Longstaff and Grinblatt (2000) find that the treasury derivatives complete markets.
True. They find that stripping activity is greater for Treasury securities with maturities for which discount bonds cannot be synthesized by alternative means.
Prices of commodities can behave very differently than those of financial assets. For
example, it may seem that there are persistent arbitrage opportunities in futures commodity markets.
True. Commodities are used not solely for investment purposes but also as input in
production. Because of the presence of the convenience yield associated with commodities,
it may seem that there are arbitrage opportunities.
If the gamma of a portfolio of options is large, the adjustments to keep a portfolio delta neutral need to be made only relatively infrequently.
False, opposite. If the gamma is large, the delta is very sensitive to the change in the
price of the underlying. To prevent big losses, one should frequently reconsider the delta hedge.
A position of shorting an underlying and buying a call option on it is long volatility,
whereas a position of buying an underlying and shorting a put option on it is short
volatility.
True. First position is long volatility, whereas the second position is short volatility.
Both put and call options have positive vega.
Theta on plain vanilla put option can be positive.
True. For deep ITM put options, theta is positive. Puts gain value because of time value of strike.
You consider an interest rate swap (fixed for floating) and expect that the swap rates
will increase in the near future. Based on your expectations, you should enter swap to
receive floating rate and pay fixed rate.
True. If future swap rates increase, this is because LIBOR likely increases. Therefore if
you receive floating, it is likely that you will get more that what you pay.
Suppose that US treasury bond futures contract is about to expire, the delivery price
is F = 90 USD. There are two bonds A and B with coupons paid semiannually eligible
for delivery. Bond A has market price of 100 USD, coupon of 6% and time to maturity
T = 2 years; bond B has market price of 110 USD, coupon of 8% and time to maturity
T = 1 year. Bond A is the cheapest to deliver bond.
True,
Compute conversion rates for both bonds:
Conversion for bond A = 1
Conversion for bond B =
Therefore the invoice prices for these two bonds are:
Invoice price for bond A = 90
Invoice price for bond B = 91.7221 (conversion factor * futures price).
Cheapest to Deliver = Current Bond Price – Settlement Price x Conversion Factor
Conversion factor:
Spot price of the stock is 100 USD, the forward price of the same stock to be delivered in one year is 110 USD. Borrowing market rate is 6% per year and lending market rate
is 4% per year. There is an arbitrage opportunity here and arbitrage profit is about 6 USD.
False. Compute no-arbitrage bound, it is [104, 106]. Therefore we deal here with an
arbitrage opportunity but we could cash out only 4 USD because the borrowing rate is
6%.
Only in the case of constant interest rates, similar futures and forward contracts (on the
same underlying, with the same delivery time) have identical prices.
False. If interest rates are deterministic, prices on similar forward and futures contracts are identical.
Skilj på "constant interest rates" och "deterministic interest rates"
Determenistic = "In deterministic models, the output of the model is
fully determined by the parameter values and the initial conditions."
If the swap at inception is at par, its value is zero until the maturity.
False. The value of a par swap is zero at inception only, after that its value may be
positive and negative for both counter parties.
Consider forward on Crude Oil. If the
interest rate is less than the convenience
yield at all maturities, the forward curve
must be in backwardation.
False. If the sum of the interest
rate and the storage cost is less
than the convenience yield, than
the forward rate is in
backwardation.
Cost of carry formula.
Consider the cheapest-to-deliver option
on the Treasury bond futures at the
CBOT. The short position would always
want to deliver the standard grade
bond, if such a bond existed.
False. The short position would
always deliver the cheapest-to-
deliver bond.
All else equal, an increase in the
risk-free rate decreases the value
of the put option.
True. Yes. If interest rate increases, the expected return on the underlying is expected to increase. Therefore, the value of the call option increases, whereas the value of the put option decreases.
Put call parity can prove this as well.
Hedging can lead to a worse outcome.
True. Remember discussion
of Ryanair hedge of the fuel
prices.
Futures and options are zero
sum games.
True. Loss of one side of the
trade is gain of the other side
of the trade. Zero-net supply.
Valuing options in a Binomial model
using risk-neutral methods is valid
only if all investors are risk-neutral.
False. It is valid only if no-arbitrage
holds. It has nothing to do with investors risk-appetite.
Suppose you are long a European
put option and short an otherwise
identical European call option on
the S&P 500 Index. Other things
being equal, an increase in interest
rates will decrease the value of your
portfolio.
True. Look at the put-call parity
condition. If interest rate increases
the value of Put-Call decreases:
A short position in a put option can
be synthetically replicated over a small time interval by buying the
stock and selling the risk less asset
(borrowing).
True. Remember binomial tree exercise. A long put position can be replicated by a short stock position and lending money. Hence, a short put position can be replicated by being the stock and borrowing money (selling the riskless asset).
