8006 Exam I: Finance Theory Financial Instruments Financial Markets - 2015 Edition Questions and Answers

Commodity futures prices can be expressed as the summation of their spot prices and the carrying costs. Therefore any changes in either of these two would be a risk to the futures prices, and Choice 'b' is the correct answer. It is common to decompose complex commodity portfolios into underlying equivalent spot positions and the carrying costs, which includes interest, convenience yield and storage costs. For liquid commodities such as gold where changes of a short squeeze are low, interest costs dominate the carryings costs. Choice 'b' is the correct answer as it is most complete and covers the elements in the other choices. The 'lease rate' for a commodity is equivalent to (Fwd Price - Spot Price)/Spot Price, and comprises the interest and storage costs and the convenience yield. The other choices do not represent complete answers.

The vast majority of exchange traded futures contracts are closed out prior to expiry by the parties acquiring offsetting positions. Very few contracts are settled by delivery. Since P&L on futures contracts is settled daily, 'cash settlement' really does not mean much as only the previous day's P&L is due or receivable on any given day.

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.

In a Dutch auction, the buyers put in their bids and the price at which the entire auction clears is the price at which each bidder receives the security. For example, if Bank A bids 5.5% for a bond, Bank B bids 5.6% and Bank C bids 5.7% and the total amounts bid by these three banks are enough to clear the auction, each of the banks will receive the bond at the price paid by the lowest successful bidder, ie Bank C. This is even though Banks A and B had bid higher (remember that bidding a higher yield means offering a lower price).

In a standard auction, every bidder pays the price they bid.

In a Dutch auction, there is an incentive to bid aggressively if a participant believes the final winning bid will be lower.

All the statements are correct, and represent the relationships between the values of the Greek variables for calls and puts for the same option. It is important to know why as the PRM exam often asks tough questions about the Greeks.

Statement I is correct because of the put-call parity. According to the put-call parity,

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

Now the delta of the spot is 1, and that of the discounted price is zero. Therefore

Delta of Call - Delta of Put = 1 - 0, or by rearranging we get the equation in statement I.

Statements II and III are correct as the gamma and vega of both the spot price and the discounted price are zero. Therefore using the put-call parity, we can say

Gamma of Call - Gamma of Put = 0 - 0, and

Vega of Call - Vega of Put = 0 - 0

Rearranging, we get statements II and III.

Statement IV is correct because of the following relationship between theta of call and theta of put:

Theta of Put = Theta of Call + rKe^(-rt).

Since rKe^(-rt) can only be a positive number, theta of put can only exceed the theta of a call. However, since theta is generally negative, it often implies that the theta of a call is the larger absolute number.

Additional explanation for the last point: Assume rKe^(-rt) =+1 and theta of a put is -5 (completely hypothetical)

Now Theta of Put = Theta of Call + rKe^(-rt)

Ie -5 = -6 + 1

Now -5 is the larger number than -6. In other words, theta of put exceeds that of the call in a pure mathematical sense, which is what I mean when I say “theta of put can only exceed the theta of call”. But if you ignore the sign, then theta of call is larger at 6 when compared to 5. Therefore the theta of the put is greater than the theta of the call – which is what the answer says.

In an interest rate swap, only the net payment is made. In this case,

- the customer pays the bank 4%*(3/12)*$100m

- the bank owes the customer (2% + 100bp))*(3/12)*$100m

Therefore the customer pays (4% - (2% + 100bp))*(3/12)*$100m. 3/12 represents the 3 month time interval. This is equal to a net payment of $250k from the customer to the bank. Therefore Choice 'c' is the correct answer and the rest are incorrect.

A portfolio loses its delta neutrality if the delta of the portfolio changes but the underlying hedge is unchanged. The portfolio that will require the most rebalancing will be one whose delta changes by larger amounts in response to a given change in the price of the underlying. The sensitivity of the changes in delta in response to changes in the price of the underlying is measured by gamma. The higher the gamma, the more the portfolio will go out of delta neutrality given a change in the price of the underlying. Therefore Choice 'b' is the correct answer.

