PRMIA 8006 - Exam I: Finance Theory Financial Instruments Financial Markets - 2015 Edition

Gamma risk is unique to options. Therefore gamma risk can only be hedged using options, and Choice 'a', ie an option position that offsets any existing gamma in the portfolio, is the only way to hedge gamma.

Positions in stock or bank deposits do not carry gamma and therefore cannot be used to hedge gamma risk.

The value of Bond 1 is $900,000 and the value of Bond 2 is $1,100,000, or their respective weights in the portfolio are 45% and 55% respectively. The combined duration is the weighted average of their individual durations, ie (45% x 5) + (55% x 10) = 7.75 years

According to ISDA, a credit event is an event linked to the deteriorating credit worthiness of an underlying reference entity in a credit derivative. The occurrence of a credit event usually triggers full or partial termination of the transaction and a payment from protection seller to protection buyer. Credit events include

- bankruptcy,

- failure to pay,

- restructuring,

- obligation acceleration,

- obligation default and

- repudiation/moratorium.

Therefore all four events listed are credit events and Choice 'b' is the correct answer.

Explanation:

The BSM formula assumes that prices are lognormally distributed. Therefore Choice 'b' does not represent a valid assumption underlying Black Scholes. All other choices represent valid assumptions.

Short calls and short puts have negative gamma, ie their delta increases at a decreasing rate as the price of the underlying change. Likewise, remember that long calls and long puts have positive gamma.

The minimum variance hedge ratio answers the question of how much of the hedge to buy to hedge a given position. It minimizes the combined volatility of the primary and the hedge position. The minimum variance hedge ratio is given by the expression 103.10.b103.10.b. Effectively, this is the same as the beta of the primary position with respect to the hedge.

All other choices are incorrect.

Intuitive understanding of the hedge ratio: There are two standard deviations in play here: one, the standard deviation of the asset to be hedged (σx), and two, the standard deviation of the asset being used as the hedge(σy).  Now assume an extreme case where both the asset to be hedged and the asset being used as the hedge have a correlation of 1.  (That keeps the calculation simple.).  Also assume that the std dev of the asset to be hedged is 10% annually, and that of the asset being used as a hedge is say 20%.  At this point, just ignore all formulae, and think intuitively about how much of the hedge should we buy to have the same risk (in the opposite direction) as the primary position but in the opposite direction so the variation in the primary position is cancelled by opposite movements in the hedge position.  Obviously, we need to buy half because the asset being used as a hedge is twice as volatile.  In other words, we divided the volatility (std dev) of the primary position by the volatility of the asset being used as a hedge.In this question, σx is the standard deviation of the asset to be hedged, and σy is the standard deviation of the asset being used to hedge.  We will need to divide σx by σy.

Tailing the hedge refers to adjusting the number of futures contracts required to hedge a position given that futures contracts result in daily cash flows from daily P&L. In the case of forward contracts, P&L is not settled daily but at the end of the contract. Because margin payments (ie the daily P&L) for futures contracts create interest cost or income (depending upon the direction of the flows), the number of contracts required to hedge a given position can fluctuate. This effect is generally small for small positions, or where the time period under consideration is not very long. But it can get large for larger positions that are open for longer time periods, therefore requiring an adjustment to 'tail the hedge'. The adjustment is straightforward - the number of contracts required needs to be divided by (1+r)t, where r is the interest and t the time period (or by ert if continuously compounded rates are being used. Therefore statement I is true and statement II is false.

Statement III is not correct as counterparty risk is certainly an issue for forward contracts, while futures contracts are guaranteed by the exchange, besides having margin requirements and daily P&L movement in place.

Statement IV is not correct as futures come in standardized contracts with standard amounts and a few standard expiry dates to pick, whereas forward contracts are customized and dates, amounts and other details of the contract can be negotiated.

Delta hedges need to be rebalanced periodically because with changes in prices, delta changes, and the amount of hedge required changes as well, requiring a readjustment. The more frequent the rebalancing, the better the performance of the hedge. However, the greater the rebalancing frequency, the greater are the number of trades required, increasing the costs from trading. The trader has to balance transaction costs against the adverse effects from poorer hedge performance. Therefore rebalancing frequency is a consideration for a trader and statement I is true.

Statement II is not true. When options are granted, their future impact on the price gets priced in into the current price, and no change occurs when these are actually exercised.

Statement III 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, a large negative theta means a large positive gamma and vice-versa. Therefore theta can be used as a proxy for gamma and this statement is correct.

Statement IV is correct. Vega is highest closest to the strike price - look at the graph for vega to see how vega behaves with changes to spot. (Do it yourself or there is a graph in the tutorial too.)

The profit potential from the conversion of convertible bonds into stock is limited by the issuer's option to call the security at short notice. Generally, the convertible debt security is convertible into a certain number of shares, and the debt holder will generally not convert the security to shares unless there is a profit to be made. The 'premium' is irrelevant, because as long as the premium exists, the debt holder has no incentive to convert, as he would be better off buying the shares in the market. It is only when share prices go beyond a level that it becomes advantageous convert the security into shares. However, the prospect of granting cheap shares to the debt holders is not too appealing to the issuer, and as soon as the share price goes beyond a point where the value of the shares exceeds the face value of the debt the issuer has an incentive to exercise its option to call the security.

Therefore the profit potential from the conversion of convertible bonds into shares is limited by the issuer's option to call the security, and Choice 'a' is the correct answer. The 'premium', or interest rates, or volatility are irrelevant.

An easy way to analyze the impact on convexity is to think about the impact on duration. If the change causes duration to increase, it would cause the convexity to increase too, and if it causes the duration to decrease, it would reduce convexity as well.An increase in yields reduces duration, and therefore reduces convexity.

An increase in maturity increases duration, and therefore increases convexity.

An increase in the coupon rate reduces duration, and therefore reduces convexity.

An increase in duration increases convexity.

Therefore statements II and IV represent situations where convexity increases.