Lecture 16 Options and Option Pricing

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Lecture 16 Options and option pricing

Lecture 16 1 / 22 In order to value them we need a stochastic model of the future behavior of the underlying.

Introduction

One of the most, perhaps the most, important family of derivatives are the options.

Lecture 16 2 / 22 Introduction

One of the most, perhaps the most, important family of derivatives are the options.

In order to value them we need a stochastic model of the future behavior of the underlying.

Lecture 16 2 / 22 The two most comman types of options are the call option and the put option.

Like futures, these type of derivatives are traded on many exchanges around the world.

Options (1)

An option is a derivative in which there is some sort of choice for the owner of the derivative.

Lecture 16 3 / 22 Like futures, these type of derivatives are traded on many exchanges around the world.

Options (1)

An option is a derivative in which there is some sort of choice for the owner of the derivative.

The two most comman types of options are the call option and the put option.

Lecture 16 3 / 22 Options (1)

An option is a derivative in which there is some sort of choice for the owner of the derivative.

The two most comman types of options are the call option and the put option.

Like futures, these type of derivatives are traded on many exchanges around the world.

Lecture 16 3 / 22 A put option is the right, but not the obligation, of the owner of the option to sell an asset at a future time T for an amount K.

Here K is known as the strike price and T as the exercise time or maturity time.

Options (2)

Deﬁnition A call option is the right, but not the obligation, of the owner of the option to buy an asset at a future time T for an amount K.

Lecture 16 4 / 22 Here K is known as the strike price and T as the exercise time or maturity time.

Options (2)

Deﬁnition A call option is the right, but not the obligation, of the owner of the option to buy an asset at a future time T for an amount K.

A put option is the right, but not the obligation, of the owner of the option to sell an asset at a future time T for an amount K.

Lecture 16 4 / 22 Options (2)

Deﬁnition A call option is the right, but not the obligation, of the owner of the option to buy an asset at a future time T for an amount K.

A put option is the right, but not the obligation, of the owner of the option to sell an asset at a future time T for an amount K.

Here K is known as the strike price and T as the exercise time or maturity time.

Lecture 16 4 / 22 The payoﬀ at time T of a put option is given by

max(K − ST , 0).

Options (3)

The payoﬀ at time T of a call option is given by

max(ST − K, 0).

Lecture 16 5 / 22 Options (3)

The payoﬀ at time T of a call option is given by

max(ST − K, 0).

The payoﬀ at time T of a put option is given by

max(K − ST , 0).

Lecture 16 5 / 22 American options Bermudan options Other: Parisian, Canadian, Russian,...

Options (4)

There are diﬀerent types of options with respect to the exercise time. European options

Lecture 16 6 / 22 Bermudan options Other: Parisian, Canadian, Russian,...

Options (4)

There are diﬀerent types of options with respect to the exercise time. European options American options

Lecture 16 6 / 22 Other: Parisian, Canadian, Russian,...

Options (4)

There are diﬀerent types of options with respect to the exercise time. European options American options Bermudan options

Lecture 16 6 / 22 Options (4)

There are diﬀerent types of options with respect to the exercise time. European options American options Bermudan options Other: Parisian, Canadian, Russian,...

Lecture 16 6 / 22 In this case the derivative’s payoﬀ X is a function F of the underlying ST :

X = F (ST ).

There are also path-dependent derivatives:

X = G((St , 0 ≤ t ≤ T )).

In this case the whole (or part) of path of the underlying is needed to know on order to calculate the payoﬀ X .

Options (5)

Call and put options are examples of simple derivatives

Lecture 16 7 / 22 :

X = F (ST ).

There are also path-dependent derivatives:

X = G((St , 0 ≤ t ≤ T )).

In this case the whole (or part) of path of the underlying is needed to know on order to calculate the payoﬀ X .

Options (5)

Call and put options are examples of simple derivatives

In this case the derivative’s payoﬀ X is a function F of the underlying ST

Lecture 16 7 / 22 There are also path-dependent derivatives:

X = G((St , 0 ≤ t ≤ T )).