A portfolio with a low gamma will maintain its delta over a larger price range for the underlying, and is therefore likely to require less frequent rebalancing. The absolute level of the delta itself does not matter, what matters is the gamma. Vega is irrelevant to the delta hedging decision. Therefore the other choices are incorrect.

The London Interbank Offered Rate (LIBOR) is published by the British Bankers' Association (BBA). The BBA polls the rates offered by at least eight banks drops the top and bottom quartile, and averages the middle two quartiles of the rates to arrive at the LIBOR.

A utility function is graphed with utility on the y-axis and the variable driving utility (generally wealth) along the x-axis.

A concave utility function, ie a function with a downward sloping curve, indicates risk aversion. A convex utility function indicates a risk seeking attitude and a straight line (ie no curvature) indicates a risk neutral attitude.

If $x be the spot price of oil, then in order for the forward price to be $75, the following relationship must hold: ($x + $4/(1.06))*(1 + 6%) = $75. Solving, we get x = $66.98

Futures contracts carry no gamma. Only options have gamma. Choice 'a' is the correct answer. Any instrument whose price varies in a linear fashion with respect to the underlying will have gamma equal to zero.

The underlying asset in a Treasury note futures contract is any long term Treasury note with a maturity of no less than 6.5 years and no more than 10 years. The underlying asset in a treasury bond futures contract is any long term US Treasury bond with a maturity of more than 15 years and not callable within 15 years.

Note the difference betwen what the underlying asset is for treasury bond futures and treasury note futures. In either case, adjustments will be made for the coupon and maturity of the actual bond delivered (the concept of the adjustment of 'cheapest-to-deliver').

Equity futures are by far the most traded futures contracts in terms of value, followed by interest rate products. Choice 'd' is the correct answer.

A deep out of the money option will not react much to the change in the price of the underlying. In fact, the more out of the money it is, the more unresponsive it will be to changes in the prices of the underlying. Therefore, its delta will approach zero. Since delta is zero, gamma, which measures the rate of change in the delta, will also be zero as delta is unchanging. Therefore Choice 'a' is the correct answer.

Statement I is true as standard deviation is the root of the squared deviations, and is always positive, and identical regardless of the positions being long or short.Statement II is false as the expected return of a long position is exactly the negative of the expected return of the short position.Statement III is correct as a correlation of +1 between two long positions implies a correlation of -1 between one long and one short position. This is the basis of hedging, for example, of spot positions with futures.Statement IV is inaccurate. The weight of an asset in a portfolio does not change the correlation of the asset with the portfolio, or with other assets.Therefore Choice 'c' is the correct answer.

The stock has a beta of 1.2, therefore intuitively it can be expected to earn more than the broad market index. It will earn the risk free rate, ie 3%, and 1.2 times the equity risk premium of 5% (8% - 3%). The expected returns from the stock therefore are 3% + (8% - 3%)*1.2 = 9%

https://riskprep.com/images/stories/questions/102.07.a.png  is the coefficient of risk aversion at x. Its inverse, ie https://riskprep.com/images/stories/questions/102.07.b.png , is called the coefficient of risk tolerance.

Risk aversion or risk tolerance is indicated in a utility function by its curvature. A concave utility function indicates risk aversion and a convex function indicates risk tolerance. The curvature is measured as a ratio of the second derivative to the first derivative of a function. A negative second derivative implies concavity. The expression - is the coefficient of risk aversion, and its inverse, which is in the same units as wealth, is called the coefficient of risk tolerance.

The loss to the fund manager is $20*100 = $2,000. The initial margin placed was $5,000. After the loss is charged to his account, the margin balance is only $3,000 ($5,000 - $2,000). The margin to be maintained is $5,000. Therefore the margin call is $2,000.

