and Petter N. Kolm. Financial Modeling of the Equity Market: From CAPM to Cointegration. Wiley, 2006. ISBN: 9780471699002. [Preview with Google Books]

Tsay, Ruey S. Analysis of Financial Time Series. Wiley-Interscience, 2001. ISBN: 9780471415442. [Preview with Google Books] Web page for Analysis of Financial Time Series.

### Financial Glossary

Alpha. Measure of risk-adjusted performance. An alpha is usually generated by regressing the security or mutual fund’s excess return on the S&P 500 excess return. The beta adjusts for the risk (the slope coefficient). The alpha is the intercept. Example: Suppose the mutual fund has a return of 25%, and the short-term interest rate is 5% (excess return is 20%). During the same time the market excess return is 9%. Suppose the beta of the mutual fund is 2.0 (twice as risky as the S&P 500). The expected excess return given the risk is 2 x 9%=18%. The actual excess return is 20%. Hence, the alpha is 2% or 200 basis points. Alpha is also known as the Jensen Index. Related: Risk-adjusted return.

Arbitrage. A transaction with zero cost initially but will end up with positive cash flow at the end.

Asset-Backed Security. A security that is collateralized by loans, leases, receivables, or installment contracts on personal property, not real estate.

Beta. The measure of a fund’s or a stock’s risk in relation to the market or to an alternative benchmark. A beta of 1.5 means that a stock’s excess return is expected to move 1.5 times the market excess returns. E.g., if market excess return is 10%, then we expect, on average, the stock return to be 15%. Beta is referred to as an index of the systematic risk due to general market conditions that cannot be diversified away.

Black–Scholes Model. A mathematical model of a financial market containing certain derivative investment instruments.

Bond. A debt security, under which the issuer owes the holders a debt and, depending on the terms of the bond, is obliged to pay them interest (the coupon) and / or to repay the principal at a later date, termed the maturity.

Bond Duration. It is the sensitivity of the bond price to the changes to the bond yield. In mathematics, this is the first derivative of the bond price with respect to the yield.

Bond Yield. The break even rate to price the bond future cash flow versus it price today.

Broker. An individual who is paid a commission for executing customer orders. Either a floor broker who executes orders on the floor of the exchange, or an upstairs broker who handles retail customers and their orders. Also, person who acts as an intermediary between a buyer and seller, usually charging a commission. A “broker” who specializes in stocks, bonds, commodities, or options acts as an agent and must be registered with the exchange where the securities are traded. Antithesis of dealer.

Call Option (Call). A financial contract between two parties, the buyer and the seller. The buyer of the call option has the right, but not obligation to buy an agreed quantity of a particular commodity or financial instrument (the underlying) from the seller of the option at a certain time (the expiration date) for a certain price (the strike price). A European call option allows the holder to exercise the option only on the option expiration date. An American call option allows exercise at any time before the maturity.

Coupon Bond. A coupon payment on a bond is a periodic interest payment that the bondholder receives during the time between when the bond is issued and when it matures.

Credit Default Swap (CDS). A financial agreement that the seller of the CDS will compensate the buyer in the event of a loan defaultt or other credit eventt. The buyer of the CDS makes a series of payments (the CDS “fee” or “spread”) to the seller and, in exchange, receives a payoff if the loan defaults.

Credit Exposure. The total amount of credit extended to a borrower by a lender. The magnitude of credit exposure indicates the extent to which the lender is exposed to the risk of loss in the event of the borrower’s default. Credit exposure can be minimized through purchasing credit default swaps or other types of financial instruments.

Cross Gamma. The rate of change of delta in one underlying to a change in the level of another underlying.

Currency Exchange Rate Exposure. Describes the influence of exchange rate movements on the value of a trade, firm, or a sector of the economy.

Dealer. An entity that stands ready and willing to buy a security for its own account (at its bid price) or sell from its own account (at its ask price). Individual or firm acting as a principal in a securities transaction. Principals are market makers in securities, and thus trade for their own account and risk. Antithesis of broker. See: Agency.

Delta. The number of units of an asset that should be purchased to hedge one unit of liability. Delta measures the rate of change of option value with respect to changes in the underlying asset’s price. Delta is the first derivative of the value of the option with respect to the underlying instrument’s price.

Derivative Instrument (Derivative). A derivative instrument is a contract between two parties that specifies conditions under which payments are to be made between the parties.

Discount Factor. The value of future one dollar today.

Drift. A statistical term referring to the bias which exists within pricing of various derivative products. The drift variable is strongly time-dependent and is often used to adjust derivative prices to try and reflect random behavior. A drift rate of 0 implies that the expected value of a variable at any time in the future is equal to its current value.

Forward Contract (Forward). A contract between two parties to buy or sell an asset at a specified future time at a price agreed upon today. The price agreed upon is called the delivery or strike price, which is equal to the forward price at the time the contract is entered into.

