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Put/Call Parity and Binomial Model (Mc. Donald, Chapters 3, 5, 10) Put/Call Parity and Binomial Model (Mc. Donald, Chapters 3, 5, 10)

Synthetic Forwards • A synthetic long forward contract – Buying a call and selling Synthetic Forwards • A synthetic long forward contract – Buying a call and selling a put on the same underlying asset, with each option having the same strike price and time to expiration – Example: buy the $1, 000 strike S&R call and sell the $1, 000 -strike S&R put, each with 6 months to expiration Figure 3. 6 Purchase of a 1000 strike S&R call, sale of a 1000 -strike S&R put, and the combined position. The combined position resembles the profit on a long forward contract. 4 -2

Synthetic Forwards (cont’d) • Differences between a synthetic long forward contract and the actual Synthetic Forwards (cont’d) • Differences between a synthetic long forward contract and the actual forward – The forward contract has a zero premium, while the synthetic forward requires that we pay the net option premium – With the forward contract, we pay the forward price, while with the synthetic forward we pay the strike price 4 -3

Equation 3. 1: Put/Call Parity 4 -4 Equation 3. 1: Put/Call Parity 4 -4

Insuring a Long Position: Floors • A put option is combined with a position Insuring a Long Position: Floors • A put option is combined with a position in the underlying asset • Goal: to insure against a fall in the price of the underlying asset 4 -5

Table 3. 1 Payoff and profit at expiration from purchasing the S&R index and Table 3. 1 Payoff and profit at expiration from purchasing the S&R index and a 1000 -strike put option. Payoff is the sum of the first two columns. Cost plus interest for the position is ($1000 + $74. 201) × 1. 02 = $1095. 68. Profit is payoff less $1095. 68. 4 -6

Insuring a Long Position: Floors (cont’d) • Example: S&R index and a S&R put Insuring a Long Position: Floors (cont’d) • Example: S&R index and a S&R put option with a strike price of $1, 000 together Figure 3. 1 Panel (a) shows the payoff diagram for a long position in the index (column 1 in Table 3. 1). Panel (b) shows the payoff diagram for a purchased index put with a strike price of $1000 (column 2 in Table 3. 1). Panel (c) shows the combined payoff diagram for the index and put (column 3 in Table 3. 1). Panel (d) shows the combined profit diagram for the index and put, obtained by subtracting $1095. 68 from the payoff diagram in panel (c)(column 5 in Table 3. 1). Buying an asset and a put generates a position that looks like a call! 4 -7

Alternative Ways to Buy a Stock • Four different payment and receipt timing combinations Alternative Ways to Buy a Stock • Four different payment and receipt timing combinations – – Outright purchase: ordinary transaction Fully leveraged purchase: investor borrows the full amount Prepaid forward contract: pay today, receive the share later Forward contract: agree on price now, pay/receive later • Payments, receipts, and their timing Table 5. 1 Four different ways to buy a share of stock that has price S 0 at time 0. At time 0 you agree to a price, which is paid either today or at time T. The shares are received either at 0 or T. The interest rate is r. 4 -8

Pricing Prepaid Forwards • Pricing by analogy – In the absence of dividends, the Pricing Prepaid Forwards • Pricing by analogy – In the absence of dividends, the timing of delivery is irrelevant – Price of the prepaid forward contract same as current stock price – (where the asset is bought at t = 0, delivered at t = T) 4 -9

Pricing Prepaid Forwards (cont’d) • Pricing by arbitrage – If at time t=0, the Pricing Prepaid Forwards (cont’d) • Pricing by arbitrage – If at time t=0, the prepaid forward price somehow exceeded the stock price, i. e. , , an arbitrageur could do the following Table 5. 2 Cash flows and transactions to undertake arbitrage when the prepaid forward price, FP 0, T , exceeds the stock price, S 0. 4 -10

