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Principles of option pricing
Option
A contract that gives the holder the right - not the obligation - to buy (call), or to sell (put) a specified amount of the underlying asset, at a set exchange rate and expiration
date.
Glossary of terms
The investor buying the option is called the buyer or holder.
The investor selling the option is called the writer or seller.
When the holder of the option decides to buy (sell) the asset at maturity, it is said that he/she is exercising the option.
The asset to be bought or sold is called the underlying asset.
Since the holder enjoys a privilege - the option to buy or sell - he/she must pay a premium to acquire the option.
The price agreed upon for buying or selling the underlying asset is called exercise price or strike price.
Glossary of terms (con’t)
Options are traded on options exchanges.
The number of outstanding option contracts at any time is called open interest.
American vs. European options
American options can be exercised at any time during their life span
European options can be exercised only at maturity
Option valuation basics
Like with any other financial asset, the option premium or market value or option price is a function of future expected cash flows.
Notation
C: price of an American call
c: price of an European call
P: price of an American put
p: price of an European put
E: exercise or strike price
S: stock price before maturity
ST: stock price at maturity
T: time to maturity
r: risk-free rate
Boundaries to option prices: Call options.
At expiration:C = max[0, (ST -E)]
Before expirationUpper bound:
A call cannot sell for more than the stock: C < S and c < S
Lower bound:
C > = max[0, (S -E)]
c > = max[0, (S - E/(1+r)T)]
What happens if this relationship is not
satisfied?
Boundaries to option prices: Arbitrage
Assume the Exxon December 26 call struck at $80 sells for $1. It is now December 17. The stock of Exxon is at $83/share. The risk-free rate is
6%.
If the option is American, buy the call for $1, exercise it and make $3
Arbitrage profit = $2
What if the option is European?
Construct an arbitrage portfolio
Arbitrage portfolio to exploit boundary violations for European call options
When Action Cash flow
Today,December 17
Short one share of Exxon
Buy one call
Lend $79.8835 at 6% for 9days
$83
-$1
-$79.8835
net CF = $2.116
Arbitrage portfolio to exploit boundary violations for European call options
When Action Cash flow
Today,
December 17
Short one share of Exxon
Buy one call
Lend $79.8835 at 6% for 9days
$83
-$1
-$79.8835
net CF = $2.116
Arbitrage portfolio to exploit boundary violations for European call options
When Action Cash flow
Today,December 17
Short one share of Exxon
Buy one call
Lend $79.8835 at 6% for 9days
$83
-$1
-$79.8835
net CF = $2.116
Arbitrage portfolio to exploit boundary violations for European call options
When Action Cash flow
Today,December 17
Short one share of Exxon
Buy one call
Lend $79.8835 at 6% for 9days
$83
-$1
-$79.8835
net CF = $2.116
Arbitrage portfolio to exploit boundary violations for European call options
When Action Cash flow
Today,
December 17
Short one share of Exxon
Buy one call
Lend $79.8835 at 6% for 9days
$83
-$1
-$79.8835
net CF = $2.116
Arbitrage portfolio to exploit boundary violations for European call options
When Action Cash flow
Today,December 17
Short one share of Exxon
Buy one call
Lend $79.8835 at 6% for 9days
$83
-$1
-$79.8835
net CF = $2.116
December 26
Collect $79.8835(1.06)0.025
Exercise your call and buyone Exxon share at $80
Return the Exxon share youborrowed
$80
-$80
0
net CF = 0
Arbitrage portfolio to exploit boundary violations for European call options
When Action Cash flow
Today,December 17
Short one share of Exxon
Buy one call
Lend $79.8835 at 6% for 9days
$83
-$1
-$79.8835
net CF = $2.116
December 26
Collect $79.8835(1.06)0.025
Exercise your call and buyone Exxon share at $80
Return the Exxon share youborrowed
$80
-$80
0
net CF = 0
Arbitrage portfolio to exploit boundary violations for European call options
When Action Cash flow
Today,December 17
Short one share of Exxon
Buy one call
Lend $79.8835 at 6% for 9days
$83
-$1
-$79.8835
net CF = $2.116
December 26
Collect $79.8835(1.06)0.025
Exercise your call and buyone Exxon share at $80
Return the Exxon share youborrowed
$80
-$80
0
net CF = 0
Arbitrage portfolio to exploit boundary violations for European call options
When Action Cash flow
Today,December 17
Short one share of Exxon
Buy one call
Lend $79.8835 at 6% for 9days
$83
-$1
-$79.8835
net CF = $2.116
December 26
Collect $79.8835(1.06)0.025
Exercise your call and buyone Exxon share at $80
Return the Exxon share youborrowed
$80
-$80
0
net CF = 0
Arbitrage portfolio to exploit boundary violations for European call options
When Action Cash flow
Today,December 17
Short one share of Exxon
Buy one call
Lend $79.8835 at 6% for 9days
$83
-$1
-$79.8835
net CF = $2.116
December 26
Collect $79.8835(1.06)0.025
Exercise your call and buyone Exxon share at $80
Return the Exxon share youborrowed
$80
-$80
0
net CF = 0
Analysis
We have created a riskless portfolio: the terminal cash flow is zero, regardless of the stock price, while the up-front cash flow is positive.
