When the forward prices are greater than the spot prices, the market is said to be in contango. When forward prices are lower than spot prices, the market is said to be backwarded. A short squeeze may contribute to backwardation. Choice 'a' is the correct answer.

The day count convention used for US treasury bonds is Act/Act. The other choices are incorrect.

The efficient frontier is plotted on a graph with portfolio return (mean) as the y-axis and portfolio volatility, or standard deviation, on the x-axis. Asset beta and standard deviation of the market portfolio have nothing to do with the determination of the efficient portfolio.Therefore Choice 'd' is the correct answer, and the rest of the choices are incorrect.

The LIBOR rate to use is the one at the beginning of the period, is 4.5%. The fixed rate payer owes 5%, and the floating rate payer owes 4.5% + 100bps. Thus the fixed rate payer will receive a payment equal to 0.5% for six months on $1m. This works out to $2500.(Recall that a fixed for floating interest rate swap exchanges fixed for floating rate payments, with only the net payment being made by either party.)

The LIBOR rate to use is the one at the beginning of the period, is 4.5%. The fixed rate payer owes 5%, and the floating rate payer owes 4.5% + 100bps. Thus the fixed rate payer will receive a payment equal to 0.5% for six months on $1m. This works out to $2500.

(Recall that a fixed for floating interest rate swap exchanges fixed for floating rate payments, with only the net payment being made by either party.)

Fixed income instruments go through the 'pull-to-par' phenomenon as they get closer to maturity. In this case, since interest rates stay constant, the bond will rise in price over the year to get closer to its redemption value. Of course, you can apply the standard discounting logic as well to say that while one year's coupon payment drops off, the remaining years' cash flows get discounted for a year less and the value would go up. However it is just easier to remember that at maturity the bond will be close to its notional or face value, and therefore if it is selling at a discount now, it will rise in value over its remaining life to get to the face value.

Exactly the reverse logic would apply to bonds sold at a premium, they would decline in value over time as to get to the par value at maturity.

Since the stock is selling for $100 spot, its 3 month forward price for you would be $100*(1 + 6%*3/12) = $101.50. Since the forward is priced at $104, it is 'rich' compared to the spot and should be sold. A profitable trade would be to buy the spot and sell the forward.

Imagine you borrow $100 and buy the stock, and simultaneously sell the forward. At the end of 3 months, you will receive $104 in return from the forward, and be obliged to return the initial $100 borrowed together with interest of $1.50 for the three months. You would make a riskless profit of $2.50 per share in three months. Thus Choice 'c' is the correct answer.

Note that the treasury yield is irrelevant, what matters is the rate at which you can finance the trade which is 6% per annum.

Statement I is not correct because the 4% bond with the same modified duration as the zero coupon bond will have a maturity much later in the future than the zero coupon, and the final repayment coming much later will cause its convexity to be greater than the equivalent duration zero coupon bond.

Statement II is not correct as the convexity of a bond increases not in a linear fashion but in proportion to the square root of time as the maturity of the bond is increased.

Statement III is not correct as mortgage based bonds bring negative convexity even in the case of long positions,

Statement IV is not correct as the duration of a zero coupon bond maturing in 10 years has a modified duration greater than that of a coupon bearing bond of the same maturity, which means its convexity is higher and not lower (note this is different from statement I where the modified durations were identical. Here, the maturity is identical which has different implications.)

Therefore none of the statements is correct and Choice 'd' is the correct answer.

Choice 'd' is the correct answer. Continuously compounded returns are given by the log of S2/S1. The expression (S2 - S1)/S1 gives discrete returns, not continuously compounded returns.

Note that the question is only asking for the return over the given time period, not annualized returns. So t is not relevant here. If we had to find annualized returns, and t were expressed in years, then the annualized return would have been (1/t)*ln(S2/S1).

Also refer to the tutorial on interest rates and compounding for more information on continuously compounded rates.

Explanation:

Remember how exchange rates are generally quoted. Most exchange rates are quoted in terms of how many foreign currencies does USD 1 buy. Therefore, a rate of 99 for the JPY means that USD 1 is equal to JPY 99. These are called 'direct rates'. However, there are four major world currencies where the rate quote convention is the other way round - these are EUR, GBP, AUD and NZD. For these currencies, the FX quote implies how many US dollars can one unit of these currencies buy. So a quote of "1.1023" for the Euro means EUR 1 is equal to USD 1.1023 and not the other way round.

A fixed for floating interest rate swap would be the most appropriate to the company's needs. It will allow it to receive a fixed rate (which will offset the fixed payment it has to make on the loan) and pay a floating rate which it expects will be lower than the fixed rates. Choice 'a' is the correct answer.