In this case the whole (or part) of path of the underlying is needed to know on order to calculate the payoﬀ X .

Options (5)

Call and put options are examples of simple derivatives

In this case the derivative’s payoﬀ X is a function F of the underlying ST :

X = F (ST ).

Lecture 16 7 / 22 Options (5)

Call and put options are examples of simple derivatives

In this case the derivative’s payoﬀ X is a function F of the underlying ST :

X = F (ST ).

There are also path-dependent derivatives:

X = G((St , 0 ≤ t ≤ T )).

In this case the whole (or part) of path of the underlying is needed to know on order to calculate the payoﬀ X .

Lecture 16 7 / 22 An up-and-in call option has payoﬀ X = max(ST − K, 0)1 St ≥ L for some t ∈ [0, T ] .

There are also up-and-out, down-and-in and down-and-out contracts.

Options (6)

Example An up-and-in contract is a derivative in which the value of the underlying must reach a given level L in order to be active.

Lecture 16 8 / 22 There are also up-and-out, down-and-in and down-and-out contracts.

Options (6)

Example An up-and-in contract is a derivative in which the value of the underlying must reach a given level L in order to be active.

An up-and-in call option has payoﬀ X = max(ST − K, 0)1 St ≥ L for some t ∈ [0, T ] .

Lecture 16 8 / 22 Options (6)

Example An up-and-in contract is a derivative in which the value of the underlying must reach a given level L in order to be active.

An up-and-in call option has payoﬀ X = max(ST − K, 0)1 St ≥ L for some t ∈ [0, T ] .

There are also up-and-out, down-and-in and down-and-out contracts.

Lecture 16 8 / 22 This can be used in order to help an investor to get the payoﬀ he wants.

Assume that you belive that a stock is going to move from its current price, but you do not know in which direction.

If this is the case, you can buy a strangle. A strangle is a a sum of a put option with strike price K1 and a call option with strike price K2 > K1.

Option strategies (1)

By combining basic options we can create new payoﬀs.

Lecture 16 9 / 22 Assume that you belive that a stock is going to move from its current price, but you do not know in which direction.

If this is the case, you can buy a strangle. A strangle is a a sum of a put option with strike price K1 and a call option with strike price K2 > K1.

Option strategies (1)

By combining basic options we can create new payoﬀs.

This can be used in order to help an investor to get the payoﬀ he wants.

Lecture 16 9 / 22 If this is the case, you can buy a strangle. A strangle is a a sum of a put option with strike price K1 and a call option with strike price K2 > K1.

Option strategies (1)

By combining basic options we can create new payoﬀs.

This can be used in order to help an investor to get the payoﬀ he wants.

Assume that you belive that a stock is going to move from its current price, but you do not know in which direction.

Lecture 16 9 / 22 A strangle is a a sum of a put option with strike price K1 and a call option with strike price K2 > K1.

Option strategies (1)

By combining basic options we can create new payoﬀs.

This can be used in order to help an investor to get the payoﬀ he wants.

Assume that you belive that a stock is going to move from its current price, but you do not know in which direction.

If this is the case, you can buy a strangle.

Lecture 16 9 / 22 Option strategies (1)

By combining basic options we can create new payoﬀs.

This can be used in order to help an investor to get the payoﬀ he wants.

Assume that you belive that a stock is going to move from its current price, but you do not know in which direction.

If this is the case, you can buy a strangle. A strangle is a a sum of a put option with strike price K1 and a call option with strike price K2 > K1.

Lecture 16 9 / 22 A bull spread: buy a call option with strike price K1 and sell a call option with strike price K2 > K1.

A bear spread: sell a put option with strike price K1 and buy a put with strike price K2 > K1. A butterﬂy: buy 1 call with strike price K − a, sell 2 calls with strike price K and buy 1 call with strike price K + a.

Option strategies (2)

Other examples of option combinations include

A straddle: this is a strangle with K1 = K2

Lecture 16 10 / 22 A bear spread: sell a put option with strike price K1 and buy a put with strike price K2 > K1. A butterﬂy: buy 1 call with strike price K − a, sell 2 calls with strike price K and buy 1 call with strike price K + a.