We made $2.116 in pure arbitrage profits.
Boundaries to option prices: Put options.
At expiration:P = max[0, (E -ST)]
Before expirationUpper bound:
A put cannot sell for more than the stock: P < S and p < S
Lower bound:
P > = max[0, (E - S)]
p > = max[0, (E/(1+r)T -S)]
Boundary violations
By now, we know that if price boundaries are violated, we might be able to construct an arbitrage portfolio.
Boundary violations: American put options
Assume the Exxon December 26 put struck at $80 sells for $2. It is now December 17. The stock of Exxon is at $75/share. The risk-free rate is 6%.
If the option is American, we can buy it for $2 and exercise it.
Arbitrage profit = $3.
Arbitrage portfolio to exploit boundary violations for European put options
When Action Cash flow
Today, December 17 Buy one share of Exxon
Buy one put
Borrow $79.8835 at 6% for 9days
-$75
-$2
$79.8835
net CF = $2.8835
Arbitrage portfolio to exploit boundary violations for European put options
When Action Cash flow
Today, December 17 Buy one share of Exxon
Buy one put
Borrow $79.8835 at 6% for 9days
-$75
-$2
$79.8835
net CF = $2.8835
Arbitrage portfolio to exploit boundary violations for European put options
When Action Cash flow
Today, December 17 Buy one share of Exxon
Buy one put
Borrow $79.8835 at 6% for 9days
-$75
-$2
$79.8835
net CF = $2.8835
Arbitrage portfolio to exploit boundary violations for European put options
When Action Cash flow
Today, December 17 Buy one share of Exxon
Buy one put
Borrow $79.8835 at 6% for 9days
-$75
-$2
$79.8835
net CF = $2.8835
Arbitrage portfolio to exploit boundary violations for European put options
When Action Cash flow
Today, December 17 Buy one share of Exxon
Buy one put
Borrow $79.8835 at 6% for 9days
-$75
-$2
$79.8835
net CF = $2.8835
Arbitrage portfolio to exploit boundary violations for European put options
When Action Cash flow
Today, December 17 Buy one share of Exxon
Buy one put
Borrow $79.8835 at 6% for 9days
-$75
-$2
$79.8835
net CF = $2.8835
December 26 Exercise your put and sellone Exxon share at $80
Pay back your loan$79.8835(1.06)0.025
$80
-$80
net CF = 0
Arbitrage portfolio to exploit boundary violations for European put options
When Action Cash flow
Today, December 17 Buy one share of Exxon
Buy one put
Borrow $79.8835 at 6% for 9days
-$75
-$2
$79.8835
net CF = $2.8835
December 26 Exercise your put and sellone Exxon share at $80
Pay back your loan$79.8835(1.06)0.025
$80
-$80
net CF = 0
Arbitrage portfolio to exploit boundary violations for European put options
When Action Cash flow
Today, December 17 Buy one share of Exxon
Buy one put
Borrow $79.8835 at 6% for 9days
-$75
-$2
$79.8835
net CF = $2.8835
December 26 Exercise your put and sellone Exxon share at $80
Pay back your loan$79.8835(1.06)0.025
$80
-$80
net CF = 0
Arbitrage portfolio to exploit boundary violations for European put options
When Action Cash flow
Today, December 17 Buy one share of Exxon
Buy one put
Borrow $79.8835 at 6% for 9days
-$75
-$2
$79.8835
net CF = $2.8835
December 26 Exercise your put and sellone Exxon share at $80
Pay back your loan$79.8835(1.06)0.025
$80
-$80
net CF = 0
Analysis
We have created a riskless portfolio: the terminal cash flow is zero, regardless of the stock price, while the up-front cash flow is positive.
