Tutorials Advanced Financial Mathematics
Summer Term 2023
PD Dr. V. Paulsen Series 01 05.04.2023
The exercises of this series deepens the contents of the chapter Informal Introduction.
It is not necessary to hand-over some written solutions. The exercises will be discussed
orally in the tutorials.
Ex. 1: Bear-Spread
A Bear-spread denotes the following trading strategy. Buy a put with strike K2 and sell
a put with strike K1 < K2 . Hold this position until maturity T . Both put-options have
the same maturity and belongs to the same underlying.
1. Plot the corresponding profit-diagram.
2. What does a trader expect, who realises a Bear-Spread?
How can a Bear spread be realised with a portfolio of call options?
Ex. 2: Exchange Option
We consider a financial market with two stocks that have initial prices S1 (0), S2 (0) and
random terminate price S1 (T ), S2 (T ) at T . An Exchange-Option gives its holder the right
to exchange stock 1 with stock 2 at T . This means that the holder would give stock 1 and
receive stock 2 at T if he expires his option. We denote by E12 the initial price of this
option.
Analogously one can also consider the corresponding option, the right to exchange stock
2 with stock 1. We denote its initial price with E21 .
1. Which payoffs in T are caused by both Options?
2. Show the following identity
E12 + S1 (0) = E21 + S2 (0).
3. In which sense is this a generalisation of the put-call parity.
Ex 3: Properties of the put price
We suppose that the financial market is free of arbitrage and consider a put option with
maturity T , strike K on an underlying S. Show the following properties of the initial put
price which is denoted by p(S0 , T, K)
1. p(S0 , T, K) ≥ max{0, KB(0, T ) − S0 }
Summer Term 2023
PD Dr. V. Paulsen Series 01 05.04.2023
The exercises of this series deepens the contents of the chapter Informal Introduction.
It is not necessary to hand-over some written solutions. The exercises will be discussed
orally in the tutorials.
Ex. 1: Bear-Spread
A Bear-spread denotes the following trading strategy. Buy a put with strike K2 and sell
a put with strike K1 < K2 . Hold this position until maturity T . Both put-options have
the same maturity and belongs to the same underlying.
1. Plot the corresponding profit-diagram.
2. What does a trader expect, who realises a Bear-Spread?
How can a Bear spread be realised with a portfolio of call options?
Ex. 2: Exchange Option
We consider a financial market with two stocks that have initial prices S1 (0), S2 (0) and
random terminate price S1 (T ), S2 (T ) at T . An Exchange-Option gives its holder the right
to exchange stock 1 with stock 2 at T . This means that the holder would give stock 1 and
receive stock 2 at T if he expires his option. We denote by E12 the initial price of this
option.
Analogously one can also consider the corresponding option, the right to exchange stock
2 with stock 1. We denote its initial price with E21 .
1. Which payoffs in T are caused by both Options?
2. Show the following identity
E12 + S1 (0) = E21 + S2 (0).
3. In which sense is this a generalisation of the put-call parity.
Ex 3: Properties of the put price
We suppose that the financial market is free of arbitrage and consider a put option with
maturity T , strike K on an underlying S. Show the following properties of the initial put
price which is denoted by p(S0 , T, K)
1. p(S0 , T, K) ≥ max{0, KB(0, T ) − S0 }
