1. Put/Call Parity and Binomial Model (McDonald, Chapters 3, 5, 10)

2. 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.

3. 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. Equation 3.1: Put/Call Parity

5. 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

6. 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 ($ $74.201) × 1.02 = $ Profit is payoff less $

7. 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 $ 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!

8. 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 S0 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.

9. 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)

10. 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
Since, this sort of arbitrage profits are traded away quickly, and cannot persist, at equilibrium we can expect:
Table 5.2 Cash flows and transactions to undertake arbitrage when the prepaid forward price, FP 0,T , exceeds the stock price, S0.

11. Pricing Prepaid Forwards (cont’d)
What if there are dividends? Is still valid?
No, because the holder of the forward will not receive dividends that will be paid to the holder of the stock 
For discrete dividends Dti at times ti, i = 1,…., n
The prepaid forward price:
For continuous dividends with an annualized yield d

12. Pricing Prepaid Forwards (cont’d)
Example 5.1
XYZ stock costs $100 today and is expected to pay a quarterly dividend of $ If the risk-free rate is 10% compounded continuously, how much does a 1-year prepaid forward cost?

13. 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?

14. Pricing Forwards on Stock
Forward price is the future value of the prepaid forward
No dividends
Continuous dividends

15. 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 – F0, T
Borrow and purchase shares as follows
Note that the total payoff at expiration is same as forward payoff
Table 5.3 Demonstration that borrowing S0e−δT to buy e−δT shares of the index replicates the payoff to a forward contract,
ST − F0,T .

16. Table 5.4 Demonstration that going long a forward contract at the price F0,T = S0e(r−δ)T and lending the present value of the forward price creates a synthetic share of the index at time T .

17. Table 5.5 Demonstration that buying e−δT shares of the index and shorting a forward creates a synthetic bond.

18. 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.

19. 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”

20. 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

21. Computing the Option Price
Next, consider two portfolios
Portfolio A: buy one call option
Portfolio B: buy 2/3 shares of XYZ and borrow $ at the risk-free rate
Costs
Portfolio A: the call premium, which is unknown
Portfolio B: 2/3  $41 – $ = $8.871

22. Computing the Option Price (cont’d)
Payoffs:
Portfolio A: Stock Price in 1 Year
$30.0 $60.0
Payoff 0 $20.0
Portfolio B: Stock Price in 1 Year
$ $60.0
2/3 purchased shares $ $40.000
Repay loan of $ – $ – $20.000
Total payoff 0 $20.000

23. 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

24. Computing the Option Price
Next, consider two portfolios
Portfolio A: buy one call option
Portfolio B: buy shares of XYZ and borrow $ at the risk-free rate
Costs
Portfolio A: the call premium, which is unknown
Portfolio B:  $41 – $ = $7.839

25. 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 per period is erh
uS0 denotes the stock price when the price goes up, and dS0 denotes the stock price when the price goes down

26. The Binomial Solution (cont’d)
Stock price tree:  Corresponding tree for the value of the option:
uS Cu
S C0
dS0 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

27. 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

28. 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

29. A Graphical Interpretation of the Binomial Formula (cont’d)
Figure The payoff to an expiring call option is the dark heavy line. The payoff to the option at the points dS and uS 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 ∆.

30. Pricing with Dividends
Equation 10.5

31. Constructing a Binomial Tree (cont’d)
With uncertainty, the stock price evolution is
(10.10)
Where  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 rewrite (10.10) as
(10.11)
We refer to a tree constructed using equation (10.11) as a “forward tree.”

32. Figure 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 = 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.

33. 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.

34. 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 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 = 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.

35. 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

36. 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)

37. Many Binomial Periods (cont’d)
The stock price and option price tree for this option
Figure Binomial tree for pricing a European call option; assumes S = $41.00, K = $40.00, σ =0.30, r = 0.08, T = 1.00 year, δ = 0.00, and h = 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.

38. Many Binomial Periods (cont’d)
Note that since the length of the binomial period is shorter, u and d are smaller than before: u = and d = (as opposed to and with h = 1)
The second-period nodes are computed as follows
The remaining nodes are computed similarly
Analogous to the procedure for pricing the 2-year option, the price of the three-period option is computed by working backward using equation (10.3)
The option price is $7.074

39. 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)

40. Put Options (cont’d)
A binomial tree for a European put option with 1-year to expiration
Figure Binomial tree for pricing a European put option; assumes S = $41.00, K = $40.00, σ = 0.30, r = 0.08, T = 1.00 year, δ = 0.00, and h = 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.

41. American Put Options
Consider an American version of the put option valued in the previous example
Figure 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 = 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.

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

43. Options on a Stock Index
A binomial tree for an American call option on a stock index:
Figure 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 = 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.

44. Options on Currency
With a currency with spot price x0, the forward price is
Where rf is the foreign interest rate, and it replaces in the dividend case.

45. Options on Currency (cont’d)
Consider a dollar-denominated American put option on the euro, where
The current exchange rate is $1.05/€
The strike is $1.10/€
The euro-denominated interest rate is 3.1%
The dollar-denominated rate is 5.5%