Pass the PRMIA PRM Certification 8013 Questions and answers with ExamsMirror

A residential mortgage carries a prepayment option for the borrower, where in the event of a decrease in interest rates they are likely to repay the mortgage and refinance at the new lower rate. If rates go up, the bank does not have a similar option. If the volatility of the underlying interest rates increases, mortgage holders are more likely to be faced with new rates that are sufficiently lower to persuade them to refinance. This reduces the ability of the bank to continue to earn the higher interest rate it was earning before.

Therefore an increase in volatility of the mortgage interest rates decreases the value of the mortgage portfolio.

The same rationale applies to callable bonds, where the borrower has the option to call the bond.

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.

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.

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 relationship between the value of an option, and its delta, gamma and theta is given by rV = θ + rSΔ + 0.5Γ(σS)2, where V is the value of the option, r the risk-free rate, S the spot price of the underlying, and θ, Δ & Γ are the respective Greeks.

For a delta neutral portfolio, Δ = 0 and this equation reduces to rV = θ + 0.5Γ(σS)2. Now rV is generally a small number, which means that if θ is large and positive, Γ must be large and negative to offset that. Therefore Choice 'b' is the correct answer.

For a deep in the money option, the option value changes in line with the value of the underlying because at this point the option is effectively behaving like the underlying. Therefore its delta is the same as the delta of the underlying, ie 1. In the same way, its gamma will be the same as the gamma of the underlying, which is 0. Therefore Choice 'b' is the correct answer.

In the US, the Federal Reserve sets a limit (eg, 2 x, so that only 50% of the money could be borrowed) as a multiple of capital posted to prevent excessive leverage or gearing in the system.

The other choices are incorrect.

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.

Contingent premium options are options where no premiums are due upfront, and are due only at expiry and only if the option finishes in the money. (If this sounds too good to be true, remember that at expiry it is possible that the premiums due exceed the payoffs from the option. So there is no free lunch.) For this reason they are also called pay-later options.

They are simple combinations of a vanilla option and the 'right' number of cash-or-nothing options. The buyer of a contingent premium option has effectively purchased a vanilla option and sold enough number of cash-or-nothing options so that the net premium due at inception is zero.

Therefore statement I, II and III are correct and Choice 'b' is the correct answer.

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.