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» Q&A: How to produce a smaller margin of error...

» Q&A: How would investors exploit the arbitrage opportunity if...

» Q&A: How would investors exploit the arbitrage opportunity if...

» Q&A: Is the distribution of this random variable X binomial?

» Q&A: List 3 characteristics of a well-functioning market.

» Q&A: List 3 different exchange memberships.

» Q&A: List 3 reasons why foreign investors would purchase...

» Q&A: List 3 things that define the normal distribution for the return of a portfolio of 10 securities.

» Q&A: should the delta of a call option be close to...

» Q&A: The lifetime of a 2-volt non-rechargeable battery in constant use has...

### Q&A: How to produce a smaller margin of error...

...given this porfolio?

*Answer* : The width of a confidence interval is reduced by

- using a lower level of confidence. ie. for a normal variable, noticed the reliability factors increase with higher level of confidence, which in turn lead bigger Point Estimate +/- Reliability Factor * Standard Error = Confidence Interval
- Increasing the sample size: the standard error of the sample mean declines as the sample size increases.

*Category: C++ Quant > Finance > Quantitative Analysis > Probability*

### Q&A: How would investors exploit the arbitrage opportunity if...

this put option is selling for $4?

*Answer* : the put option is underpriced, so buy the option and the underlying. Suppose an investor buys 150 puts and 100 shares.

- The number of units of the underlying purchased for each option sold would be the hedge ratio: n = (p+ - p-) / (S+ - S-) = 0.66.
- The initial outlay would be 100 * $80 + 150 * $4 = $8600.
- 6 months later, the portfolio value will be
- S- + p- * n = 100 * $75 + 10 * 150 = $9000 * S+ + p+ * n = 100 x $90 + 0 * 150 = $9000

- the six-month return is 9000/8600 - 1 = 4.6%, and the annualized return is (1.046)^2 - 1 = 9.4% > the actual risk-free return of 6%

*Bonus Points*

- Unlike a call option, the arbitrage strategy for a under-priced put is to long positions in BOTH instruments.

*Category: C++ Quant > Finance > Derivatives > Valuation*

### Q&A: How would investors exploit the arbitrage opportunity if...

this call option is selling for $3?

*Answer* : the call option is overpriced, so sell the option and buy the underlying. Suppose an investor sells 300 calls and buys 100 shares

- The number of units of the underlying purchased for each option sold would be the hedge ratio: n = (c+ - c-) / (S+ - S-) = 0.3333.
- The initial outlay would be 100 * $80 - 300 * $3 = $7100.
- 6 months later, the portfolio value will be
- S- - c- * n = 100 x $75 - 0 = $7500 * S+ - c+ * n = 100 x $90 - 300 * $5 = $7500

- the six-month return is 7500/7100 - 1 = 5.63%, and the annualized return is (1.0563)^2 - 1 = 11.58% > the actual risk-free return of 6%

*Bonus Points*

- If the call option is underpriced (selling for < $2.38), an investor would buy the option and sell short the underlying, which would generate cash up front. At expiration, the investor would have to pay back an amount less than 7%

*Category: C++ Quant > Finance > Derivatives > Valuation*

### Q&A: Is the distribution of this random variable X binomial?

A set of four credit cards consist of two Visa cards and two Master cards. The cards are shuffled thoroughly, and I am dealt two cards. X is the number of Visa cards in these two cards.

*Answer*

- n = 2: since two cards are being selected, the number of observations is two.
- If the observations were independent, the distribution would be binomial. But they are NOT.
- If the first card is Visa, the probability that the second card dealt is Visa is 1/3.
- If the first card is Master, the probability that the second card dealt is Master is 2/3.

- These two probabilities would have to be the same if the observations were independent.

*Category: C++ Quant > Finance > Probability > Distributions*

### Q&A: List 3 characteristics of a well-functioning market.

*Answer*

- Availability of Information (ie. on the price and volume of past transactions and the prevailing bid and ask prices.
- Low transaction cost (% of the value of the trade, including the cost of reaching the market, the actual brokerage costs, and the cost of transferring the asset)
- Informational efficiency: prices rapidly adjust to new information (to keep the prevailing price fail)
- Liquidity: marketability (can be bought and sold quickly), price continuity (at a price close to the prices for previous transactions, assuming no new information has been received), and depth (ie. potential buyers and sellers willing to trade at prices above and below the current market price.)