Option strategies (2)

Other examples of option combinations include

A straddle: this is a strangle with K1 = K2

A bull spread: buy a call option with strike price K1 and sell a call option with strike price K2 > K1.

Lecture 16 10 / 22 A butterﬂy: buy 1 call with strike price K − a, sell 2 calls with strike price K and buy 1 call with strike price K + a.

Option strategies (2)

Other examples of option combinations include

A straddle: this is a strangle with K1 = K2

A bull spread: buy a call option with strike price K1 and sell a call option with strike price K2 > K1.

A bear spread: sell a put option with strike price K1 and buy a put with strike price K2 > K1.

Lecture 16 10 / 22 Option strategies (2)

Other examples of option combinations include

A straddle: this is a strangle with K1 = K2

A bull spread: buy a call option with strike price K1 and sell a call option with strike price K2 > K1.

Lecture 16 10 / 22 This is true since

max(ST − K, 0) − max(K − ST , 0) = ST − K.

It follows from linear pricing that

Price of call at 0 − Price of put at 0 = S0 − K · d(0, T ).

This relation is called the put-call parity.

The put-call parity

If we buy 1 call option and sell 1 put option both having the same strike price K and maturity time T , then the resulting payoﬀ at T will be

ST − K.

Lecture 16 11 / 22 It follows from linear pricing that

Price of call at 0 − Price of put at 0 = S0 − K · d(0, T ).

This relation is called the put-call parity.

The put-call parity

If we buy 1 call option and sell 1 put option both having the same strike price K and maturity time T , then the resulting payoﬀ at T will be

ST − K.

This is true since

max(ST − K, 0) − max(K − ST , 0) = ST − K.

Lecture 16 11 / 22 This relation is called the put-call parity.

The put-call parity

If we buy 1 call option and sell 1 put option both having the same strike price K and maturity time T , then the resulting payoﬀ at T will be

ST − K.

This is true since

max(ST − K, 0) − max(K − ST , 0) = ST − K.

It follows from linear pricing that

Price of call at 0 − Price of put at 0 = S0 − K · d(0, T ).

Lecture 16 11 / 22 The put-call parity

If we buy 1 call option and sell 1 put option both having the same strike price K and maturity time T , then the resulting payoﬀ at T will be

ST − K.

This is true since

max(ST − K, 0) − max(K − ST , 0) = ST − K.

It follows from linear pricing that

Price of call at 0 − Price of put at 0 = S0 − K · d(0, T ).

This relation is called the put-call parity.

Lecture 16 11 / 22 Two popular models are The binomial model The Black-Scholes(-Samuelson) model

Pricing options

In order the ﬁnd the price (or value) of an option we need to construct a stochastic model.

Lecture 16 12 / 22 The Black-Scholes(-Samuelson) model

Pricing options

In order the ﬁnd the price (or value) of an option we need to construct a stochastic model.

Two popular models are The binomial model

Lecture 16 12 / 22 Pricing options

In order the ﬁnd the price (or value) of an option we need to construct a stochastic model.

Two popular models are The binomial model The Black-Scholes(-Samuelson) model

Lecture 16 12 / 22 The setup is as follows. Two times: today (t = 0) and tomorrow (t = 1). Two outcomes: ‘up’ and ‘down’. The probability of up and down is p and 1 − p respectively. Two assets: one riskless and one risky.

Single-period option pricing (1)

This is the binomial model with only one time period.

Lecture 16 13 / 22 Two outcomes: ‘up’ and ‘down’. The probability of up and down is p and 1 − p respectively. Two assets: one riskless and one risky.

Single-period option pricing (1)

This is the binomial model with only one time period.

The setup is as follows. Two times: today (t = 0) and tomorrow (t = 1).

Lecture 16 13 / 22 The probability of up and down is p and 1 − p respectively. Two assets: one riskless and one risky.