Answer:

Implied volatilities are the same for calls and puts with similar exercise and strike prices. If not, it would offer an arbitrage opportunity. Therefore Choice 'b' is the correct answer.

A zero coupon bond has a Macaulay duration equal to its maturity, ie in this case 5 years. We can calculate modified duration from Macaulay duration by using the following relationship:

Modified duration = Macaulay duration/(1 + y) where y is the yield.

Therefore the correct answer is 5 / (1.05) = 4.76.

Or intuitively, all the other answers appear clearly incorrect: 5 and 5.25 are too high, and 4 is too low. Therefore the only reasonable choice is 4.76.

Statement I is true. Consider the Black Scholes PDE 202.23.e1

and substitute delta = 0 (as this is a delta neutral portfolio), and we get the result 202.23.e.

Since r is generally small, and σ and S are positive, theta will be negative when gamma is positive and vice versa.

Statement II is incorrect. The gamma of a call and a put are equal and do not add to 1. The relationship described applies to delta, and not gamma.

Statement III is correct because positive gamma means that the portfolio gains both from an increase and a decrease in the value of the underlying, given delta neutrality. (Generally, a positive gamma is a good thing, but like everything else, it does not come free. A trader can choose to keep their portfolio gamma positive, but in doing so he or she would be giving up premiums from options they would have sold to neutralize the gamma.)

Statement IV is correct because the only way to hedge gamma and vega is through other options positions as only options have gamma and vega. If only one tradeable option is available, it would be possible to hedge either the gamma or the vega, but not both, as achieving neutrality in one will upset the neutrality of the other. The only way to simultaneously hedge the two would be to use two different options on the underlying, and determine the number of options to be traded (using a system of simultaneous equations).

The crack spread is the difference between the price of refined products and crude oil. An option on the crack spread can protect a refiner from both a fall in the price of its output and a rise in the price of its inputs. Calendar spreads are options with different maturities. Crude oil futures and swaps only protect against an adverse change in the price of crude, and not that of refined products. Choice 'b' is the correct answer.

The duration of a perpetuity is 1/running yield. Therefore the correct answer is 1/.05 = 20 years

A short squeeze results when short sellers are trying to cover their short positions by buying in the spot markets, which do not have adequate supply. This results in sharp spikes in spot prices, which further forces any other shorts to try cut their losses. The result is a sharp rise in spot prices.

Choice 'a' is the correct answer, the other choices do not describe a short squeeze.

A higher coupon brings forward the cash flows of a bond, thereby lowering duration. Therefore statement I is true. The lower the duration the lower the convexity. Therefore a higher coupon leads to lower duration and therefore lower convexity. Thus statement II is true.

Bonds do not have negative modified duration, therefore statement III is false.

If the bond is callable, it leads to negative convexity as a fall in interest rates is likely to induce the issuer to call the bond and re-issue at the new lower interest rates (much like refinancing mortgages when interest rates fall), therefore the upside from a fall in rates is limited. This creates 'negative convexity'. Thus statement IV is correct.

To answer this question, we need to look at the put-call parity, which can be expressed as:

Value of call - Value of put = Spot price - Exercise price discounted to the present

or, Value of call - Value of put = Stock - Bond with a future value equal to exercise price

Therefore, a long call and a short put is equivalent to a long stock position and a short bond.

Choice 'a' is therefore the correct answer. (Alternatively, we could also have constructed a graph of the payoff profiles to arrive at the same answer).

The prices must satisfy the put-call parity, and if they do not, it means there is an arbitrage opportunity, The put-call parity is as follows. We plug in the values to check if the parity is maintainted.

Buying a call + Bank Deposit (PV of exercise price) = Buying the stock + Buying a put

Thus the LHS = $10 + $100/(1 + 5%) = $105.24

and RHS = $100 + $10 = $110, ie the equality does not hold.

Since the trader can make a profit by buying low and selling high, and the set of positions on the left should be brought, and those on the RHS should be sold. Thus the trader should buy the call, sell the put and sell the stock to make a risk free profit. (There is no need to explicitly place a bank deposit at the risk-free rate).

The all in price would be equal to the clean price plus accrued interest. The accrued interest for the 60 days that have passed since the last coupon payment is $5 x 60/360 = $0.83. Therefore the dirty price, or the all in price, will be $98.83. (Note that the $5 is the annual 5% coupon on the $100 face.)

Explanation:

The correct answer is the index value times the contract size, in this case 1500 x 50.

One way to think about index futures is this: Consider equity index trading as trading in the shares of a company whose share price is equal to a number of dollars which is the same as the index. If the 'contract multiplier' for a index futures contract is 50, that means the futures contract is for 50 shares of such a fictitious company. Therefore the notional value of the contract will be 15000 x 50, and Choice 'a' is the correct answer.