*Bonus Points*

- Violations of any of these characteristics will lead to higher costs and a lesser ability of investors to make quick, intelligent decisions.
- Price continuity: quoting security prices in decimals created smaller bid and asked spreads, allowing more granular price differentiation between competing intermediaries.

_Category: C++ Quant > Finance > Financial Markets _

### Q&A: List 3 different exchange memberships.

*Answer*

- a Registered trader is one who trades on his own account
- a Floor broker transacts for others
- a Commission broker trades for her/his firm, and
- a Specialist makes a market in specific securities.

*Bonus Points*

- Specialists maintain a market in one or more listed securities, with two major functions
- Serve as a broker to match buy and sell orders and to handle special limit orders placed with member brokers.
- Act as a dealer to maintain a fair and orderly market by dealing personally in the stock. Provides liquidity to the market by standing ready to trade at quoted bid and asked prices. For example, if there is an inadequate flow of orders, specialists buy and sell shares for their own accounts to narrow the bid ask spread and improve the price continuity.

- The specialist derives income from the broker (commissions) and the dealer (spread between the bid and asked prices at which they buy and sell securities) functions. It also appears that specialists' access to their book of limit orders gives them unique knowledge about the probable direction of price movement over short period of time.

*Category: C++ Quant > Finance > Financial Markets*

### Q&A: List 3 reasons why foreign investors would purchase...

...CHF denominated bonds issued by Japanese firms that are convertible into common stock shares.

*Answer* : when investors take a view over the longer term in the foreign exchange markets

- drop in the market interest rate on Swiss franc bonds;
- rise in the JPY relative to the CHF
- rise in the price of the company's stock

*Category: C++ Quant > Finance > Derivatives > Options*

### Q&A: List 3 things that define the normal distribution for the return of a portfolio of 10 securities.

*Answer*: 10 means, 10 variances and 10*(10-1)/2=45 correlations of the component securities.

*Bonus Points*

- The need to specify correlations is a distinguishing feature of the multivariate normal distribution in contrast to the univariate normal distribution.
- A multivariate normal distribution is completely specified if one has the lists for the mean returns, variances, and correlations or covariances between the variables.

- Portfolio return is a normally distributed if the individual security returns are (joint) normally distributed
- Portfolio return is a weighted average (ie. a linear combination)of the returns on the component securities

- When we have a group of assets, we can model the distribution of returns on each asset individually, or the distribution of returns on the assets as a group.

*Category: C++ Quant > Finance > Quantitative Analysis > Probability > Distribution*

### Q&A: should the delta of a call option be close to...

...-1, 0, or 1 if it's on a stock trading at $100. The continuously compounding risk-free interest rate is 6%. The option has an exercise price of $50 and expires in one year. The standard deviation of the stock's returns is 0.1.

*Answer* : Without no calculation, one knows its delta should be close to 1. The giveaway is the option is deep-in-the-money.

*Bonus Points*

- When the underlying is at-the-money (ie. near the exercise price), delta is most sensitive to a change in the underlying price.
- When the call/put option is deep-out-of-the-money, its delta approaches 0.
- When the option is deep-in-the-money: Call delta approaches 1, Put delta approaches -1

*Category: C++ Quant > Finance > Derivatives > Options*

### Q&A: The lifetime of a 2-volt non-rechargeable battery in constant use has...

...a mean of 516 hours and a standard deviation of 60 hours. A box of 100 of these batteries is considered a random sample, what is the probability the average lifetime of the batteries in the box are shorter than 505 hours?

*Answer* : make use of the central limit theorem.

- compute sample standard deviation: because the sample size is > 30, the distribution of the x-bars is approximately normal with mean 516 and stanadrd deviation s/N^0.5 = 60/sqrt(100) = 6
- compute the z-score for 500: (505 - 516)/6 = -1.83
- P(Z<-1.83) = 3.3% = P(X-bar < 505)

*Bonus Points*

- The central limit theorem states that given a distribution with a mean m and variance s^2, the sampling distribution of the mean approaches a normal distribution with a mean (m) and a variance s^2/N as N increases.
- N, the sample size, in general has to be > 30.
- N is the sample size for each mean and not the number of samples taken.
- As N grows bigger, the distributions become more and more normal, the spread of the distributions decreases.

- the theorem can be applied on a population with any probability distribution.

*Category: C++ Quant > Finance > Quantitative Analysis > Probability*