Note that there would have been no margin call till the balance in the margin account had reduced to the 'maintenance' level of $4,000. So if the price had fallen by $5, and the loss had been $500, there would have been no margin call. But when the margin call would be made, it would require the balance to be brought up to the initial margin of $5,000 (and not the maintenance margin of $4,000).

Implied volatility is the volatility that is priced in the current market prices. In other words, we know the formula for pricing an option, and we find the value of the volatility for which we get the current market price. It is the volatility that the market is ascribing to that security. Therefore statement I is not correct.

Implied volatility in the market tends to vary across strikes, even with the same underlying. There is a certain level of demand for out of the money puts, for example, from investors trying to set a floor for their losses. At the same time, there is a natural supply of out of the money calls from investors trying to earn some relatively risk free premiums. These factors cause options with different strikes to be priced in a way as to give different estimates of market volatility at each strike *& resulting in the 'volatility smile'). Therefore statement II is not correct.

The shape of the implied volatility curve is called the 'volatility smile', because of the shape it sometimes takes. Statement III is correct.

Theta is the rate of change of an option's price with the passage of time. It causes the value of the option to decrease over time, such that if it is out of the money and close to expiry, the price of the option approaches zero. Theta is therefore negative by definition. Statement IV is not correct. Remember that theta will be positive for short positions.

The buyer of the FRA gets to borrow $100m at 10% per annum for 6 months at the end of 6 months from the contract date. Thus, the interest due is $100m * 10% * 6/12 = $5m. However, at the end of the 6 months (when the notional borrowing period begins), the spot rate is 12%. The interest due on a borrowing at the spot rate would be $100m * 12% * 6/12 = $6m. Since the buyer gets to borrow cheaper than the going rate, he or she has made a gain of $1m on the FRA. However, this amount is not due immediately, it is due at the end of the 6 month borrowing period (ie, 12 months from the date the contract was entered into). The seller of the FRA can make the buyer whole by paying the buyer the present value of $1m due in 6 months time, which is $1m/(1 + 12%*6/12) = $1m/1.06 = $943,396.

Thus, Choice 'd' is the correct answer. Remember that the FRA gets settled at the beginning of the notional borrowing period as all future cash flows, and the applicable discount rates are known with certainty. The buyer and the seller do not need to wait for the entire period to get over first. The cash settlement allows the buyer to borrow at the market rate and still have the same net borrowing cost as he or she had initially contracted for in the FRA.

Analytical methods are more 'efficient' than numerical methods in terms of computing power required, consistency of results and providing clarity on the inputs driving the results. Therefore 'efficient valuation' is not a reason to adopt numerical pricing methods over analytical methods.

It is much easier to incorporate changing stochastic volatility, discontinuous asset prices or other complex payoffs using numerical pricing methods. So these certainly represent good reasons to use numerical pricing methods for valuing exotics.

Therefore Choice 'b' is the correct answer.

Choice 'a' correctly describes a holder extendible option. Choice 'd' describes a writer extendible option. The other choices are meaningless.

The equivalent continuously compounded rate in this case can be calculated as ln(1+5.2%) = 5.07%. The other answers are incorrect.

Refer to the tutorial on interest rates for more details on how continuously compounded rates work.

A total return swap allows an investor or a financial institution to gain exposure to an asset or a portfolio without actually having to go long in those positions. The counterparty provides the return from the exposure and receives LIBOR plus a spread in exchange. Effectively, the counterparty has funded the investor's position for a fixed interest rate, therefore a TRS is essentially a funding arrangement. However, the structure of a TRS may also have other consequences that are relevant for tax and accounting - the assets under a TRS are not held on the balance sheet of the investor receiving the total return. Statement I is correct as the TRS helps gain an exposure without having to fund the position with cash.

Repos are useful for shorting corporate bonds. The investor desirous of shorting a corporate bond would sell the bond in the market, and immediately borrow the bond from someone else using a repo to deliver to the party he has sold the bond to. Repos are used extensively to cover short bond positions, and therefore statement II is correct.