The 'running yield' refers to the coupon rate divided by the current price. In this case, it is 6/96 = 6.25%. Remember that the running yield is also called the current yield.

A normal yield curve is generally upward sloping. Downward sloping, humped or flat yield curves are less common and indicate exceptional market conditions.

The put-call parity can be expressed as:

Call – Put = Spot – PV of exercise price

Note that a negative sign above means a short position. The 'term PV of exercise price' is the same as a bank deposit placed today equivalent to the PV of the exercise price so that we will have the cash flow on the exercise date to exercise the option.

Therefore only Choice 'd' is the correct answer as rearranging the above gives us Buying a call + Bank Deposit = Buying the stock + Buying a put.

Choice 'd' is therefore the correct answer.

A forward contract is a derivative contract, and has a delta of 1. Therefore Choice 'd' is the correct answer. This is because the value of a forward contract is given by S - Ke^(-rt), where S is the current spot price, r the risk free rate, K the forward price, and t the time to maturity. As S changes from S to (S + δS), the value changes to (S + δS) - Ke^(-rt), ie the change in value is exactly δS in response to a change in the price of the underlying by δS. Therefore the forward contract has a delta of 1.

All other choices are incorrect.

Note that this is different from the delta of a futures contract which is different from 1, and equal to e^(rt), a number greater than 1.

Choice 'c' correctly describes a 'holder extendible option'. Choice 'd' describes a 'writer extendible option'. Choice 'a' describes a 'shout option'. Choice 'b' describes a 'knock in' option.

Since $100 is convertible into two shares, the price upon conversion per share is $50, which is the same as the current share price. Therefore there is no premium, and Choice 'c' is the correct answer.

Backwardation is said to occur when futures prices are lower than the current spot prices. This would happen only when carrying costs are negative. Carrying costs are equal to interest, plus storage costs and less any 'convenience yield'. The existence of large convenience yields may explain backwardation in commodity futures prices. Therefore Choice 'd' is the correct answer.

Contango is the 'normal' market situation where forward prices are higher than spot prices. Storage costs explain contango, not backwardation. Risk free rates, or the cost of funding for the futures position, are always positive and do not explain backwardation.

To solve this question, we need to revisit how delta is calculated in a single step binomial model:

Consider a portfolio with just two positions: 1 Long Call option, and Δ Short Stock. We do not know what Δ (Delta) is.

Now if the current price of the stock is P, which can take the value P2 (higher value) and P1 (lower value) in the future at time T, then Δ needs to be such that the value of the portfolio is equal in both the cases. If the price goes up, the value of the option will be $1 (given) and the value of the short stock will be -Δ P2. If the price goes down, the value of the option will be $0, and the value of the stock will be -Δ P1.

Therefore, $1 - Δ P2 = $0 - Δ P1.

Solving for Δ, we get Δ = $1 /(P2 - P1) = $1 / ($102-$98) = 1/4.

Mortgage backed securities carry prepayment risk as borrowers tend to prepay mortgages when rates fall, and substitute it with newer cheaper mortgages. This creates the issue of 'negative convexity' for mortgages, ie, they lose value when rates rise, but do not gain in value when rates fall.

Prepayment risk can be offset by instruments that also carry negative convexity. A swaption is an option to borrow in the future at an agreed rate, which may be fixed or floating. An option to borrow in the future paying floating and receiving fixed guards against losses when rates fall, as the option can be exercised for a profit when rates are declining and the mortgage portfolio is being prepaid. A callable bond is very similar to an MBS in that the issuer can call it back when rates fall. Thus a long position in an MBS can be offset by a short position in a callable bond. Thus I and III are valid choices.

A cap allows exchanging fixed for floating when interest rates rise above an agreed rate. A long cap position allows borrowing at a fixed rate in exchange for floating, and a short cap implies receiving fixed and paying floating when rates go above the strike rate. Prepayment risk arises from falling interest rates, therefore a short cap will not protect against such a risk as falling interest rates would mean that no payments would be exchanged. Thus II does not help hedge against the risk in question.

A long position in a fixed for floating swap would require paying fixed and receiving floating when rates are falling. This would just make the problem from prepayments worse as the position would pay fixed and receive a falling rate. Thus IV is not an appropriate way to hedge prepayment risk.

From the put-call parity, we know that Call - Put = Stock - Bank deposit. Since the bank deposit has a zero Gamma, and the Gamma of the Stock itself is also 0, we get the relationship Gamma of Call - Gamma of Put = 0. Therefore, if the Gamma of a call option is 0.08, the Gamma of the corresponding put option is also 0.08.