In order to replicate the option in a longer period we would need to dynamically replicate the portfolio.
The Black-Scholes model underprices
out-of-the money options.
True. The Black-Scholes model is based on the assumption of constant volatility, whereas in the data, out-of-the money options have higher implied volatility than at-the-money options.
Lookback options are less expensive,
whereas barrier options are more
expensive than plain vanilla options.
False. The opposite is true. The plain vanilla options give the option holder a better chance to receive a positive payoff compared to the barrier options that cease to exist under some circumstances.
At the same time, lookback options are associated with a not smaller payoff to the option holder compared to the payoff associated with plain vanilla options.
Lookback option:
A lookback option is an exotic option that allows investors to "look back" at the underlying prices occurring over the life of the option and then exercise based on the underlying asset's optimal value. This type of option reduces uncertainties associated with the timing of market entry.
There are three- and six-month American calls with the same strike on ABC stock.
Suppose the three-month option costs 6 USD and the six month option costs 3 USD.
Buying the six-month call and selling the three-month call is an arbitrage. If it is an
arbitrage, specify how to exploit the corresponding arbitrage opportunity.
True. Receive 3 USD at initiation and exercise the six-month call whenever the three-month call is exercised.
In the binomial model, consider a stock with current value 100. At the end of the first
period in the tree, there will be a dividend payment of 5. The ex-dividend value of the
stock will be either 110 or 90. The risk-free rate is zero. The risk-neutral probability of
an up-move is 0.25.
True.
u = 115/100 = 1.15, and
d = 95/100 = 0.95,
q = (1 − 0.95)/(1.15 − 0.95) = 0.25.
We do not have to worry about discounting since risk-free rate it zero.
In the real world, where convenience yields are unpredictable, since we cannot value
commodity forwards by no-arbitrage, we also cannot value commodity swaps by no-
arbitrage. Assume that credit risk is not an issue.
False. Ignoring the credit risk, the swap is a string of forward contracts by no-arbitrage.
So we can use market prices of forward contracts to value the swap even though we
cannot value forward contract by no-arbitrage.
Forward price of one Swedish Krona increases if the Swedish interest rate decreases.
True. Covered interest rate parity.
When the no-arbitrage condition is satisfied with the use of a forward contract to hedge against exposure to exchange rate risk, interest rate parity is said to be covered.
where i_d = domestic interest rate
i_f = foreign interest rate
F_t/S_t is the forward hedge. So if the domestic rate decreases (SEK) then
A repurchase agreement is equivalent to a sale of the underlying asset coupled with a
short forward contract on the underlying.
False. A repo is equivalent to a sale of the underlying asset coupled with a long
forward contract on the underlying
Typically, on-the-run treasure bonds have lower yields than their off-the-run counter-
parts.
True. The on-the-run bonds have higher prices and lower yields.
The value of the forward contract is always zero.
False. The value of the forward contract is zero only at inception.
Swaps and options can be always priced by designing a corresponding static replicating
strategy.
False. Swaps can be replicated by a static strategy, whereas options require a
dynamic strategy.
Prices of the futures and forward contracts on the same underlying with the same delivery
date are the same.
False. They are the same only if there is no reinvestment risk and no coun-terparty risk.
Using derivative instruments, it is possible to design a strategy which is profitable if
the volatility of the underlying instrument is high, regardless of the future price of the
underlying.
True. Straddle is a good example of such a strategy.
Equity of a corporation can be interpreted as an option.
True. Equity is a call option on the assets of the firm with a strike price equal
to the debt of the firm.
The price of the call option is increasing in the expected return on the underlying.
False. Prices of options do not depend on the expected return on the underlying. (Only the actual price and the risk-neutral rate etc.) So the expected value is using real probabilities, which is not interesting for options.
In financial markets, implied volatility can be precisely measured by the realized volatility, i.e., historical volatility of the underlying.
False. Implied volatility tends to be larger than realized volatility. People make
money on this difference.
Barrier options are always cheaper than their plain vanilla counterparts.
True. Barrier options are equivalent to plain vanilla options upon some contingency. If contingency is not met the option does not exist.
There are three- and six-month American calls with the same strike on ABC stock.
Suppose the three-month option costs 6 USD and the six month option costs 3 USD.
Buying the six-month call and selling the three-month call is an arbitrage. If it is an
arbitrage, specify how to exploit the corresponding arbitrage opportunity.
True. Receive 3 USD at initiation and exercise the six-month call whenever the three-month call is exercised.
The convenience yield of gold is likely to be higher that the convenience yield of crude
oil.
False. Gold is an investment commodity. Its convenience yield is almost zero. Crude oil
is consumption commodity, hence its convenience yield is high. Double check the formula sheet.
Consider an existing fixed for floating interest rate swap immediately after a floating
rate payment. The principals are both $10M and the swap rate is 4%. If the LIBOR
term structure is flat at 3%, the Mark-to-Market value of the swap is positive in favor
of the party receiving floating.