Statement III is not correct as counterparty risk continues to exist with a TRS. The counterparty may fail to provide the agreed returns, and the risk exists.

Statement IV is correct as the bank that borrows funds using a repo continues to hold the underlying assets on its balance sheet. Therefore Choice 'd' is the correct answer.

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.

Since AUD 1 = USD 0.6831, and USD 1 = SEK 8.1329, AUD 1 = SEK 8.1329*0.6831 = 5.5556.

It is important to 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. 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.

When calculating cross rates, it is important to pay attention to how the rates are quoted. This particular question is quoted in a very straightforward way because it specifies exactly what the rate means, eg by saying USD/AUD it clarifies that the rate is the number of USDs per AUD. If the question is not clear, remember how exchange rates are quoted - all are against 1 USD, except for EUR, GBP, AUD and NZD where it is the other way round.

When 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 as shorts try to buy in the spot market to cover their short positions. Similarly, expectations of oversupply in the future, for example due to a bumper harvest may create situations where the forward prices fall below spot prices. Convenience yield is the benefit from having access to the commodity - and if the convenience yield is very high, for example in a market where manufacturers must never run out of a particular raw material, then these can switch the costs of carry (which include interest and storage costs, less convenience yields) to being negative.

Since all these factors can contribute to backwardation in the market, Choice 'd' is the correct answer.

The put-call parity applies to European options and does not hold for American options because the latter can be exercised at any time up to expiry. Therefore statement I is false.

An American put can not be cheaper than a European put is correct. This is because the American put will always be valued at at least its intrinsic value whereas a European value may be valued at less than intrinsic value if the exercise date is too far away in the future. This is because a long time interval will increase the chance that the intrinsic value may not be realized at expiry. Because the American put can be exercised anytime, it will always be at least equal to its intrinsic value because if it were to be cheaper than that, investors would immediately buy and exercise it, creating a riskless profit. Therefore statement II is correct, and statement IV is correct as well.

It may be optimal to exercise an American put (unlike an American call - note the difference!) before its expiry on a non dividend paying stock. This would be the case only when the put is deep in the money. If the option is really deep in the money (say, when the underlying's price is down to zero), then it may be worthwhile to cash the option out prior to expiry as the upside might be lost if time were to be allowed to pass. Additionally, any further upside in the case of a deep in the money option may be zero or very little, and the time value of money would also make immediate exercise prefereable. Therefore statement III is not correct.

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 swap can be valued by valuing the two individual components of the swap.

The fixed rate bond equivalent in the swap is valued at =4/1.05 + 104/(1.06^2) = $96,369,154.

The FRN component will be valued at par as we are at a point where the rate has just been reset, ie $100m.

The investor is paying the fixed rate, and is therefore short the bond. He/she is receiving LIBOR, and is therefore long the FRN. The value of the swap to the investor therefore is +$100,000,000-$96,369,154 = $3,630,846

Detailed explanation:

An Interest Rate Swap exchanges fixed interest flows for floating rate flows. The floating rate leg is tied to some reference rate, such as LIBOR. The parties exchange net cash flows periodically. Conceptually, an interest rate swap is the combination of a fixed coupon bond and a floating rate note. The party receiving the fixed rate is long the fixed coupon bond and short the FRN, and the party receiving the floating rate is long the FRN and short the fixed coupon bond.

An interest rate swap can be valued as the difference between the two hypothetical bonds. FRNs sell for par at issue time as they pay whatever the current rate is, subject to periodic resets. Therefore immediately after a payment is made on a swap, the value of the FRN component is equal to its par value. The bond can be valued by discounting its cash flows. The difference between the two represents the value of the swap. When the swap is entered into, the fixed rate leg is set in such a way that the value of the hypothetical bond is equal to that of the FRN, and therefore the swap is valued at zero. The rate at which the fixed rate leg is set is called the swap rate. Over its life, market rates change and the value of the fixed coupon bond equivalent in our swap diverges from par (whereas the FRN stays at par - at least right after payments are exchanged and the new floating rate is set for the next period). Thus the swap acquires a non-zero value.

