FN 913
Fall 2022
HW 3
Due Date: Nov 4
1. Please read Investopedia’s lesson on the box spread:
https:
www.investopedia.com/terms
oxspread.asp
and watch the following Youtube video:
https:
www.youtube.com/watch?v=3zW86yXg7RM&t=209s
(a) Compare Investopedia’s definition with the definition we gave in class. Show they
are the same.
(b) Consider the “Box Spread Example” on the Investopedia webpage. For each
stock price value at expiration listed below, state which options get exercised fo
or against the investor, what his payoff is for each of the options, and what his
total profit is (ignoring commissions):
i. $40
ii. $52
iii. $60
(c) Next read the article “1RONYMAN.doc” posted on Blackboard. Verify that his
trades netted him $287,500 up front and that he would owe $250,000 at expiration
if the options were European.
(d) As noted in the article, the options were American and not European. What was
the potential (and ultimately realized) risk? How did this affect his “guaranteed”
profit? Do a ‘back of the envelope’ calculation to estimate the underlying asset
price when he made the trades–assume the risk-free rate is zero and use Put-Call
parity to back out S. Is it surprising the calls he sold were exercised?
2. Consider a European put option on a stock whose cu
ent price is $52 and has volatility
30%. The put expires in two months and has strike price $55. The risk-free rate is 5%.
(a) Find the price of the put using a two-step binomial tree.
(b) Find the price of the put using the Black-Scholes formula.
(c) Repeat part (a) to find the price if the put option were American.
1
Topic 1: Introduction
Topic 2:
Options
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Options Basics
The 2 basic types of options:
A call option gives the holder the right (but not the obligation) to buy an asset by a certain date for a certain price.
A put option gives the holder the right (but not the obligation) to sell an asset by a certain date for a certain price.
Furthermore, options can be either:
European– can be exercised only on the expiration date
American– can be exercised at any time up to the expiration date
Most options traded on exchanges are American, but European options are easier to analyze.
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Options Basics
What is the payoff to the holder of a European call option with maturity date T and strike price K=80?
If , won’t exercise and payoff = 0
, will exercise and payoff =
Don’t confuse the payoff with the profit. The profit to the holder will be less than the payoff, because of the original cost of the option (the premium), which is paid up front.
Note that this was i
elevant for futures/forwards, because those don’t cost anything
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Options Basics
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Blue = payoff
Red = profit
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Options Basics
What if you had sold, or written, this call option?
Now you have the short side; you don’t own the option of whether or not to exercise—that belongs to the holder.
So your payoff and profit are exactly the opposite (i.e., the negative) of the payoff and profit of the holder.
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Options Basics
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Black = payoff
Red = profit
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Options Basics
What is the payoff to the holder of a European put option with maturity date T and strike price K=80?
If , won’t exercise and payoff = 0
, will exercise and payoff =
So now the payoff and profit will be a line with slope -1 if ST is less than K, and will have slope 0 otherwise.
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Options Basics
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Blue = payoff
Red = profit
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Options Basics
Again, if you had the short put position then you don’t own the option of whether or not to exercise-- that belongs to the long position.
So your payoff and profit are exactly the opposite of those of the long position.
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Options Basics
REMARK: Technically, we should be accounting for the time value of money when we consider the profit diagram.
The option premium is paid up front
The payoff from exercising is paid at the exercise date
But it’s standard to ignore this issue when discussing the profit diagrams and we will follow convention.
Actually, most options have relatively short expiration dates anyway, so it doesn’t matter too much in practice.
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Options Basics
Terminology:
Intrinsic value – The payoff to an option if it were exercised immediately.
Call intrinsic value at time t:
Put intrinsic value at time t :
Out of the money: when an option has a negative intrinsic value.
At the money: when an option has exactly a zero intrinsic value.
In the money: when an option has a positive intrinsic value.
Time value: the difference between the market value of an option and its intrinsic value.
Market Value = Intrinsic value + Time value
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Options Basics
These notions of intrinsic value and time value are useful for thinking about the value of options.