Futures Contract (Futures). A standardized contract between two parties to buy or sell a specified asset of standardized quantity and quality for a price agreed today (the futures price or strike price) with delivery and payment occurring at a specified future date, the delivery date. The contracts are negotiated at a futures exchange, which acts as an intermediary between the two parties. The exchange requires both parties to put up an initial amount of cash, the margin. Additionally, since the futures price will generally change daily, the difference in the prior agreed-upon price and the daily futures price is settled daily also (variation margin). The exchange will draw money out of one party’s margin account and put it into the other’s so that each party has the appropriate daily loss or profit. A forward is like a futures in that it specifies the exchange of goods for a specified price at a specified future date. However, a forward is not traded on an exchange and thus does not have the interim partial payments due to marking to market. Nor is the contract standardized, as on the exchange.

Gamma. The rate of change in the delta with respect to changes in the underlying price. Gamma is the second derivative of the value function with respect to the underlying price.

Hedge. A hedge is an investment position intended to offset potential losses/gains that may be incurred by a companion investment.

Implied Volatility. The implied volatility of an option contract is the volatility of the price of the underlying security that is implied by the market price of the option based on an option pricing model. In other words, it is the volatility that, when used in a given pricing model (such as Black-Scholes), yields a theoretical value for the option equal to the current market price of that option.

Initial Public Offering (IPO). A company’s first sale of stock to the public. Securities offered in an IPO are often, but not always, those of young, small companies seeking outside equity capital and a public market for their stock. Investors purchasing stock in IPOs generally must be prepared to accept considerable risks for the possibility of large gains. IPOs by investment companies (closed-end funds) usually include underwriting fees that represent a load to buyers.

Interest Rate Exposure. The amount of financial loss which can be incurred as a result of adverse changes in interest rates.

Interest Rate Swap. The contract to exchange fixed cash flow versus floating cash flows in the future.

Intrinsic Value. The intrinsic value of an option is its value assuming it were exercised immediately. Thus if the current (spot) price of the underlying security (or commodity etc.) is above the agreed (strike) price, a call has positive intrinsic value (and is called “in the money”), while a put has zero intrinsic value (and is “out of the money”).

Maturity Date (Maturity). The final payment date of a loan or other financial instrument, at which point the principal (and all remaining interest) is due to be paid.

Over-the-Counter (OTC). A decentralized market (as opposed to an exchange market) where geographically dispersed dealers are linked by telephones and computer screens. The market is for securities not listed on a stock or bond exchange. The NASDAQ market is an OTC market for U.S. stocks. Antithesis of listed.

Par Bond. A coupon bond with fixed rate whose price today equals to par amount.

PnL. The financial term for Profit & Loss, sometimes written P&L.

Put Option (Put). A contract between two parties to exchange an asset (the underlying), at a specified price (the strike), by a predetermined date (the expiry or maturity). One party, the buyer of the put, has the right, but not an obligation, to sell the asset at the strike price by the future date, while the other party, the seller of the put, has the obligation to buy the asset at the strike price if the buyer exercises the option. A European put option allows the holder to exercise the option only on the option expiration date. An American call option allows exercise at any time before the maturity.

Replicating Portfolio. A portfolio of assets for which changes in value match those of a target asset. For example, a portfolio replicating a standard option can be constructed with certain amounts of the asset underlying the option and bonds. Sometimes referred to as a Synthetic Asset.

Risk Neutral Valuation. The process by which options and other derivatives are priced by treating investors as though they were risk neutral.

SABR Model. A stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. The name stands for “stochastic alpha, beta, rho”, referring to the parameters of the model.

Strike. The strike price (or exercise price) is the fixed price at which the owner of an option can purchase (in the case of a call), or sell (in the case of a put), the underlying security or commodity.

Structured Note. A derivative investment that will change in value with movements of an underlying index; or a note whose issuer makes swap arrangements to alter its required cash flows.

Swaption. An option granting its owner the right but not the obligation to enter into an underlying swap. Although options can be traded on a variety of swaps, the term “swaption” typically refers to options on interest rate swaps.

Swap Rate. The fixed rate in the interest rate swap contract.

Theta. The ratio of the change in an option price to the decrease in time to expiration. Also called time decay.

Value-at-Risk Model (VaR). Procedure for estimating the probability of portfolio losses exceeding some specified proportion based on a statistical analysis of historical market price trends, correlations, and volatilities.

Volatility. A measure of risk based on the standard deviation of the asset return. Volatility is a variable that appears in option pricing formulas, where it denotes the volatility of the underlying asset return from now to the expiration of the option. There are volatility indexes. Such as a scale of 1-9; a higher rating means higher risk.

Volatility Skew/Smile. The phenomena of options’ implied volatilities varying with strike price.

Yield Curve. The change of the swap rates / zero rates over time horizon.

Zero Coupon Bond. A zero-coupon bond (also called a discount bond or deep discount bond) is a bond bought at a price lower than its face value, with the face value repaid at the time of maturity.

Zero Rate. The annualized rates of the zero coupon bond yield.

Zero Recovery Swap. A swap that terminates with no further payments should a specified reference party, often one of the counterparties, experience a default event. Dealers offer Zero Recovery Swaps as an alternative to requiring collateral.

## Course Info

Fall 2013
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