There are two ways to value a swap. If interest rates for the future are known, the bond and the FRN can be valued and their difference will be equal to the value of the swap. Sometimes, the current swap rates are known. In such a case, the swap can be valued by imagining entering into an opposite swap at the new swap rate, which will leave a residual fixed cash flow for the remaining life of the swap. This residual cash flow can be valued and that represents the value of the swap. For example, if a 4 year swap was entered into exchanging an annual fixed 5% payment on a notional of $100m for a floating payment equal to LIBOR, and at the end of year 1 the swap rate is 6%, then the party paying fixed can choose to enter into a new swap to receive 6% and pay LIBOR. All cash flows between the old and the new swap will offset each other except a net receipt of 1% for the next 3 years. This cash flow can be valued using the current yield curve and represents the value of the swap.

Explanation:

Straddles and strangles are strategies that would benefit from sharp movement in option prices, regardless of direction. These comprise a long call and a long put, which would benefit regardless of whether prices rise or fall. The only time they would lose money would be when prices stay constant.

A collar would gain when stock prices fall, and not when they rise. Since our investor does not have a view on the direction of the movement, this strategy will not work for him.

A butterfly spread or a condor would gain when prices stay range-bound, so that cannot be a useful strategy.

Therefore Choice 'd' 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.

The correct answer is Choice 'b', as this question is merely asking for the forward rate based on known spot rates. The forward rate applicable to the three month period commencing in 3 months time is given by [(1 + 6%*6/12)/(1 + 5%*3/12) - 1]*4 = 6.91%. Thus Choice 'b' is the correct answer.

Here is a step by step way to think about it: $1 invested now at 6% for 6 months grows to (1 + 6%*6/12)=1.03. At the same time, using the 3 month rate, $1 invested now at 5% for 3 months grows to (1 + 5%*3/12)=1.0125. Effectively, this means that the 1.0125 at the end of 3 months grow to 1.03 at the end of 6 months, implying the rate of interest during the 3 months from 3 to 6 months is (1.03/1.0125 - 1)*4 = 6.91%.

Treasury bond futures do not specify which bond can be used to effect delivery, but allow the seller to pick from a number of available bonds. As a result, one of these eligible bonds emerges as being the 'cheapest' to deliver, and this CTD bond is determined by the basis between the cash price of the bond and the futures spot price as adjusted by the conversion factor for this specific bond. (ie, basis = Cash Price of the Bond - (Futures Price x Conversion Factor).)

The bond with the lowest basis is generally the CTD. In the given question, the three basis calculations are:

1) 97 - (0.9 x 105) = 2.5

2) 101 - (1 x 105) = -4

3) 106 - (1.1 x 105) = -9.5

Since the third bond with a quote of 106 has the lowest basis, Choice 'b' is the correct answer.

Once the price of the stock exceeds $100, the investor would forego any additional gains as the call would be exercised. Therefore the maximum gain to the investor is $10, plus the premium earned, which is $3, making for a maximum gain of $13.

In the event the stock price declines, the investor would lose on his long stock position an amount equal to the decline, less any premium earned. The worst case price for a stock is 0, and in this case the investor would lose his $90 in the stock, offset by $3 in premiums earned (the written call would expire worthless), thereby limiting his total loss to $87.

Choice 'c' is therefore the correct answer.