Single-period option pricing (1)

This is the binomial model with only one time period.

The setup is as follows. Two times: today (t = 0) and tomorrow (t = 1). Two outcomes: ‘up’ and ‘down’.

Lecture 16 13 / 22 Two assets: one riskless and one risky.

Single-period option pricing (1)

This is the binomial model with only one time period.

The setup is as follows. Two times: today (t = 0) and tomorrow (t = 1). Two outcomes: ‘up’ and ‘down’. The probability of up and down is p and 1 − p respectively.

Lecture 16 13 / 22 Single-period option pricing (1)

This is the binomial model with only one time period.

The setup is as follows. Two times: today (t = 0) and tomorrow (t = 1). Two outcomes: ‘up’ and ‘down’. The probability of up and down is p and 1 − p respectively. Two assets: one riskless and one risky.

Lecture 16 13 / 22 1 + rf % 1 & 1 + rf

t = 0 t = 1

Single-period option pricing (2)

The riskless asset has price 1 today and price 1 + rf tomorrow.

Lecture 16 14 / 22 Single-period option pricing (2)

The riskless asset has price 1 today and price 1 + rf tomorrow.

1 + rf % 1 & 1 + rf

t = 0 t = 1

Lecture 16 14 / 22 uS % S & dS

t = 0 t = 1

We assume that d < u.

Single-period option pricing (3)

The risky asset has price S today and either uS or dS tomorrow.

Lecture 16 15 / 22 We assume that d < u.

Single-period option pricing (3)

The risky asset has price S today and either uS or dS tomorrow.

uS % S & dS

t = 0 t = 1

Lecture 16 15 / 22 Single-period option pricing (3)

The risky asset has price S today and either uS or dS tomorrow.

uS % S & dS

t = 0 t = 1

We assume that d < u.

Lecture 16 15 / 22 Note the strict inequalities here.

The reason for this is that if 1 + rf ≥ u, then we can create an arbitrage opprortunity by selling a stock and putting the money in the bank.

If 1 + rf ≤ d, then borrowing money in the bank and buying stocks for it will create an arbitrage opprtunity

Single-period option pricing (4)

In order to rule out arbitrage opportunities we must have

d < 1 + rf < u

Lecture 16 16 / 22 The reason for this is that if 1 + rf ≥ u, then we can create an arbitrage opprortunity by selling a stock and putting the money in the bank.

If 1 + rf ≤ d, then borrowing money in the bank and buying stocks for it will create an arbitrage opprtunity

Single-period option pricing (4)

In order to rule out arbitrage opportunities we must have

d < 1 + rf < u

Note the strict inequalities here.

Lecture 16 16 / 22 If 1 + rf ≤ d, then borrowing money in the bank and buying stocks for it will create an arbitrage opprtunity

Single-period option pricing (4)

In order to rule out arbitrage opportunities we must have

d < 1 + rf < u

Note the strict inequalities here.

The reason for this is that if 1 + rf ≥ u, then we can create an arbitrage opprortunity by selling a stock and putting the money in the bank.

Lecture 16 16 / 22 Single-period option pricing (4)

In order to rule out arbitrage opportunities we must have

d < 1 + rf < u

Note the strict inequalities here.

The reason for this is that if 1 + rf ≥ u, then we can create an arbitrage opprortunity by selling a stock and putting the money in the bank.

If 1 + rf ≤ d, then borrowing money in the bank and buying stocks for it will create an arbitrage opprtunity

Lecture 16 16 / 22 To solve this problem we create a replicating portfolio.

Assume that we can construct a portfolio of x number of stocks and y units of money in the bank.

This portfolio will have the following payoﬀ:

In the up state: x · uS + y · (1 + rf )

In the down state: x · dS + y · (1 + rf ).

Single-period option pricing (5)

Now let us try to ﬁnd the price today of a derivative that has payoﬀ

Cu in the up state and Cd in the down state.

Lecture 16 17 / 22 Assume that we can construct a portfolio of x number of stocks and y units of money in the bank.