Pricing Prepaid Forwards (cont’d) • 4 -11 Pricing Prepaid Forwards (cont’d) • 4 -11

Pricing Prepaid Forwards (cont’d) • Example 5. 1 – XYZ stock costs $100 today Pricing Prepaid Forwards (cont’d) • Example 5. 1 – XYZ stock costs $100 today and is expected to pay a quarterly dividend of $1. 25. If the risk-free rate is 10% compounded continuously, how much does a 1 -year prepaid forward cost? 4 -12

Pricing Prepaid Forwards (cont’d) • Example 5. 2 – The index is $125 and Pricing Prepaid Forwards (cont’d) • Example 5. 2 – The index is $125 and the dividend yield is 3% continuously compounded. How much does a 1 -year prepaid forward cost? 4 -13

Pricing Forwards on Stock • Forward price is the future value of the prepaid Pricing Forwards on Stock • Forward price is the future value of the prepaid forward – No dividends – Continuous dividends 4 -14

Creating a Synthetic Forward • One can offset the risk of a forward by Creating a Synthetic Forward • One can offset the risk of a forward by creating a synthetic forward to offset a position in the actual forward contract • How can one do this? (assume continuous dividends at rate d) – Recall the long forward payoff at expiration: = ST – F 0, T – Borrow and purchase shares as follows Table 5. 3 Demonstration that borrowing S 0 e−δT to buy e−δT shares of the index replicates the payoff to a forward contract, ST − F 0, T. – Note that the total payoff at expiration is same as forward payoff 4 -15

Table 5. 4 Demonstration that going long a forward contract at the price F Table 5. 4 Demonstration that going long a forward contract at the price F 0, T = S 0 e(r−δ)T and lending the present value of the forward price creates a synthetic share of the index at time T. 4 -16

Table 5. 5 Demonstration that buying e−δT shares of the index and shorting a Table 5. 5 Demonstration that buying e−δT shares of the index and shorting a forward creates a synthetic bond. 4 -17

Creating a Synthetic Forward (cont’d) • The idea of creating synthetic forward leads to Creating a Synthetic Forward (cont’d) • The idea of creating synthetic forward leads to following – Forward = Stock – zero-coupon bond – Stock = Forward + zero-coupon bond – Zero-coupon bond = Stock – forward • Cash-and-carry arbitrage: Buy the index, short the forward Figure 5. 6 Transactions and cash flows for a cash-and-carry: A marketmaker is short a forward contract and long a synthetic forward contract. 4 -18

Introduction to Binomial Option Pricing • Binomial option pricing enables us to determine the Introduction to Binomial Option Pricing • Binomial option pricing enables us to determine the price of an option, given the characteristics of the stock or other underlying asset • The binomial option pricing model assumes that the price of the underlying asset follows a binomial distribution—that is, the asset price in each period can move only up or down by a specified amount • The binomial model is often referred to as the “Cox-Ross. Rubinstein pricing model” 4 -19

A One-Period Binomial Tree • Example – Consider a European call option on the A One-Period Binomial Tree • Example – Consider a European call option on the stock of XYZ, with a $40 strike and 1 year to expiration – XYZ does not pay dividends, and its current price is $41 – The continuously compounded risk-free interest rate is 8% – The following figure depicts possible stock prices over 1 year, i. e. , a binomial tree $60 $41 $30 4 -20

Computing the Option Price • Next, consider two portfolios – Portfolio A: buy one Computing the Option Price • Next, consider two portfolios – Portfolio A: buy one call option – Portfolio B: buy 2/3 shares of XYZ and borrow $18. 462 at the risk-free rate • Costs – Portfolio A: the call premium, which is unknown – Portfolio B: 2/3 $41 – $18. 462 = $8. 871 4 -21