Some parts of this question draw upon contingent claims framework to the value of a firm. According to this framework, the relationship between the debt and equity holders of a firm can be viewed as follows: The equity holders have a call option on the assets of the firm with a strike price equal to the value of the debt, and the debt holders have sold them this call. This is so because should the value of the assets of the firm fall below the value of the debt, the equity holders can walk away by handing over the assets to the debt holders in full extinguishment of their claims. If the value of the firm's assets is greater than the value of debt, the equity holders will exercise their call option. At the same time, it is also possible to view the debtholders as holding an asset and having sold a put on the assets of the firm with a strike price equal to the value of the debt. If the value of the assets of the firm were to fall below the value of the debt, they will end up buying the assets at a price equal to the value of the debt.

An increase in the volatility of returns on the assets of a firm increases the volatility of the asset value. This means the likelihood that the asset value will go below the value of the debt of the firm will increase. Callable debt can be considered to be a summation of two separate securities: a regular debt, for which the firm pays interest and receives a principal loan; and a call option that the debt holders have sold to the firm allowing the firm to buy back the debt. In return, the firm pays an implied 'premium' to the debt holders who have agreed to buy the callable debt. Therefore the total payments by the company to callable debt holders is equal to the interest payments plus the premium payment for the option. An increase in asset volatility will increase the value of this option as it is likely that the assets will increase in value, and strengthen the company's credit, and lower its spread at which time it would like to repay the debt and refinance/roll over at the new lower rate. Therefore an increase in volatility will increase the 'premium' demanded by the callable debt holders, thereby increasing the total yield and lowering the value of the callable debt. Therefore Choice 'b' (An increase in the value of the callable debt of the firm) is incorrect.

An increase in asset volatility will decrease the value of the firm as it is now riskier than before (higher standard deviation, same expected returns). Therefore Choice 'a' (An increase in the value of the equity of the firm) is false too.

The value of the implicit put in the debt of the firm will increase and not decrease as the volatility of the underlying assets increases. Therefore Choice 'c' (A decrease in the value of the implicit put in in the debt of the firm) is incorrect too.

Choice 'd' (A decrease in the value of the non-callable debt issued by the firm) is correct because higher asset volatility will increase the riskiness of the company's debt, making the required yield higher and decreasing its value.

Investors in CDOs bear credit risk. The SPV is merely a conduit that owns the underlying assets on which the sponsoring institution has bought protection. The investors have sold them this protection, and are on the hook for defaults or other credit events. The reference entity is relevant only to CDSs, not CDOs. Choice 'b' is the correct answer.

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.

All the statements are correct, therefore Choice 'd' is the correct answer. Let us look at each of these statements one by one.

I. A deep in-the-money call option has a value very close to that of a forward contract with a forward price equal to the exercise price. This is true because a deep in the money call option is most likely to be exercised, and is therefore effectively like a forward contract to buy the stock at the exercise price.

We can also look at this using the BSM formula for a call option. If c be the value of a call option, and all other variables have their usual meaning (S0 is the spot price, K is exercise price, and t is time to expiry), then according to the Black Scholes model the value of a call is given by the following expression:

202.22.e1

As S0 becomes large, d1 and d2 become large, and therefore N(d1) and N(d2) approach 1, leaving the value of the call to be equal to 202.22.e2, which is the formula for a forward contract.

II. If the volatility of a stock goes down to zero, the value of a call option on the stock will tend to be close to that of a forward contract so long as the option is in the money.

Again, this is true because if volatility is low or zero, the stock price will grow at its expected rate, and end up to be what the forward price is (S0 ert). If the option is out of the money, the value of the option will tend to 0.

III. All other things remaining the same, the issue of stock warrants exercisable at a future date will cause a decline in the current stock price.

This is true because the stock warrants are likely to be exercised only when they are in the money, ie when their exercise price is less than the going stock price, and at that time it will dilute the value of the existing shares. However, the reduction in the price is priced into the share price at the time of the issue of the warrants, and it is not that the share price falls the day they are exercised.

IV. Implied volatilities are calculated from market prices of options and are forward looking. This statement is true: historical volatilities calculated from past prices are backward looking, while 'implied volatility' is the volatility implied from market prices, and is forward looking as it encapsulates the market's view of how volatile the future is likely to be.