options - introduction to option valuation
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Introduction
1.
Welcome to the Knowledge Check.
If you have prior knowledge of Options
Introduction to Option Valuation, try the
Knowledge Check. A perfect score is no
guarantee that you know everything covered
in the tutorial, but a less than perfect score
will help you identify any knowledge gaps.
If the subject of this tutorial is new to you,the Knowledge Check will indicate the level of
the information that you're about to
encounter. You may think you don't know
much about this area, but you might surprise
yourself!
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Question 1 of 5
A call option on widgets expires tomorrow. Ithas a strike price of EUR 100. The current
market price of widgets is EUR 90.
Assume that interest rates are zero, widgets
are non-income producing, and there is no
holding cost.
Which of the following statements is true?
The option is in the money.
The option is at the money.
The option is out of the
money.
3.
Question 2 of 5
An agent owns a USD 110 call option on
widget futures. The price of the widget
futures contract is USD 100. The option costs
USD 3.
What is the time value of this option?
Zero
USD 10
-USD
10
USD 3
4.
Question 3 of 5
All other things remaining equal, which of
the following would increase the value of a
put option on a futures contract?
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A decrease in the maturity of the option
An increase in the underlying price of the futures
contract
An increase in the volatility of the futures contract
Question 4 of 5
An American-style call option has a strike
price of EUR 50, and three months to expiry.
Interest rates for the period are 10%, and
volatility is 5%. The current price of the
underlying asset is EUR 75.
Should the option be exercised early?
Yes
No
6.
Question 5 of 5
Which of the following is the correct put-call
parity equation?
7.
OBJECTIVES
On completion of this tutorial, you will be
able to:
explain when an option is 'in' or 'out'of the money
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show how an option price is brokeninto two components: intrinsic value
and time value
describe the major influences onoption values
outline the upper and lowerboundaries of option prices
and explain the factors affecting the
exercise decision
describe the 'put-call' parityrelationship
Prerequisite Knowledge
Prior to studying this tutorial, you should
have simple familiarity with the discounting
of future values, and a good understanding of
the concepts outlined in the following
tutorial:
OptionsAn Introduction
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Forward Prices
How do forward prices affect 'moneyness'?
Consider a one-year European call option
struck on widgets at USD 105. The underlying
price is currently USD 100, while the forward
price is USD 110. The 'moneyness' of theoption can be expressed relative to the spot
and the forward price.
Relative to the spot
price
Relative to the
forward price
This option is 'out of
the money' (OTM),
because it would not
make sense to
exercise at USD 105when the market
price was only USD
100. However, this
option is not
exercisable now; it is
exercisable in one
years time.
This option is 'in the
money' (ITM)it
would make sense to
buy widgets at USD
105 if the price wasactually USD 110.
Different markets have different conventions;equity markets tend to relate option strike
prices to spot prices, while interest rate
markets tend to focus on forward prices. In
the foreign exchange market, American-style
options usually relate the strike to the spot FX
rate, while European-style options usually
relate the strike to the forward FX rate.
Rather than attempting to use different
practices in different markets, it is best to be
precise; rather than referring to an at-the-
money option, it would be better to use the
terms 'at-the-money forward' or 'at-the-
money spot'.
Note that for options on futures, the debate
is irrelevant; futures prices are already
forward prices so 'moneyness is a simple
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comparison between strike price and market
price. We will sometimes take advantage of
this by using imaginary futures contracts in
our example.
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Option Moneyness
Option moneyness is summarized in the
following table:
13.
Moneyness Example 1
A call option is written on a futures contract.
It has a three-month maturity and a strike
price of EUR 100. Three-month interest rates
are 4%. The current futures price is EUR 100.
What is the option moneyness?
In the money
At the money
Out of the
money
14.
Moneyness Example 2
A put option is written on an underlying
asset. It has a one-year maturity and a strike
price of EUR 100. The asset generates no
income, and has no ownership or storagecosts. One-year interest rates are 5%. The
current asset price is EUR 100.
Which of the following statements is true?
The option is in the money
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forward.
The option is at the money
forward.
The option is out of the money
forward.
Components of Option Value 16.
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Intrinsic Value
A 3-month American call option on a
hypothetical futures contract has a strike
price of EUR 100. The current price of a
futures contract is EUR 90. The option costs
EUR 5.
Assuming that interest rates are zero, what is
the intrinsic value of this option?
EUR 5
-EUR
10
Zero
-EUR 5
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Factors Affecting Option Value An
Overview
It is comparatively easy to calculate the
intrinsic value of an option, but much harder
to estimate the option's time value. This is
because time value includes the value of'optionality' (considered either as the
additional premium required by an option
seller to compensate for the risk, or the
amount a buyer is willing to pay for the
possibility of future payoffs).
So how are fair prices for options obtained?
