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» Q&A: For a call option with a delta of 0.5...

» Q&A: For normal and lognormal distributions, list some of their similarities...

» Q&A: Give an example of a call market.

» Q&A: Given two crude futures contracts...

» Q&A: How does buying a put option effect...

» Q&A: How does the investment strategy of a speculator...

» Q&A: How is a future contract different from a forward contract?

» Q&A: How is an interest rate swap different from a package of forward contracts?

» Q&A: How is Monte Carlo Simulation different from...

» Q&A: How to create a delta-neutral porfolio with...

### Q&A: For a call option with a delta of 0.5...

...a $0.5 increase in the underlying price (current price: $100) will cause the price of the call option to move in which direction & by how much?

*Answer* : increase by 0.5 * $0.5 = $0.25.

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

### Q&A: For normal and lognormal distributions, list some of their similarities...

...as well as differences.

*Answer*

- A random variable Y follows a lognormal distribution if its natural logarithm, lnY, is normally distributed.
- The reverse is also true: if a random variable Y follows a lognormal distribution, then its natural logarithm, lnY, is normally distributed.

- Like the normal distribution, the lognormal distribution is completely described by mean and variance. Unlike the normal distribution, the lognormal distribution is defined in terms of the parameters of the associated normal distribution.
- In contrast, the normal distribution is defined by its own mean and variance. * the mean of Y is not equal to the mean of X, and the variance of Y is not equal to the variance of X.

- The lognormal distribution is bounded below by 0. In contrast, the normal distribution extends to - infinite without limit.
- The lognormal distribution is skewed to the right (that is, it has a long right tail). In contrast, the normal distribution is bell-shaped (i.e. it's symmetrical).

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

### Q&A: Give an example of a call market.

*Answer*:

- During the early stages of development of an exchange when there are few stocks listed or a small number of active investors / traders.
- At the opening for stocks on the NYSE if there is an overnight buildup of buy and sell orders, in which case the opening price can differ from the prior day's closing price.
- If trading is suspended during the day because of some significant new information. The mechanism is considered to contribute to a more orderly market and less volatility in such instances because it attempts to avoid major up and down price swings.

*Bonus Points*

- In a call markets, trading for individual stocks takes place at specified times. The intent is to gather all the bids and asks for the stock and attempt to arrive at a single price where the quantity demanded is as close as possible to the quantity supplied. In a continuous market, trading occurs any time the market is open to buyers and sellers.
- In a continuous market, trades occur at any time the market is open. Stocks are priced either by auction or by dealers. In an auction market, there are sufficient willing buyers and sellers to keep the market continuous. In a dealer market, enough dealers are willing to buy or sell the stock.
- Although many exchanges are considered continuous, they (i.e. NYSE) also employ a call-market mechanism on specific occasions.

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

### Q&A: Given two crude futures contracts...

...what items on the contracts are most likely to be identical and what difference?

*Answer* : Most likely identical

- the amount of crude oil
- the quality or grade of crude oil
- the mechanism for delivery of crude oil

Most likely different

- expiration date
- settlement date
- price

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

### Q&A: How does buying a put option effect...

...the return distribution of a stock, which is a bell shaped curve?

*Answer* : Truncates the leftside of the curve and moves the whole bell curve to the left.

- Buy a protective put option protects you from severe drops in the value, so truncates the leftside of the curve
- The curve moves to the left by the amount of the premium you paid for such protect (i.e., the price of the put option).

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

### Q&A: How does the investment strategy of a speculator...

...differ from that of a hedger?

*Answer* : A speculator increases expected return by increasing risks. In a well-functioning market, increased risk is associated with increased expected return.

A hedger increases expected return by dereasing pre-existing risks even though doing so may decrease returns. For example, supports Microsoft, who accounts for its profit and loss in US dollars, is expecting to receive 1 billion British pounds in three months. To hedge the risk of dollar appreciation, it can lock in the exchange rate by engaging in a forward contract with an investment bank (who in turn can offset its new risk with a transaction with another party.)

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

### Q&A: How is a future contract different from a forward contract?

*Answer*: A futures contract is a forward contract with some additional features, mainly in the institutional settings in which they trade.