This portfolio will have the following payoﬀ:

In the up state: x · uS + y · (1 + rf )

In the down state: x · dS + y · (1 + rf ).

Single-period option pricing (5)

Now let us try to ﬁnd the price today of a derivative that has payoﬀ

Cu in the up state and Cd in the down state.

To solve this problem we create a replicating portfolio.

Lecture 16 17 / 22 This portfolio will have the following payoﬀ:

In the up state: x · uS + y · (1 + rf )

In the down state: x · dS + y · (1 + rf ).

Single-period option pricing (5)

Now let us try to ﬁnd the price today of a derivative that has payoﬀ

Cu in the up state and Cd in the down state.

To solve this problem we create a replicating portfolio.

Assume that we can construct a portfolio of x number of stocks and y units of money in the bank.

Lecture 16 17 / 22 Single-period option pricing (5)

Now let us try to ﬁnd the price today of a derivative that has payoﬀ

Cu in the up state and Cd in the down state.

To solve this problem we create a replicating portfolio.

Assume that we can construct a portfolio of x number of stocks and y units of money in the bank.

This portfolio will have the following payoﬀ:

In the up state: x · uS + y · (1 + rf )

In the down state: x · dS + y · (1 + rf ).

Lecture 16 17 / 22 Hence, the value, or price, of the derivative with payoﬀ (Cu, Cd ) is given by C − C uC − dC V = x · S + y = u d + d u u − d (1 + rf )(u − d)

Single-period option pricing (6)

To ﬁnd the replicating portfolio, we solve the following linear system of equations: xuS + y(1 + rf ) = Cu xdS + y(1 + rf ) = Cd . The solution is given by

( Cu−Cd x = (u−d)S y = uCd −dCu (1+rf )(u−d)

Lecture 16 18 / 22 C − C uC − dC = u d + d u u − d (1 + rf )(u − d)

Single-period option pricing (6)

To ﬁnd the replicating portfolio, we solve the following linear system of equations: xuS + y(1 + rf ) = Cu xdS + y(1 + rf ) = Cd . The solution is given by

( Cu−Cd x = (u−d)S y = uCd −dCu (1+rf )(u−d)

Hence, the value, or price, of the derivative with payoﬀ (Cu, Cd ) is given by

V = x · S + y

Lecture 16 18 / 22 Single-period option pricing (6)

To ﬁnd the replicating portfolio, we solve the following linear system of equations: xuS + y(1 + rf ) = Cu xdS + y(1 + rf ) = Cd . The solution is given by

( Cu−Cd x = (u−d)S y = uCd −dCu (1+rf )(u−d)

Hence, the value, or price, of the derivative with payoﬀ (Cu, Cd ) is given by C − C uC − dC V = x · S + y = u d + d u u − d (1 + rf )(u − d)

Lecture 16 18 / 22 The probabilities of up and down moves does not enter the valuation formula.

Let 1 + r − d q = f . u − d

It follows from the condition d < 1 + rf < u that q ∈ (0, 1).

Single-period option pricing (7)

Note that

There exists a unique value of every derivative

Lecture 16 19 / 22 Let 1 + r − d q = f . u − d

It follows from the condition d < 1 + rf < u that q ∈ (0, 1).

Single-period option pricing (7)

Note that

There exists a unique value of every derivative The probabilities of up and down moves does not enter the valuation formula.

Lecture 16 19 / 22 Single-period option pricing (7)

Note that

There exists a unique value of every derivative The probabilities of up and down moves does not enter the valuation formula.

Let 1 + r − d q = f . u − d

It follows from the condition d < 1 + rf < u that q ∈ (0, 1).

Lecture 16 19 / 22 1 (1 + r )C − (1 + r )C + uC − dC = · f u f d d u 1 + rf u − d 1 1 + rf − d u − (1 + rf ) = Cu + Cd 1 + rf u − d u − d 1 = (qCu + (1 − q)Cd ) 1 + rf 1 = Eˆ(C). 1 + rf

Again, Eˆ means expected value with respect to the risk-neutral (q, 1 − q)-probabilities.