A Course in
Financial Calculus
ANSWERS TO EXERCISES
Alison Etheridge
University of Oxford
1
,1 Single period models
01.
Each of the strategies you mentioned reflects a different view about the market
based on the expected price movement of the underlying asset. Let's break down
the market views associated with each strategy:
(a) Bullish Vertical Spread:
In a bullish vertical spread, an investor expects the price of the underlying asset to
rise moderately. By buying one European call option and selling another call
option with a higher strike price (both with the same expiration date), the
investor aims to profit from a moderate increase in the underlying asset's price.
This strategy is suitable when the investor believes the market will trend upward,
but not drastically.
(b) Bearish Vertical Spread:
A bearish vertical spread is employed when an investor anticipates a moderate
decline in the underlying asset's price. By purchasing one European call option
and simultaneously selling another call option with a lower strike price (both with
the same expiration date), the investor aims to benefit from a moderate decrease
in the underlying asset's price. This strategy is used when the investor expects the
market to trend downward moderately.
(c) Strip:
The strip strategy involves buying one European call option and two European put
options with the same exercise date and strike price. This strategy is typically
employed when an investor expects a significant downward movement in the
underlying asset's price. The investor is essentially more bearish and is preparing
for a substantial decrease in the asset's value. The two puts provide higher profit
potential in case of a sharp decline.
2
,(d) Strap:
The strap strategy, on the other hand, involves buying two European call options
and one European put option with the same exercise date and strike price. This
strategy is used when an investor is highly bullish and expects a substantial
upward movement in the underlying asset's price. The two calls provide increased
profit potential in the event of a significant price increase.
(e) Strangle:
A strangle strategy involves buying a European call option and a European put
option with the same expiry date but different strike prices. This strategy is
employed when an investor anticipates a significant price movement in the
underlying asset but is uncertain about the direction (up or down). There are
three possible cases for a strangle:
- Long Call, Long Put: The investor expects a large price movement in either
direction and aims to profit from the volatility.
- Long Call, Short Put: The investor expects a moderate price increase but also
wants to profit if the price drops significantly.
- Short Call, Long Put: The investor expects a moderate price decrease but also
wants to profit if the price increases significantly.
Each of these cases reflects a different assessment of potential price movements
and volatility.
Remember that options trading can be complex and involves risks. It's important
to have a solid understanding of the underlying market dynamics and the
potential outcomes of each strategy before implementing them.
3
, 02.
A butterfly spread is indeed a complementary strategy to a straddle. It involves
using three options with the same expiration date but different strike prices to
create a specific payoff pattern. The payoff of a butterfly spread can be
represented as follows:
Payoff = Max(0, S - E1) - 2 * Max(0, S - E2) + Max(0, S - E3)
Where:
- S is the price of the underlying asset at expiry.
- E1 is the lower strike price.
- E2 is the middle strike price.
- E3 is the higher strike price.
To construct a portfolio with the same payoff using European calls and puts, you
can use the following combination:
1. Buy one European call option with strike price E1.
2. Sell two European call options with strike price E2.
3. Buy one European call option with strike price E3.
This portfolio replicates the payoff of a butterfly spread. Let's break down how
the portfolio's payoff matches the butterfly spread's payoff:
1. Buy one European call with strike E1:
Payoff = Max(0, S - E1) if S > E1, otherwise 0.
4
Financial Calculus
ANSWERS TO EXERCISES
Alison Etheridge
University of Oxford
1
,1 Single period models
01.
Each of the strategies you mentioned reflects a different view about the market
based on the expected price movement of the underlying asset. Let's break down
the market views associated with each strategy:
(a) Bullish Vertical Spread:
In a bullish vertical spread, an investor expects the price of the underlying asset to
rise moderately. By buying one European call option and selling another call
option with a higher strike price (both with the same expiration date), the
investor aims to profit from a moderate increase in the underlying asset's price.
This strategy is suitable when the investor believes the market will trend upward,
but not drastically.
(b) Bearish Vertical Spread:
A bearish vertical spread is employed when an investor anticipates a moderate
decline in the underlying asset's price. By purchasing one European call option
and simultaneously selling another call option with a lower strike price (both with
the same expiration date), the investor aims to benefit from a moderate decrease
in the underlying asset's price. This strategy is used when the investor expects the
market to trend downward moderately.
(c) Strip:
The strip strategy involves buying one European call option and two European put
options with the same exercise date and strike price. This strategy is typically
employed when an investor expects a significant downward movement in the
underlying asset's price. The investor is essentially more bearish and is preparing
for a substantial decrease in the asset's value. The two puts provide higher profit
potential in case of a sharp decline.
2
,(d) Strap:
The strap strategy, on the other hand, involves buying two European call options
and one European put option with the same exercise date and strike price. This
strategy is used when an investor is highly bullish and expects a substantial
upward movement in the underlying asset's price. The two calls provide increased
profit potential in the event of a significant price increase.
(e) Strangle:
A strangle strategy involves buying a European call option and a European put
option with the same expiry date but different strike prices. This strategy is
employed when an investor anticipates a significant price movement in the
underlying asset but is uncertain about the direction (up or down). There are
three possible cases for a strangle:
- Long Call, Long Put: The investor expects a large price movement in either
direction and aims to profit from the volatility.
- Long Call, Short Put: The investor expects a moderate price increase but also
wants to profit if the price drops significantly.
- Short Call, Long Put: The investor expects a moderate price decrease but also
wants to profit if the price increases significantly.
Each of these cases reflects a different assessment of potential price movements
and volatility.
Remember that options trading can be complex and involves risks. It's important
to have a solid understanding of the underlying market dynamics and the
potential outcomes of each strategy before implementing them.
3
, 02.
A butterfly spread is indeed a complementary strategy to a straddle. It involves
using three options with the same expiration date but different strike prices to
create a specific payoff pattern. The payoff of a butterfly spread can be
represented as follows:
Payoff = Max(0, S - E1) - 2 * Max(0, S - E2) + Max(0, S - E3)
Where:
- S is the price of the underlying asset at expiry.
- E1 is the lower strike price.
- E2 is the middle strike price.
- E3 is the higher strike price.
To construct a portfolio with the same payoff using European calls and puts, you
can use the following combination:
1. Buy one European call option with strike price E1.
2. Sell two European call options with strike price E2.
3. Buy one European call option with strike price E3.
This portfolio replicates the payoff of a butterfly spread. Let's break down how
the portfolio's payoff matches the butterfly spread's payoff:
1. Buy one European call with strike E1:
Payoff = Max(0, S - E1) if S > E1, otherwise 0.
4
