Derivatives (ECONM3017)

Lecture Seven: Options II

(Option Properties)

Nick Taylor
nick.taylor@bristol.ac.uk

University of Bristol

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Table of contents

1 Learning Outcomes
2 General Information
3 Factors Affecting Option Prices
4 Option Bounds
5 The Put-Call Relationship
6 Early Exercise
7 The Effects of Dividends
8 Trading Strategies
9 Summary
10 Reading

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Learning Outcomes

At the end of this lecture you will be able to:

1 Understand the determinants of option prices.
2 Prove the existence of lower and upper bounds on option prices (premia),
and a relationship between call and put option premia.
3 Demonstrate familiarity with a variety of option trading strategies.

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General Information

Key Assumptions:
There are no transaction costs.
All trading profits (net of trading losses) are subject to the same tax
rate.
Borrowing and lending are possible at the risk-free interest rate.
Arbitrage opportunities are taken immediately (and thus removed
immediately). Equivalently, no arbitrage opportunities occur.

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General Information (cont.)

Key Notation:
Current stock price: S.
Strike price of option: K .
Time to expiration of option: T .
Stock price on expiration date: ST .
Continuously compounded risk-free interest rate: r .
Value of American call option to buy one share: C .
Value of American put option to buy one share: P.
Value of European call option to buy one share: c.
Value of European put option to buy one share: p.

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Factors Affecting Option Prices

The Six Factors:

1 Current stock price, S.
2 Strike price, K .
3 Time to expiration, T .
4 Stock price volatility, σ.
5 Risk-free interest rate, r .
6 Expected dividends, D.

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Factors Affecting Option Prices (cont.)

Affect Directions:
European European American American
Variable Call Put Call Put
Current stock price + − + −
Strike price − + − +
Time to expiration ? ? + +
Volatility + + + +
Risk-free rate + − + −
Expected dividends − + − +

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Option Bounds

All Bounds
Upper Bounds (Calls): c ≤ S, C ≤ S.
Upper Bound (Am. Put): P ≤ K .
Upper Bound (Euro. Put): p ≤ Ke −rT .
Lower Bound (Euro. Call): c ≥ max(S − Ke −rT , 0).
Lower Bound (Euro. Put): p ≥ max(Ke −rT − S, 0).

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Option Bounds (cont.)

Proofs
Proof that p ≤ Ke −rT :
Terminal Value
Action Initial Value ST ≤ K ST > K
Write Put p −(K − ST ) 0
Lend −Ke −rT K K
Total p − Ke −rT ST K

As the terminal value of the

investment is non-negative, the initial
investment profit must be non-positive.

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Option Bounds (cont.)

Proofs (cont.)
Proof that c ≤ S:
Terminal Value
Action Initial Value ST ≤ K ST > K
Write Call c 0 −(ST − K )
Buy Stock −S ST ST
Total c −S ST K

As the terminal value of the

investment is non-negative, the initial
investment profit must be non-positive.

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Option Bounds (cont.)

Proofs (cont.)
Proof that c ≥ max(S − Ke −rT , 0):
Terminal Value
Action Initial Value ST ≤ K ST > K
Buy Call −c 0 ST − K
Sell Stock S −ST −ST
Lend −Ke −rT K K
Total S − Ke −rT − c −ST + K 0

As the terminal value of the

investment is non-negative, the initial
investment profit must be non-positive.

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Option Bounds (cont.)

Proofs (cont.)
Proof that p ≥ max(Ke −rT − S, 0):
Terminal Value
Action Initial Value ST ≤ K ST > K
Buy Put −p K − ST 0
Buy Stock −S ST ST
Borrow Ke −rT −K −K
Total Ke −rT − S − p 0 ST − K

As the terminal value of the

investment is non-negative, the initial
investment profit must be non-positive.

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Option Bounds (cont.)

Examples

Example (1)
Consider a European call option on a non-dividend paying stock. The stock
price is $51, the strike price is $50, the time to maturity is 6 months, and the
risk-free rate of interest is 12% per annum. In this case, S = 51, K = 50,
T = 0.5, r = 0.12, and the lower bound for the option price is

S − Ke −rT = 51 − 50e −0.12×0.5 = $3.91.

What is the upper bound for the price of this call option?

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Option Bounds (cont.)

Examples (cont.)

Example (2)
Consider a European put option on a non-dividend paying stock. The stock
price is $38, the strike price is $40, the time to maturity is 3 months, and the
risk-free rate of interest is 10% per annum. In this case, S = 38, K = 40,
T = 0.25, r = 0.10, and the lower bound for the option price is

Ke −rT − S = 40e −0.1×0.25 − 38 = $1.01.

What is the upper bound for the price of this put option?

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The Put-Call Relationship

Put-Call Parity
For non-dividend paying stock we have: c + Ke −rT = p + S.
Proof:
Terminal Value
Action Initial Value ST ≤ K ST > K
Write Call c 0 −(ST − K )
Buy Put −p K − ST 0
Buy Stock −S ST ST
Borrow Ke −rT −K −K
Total c + Ke −rT − p − S 0 0

As the terminal value of the

investment is zero, the initial
investment profit must also be zero.

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The Put-Call Relationship (cont.)