We made $2.8835 in pure arbitrage profits.
Three important concepts
Intrinsic value - how much the call is worth if exercised.
Market value, price, or premium - the price at which the call can be sold/purchased in the market.
Time value - the difference between premium and intrinsic value
A Snapshot of the intrinsic value of the American Call: S - E
S
S - E
E
A Snapshot of the intrinsic value of the American Call: S - E
S
S - E
E
A Snapshot of the intrinsic value of the American Call: S - E
S
S - E
E
A Snapshot of the intrinsic value of the American Call: S - E
S
S - E
E
A Snapshot of the intrinsic value of the American Call: S - E
S
S - E
E
A Snapshot of the intrinsic value of the American Call: S - E
S
S - E
E
A Snapshot of the intrinsic value of the American Call: S - E
S
S - E
E
A Snapshot of the intrinsic value of the American Call: S - E
S
S - E
E
A Snapshot of the intrinsic value of the American Call: S - E
S
S - E
E
A Snapshot of the intrinsic value of the American Put: E - S
S
E - S
E
E
A Snapshot of the intrinsic value of the American Put: E - S
S
E - S
E
E
A Snapshot of the intrinsic value of the American Put: E - S
S
E - S
E
E
A Snapshot of the intrinsic value of the American Put: E - S
S
E - S
E
E
A Snapshot of the intrinsic value of the American Put: E - S
S
E - S
E
E
A Snapshot of the intrinsic value of the American Put: E - S
S
E - S
E
E
A Snapshot of the intrinsic value of the American Put: E - S
S
E - S
E
E
A Snapshot of the intrinsic value of the American Put: E - S
S
E - S
E
E
The market value of the American Call as expiration approaches
S
S - E
E
The market value of the American Call as expiration approaches
S
S - E
E
The market value of the American Call as expiration approaches
S
S - E
E
The market value of the American Call as expiration approaches
S
S - E
E
The market value of the American Call as expiration approaches
S
S - E
E
The market value of the American Call as expiration approaches
S
S - E
E
The market value of the American Call as expiration approaches
S
S - E
E
Remark
As expiration approaches, the market value of the option converges to its intrinsic value
At the same time, time value converges to zero
Relationship between several variables and the market value of options
Variable European call European put American call American put
Stockprice
+ - + -
Strikeprice
- + - +
Time toexpiration
? ? + +
Stockvolatility
+ + + +
Risk-freerate
+ - + -
Dividends - + - +
Relationship between several variables and the market value of options
Variable European call European put American call American put
Stockprice
+ - + -
Strikeprice
- + - +
Time toexpiration
? ? + +
Stockvolatility
+ + + +
Risk-freerate
+ - + -
Dividends - + - +
Relationship between several variables and the market value of options
Variable European call European put American call American put
Stock price + - + -
Strike price - + - +
Time toexpiration
? ? + +
Stockvolatility
+ + + +
Risk-freerate
+ - + -
Dividends - + - +
Relationship between several variables and the market value of options
Variable European call European put American call American put
Stock price + - + -
Strike price - + - +
Time toexpiration
? ? + +
Stockvolatility
+ + + +
Risk-freerate
+ - + -
Dividends - + - +
Relationship between several variables and the market value of options
Variable European call European put American call American put
Stockprice
+ - + -
Strikeprice
- + - +
Time toexpiration
? ? + +
Stockvolatility
+ + + +
Risk-freerate
+ - + -
Dividends - + - +
Relationship between several variables and the market value of options
Variable European call European put American call American put
Stockprice
+ - + -
Strikeprice
- + - +
Time toexpiration
? ? + +
Stockvolatility
+ + + +
Risk-freerate
+ - + -
Dividends - + - +