False. The value of the floating leg immediately after a floating rate payment is the value of the principal. The value of the fixed leg is higher, because fixed rate is 4% that is larger than 3%.
If the minimum-variance hedge ratio is calculated at 1.0, the hedge must be perfect.
False. The minimum-variance hedge is it could only be one if p < 1, hence the hedge clearly is not perfect.
Consider an American call option on a stock that will pay a $5 dividend per share in
three weeks. The option matures in two months, and there will be no further dividend
payments. If the call is fairly priced, it may be optimal to exercise the call at a point
after the dividend payment but before maturity.
False. American call options on non-dividend paying stock have zero exercise value
premium. Note that in three weeks after the dividend has been paid out, the stock can be
considered as non-dividend paying underlying because no other dividends are expected.
Never exercise an American call on NDP UL!
Commodity prices can behave very differently than those of financial assets. For example,
it may seem that there are persistent arbitrage opportunities in futures commodity
markets
True. Prices of consumption commodities could behave quite differently. Owners of
these commodities usually plan to use them not for speculative purposes. Consumption
commodities..
If you are long European put option, you lose when the volatility of the underlying is
high.
False. You are long volatility, long vega. Thus, you are benefiting from higher volatility
as your option grows in price.
If the Black-Scholes model is true then you should expect to observe higher implied
volatility on the out-of-money options than on at-the-money options, i.e., implied volatility
smile. Note that we consider options on the same underlying with the same maturity
but different strikes.
False. The Black-Scholes model assumes that the volatility is constant across all strikes.
Suppose you have two put options on the same stock and the strike of the first option
(K1) is greater than the strike of the second option (K2). In a one-period binomial
model, assuming that uS > K1 > K2 > dS and that u > 1 + r > d, the delta for the
first option is always less than the delta for the second option.
True. The delta for the call option in the binomial model is ∆ = (Pu−Pd)/(uS−dS). Because Pu = 0 for both options and Pd_K1 > Pd_K2 the numerator is smaller for the option with the strike K1.
You consider an interest rate swap (fixed for floating) and expect that the swap rates
will increase in the near future. Based on your expectations, you should enter swap to
receive floating rate and pay fixed rate.
True. If future swap rates increase, this is because LIBOR likely increases. Therefore if
you receive floating, it is likely that you will get more than what you pay.
Consider forward contracts on Crude Oil. If the interest rate at all maturities is lower
than the convenience yield, the forward curve must be in backwardation.
False. If the sum of storage costs and interest rate is less than the convenience yield
then the forward rate is in backwardation.
Put-call parity for European options is true only under assumption that the underlying
price follows the log-normal distribution.
False. There is no need to make any distributional assumption in order to derive the
put-call parity. It is based solely on the absence of arbitrage.
Theta on plain vanilla put option can be positive.
True. For deep ITM put options, theta is positive. Puts gain value because of time value
of strike
To replicate a long position in any commodity swap, it is enough to form a portfolio of
forward or futures contracts on the same commodity of various maturities corresponding
to delivery dates of commodity in the swap contract.
False. You also need to borrow and lend to account for differences in cash flows. This is because the swap will not have a 0 cash flow at each period if you replicate. You will probably need to borrow money for the first maturities and lend money for the last ones in order to fully replicate the position.
It is never optimal to exercise an American put option on a non-dividend paying stock
early.
False. It may be optimal to exercise an american put option on a non-dividend stock
early if it is very deep in-the-money.
4. LTCM, convergence trade, 5 points
Please explain in detail how LTCM was taking advantage from the price difference between
on-the-run and off-the-run bonds. Please provide the list of all the instruments that were
used in the trade and explain how the trade was executed.
LTCM was buying ”cheap” off-the-run bonds and selling ”expensive” on-the-run bonds. To
execute the trade without initial capital, they used repo and reverse repo agreements. On the
one side of the trade, they borrowed on-the-run bonds by entering a reverse repo agreement
and sold them in the market. On the other side of the trade, they entered a repo agreement:
they bought off-the-run bonds (used money obtained from selling the on-the run bonds),
delivered them to their counterparty in a repo contract, got a loan that they used in a
reverse-repo agreement. They managed to match the loans received and given away as a part of the reverse repo and repo agreements. At the maturity of the contracts, they delivered
a loan back (repo agreement) that they received in a reverse repo agreement, they got an off-the-run bond of longer maturity as a part of the repo agreement, sold it in the market,
bought an off-the-run bond of a shorter maturity and delivered as a part of the reverse repo agreement. As a result, they cashed out the difference in price between the on-the-run and
off-the-run bonds at inception and did not need to pay anything at the delivery. The short
description of the deal: (1) buy off-the-run bond and repo it, (2) borrow on-the-run bond by
