Mathematical Finance II (6CCM338A) Jan-Mar 2026
Homework Problems 6CCM338A- Week 2
On the Keats page “Tutorial 2: Suggestion” section, please indicate two questions from below which
you would like to be covered in the next tutorial (please do this before 31th Jan).
Problem 1. An agent with a bounded and twice differentiable utility function U is risk-seeking if
and only if the agent is willing to accept a fair game, if and only if Wc ≥ E[W ] for every random
variable W representing terminal wealth.
Example solution. Since U is a utility function, U is strictly increasing.
The first if and only if: Assume U is the utility for a risk-seeking investor. This is equivalent
to (since U is twice differentiable) U ′ > 0 and U ′′ ≥ 0. So for any p ∈ [0, 1], W0 + h0 , W0 + h1 ∈ R
we get (following the notation from class):
U (p(W0 + h0 ) + (1 − p)(W0 + h1 )) ≤ pU ((W0 + h0 )) + (1 − p)U (W0 + h1 ).
Now, if we further assume ph0 + (1 − p)h1 = 0, i.e. the game where one starts with initial wealth
W0 and at the end of period left with W = W0 + h0 with probability p and W = W0 + h1 with
probability 1 − p is a fair game, then the above inequality reduces to:
⇐⇒ U (W0 ) ≤ pU ((W0 + h0 )) + (1 − p)U (W0 + h1 )
⇐⇒ U (W0 ) ≤ EU (W )
⇐⇒ EU (W0 ) ≤ EU (W )
⇐⇒ The agent is willing to accept any such fair game.
The second if and only if: Assume this investor always accepts a fair game. Using the same
notation from the class note, we can write
E(U (W0 )) ≤ E(U (W ))
⇐⇒ U (E(W0 )) ≤ E(U (W ))
⇐⇒ U (E(W )) ≤ U (Wc )
⇐⇒ E(W ) ≤ Wc .
The last line follows from the strictly increasing and bounded condition of U .
Problem 2. Consider the following utility functions (defined over wealth W ):
−1
1. U (W ) = W
2. U (W ) = ln W
3. U (W ) = −W −γ + W
4. U (W ) = −e−γW
Wγ
5. U (W ) = γ
, 6. U (W ) = αW − βW 2 .
a. Show that 1-5 are indeed a utility function for a risk-averse person. For (6), what are the
conditions on α, β that make this a utility function?
b. Compute the absolute and relative risk aversion coefficients.
c. Classify the functions as increasing/decreasing risk aversion utility functions (absolute and
relative) as their wealth changes.
Example solution. .
Problem 3. Consider the coin-flipping game we discussed in class. You toss a coin, if Head occurs,
you win £2. If tail occurs, you flip the coin again. In this second flip if you get Head you win £4,
if you get tail you flip the coin again. Every additional time you flip the coin, your winning bet is
doubled than last time.
1. Calculate the expected payoff of this game. (We briefly discussed this in class)
2. If you follow the following utility function, U (w) = log2 (w). Show that you are a risk-averse
person.
3. Calculate the expected utility and the certainty equivalence of this game.
Example solution. 1. The expected payoff of the game is given as follows.
1 1 1 1
E( Payoff ) = 2 + 4 + 8 + 16 + · · ·
2 4 8 16
= 1 + 1 + 1 + 1 + ··· = ∞
Homework Problems 6CCM338A- Week 2
On the Keats page “Tutorial 2: Suggestion” section, please indicate two questions from below which
you would like to be covered in the next tutorial (please do this before 31th Jan).
Problem 1. An agent with a bounded and twice differentiable utility function U is risk-seeking if
and only if the agent is willing to accept a fair game, if and only if Wc ≥ E[W ] for every random
variable W representing terminal wealth.
Example solution. Since U is a utility function, U is strictly increasing.
The first if and only if: Assume U is the utility for a risk-seeking investor. This is equivalent
to (since U is twice differentiable) U ′ > 0 and U ′′ ≥ 0. So for any p ∈ [0, 1], W0 + h0 , W0 + h1 ∈ R
we get (following the notation from class):
U (p(W0 + h0 ) + (1 − p)(W0 + h1 )) ≤ pU ((W0 + h0 )) + (1 − p)U (W0 + h1 ).
Now, if we further assume ph0 + (1 − p)h1 = 0, i.e. the game where one starts with initial wealth
W0 and at the end of period left with W = W0 + h0 with probability p and W = W0 + h1 with
probability 1 − p is a fair game, then the above inequality reduces to:
⇐⇒ U (W0 ) ≤ pU ((W0 + h0 )) + (1 − p)U (W0 + h1 )
⇐⇒ U (W0 ) ≤ EU (W )
⇐⇒ EU (W0 ) ≤ EU (W )
⇐⇒ The agent is willing to accept any such fair game.
The second if and only if: Assume this investor always accepts a fair game. Using the same
notation from the class note, we can write
E(U (W0 )) ≤ E(U (W ))
⇐⇒ U (E(W0 )) ≤ E(U (W ))
⇐⇒ U (E(W )) ≤ U (Wc )
⇐⇒ E(W ) ≤ Wc .
The last line follows from the strictly increasing and bounded condition of U .
Problem 2. Consider the following utility functions (defined over wealth W ):
−1
1. U (W ) = W
2. U (W ) = ln W
3. U (W ) = −W −γ + W
4. U (W ) = −e−γW
Wγ
5. U (W ) = γ
, 6. U (W ) = αW − βW 2 .
a. Show that 1-5 are indeed a utility function for a risk-averse person. For (6), what are the
conditions on α, β that make this a utility function?
b. Compute the absolute and relative risk aversion coefficients.
c. Classify the functions as increasing/decreasing risk aversion utility functions (absolute and
relative) as their wealth changes.
Example solution. .
Problem 3. Consider the coin-flipping game we discussed in class. You toss a coin, if Head occurs,
you win £2. If tail occurs, you flip the coin again. In this second flip if you get Head you win £4,
if you get tail you flip the coin again. Every additional time you flip the coin, your winning bet is
doubled than last time.
1. Calculate the expected payoff of this game. (We briefly discussed this in class)
2. If you follow the following utility function, U (w) = log2 (w). Show that you are a risk-averse
person.
3. Calculate the expected utility and the certainty equivalence of this game.
Example solution. 1. The expected payoff of the game is given as follows.
1 1 1 1
E( Payoff ) = 2 + 4 + 8 + 16 + · · ·
2 4 8 16
= 1 + 1 + 1 + 1 + ··· = ∞