Precise calculations may involve the use of a
sophisticated pricing model. There are many
such models, all of which will have to take
into account the following:
Interest rates
Relationship between the strike price & the
asset price
Maturity
Volatility of the underlying asset
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Relationship Between theStrike Price & the
Asset Price
23.
Maturity
An increase in the maturity of an option will affect the forward prices of the underlying; these will
generally rise as a function of interest rates and other cash flows associated with the assets. Thiswill have the effect on prices noted previously. However, a greater effect will generally be the
fact that a longer maturity allows prices to move more. Consequently, the possible future payoffs
from an option will increase and the option value will rise. The graph below shows the effect of
time to maturity on put and call prices (both are at the money with a strike price of USD 5 and
volatility of 10%).
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The value of both options will increase as prices have 'more time' to move and hence generate a
return to the option holder. However, because forward prices will increase over time, the value
of the put rises less steeply than that of the call.
Volatility of the Underlying Asset
The amount of fluctuation of an asset price is known as its volatility the higher the volatility, the
more a vanilla option is worth.
To see this, assume two underlying assets, A and B, have an underlying price of USD 100.However, the price of B changes by more, on average, than that of A. Would you rather own a
USD 100 call option on asset A, or one on asset B? The potential losses on both options are the
samethe price might fall below USD 100, and the option would be worthless. However, the
potential payoffs are greater for asset B than for asset A. Therefore, the call option on asset B will
be higher.
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Effect of a Rise in Volatility
Returning to our simple example (3-month call optionstrike price USD 5) we can show theeffect of an increase of volatility, using our particular model, in this graph:
A rise in volatility leads to an increase in the value of our sample call option. Note that a change
in volatility has no effect on intrinsic value; it only influences the potential 'optionality'. Just as
time value is at its maximum for an option that is at the money, so too any price increases due to
higher volatility are at their maximum for ATM options.
If volatility is zero, then we simply have a payoff diagram; there is no 'optionality' because the
price does not change.
It is also worth noting that the increase in volatility would give additional value to this option
even when the underlying market is some way from the strike price. For instance, if the
underlying market is trading at USD 4.60, then a USD 5 call is almost worthless if priced using 10%
volatility; however, at 30% volatility, this option is worth around USD 0.13.
26.
Intrinsic Value
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A 3-month call option on a fictional futures
contract is struck at GBP 50. The contract is
trading at a price of GBP 60. The price of the
option is GBP 11. Interest rates are zero.
What is the intrinsic value?
GBP 11
GBP 10
Zero
GBP 1
28.
Option Price Limits & Exercise Decisions
Upper Boundary for the Price of a Call Option
We can quickly establish simple maximum and
income-bearing assets. For a call option, whet
never be worth more than the current asset p
If the price of the call option was greater than
option, buy the stock, and invest the differenc
deliver the stock and be left with the invested
they would be left with the invested cash, plu
riskless profit would be obtained.
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Lower Boundary for the Price of a Call Option
(American)
An American-style call option can never sell for less than
its intrinsic value, otherwise an option could simply be
bought, exercised immediately, and the asset that had
been bought through the option would then be sold into
the market.
For example, if a USD 100 call option cost USD 7 when
the underlying asset had a market price of USD 110, then
by purchasing the call, exercising it immediately, and
then selling the asset, a riskless profit could be obtained:
Risk-less profit = (USD 110 - USD 100) - USD 7
= USD 3
The option price will naturally disallow such
'easy' money.
Furthermore, we know that no option can have a
negative value; at worst an option holder can simply walk
away. The relationship is thus:
31.
Lower Boundary for the Price of a Call Option (European)
For a European-style call option the relationship needs to be adjusted since the
forward price is not equal to the spot price. To illustrate the difference, imagine
two portfolios:
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At expiry time T, the share is trading at a price AT. Consider portfolio 1:
If K < AT, then the option will be exercised at the price K, using the money whichhad been previously invested. The share you have purchased is worth AT.
If K > AT, then the option expires worthless, and the portfolio is simply worth themoney invested, which will generate the amount K. So at time T Portfolio 1
is worth the greater of ATand K.
Portfolio 2 will always be worth the future share price, AT; the value of Portfolio 1
is always higher than or equal to the value of Portfolio 2.
Using the terminology above, we get:
Options cannot have a negative value; consequently the lower bound must be:
Lower Boundary for the Price of a Put Option
American options are straightforward, as once again the value can never be less than intrinsic
value.
To illustrate the lower bound for a European option we will once again examine a pair of
imaginary portfolios.
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If, at time T, AT < K, then the option would be exercised. The underlying share would be
delivered and an amount K would be received. The portfolio would be worth K.
If, at time T, AT > K, then the option would expire worthless and the portfolio would be worth the
value of the stock, that is, AT.
At expiry, Portfolio 1 is worth the greater of ATand K. Portfolio 2 is only ever worth K. The value
of Portfolio 1 is always higher than or equal to the value of Portfolio 2;
Options cannot have a negative value; consequently the lower bound must be:
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