Computing the Option Price (cont’d) • Payoffs: – Portfolio A: Stock Price in 1 Computing the Option Price (cont’d) • Payoffs: – Portfolio A: Stock Price in 1 Year $30. 0$60. 0 Payoff 0 $20. 0 – Portfolio B: 2/3 purchased shares Repay loan of $18. 462 Total payoff Stock Price in 1 Year $30. 0 $60. 0 $20. 000 $40. 000 – $20. 000 –$20. 000 0 $20. 000 4 -22

A One-Period Binomial Tree • Another Example – Consider a European call option on A One-Period Binomial Tree • Another Example – Consider a European call option on the stock of XYZ, with a $40 strike and 1 year to expiration – XYZ does not pay dividends, and its current price is $41 – The continuously compounded risk-free interest rate is 8% – The following figure depicts possible stock prices over 1 year, i. e. , a binomial tree 4 -23

Computing the Option Price • Next, consider two portfolios – Portfolio A: buy one Computing the Option Price • Next, consider two portfolios – Portfolio A: buy one call option – Portfolio B: buy 0. 7376 shares of XYZ and borrow $22. 405 at the risk-free rate • Costs – Portfolio A: the call premium, which is unknown – Portfolio B: 0. 7376 $41 – $22. 405 = $7. 839 4 -24

The Binomial Solution • How do we find a replicating portfolio consisting of shares The Binomial Solution • How do we find a replicating portfolio consisting of shares of stock and a dollar amount B in lending, such that the portfolio imitates the option whether the stock rises or falls? – Suppose that the stock has a continuous dividend yield of , which is reinvested in the stock. Thus, if you buy one share at time t, at time t+h you will have e h shares – If the length of a period is h, the interest factor period is erh – u. S 0 denotes the stock price when the price goes up, and d. S 0 denotes the stock price when the price goes down 4 -25

The Binomial Solution (cont’d) • Stock price tree: u. S 0 d. S 0 The Binomial Solution (cont’d) • Stock price tree: u. S 0 d. S 0 Corresponding tree for the value of the option: Cu C 0 Cd – Note that u (d) in the stock price tree is interpreted as one plus the rate of capital gain (loss) on the stock if it foes up (down) • The value of the replicating portfolio at time h, with stock price Sh, is 4 -26

Arbitraging a Mispriced Option • If the observed option price differs from its theoretical Arbitraging a Mispriced Option • If the observed option price differs from its theoretical price, arbitrage is possible – If an option is overpriced, we can sell the option. However, the risk is that the option will be in the money at expiration, and we will be required to deliver the stock. To hedge this risk, we can buy a synthetic option at the same time we sell the actual option – If an option is underpriced, we buy the option. To hedge the risk associated with the possibility of the stock price falling at expiration, we sell a synthetic option at the same time 4 -27

A One-Period Binomial Tree • Example – Consider a European call option on the A One-Period Binomial Tree • Example – Consider a European call option on the stock of XYZ, with a $40 strike and 1 year to expiration – XYZ does not pay dividends, and its current price is $41 – The continuously compounded risk-free interest rate is 8% – The following figure depicts possible stock prices over 1 year, i. e. , a binomial tree $60 $41 $30 4 -28

A Graphical Interpretation of the Binomial Formula (cont’d) Figure 10. 2 The payoff to A Graphical Interpretation of the Binomial Formula (cont’d) Figure 10. 2 The payoff to an expiring call option is the dark heavy line. The payoff to the option at the points d. S and u. S are Cd and Cu (at point D). The portfolio consisting of ∆ shares and B bonds has intercept erh B and slope ∆, and by construction goes through both points E and D. The slope of the line is calculated as Rise/Run between points E and D, which gives the formula for ∆. 4 -29

Pricing with Dividends Equation 10. 5 4 -30 Pricing with Dividends Equation 10. 5 4 -30

