# PRMIA 8002 practice test

## Mathematical Foundations of Risk Measurement :II Exam

##### Question 1

In a 2-step binomial tree, at each step the underlying price can move up by a factor of u = 1.1 or
down by a factor of d = 1/u. The continuously compounded risk free interest rate over each time
step is 1% and there are no dividends paid on the underlying. Use the Cox, Ross, Rubinstein
parameterization to find the risk neutral probability and hence find the value of a European put
option with strike 102, given that the underlying price is currently 100.

• A. 5.19
• B. 5.66
• C. 6.31
• D. 4.18

C

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##### Question 2

A 2-step binomial tree is used to value an American put option with strike 105, given that the
underlying price is currently 100. At each step the underlying price can move up by 10 or down by 10
and the risk-neutral probability of an up move is 0.6. There are no dividends paid on the underlying
and the continuously compounded risk free interest rate over each time step is 1%. What is the value
of the option in this model?

• A. 7.12
• B. 6.59
• C. 7.44
• D. 7.29

A

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##### Question 3

In a binomial tree lattice, at each step the underlying price can move up by a factor of u = 1.1 or
down by a factor of . The continuously compounded risk free interest rate over each time step is 1%
and there are no dividends paid on the underlying. The risk neutral probability for an up move is:

• A. 0.5290
• B. 0.5292
• C. 0.5286
• D. 0.5288

D

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##### Question 4

A 2-step binomial tree is used to value an American put option with strike 104, given that the
underlying price is currently 100. At each step the underlying price can move up by 20% or down by
20% and the risk-neutral probability of an up move is 0.55. There are no dividends paid on the
underlying and the discretely compounded risk free interest rate over each time step is 2%. What is
the value of the option in this model?

• A. 11.82
• B. 12.33
• C. 12.49
• D. 12.78

C

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##### Question 5

Variance reduction is:

• A. A technique that is applied in regression models to improve the accuracy of the coefficient estimates
• B. A numerical method for finding portfolio weights to minimize the variance of a portfolio that has a given expected return
• C. A numerical method for finding the variance of the underlying that is implicit in a market price of an option
• D. A method for reducing the number of simulations required in a Monte Carlo simulation

D

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##### Question 6

The gradient of a function f(x, y, z) = x + y2 - x y z at the point x = y = z = 1 is

• A. (0, 2, 1)
• B. (0, 0, 0)
• C. (1, 1, 1)
• D. (0, 1, -1)

D

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##### Question 7

The gradient of a smooth function is

• A. a vector that shows the direction of fastest change of a function
• B. matrix of second partial derivatives of a function
• C. infinite at a maximum point
• D. a matrix containing the function's second partial derivatives

A

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##### Question 8

The Newton-Raphson method

• A. is based on finding a middle point between left and right end of the search interval
• B. is based on Taylor series and uses the first derivative
• C. can be used for continuous but not differentiable functions
• D. does provide an error bound along with every iteration

B

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##### Question 9

What is a Hessian?

• A. Correlation matrix of market indices
• B. The vector of partial derivatives of a contingent claim
• C. A matrix of second derivatives of a function
• D. The point at which a minimum of a multidimensional function is achieved

C

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##### Question 10

The bisection method can be used for solving f(x)=0 for a unique solution of x, when

• A. The function f(x) is continuous and monotonic
• B. The function f(x) is differentiable
• C. The function f(x) is differentiable and we have an explicit expression for the derivative
• D. The function f(x) is continuous