Note:

Revised Schedule

Eco 275, Options and Futures

K Foster, CCNY, Spring 2010

 

 

10

Apr 19, 21

Option Details

14, 15, 16

11

Apr 26, 28

The Greeks

17

12

May 3, 5

Value at Risk (VaR)

20

13

May 10, 12

Credit Risk & Credit Derivatives

22, 23

14

May 17

Derivatives Mistakes

34

 

sometime May 19-25

Final Exam

comprehensive

 

My Mistake Last Week:

I think I forgot a term in the Bl-S-M formulas that we worked through,

call price:          

put price:           ,

 


where  and  or .

 

 

 

Lecture Notes 9, Option Valuation Details

K Foster, CCNY, Spring 2010

 

 

Learning Outcomes (from CFA exam)

Students will be able to:

§  indicate the directional effect of an interest rate change or volatility change on an option’s price.

§  explain how an option price, as represented by the Black–Scholes–Merton model, is affected by each of the input values (the option Greeks);

§  explain the delta of an option and demonstrate how it is used in dynamic hedging;

§  explain the gamma effect on an option’s price and delta and how gamma can affect a delta hedge;

 

From Hull Chapter 14, 15, 17

 

 

 

Hull Chapter 14 Employee Stock Options

Read the chapter.  Accounting for employee stock options has changed after the dot-com bubble and corporate scandals about the dating of the options' exercise prices.  Clearly (from the formulas or just from intuition) a lower strike price favors the employee (holding all else constant).  Some firms seem to have looked backwards over stock prices for the past month to decide that their employee's options would be granted at an exercise price equal to the lowest recent stock price.

 

 

Hull Chapter 15

 

This chapter is a cleanup of the previous chapter – a good way to review but it doesn't provide much new material.  It just provides some of the details for other common option products.

 

Dividends

Suppose a stock pays dividends at a known continuous rate q.  Of course stocks don't pay dividends continuously, they pay them in lumps.  But we know how this lumpiness affects a stock price: moments before a dividend payment is issued, say $1 per share, the stock price is S + 1; moments after the dividend has been paid the stock price is just S.  So paying dividends lowers stock price returns.  This matches what we know about corporate finance: a firm that didn't pay out dividends must reward shareholders somehow, with capital gains (returns on the stock).

 

So we can model a stock that pays dividends at a known rate, q, by just reducing the initial price, S0, to S0e-qT.  This changes the Black-Scholes-Merton formula as well as the boundaries and put-call parity formulas. 

 

The lower bound for a call is now ; for a put the lower bound is .

 

Put-call parity is now that a put and a call with the same exercise price must satisfy .

 

The Black-Scholes-Merton formula is now

where now both d1 and d2 subtract q from r, so that  and .  Note that the Ke-rt does not change – only the S0 is discounted at (r – q).

 

The binomial tree formulas change to keep the expected return at (r – q) instead of r.

 

The Black-Scholes differential equation becomes

where, again, S is discounted at (r – q) but f is still multiplied by r not (r – q).

 

Options on Indices

Now the formula s and q are the average values for all of the underlying stocks in the index.  Most index options are European although the S&P 100 index has an American option.  (Although there are flex options on some.)

 

These index options can be used by portfolio managers to change their beta values or provide some insurance against particularly severe drops.  For example, suppose a fund takes $1000 from 100 investors and buys the S&P 500 index.  There are different ways to guarantee the investors that they will always be able to draw out their original investment (no returns, but this limits the downside risk).  How much would this cost?  How does this cost change with the beta of the portfolio?  If the investment were put into a single stock, with beta of 0.5 against the market, how much would the insurance cost?

 

As of April 16, the SPX (S&P 500 index) was 1192.13.  Put options for May (with T=35 days) are traded at the following prices:

K=

p=

925

.75

950

.95

975

1.10

1000

1.60

1025

2.10

1050

2.70

1075

3.50

1100

5.00

1125

7.5

1150

10.7

1175

17.4

1200

26

 

So if the wealth is 100,000, then the fund could take 100,000/1192.13 = 83.88 contracts.  But it takes some money to buy options for 'insurance' so suppose it bought 82 options on the SPX and 82 put options with strike price of 1200.  The puts cost 82*26 = 2132.  The returns are in this payoff table:

 

SPX

Option Payoff

Wealth from SPX

Wealth from Put

Wealth

Return on SPX

Return on Portfolio

1000

200

 $    82,000.00

 $    16,400.00

 $    98,400.00

-16%

-2%

1025

175

 $     84,050.00

 $    14,350.00

 $    98,400.00

-14%

-2%

1050

150

 $     86,100.00

 $    12,300.00

 $    98,400.00

-12%

-2%

1075

125

 $     88,150.00

 $    10,250.00

 $    98,400.00

-10%

-2%

1100

100

 $    90,200.00

 $      8,200.00

 $    98,400.00

-8%

-2%

1125

75

 $     92,250.00

 $       6,150.00

 $    98,400.00

-6%

-2%

1150

50

 $     94,300.00

 $      4,100.00

 $    98,400.00

-4%

-2%

1175

25

 $     96,350.00

 $      2,050.00

 $    98,400.00

-1%

-2%

1200

0

 $    98,400.00

 $                     -  

 $    98,400.00

1%

-2%

1225

0

 $  100,450.00

 $                     -  

 $  100,450.00

3%

0%

1250

0

 $  102,500.00

 $                     -  

 $  102,500.00

5%

3%

1275

0

 $   104,550.00

 $                     -  

 $   104,550.00

7%

5%

1300

0

 $  106,600.00

 $                     -  

 $  106,600.00

9%

7%

 

If beta of a portfolio were not 1, then could buy b puts per (Wealth/(index*100)) not just 1.

 

Then, since returns are lower, could "sex up" the upside risk with some call options…

 

It should make sense that, if the underlying position has a beta greater than one (so it moves more than the market return), then it will take more options to hedge the position while, if the beta is less than one (it moves less than the market return), fewer options are necessary.  Since options on indices are more liquid than options on particular stocks, hedging a position might be cheaper using index options than pushing against a thin market.

 

Currency Options – FX

This is just like the case of a stock paying a known dividend rate, but where q = rf, the foreign risk-free rate.  So,

,

where

.

 

Currency options are often used in "range forward" contracts where a company might buy a put and sell a call (a short range forward), with different strike prices K1 and K2, so that the net cost is zero, p = -c.  A long range forward would sell a put and buy a call.

 

We can use a range of different call and put prices (at matched strike prices) to back out market estimates of either q, the dividend rate, or even the foreign implied rate for FX options.

 

Can also use forward prices not strike prices, since  and for currency, .

 

Hull Chapter 16 Options on Futures

These are typically American options so we need to worry about early exercise.  The most common are options on interest rate futures (Treasury bonds and bills or Eurodollars, most often).  In both cases exercising the option entitles delivery of the option payoff and the futures position (although cash settlement is common – for many commodities the underlying physical good never is delivered!