Only exercise an American option early if its European counterpart has negative time value
So even if option is in the money, it may be more beneficial to wait rather than exercise immediately.
In other words, need to also consider the time value.
Whenever the time value is positive, it is more valuable to hold on to the option rather than exercise today.
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Options Properties
We now begin to examine options in an analytic manner.
First we discuss the general properties that options have, and the assumptions and notations that we will consistently use throughout the rest of our options discussions.
The laws and general inequalities we derive here are model independent; they hold no matter what assumptions we make about the underlying price process.
To get exact values for option prices we need stronger assumptions
For example, the Black-Scholes formula gives an analytic, precise value for options prices BUT it makes very specific (and possibly unrealistic) assumptions about how the price of the underlying asset evolves over time.
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No transactions costs
No shorting constraints
All profits are subject to same tax rate.
Bo
owing and lending at the risk-free rate is possible.
All “ca
y costs” are known in advance.
Ie, the risk-free rate, storage costs (if any), payouts, etc over the relevant time period are all known in advance.
For each of the conclusions we draw, you should think about how to work out the effect of relaxing each of these assumptions.
Assumptions
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Notation
K - strike (exercise) price
T - maturity or expiration date
t - cu
ent date, usually 0
S - cu
ent stock price
- risk free rate.
σ - volatility of stock price
C - value of American call
c - value of European call
P - value of American put
p - value of European put
Notation
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Fundamental Pricing Principals
First Principal: Law of One Price
If two securities have the same payoffs (in each state of the world), then they must have the same price.
Second Principal (slight extension of Law of One Price):
If the payoff of Security 1 is greater than or equal to the payoff of Security 2, then the cu
ent price of Security 1 is greater than or equal to the cu
ent price of Security 2.
Third Pricing Principal:
The value of an option is always the value if it were exercised under the optimal strategy.
So if there were an exercise strategy that always results in at least P dollars today, then the value of the option would be at least P.
But be careful! The strategy must be ex-ante and not ex-post; ie, the strategy must account for all possible outcomes.
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Basic A
itrage Relations
All of the following relations were proved by Robert Merton in 1973, and basically all follow from our three Pricing Principals.
(1) Options never have a negative value
Obvious, yes? Note that this follows from the Pricing Principals: since the payoff is always greater than or equal to zero, the price must be greater than or equal to zero.
(2) An American option is at least as valuable as the co
esponding European one.
For the simple reason that you could always treat an American option like a European one.
When is an American option equal in value to its European counterpart?
When and only when it is the case that it is never optimal to exercise the American option early!
Thus the price of an American option equals the price of its European counterpart only if it is not optimal to early exercise
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Basic A
itrage Relations
(3) Calls with a lower strike are more valuable. Puts with a higher strike are more valuable.
The payoffs are greater and therefore must be more valuable.
(4) Options are more valuable the more uncertainty there is in the future underlying spot price. I.e., calls and puts on stocks with higher volatility are more valuable (all else held fixed).
This is because calls and puts have higher payoffs when the spot price winds up farther away from the strike price; on the other hand their payoffs are always non-negative.
The payoffs are not symmetric. The downside is capped by zero.
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Basic A
itrage Relations
(5) American options with longer maturities are at least as valuable as those with shorter maturities (all else equal).
The main reason is the longer maturity option gives the holder all the privileges and more than the shorter maturity option.
Does this hold for European options?
Not necessarily: the holder of the longer option does NOT have the same privileges because the option can’t be exercised early.
For example, consider two European calls on a dividend paying stock that have different maturities but all else equal. If a dividend is expected after the shorter maturity call expires but before the longer maturity call expires, then because the stock price will drop after the dividend is paid, which tends to decrease the call value, the longer maturity call value may be less than the shorter maturity call value.
As another example, consider two European puts that have different maturities but all else equal. The present value of the strike price is lower for the longer maturity put, and this might offset the gain in value due to higher uncertainty.
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Basic A
itrage Relations
(6) A call is never worth more than the stock and a put is never worth