- Futures contracts always trade on an organized exchange.
- Futures contracts are always highly standardized with specified underlying goods, quantity (contract size), delivery date, trading hours and trading area. Forward contracts are customized and therefore usually do not trade after being created.
- Performance on futures contract is guaranteed by a clearing house: every trader in the futures markets has obligations only to the clearing house, and the clearing house guarantees fulfillment of the contract of the trading parties. Traders in the forward market have direct obligations to each other (ie. credit risk)
- All futures contracts are settled daily. forward contracts are typically settled at expiration.
- Futures markets are regulated by an identifiable government agency, while forward contracts in general trade in an unregulated market. A futures transaction must be reported to: the futures exchanges, the clearinghouse, and at least one regulatory agency.

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

### Q&A: How is an interest rate swap different from a package of forward contracts?

*Answer*:

- Maturities for forward or futures contracts do not extend out as far as those of an interest rate swap. An interest rate swap with a term of 20 years or longer can be obtained.
- An interest rate swap is a more transactionally efficient instrument.
- Interest rate swaps provide more liquidity than forward contracts, particularly long-dated forward contracts.

*Bonus Points*

- A swap position can be interpreted as a package of forward (futures) contracts: for each swap period, the fixed rate payer has agreed to buy a LIBOR based commodity (ie. 1-year LIBOR + 1%) for a fixed amount. This is effectively a 1-year forward contract where the fixed payer agrees to pay a fixed amount in exchange for delivery of the commodity.
- A swap position can be interpreted as a package of cash market instruments. From the perspective of the fixed-rate payer, it is equivalent to buying a floating-rate note (with the reference rate for the note being the reference rate for the swap) and funding by issuing a fixed-rate bond (with the coupon rate for the bond being the swap rate).

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

### Q&A: How is Monte Carlo Simulation different from...

...analytical methods? Historical simulation?

*Answer* : Monte Carlo simulation in finance involves identifying the risk factors associated with a problem, and the use of a computer to represent the operation of a complex financial system.

- It provides only statistical estimates, not exact results: It is a complement to analytical methods.
- It does not directly provide precise insights as analytical methods do. For example, it cannot reveal cause-and-effect relationships.
- An integral part of simulation is the generation of a large number of random samples from a probability distribution. Monte Carlo simulation uses a random number generator with a specified distribution. In contrast, Historic simulation samples from a historical record of returns to simulate a process because the historical record provides the most direct evidence on distributions (and that the past applies to the future).

*Bonus Points*

- A drawback of historical simulations is that any risk not represented in the time period selected will not be reflected in the simulation. For example, if a stock market crash did not take place in the sample period, such a risk will not be reflected in the simulation. Also, it does not lend itself to "what-if" analysis.
- Monte Carlo simulation allows us to experiment with a proposed policy before actually implementing it to assess the risks. For example, it is used to simulate the interaction of pension assets and liabilities of defined benefit pension plans.
- MC is widely used to develop estimates of Value at Risk (VAR). VAR involves estimating the probability that portfolio losses exceed a predefined level.
- MC is used to value complex securities such as European options, mortgage-backed securities with complex embedded options.

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

### Q&A: How to create a delta-neutral porfolio with...

...call options and stocks given in this Q&A?

*Answer* :

- S0 = $100, X = $100, r = 0.06, T = 1, and stderr = 0.1. The call price is $7.46 and the call delta N(d1) = 0.7422.
- Consider a portfolio, P, of a short position of one call on the stock, combined with a long position of 0.7422 units of the stock
- The portfolio would have the value: P = -c + N(d1) S0 = -7.46 + 0.7422 x 100 = $66.76.
- If the stock price were up by $1, the portfolio's value would be -8.21 + 0.7422 x 101 = $66.7522.
- if the stock price were up by $10, the portfolio's value would be -16.09 + 0.7422 x 110 = $65.552.
- If the change in the stock price were larger, the change of the portfolio would be larger, but it would still be quite small relative to the change in the stock price.
- If the change in the stock price were infinitesimal, the price of the portfolio would not change at all.

*Bonus Points*

- the delta of this portfolio is zero: the value of the portfolio is insensitive to the value of the stock.
- Investors often use delta to construct hedges to offset the risk they have assumed by buying and selling options. For example, if a dealer sells 1,000 call options discussed above, he would buy 742 shares of the stock to construct a delta-neutral portfolio.
- An option is delta-hedged when a position has been taken in the underlying which matches its delta. Such a hedge is only effective instantaneously, because the option's delta is itself altered by changes in the price of the underlying, interest rates, the option's volatility and time to expiry.
- A delta-hedge must, therefore, be rebalanced continuously to be effective. aka dynamic hedging.

*Category: C++ Quant > Finance > Portfolio*