To calculate the Modified Duration from Macaulay's duration, we use the relationship MD = D/(1+r), where MD is the modified duration and D the Macaulay Duration. Therefore Choice 'c' is the correct answer.

Overcollateralization is when the notes issued by the special purpose vehicle are less in value compared to the underlying pool of assets, thereby providing a buffer to absorb losses. Excess spread implies that the notes issued carry a lower interest rate than the interest rate received on the underlying assets. Cash reserves are reserves intended to take first hits when losses happen. All of these are valid credit enhancements for structured products. Additionally, 'insurance wraps' are also used as a credit enhancement. Choice 'c' is the correct answer.

'Margin requirements' do not mean anything in this context and are not a valid credit enhancement used for credit derivatives.

In this case, we need a portfolio that is "dollar duration" neutral, ie the net effect of any change in interest rates creates offsetting gains and losses in the asset and liability portfolios. The $125m asset portfolio with a duration of 15 years has a dollar-duration of $125*15/100=$18.75m, implying that a 1% change in interest rates will create a $18.75m change in the value of the asset portfolio. Similarly, the liabilities of $100m with a duration of 20 years mean that a 1% change in interest rates will lead to a $20m change in their value. Since $18.75m<$20m, the pension fund is short duration, and needs a strategy that will give it more duration. It can do so by buying more bonds, or swapping out some of its shorter duration bonds with longer duration bonds, or doing other things that will increase the duration of the portfolio so it becomes duration neutral.

Entering into an interest rate swap to receive fixed and pay floating is like a long position in a bond, which will increase duration. Thus Choice 'b' is the correct answer.

A swap to pay fixed and receive floating will have the opposite effect, and so will selling bond futures. Choice 'c' and Choice 'a' are therefore incorrect. As we have seen, Choice 'd' is incorrect as the fund does have interest rate exposure.

The dates on which the interest rates applicable to the floating rate leg of an IRS are determined are called 'reset dates'. Choice 'c' is the correct answer, the other choices are incorrect. Settlement date is the date a transaction settles, normally a few days after the trade is done depending upon the market. Trade date is the date the terms of the transaction are agreed. (Settlement normally happens on T+n, where T is the trade date, and n the number of days to settlement as decided by agreement or market convention.)

As a bond approaches maturity, its value will be driven less by the time value of money or its coupons, and more by its redemption value, ie the par. Therefore as maturity approaches, its value converges to its par value and this phenomenon is called pull to par.

The price of an interest rate cap is affected by all of the listed choices except the risk free rate. The risk free rate does not come into play in the pricing of caps, and therefore Choice 'b' is the correct answer.

The effectiveness of the hedge is solely determined by the correlation between the position being hedged, and the position being used as the hedge. A hedge can be perfect when correlation is 1, and will be less than perfect when it is anything other than 1.The effectiveness of the optimal hedge is given by the formula √(1-ρ2), where ρ is the correlation between the two. Therefore Choice 'c' is the correct answer. Standard deviations affect the hedge ratio, not the effectiveness of the hedge.

The Sharpe ratio does not require investors to be risk neutral, only that for a given level of returns they prefer less risk to more risk. (Risk neutral means that investors are indifferent to the level of risk, and are only driven by a desire to maximizing expected value, regardless of risk levels.)

The ability to borrow and lend any amounts of money at the risk free rate is a fundamental assumption for the rule that optimal portfolios will maximize the Sharpe ratio.

This rule also assumes that risk preferences are completely described by return and standard deviation of returns.

Therefore Choice 'd' is the correct answer as statements I and II are correct.

Basis risk between spot and futures prices is the risk that the prices of the futures contract will not move in lockstep with the spot prices. Basis risk creates a problem when investors are trying to get exposure to a stock price through a futures contract, or are trying to hedge a position using a futures contract, and over time the prices of these two diverge.

To understand why basis risk arises, we need to think about how futures prices are determined. Futures prices are nothing but the spot prices plus the carrying cost, ie what it would cost for the seller of the contract to buy the spot security and hold it till the time of delivery in the future. During this holing period, the seller of the contract has to pay interest to fund his purchase in the spot market, and also receives and distributions such as dividends. In the case of commodity futures, he may also have to pay storage charges. When determining the futures prices, it is not known with complete certainty as to what the carrying costs will ultimately turn out to be - interest rates may change, dividends may turn out to be more or less, leading to changes in the futures prices without changes in the spot prices. This gives rise to basis risk.

Therefore changes to the risk free rate, which is a proxy for the true funding cost of the futures position and changes in expected dividends creates basis risk. Volatility of the underlying is irrelevant for futures.

Therefore Choice 'b' is the correct answer.