Lecture 16 Options and option pricing Lecture 16 1 / 22
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
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
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
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
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
Options (3) The payoff at time T of a call option is given by max(s T K, 0). Lecture 16 5 / 22
Options (3) The payoff at time T of a call option is given by max(s T K, 0). The payoff at time T of a put option is given by max(k S T, 0). Lecture 16 5 / 22
Options (4) There are different types of options with respect to the exercise time. European options Lecture 16 6 / 22
Options (4) There are different types of options with respect to the exercise time. European options American 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 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
Options (5) Call and put options are examples of simple derivatives 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 S T 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 S T : X = F (S T ). 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 S T : X = F (S T ). There are also path-dependent derivatives: X = G((S t, 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
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
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(s T K, 0)1 ( S t 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(s T K, 0)1 ( S t L for some t [0, T ] ). There are also up-and-out, down-and-in and down-and-out contracts. Lecture 16 8 / 22
Option strategies (1) By combining basic options we can create new payoffs. 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. 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. 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. 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 K 1 and a call option with strike price K 2 > K 1. Lecture 16 9 / 22
Option strategies (2) Other examples of option combinations include A straddle: this is a strangle with K 1 = K 2 Lecture 16 10 / 22
Option strategies (2) Other examples of option combinations include A straddle: this is a strangle with K 1 = K 2 A bull spread: buy a call option with strike price K 1 and sell a call option with strike price K 2 > K 1. Lecture 16 10 / 22
Option strategies (2) Other examples of option combinations include A straddle: this is a strangle with K 1 = K 2 A bull spread: buy a call option with strike price K 1 and sell a call option with strike price K 2 > K 1. A bear spread: sell a put option with strike price K 1 and buy a put with strike price K 2 > K 1. Lecture 16 10 / 22
Option strategies (2) Other examples of option combinations include A straddle: this is a strangle with K 1 = K 2 A bull spread: buy a call option with strike price K 1 and sell a call option with strike price K 2 > K 1. A bear spread: sell a put option with strike price K 1 and buy a put with strike price K 2 > K 1. 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. Lecture 16 10 / 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 S T K. 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 This is true since S T K. max(s T K, 0) max(k S T, 0) = S T K. 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 This is true since S T K. max(s T K, 0) max(k S T, 0) = S T K. It follows from linear pricing that Price of call at 0 Price of put at 0 = S 0 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 This is true since S T K. max(s T K, 0) max(k S T, 0) = S T K. It follows from linear pricing that Price of call at 0 Price of put at 0 = S 0 K d(0, T ). This relation is called the put-call parity. Lecture 16 11 / 22
Pricing options In order the find the price (or value) of an option we need to construct a stochastic 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 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
Single-period option pricing (1) This is the binomial model with only one time period. 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). 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. 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. 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
Single-period option pricing (2) The riskless asset has price 1 today and price 1 + r f tomorrow. Lecture 16 14 / 22
Single-period option pricing (2) The riskless asset has price 1 today and price 1 + r f tomorrow. 1 1 + r f 1 + r f t = 0 t = 1 Lecture 16 14 / 22
Single-period option pricing (3) The risky asset has price S today and either us or ds tomorrow. Lecture 16 15 / 22
Single-period option pricing (3) The risky asset has price S today and either us or ds tomorrow. S us 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. S us ds t = 0 t = 1 We assume that d < u. Lecture 16 15 / 22
Single-period option pricing (4) In order to rule out arbitrage opportunities we must have d < 1 + r f < u Lecture 16 16 / 22
Single-period option pricing (4) In order to rule out arbitrage opportunities we must have d < 1 + r f < u Note the strict inequalities here. Lecture 16 16 / 22