Examples
Example
Suppose that the stock price is $31, the strike price is $30, the risk-free interest
rate is 10% per annum, and the price of a 3-month European call option on the
stock is $3. In this instance, the theoretical price of a put option (on the same
stock, with the same strike price) will be

p = c + Ke −rT − S = 3 + 30e −0.1×0.25 − 31 = $1.26.

Example (cont.)
Continuing the above example, what arbitrage profits are available if the put
option (on the same stock, with the same strike price) has a market value of $1?
This would imply that the market price of the put option is too low, and that

c + Ke −rT > p + S,

which in turn implies that an arbitrageur should ‘sell’ the LHS and ‘buy’ the
RHS.

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The Put-Call Relationship (cont.)

Examples (cont.)

Example (cont.)
Continuing the above example, the arbitrageur should take the following actions:
Terminal Value
Action Initial Value ST ≤ 30 ST > 30
Write Call 3 0 −(ST − 30)
Buy Put −1 30 − ST 0
Buy Stock −31 ST ST
Borrow 29.26 −30 −30
Total 0.26 0 0
Thus yielding a risk-free profit of $0.26 (per option traded).

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Early Exercise

Calls on Non-dividend Paying Stock

It is never optimal to exercise an American call option on a
non-dividend paying stock before expiration.
Puts on Non-dividend Paying Stock
It can be optimal to exercise an American put option on a non-dividend
paying stock before expiration.
Indeed, it should be exercised early if it is deep in the money.

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The Effects of Dividends

Lower Bound for Calls

It can be shown that the lower bound for a European call is
c ≥ max(S − Ke −rT − D, 0), where D is the present value of expected
dividends.
Proof:
Terminal Value
Action Initial Value ST ≤ K ST > K
Buy Call −c 0 ST − K
Sell Stock S −ST − De rT −ST − De rT
Lend −Ke −rT − D K + De rT K + De rT
Total S − Ke −rT − D − c −ST + K 0

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The Effects of Dividends (cont.)

Lower Bound for Puts

It can be shown that p ≥ max(Ke −rT + D − S, 0).
Proof:
Terminal Value
Action Initial Value ST ≤ K ST > K
Buy Put −p K − ST 0
Buy Stock −S ST + De rT ST + De rT
Borrow Ke −rT + D −K − De rT −K − De rT
Total Ke −rT + D − S − p 0 ST − K

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The Effects of Dividends (cont.)

Put-Call Parity
It can be shown that c + Ke −rT + D = p + S.
Proof:
Terminal Value
Action Initial Value ST ≤ K ST > K
Write Call c 0 −(ST − K )
Buy Put −p K − ST 0
Buy Stock −S ST + De rT ST + De rT
Borrow Ke −rT +D −K − De rT −K − De rT
Total c + Ke −rT +D −p−S 0 0

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The Effects of Dividends (cont.)

Early Exercise
It may be optimal to exercise an American call option on a dividend
paying stock early.
If it is optimal, the exercise will occur immediately prior to an
ex-dividend date.

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Trading Strategies

Basic Strategies
Writing a Covered Call:
Long stock plus short call.
Protective Put Strategy:
Long stock plus long put.

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Trading Strategies (cont.)

Spreads (positions in two or more options of the same class)

Bull Spreads:
Buying a call option with a low strike price, and selling a call option
with a high strike price.
Bear Spreads:
Buying a call option with a high strike price, and selling a call option
with a low strike price.
Which stock price direction does the
investor expect in each case?

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Trading Strategies (cont.)

Spreads (cont.)
Box Spreads:
Buying a bull call spread and a bear put spread with the same strike
prices.
Butterfly Spreads:
Buying one call option with a low strike price, buying one call option
with a high strike price, and selling two call options with a strike price
between the low and high strike prices. In this instance, the investor is
hoping that the stock price remains stable (low volatility).

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Trading Strategies (cont.)

Spreads (cont.)
Calender Spreads:
Selling a call option with a certain strike price, and buying a call option
with the same strike price but with a longer maturity. Payoff is similar
to that observed with a butterfly spread.
Diagonal Spreads:
Taking positions in options with different expirations dates and strike
prices. This leads to a wide variety of profit profiles.

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Trading Strategies (cont.)

Combinations (positions in both calls and puts with the same

underlying asset)
Straddle:
Buying a call and put option with the same strike price and expiration
date. This strategy (‘bottom straddle’) is useful when the investor
expects large price movements.
Strangle (aka bottom vertical combination):
Buying a call and put option with the same expiration date but with
different strike prices (the call has the higher strike price). This
strategy is similar to a straddle except that the stock price has to move
further to yield a profit.

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Trading Strategies (cont.)

Combinations (cont.)
Strips:
Buying one call option and two put options with the same strike prices
and expiration dates. The investor is hoping that there is a large stock
price change and considers a decrease more likely than an increase.
Straps:
Buying two call options and one put option with the same strike prices
and expiration dates. The investor is hoping that there is a large stock
price change and considers an increase more likely than a decrease.

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Summary

Determining Factors
Current stock price, strike price, maturity, volatility, interest rate, and
dividends.
Option Bounds and Parity Conditions
Proofs of lower and upper bounds, and put-call parity.
Trading Strategies
Spreads and combinations.

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Reading

Essential Reading
Chapters 11 and 12, Hull (2015).
Further Reading
Chaput, J., and L. Ederington, 2003, Option spread and combination
trading, Journal of Derivatives 10, 70-88.

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