Constructing a Binomial Tree (cont’d) • The stock price evolution can be estimated by Constructing a Binomial Tree (cont’d) • The stock price evolution can be estimated by assuming an evolution that compensates investors for time, but also uncertainty. is the annualized standard deviation of the continuously compounded return, and h is standard deviation over a period of length h. • If we divide both sides by initial stock price, we can calculate the jump process. – We refer to a tree constructed using equation (10. 11) as a “forward tree. ” 4 -31

Figure 10. 3 Binomial tree for pricing a European call option; assumes S = Figure 10. 3 Binomial tree for pricing a European call option; assumes S = $41. 00, K = $40. 00, σ = 0. 30, r = 0. 08, T = 1. 00 years, δ = 0. 00, and h = 1. 000. At each node the stock price, option price, ∆, and B are given. Option prices in bold italic signify that exercise is optimal at that node. 4 -32

Summary • In order to price an option, we need to know – – Summary • In order to price an option, we need to know – – – Stock price Strike price Standard deviation of returns on the stock Dividend yield Risk-free rate • Using the risk-free rate and , we can approximate the future distribution of the stock by creating a binomial tree using equation (10. 11) • Once we have the binomial tree, it is possible to price the option using the regular equations. 4 -33

A Two-Period European Call • We can extend the previous example to price a A Two-Period European Call • We can extend the previous example to price a 2 year option, assuming all inputs are the same as before Figure 10. 4 Binomial tree for pricing a European call option; assumes S = $41. 00, K = $40. 00, σ =0. 30, r = 0. 08, T = 2. 00 years, δ = 0. 00, and h = 1. 000. At each node the stock price, option price, ∆, and B are given. Option prices in bold italic signify that exercise is optimal at that node. 4 -34

Pricing the Call Option (cont’d) • Notice that – The option was priced by Pricing the Call Option (cont’d) • Notice that – The option was priced by working backward through the binomial tree – The option price is greater for the 2 -year than for the 1 -year option – The option’s and B are different at different nodes. At a given point in time, increases to 1 as we go further into the money – Permitting early exercise would make no difference. At every node prior to expiration, the option price is greater than S – K; thus, we would not exercise even if the option was American 4 -35

Many Binomial Periods • Dividing the time to expiration into more periods allows us Many Binomial Periods • Dividing the time to expiration into more periods allows us to generate a more realistic tree with a larger number of different values at expiration – Consider the previous example of the 1 -year European call option – Let there be three binomial periods. Since it is a 1 -year call, this means that the length of a period is h = 1/3 – Assume that other inputs are the same as before (so, r = 0. 08 and = 0. 3) 4 -36

Put Options • We compute put option prices using the same stock price tree Put Options • We compute put option prices using the same stock price tree and in the same way as call option prices • The only difference with a European put option occurs at expiration – Instead of computing the price as max (0, S – K), we use max (0, K – S) 4 -37

American Put Options • Consider an American version of the put option valued in American Put Options • Consider an American version of the put option valued in the previous example Figure 10. 7 Binomial tree for pricing an American put option; assumes S = $41. 00, K = $40. 00, σ =0. 30, r = 0. 08, T = 1. 00 year, δ = 0. 00, and h = 0. 333. At each node the stock price, option price, ∆, and B are given. Option prices in bold italic signify that exercise is optimal at that node. 4 -38

Options on Other Assets • The model developed thus far can be modified easily Options on Other Assets • The model developed thus far can be modified easily to price options on underlying assets other than nondividendpaying stocks. • The difference for different underlying assets is the construction of the binomial tree and the risk-neutral probability. 4 -39

Options on a Stock Index • A binomial tree for an American call option Options on a Stock Index • A binomial tree for an American call option on a stock index: Figure 10. 8 Binomial tree for pricing an American call option on a stock index; assumes S = $110. 00, K = $100. 00, σ = 0. 30, r = 0. 05, T = 1. 00 year, δ = 0. 035, and h = 0. 333. At each node the stock price, option price, ∆, and B are given. Option prices in bold italic signify that exercise is optimal at that node. 4 -40