Single-period option pricing (4) In order to rule out arbitrage opportunities we must have d < 1 + r f < u Note the strict inequalities here. The reason for this is that if 1 + r f 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 + r f < u Note the strict inequalities here. The reason for this is that if 1 + r f u, then we can create an arbitrage opprortunity by selling a stock and putting the money in the bank. If 1 + r f d, then borrowing money in the bank and buying stocks for it will create an arbitrage opprtunity Lecture 16 16 / 22
Single-period option pricing (5) Now let us try to find the price today of a derivative that has payoff C u in the up state and C d in the down state. Lecture 16 17 / 22
Single-period option pricing (5) Now let us try to find the price today of a derivative that has payoff C u in the up state and C d in the down state. To solve this problem we create a replicating portfolio. Lecture 16 17 / 22
Single-period option pricing (5) Now let us try to find the price today of a derivative that has payoff C u in the up state and C d 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 C u in the up state and C d 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 + r f ) In the down state: x ds + y (1 + r f ). Lecture 16 17 / 22
Single-period option pricing (6) To find the replicating portfolio, we solve the following linear system of equations: { xus + y(1 + rf ) = C u The solution is given by xds + y(1 + r f ) = C d. { x = C u C d y = (u d)s uc d dc u (1+r f )(u d) 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 ) = C u The solution is given by xds + y(1 + r f ) = C d. { x = C u C d y = (u d)s uc d dc u (1+r f )(u d) Hence, the value, or price, of the derivative with payoff (C u, C d ) 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 ) = C u The solution is given by xds + y(1 + r f ) = C d. { x = C u C d y = (u d)s uc d dc u (1+r f )(u d) Hence, the value, or price, of the derivative with payoff (C u, C d ) is given by V = x S + y = C u C d u d + uc d dc u (1 + r f )(u d) Lecture 16 18 / 22
Single-period option pricing (7) Note that There exists a unique value of every derivative 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. 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 q = 1 + r f d. u d It follows from the condition d < 1 + r f < u that q (0, 1). Lecture 16 19 / 22
Single-period option pricing (8) Using this q we can write V = C u C d u d + uc d dc u (1 + r f )(u d) Lecture 16 20 / 22
Single-period option pricing (8) Using this q we can write V = C u C d u d + uc d dc u (1 + r f )(u d) 1 = (1 + r f )C u (1 + r f )C d + uc d dc u 1 + r f u d Lecture 16 20 / 22
Single-period option pricing (8) Using this q we can write V = C u C d u d + uc d dc u (1 + r f )(u d) 1 = (1 + r f )C u (1 + r f )C d + uc d dc u 1 + r f u d [ 1 1 + rf d = C u + u (1 + r ] f ) C d 1 + r f u d u d Lecture 16 20 / 22
Single-period option pricing (8) Using this q we can write V = C u C d u d + uc d dc u (1 + r f )(u d) 1 = (1 + r f )C u (1 + r f )C d + uc d dc u 1 + r f u d [ 1 1 + rf d = C u + u (1 + r ] f ) C d 1 + r f u d u d 1 = (qc u + (1 q)c d ) 1 + r f Lecture 16 20 / 22
Single-period option pricing (8) Using this q we can write V = C u C d u d + uc d dc u (1 + r f )(u d) 1 = (1 + r f )C u (1 + r f )C d + uc d dc u 1 + r f u d [ 1 1 + rf d = C u + u (1 + r ] f ) C d 1 + r f u d u d 1 = (qc u + (1 q)c d ) 1 + r f 1 = Ê(C). 1 + r f Lecture 16 20 / 22
Single-period option pricing (8) Using this q we can write V = C u C d u d + uc d dc u (1 + r f )(u d) 1 = (1 + r f )C u (1 + r f )C d + uc d dc u 1 + r f u d [ 1 1 + rf d = C u + u (1 + r ] f ) C d 1 + r f u d u d 1 = (qc u + (1 q)c d ) 1 + r f 1 = Ê(C). 1 + r f Again, Ê means expected value with respect to the risk-neutral (q, 1 q)-probabilities. Lecture 16 20 / 22
Multiperiod models (1) Now add more time steps, and assume the standard lattice (=recombining tree) model with independent ups and downs. Lecture 16 21 / 22
Multiperiod models (1) Now add more time steps, and assume the standard lattice (=recombining tree) model with independent ups and downs. In this case we can think of the model as a lot af one-period models glued together. Lecture 16 21 / 22
Multiperiod models (2) Using risk-neutral probabilities the random variable S T can be written S T = S 0 u X d T X Lecture 16 22 / 22
Multiperiod models (2) Using risk-neutral probabilities the random variable S T can be written S T = S 0 u X d T X, where X Bin(T, q). Lecture 16 22 / 22
Multiperiod models (2) Using risk-neutral probabilities the random variable S T can be written where S T = S 0 u X d T X, X Bin(T, q). Hence, the value at time 0 of a European derivative with payoff function F and exercise time T is V = 1 (1 + r f ) T E Q [F (S T )] Lecture 16 22 / 22
Multiperiod models (2) Using risk-neutral probabilities the random variable S T can be written where S T = S 0 u X d T X, X Bin(T, q). Hence, the value at time 0 of a European derivative with payoff function F and exercise time T is V = = 1 (1 + r f ) T E Q [F (S T )] 1 (1 + r f ) T T F k=0 ( S 0 u k d T k) ( ) T q k (1 q) T k. k Lecture 16 22 / 22