1. Topic 3: Derivatives Options: puts and calls
How much is an option worth?
[Forwards and futures]
[Swaps]
[Structured finance]

2. A call option
Gives the holder the right* to purchase a security at the exercise price
(a) on or before the expiry date (American option)
(b) on the expiry date (European option)
Generally American options are more valuable than European (why?).
But in the case of a non-dividend paying stock, an American option may be sold, but will never be exercised before the expiry date.
*but no obligation

3. Other terms A put option: …the right to sell a security…
Don’t forget: you can sell or buy an option
At the money: price of security=exercise price
In the money: price of security>exercise price
Out of the money: price of security<exercise price
The counterparty to an option is said to have “written” the option
Selling a call option and writing an option are different – the latter leaves you with a commitment

4. as a function of the stock price
Pay-off at maturity/expiry from holding a call option
as a function of the stock price
Payoff of call
Payoff from call at expiry X
Stock price

5. Profit from strategy: buy one European call option at a cost of $3
Strike (exercise) price X = $80
Profit $5
X = $80
$83
$88
Call premium
ST=Asset price at expiry
-$3
NB: “call premium”=cost of call option

6. Profit from strategy: write (short) a call option
Strike price X = $80
Profit $3
Call premium
$83
X = $80
$88 ST
-$5

7. Strike (exercise) price X = $70
Buy (long) a put option
Strike (exercise) price X = $70
Profit $3
$68 ST
Put
premium
$65
X = $70
-$2

8. Write a put option Profit $2 $65 $68 X = $70 ST -$3
Strike (exercise) price X = $70
Profit $2
Put premium
$65
$68
X = $70 ST
-$3

9. Some live option prices
International securities exchange
>>Market trading and data tools
>> Quotes, volatility and calculator

10. Why hold a call or put in your portfolio?
Although often seen (and used) as risky, speculative instruments, options can be a valuable part of a risk management strategy e.g.:
Protective put:
You’d like to invest in a stock, but want to limit your downside risk
Note that the same can be achieved by holding a call plus a bond
Fraudster Madoff’s supposed plan: “split-strike”

11. In algebra If S is the market price of the security
X the exercise price
and T the expiry date.
Payoff to call holder:
ST – X if ST > X ; 0 otherwise
Payoff to call writer:
– (ST – X) if ST > X ; 0 otherwise

12. Put-call parity: The value of a call must equal:
the value of the underlying stock plus
the value of a put with the same exercise price etc minus
A bond with the same maturity value as the exercise price of the stock

15. Why hold a call or put in your portfolio? (2)
Solutions to views about risk patterns:
Problem: You might think a security is headed for a period of volatility: but you don’t know if it will go up or down:
Solution: Hold a “straddle”, i.e. a call and a put simultaneously (with the same exercise price and same maturity
Problem: You think the security will go up but not all that much, so would like to benefit from the upside, but not pay the full call option premium
Solution: Hold a spread, i.e. hold a call and write another call with a higher exercise price:

16. Step by step to finding the value of a call option
Value of call
Payoff from call at expiry X
Stock price

17. Step by step to finding the value of a call option
Value of call
Payoff from call at expiry
Value of call prior to expiry X
Stock price

18. Step by step to finding the value of a call option
Value of call
Payoff from call at expiry
Value of call prior to expiry Ct X St
Stock price

19. Step by step to finding the value of a call option
Value of call
Payoff from call at expiry
Value of call prior to expiry B Ct
BC = time value C
CD = intrinsic value D X St
Stock price

20. Options in unexpected places
An equity claim on a leveraged firm is an option
A collateralized loan is (for the lender) like a call on the collateral
An investment idea which can be implemented sometime in the future is like an option: when to exercise it?

21. How much should you pay for an option?
Simple case: two period, two states of world (u and d)
Find a portfolio that perfectly hedges the option
Payoffs equal in both states
One holds the stock (and some borrowing)
The other holds the option
This portfolio must have the same value as the stock

22. How much should you pay for an option? (2)
We find that the price does depend on the interest rate for borrowing
But not on the final stock price or the beta of the stock
Hedge ratio: how many units of the stock needed to hedge the option?

23. How much should you pay for an option? (3)
More general: two period, three states of world
could come from two half-steps (uu, ud, du and dd)
And we can continue the subdivision
….to continuous time
….and the binomial gets closer to a Normal distribution
The maths gets more complicated
(Stochastic calculus)
(Black and Scholes)

24. How much should you pay for an option? (4)
N(d) = The probability that a random draw from a standard Normal (Gaussian) distribution will be less than d.

25. How much should you pay for an option? (4)
Price of stock
Exercise price
N(d) = The probability that a random draw from a standard Normal (Gaussian) distribution will be less than d.
Price of call
Interest rate
Time to maturity
Stock price volatility

26. How much should you pay for an option? (5)
Surprising feature of the Black-Scholes solution:
Option value does not depend on the expected rate of return on the stock
If N close to 1, then stock price is well above exercise price (relative to the volatility), so option almost certain to be exercised

27. How much should you pay for an option? (6)
Surprising feature of the Black-Scholes solution:
Option value does not depend on the expected rate of return on the stock
The data needs: if we believe Black-Scholes and want to know option prices, hardest part is stock volatility σ
Implied volatility: if we know the option price and believe Black-Scholes, then we can solve back for σ
If N close to 1, then stock price is well above exercise price (relative to the volatility), so option almost certain to be exercised

28. The implied volatility of US stocks, 2004-yesterday
Note: log scale

29. The implied volatility of US stocks, 1990-yesterday
Note: log scale

30. Hedge ratio (“delta”) (Already defined for the two-state model)
Is the change in the price of the option for a $1 increase in the stock price
i.e. the slope of the option price curve
Is less than one (for call option)
But note that elasticity of option price with respect to stock price is higher
Equals N(d1) in the Black Scholes formula
Think of it as the defining the proportions in which options and stocks must be held to hedge the risk

31. Assumptions underlying Black-Scholes and how well does it fit?
“Brownian motion” – normally distributed continuous price changes – no jumps
Fits quite well
But, especially after 1987, implied volatility seems systematically related to exercise price
Deep in the money puts are expensive
The volatility “smirk”

32. Out-of-the-money puts have high implicit volatility
Put options on S&P 500, 21 days, 1995
Bates, 2005

33. The Smirk

34. Out-of-the-money puts have high implicit volatility
The Smirk
Put options on S&P 500, 21 days, 1995
Bates, 2005

35. True hedge ratio delta deviates a little from Black-Scholes formula
Put options on S&P 500, 21 days, 1995
Bates, 2005

36. The Crash of ’87 – Oops, I thought I was insured

37. The Crash of ’87 – Oops, I thought I was insured

38. Dynamic hedging and the crash of October 19, 1987
Portfolio insurance products used dynamic hedging to try to eliminate downside risk
(using “protective” puts)
But market movements were too sudden
dynamic hedging adjustments couldn’t keep up with price movements
deltas increased and insurers needed to buy more puts and/or sell stock
Information systems couldn’t keep up – price information was out of date
Trading stopped for a while in some stocks and derivatives
So the insurance did not work properly
And indeed helped to destabilize the market
One-day fall in US stock prices of 20%

39. The Crash of ’87 --- Hmm, not so bad on the longer view

40. The Crash of ’87 --- What crash?
Shiller: Irrational Exuberance

41. The Crash of ’87 --- What crash?
Shiller: Irrational Exuberance

42. Forward and futures markets
Forwards useful for hedging a known commitment due in the future.
Forward transactions and futures markets do much the same thing
Forward transactions are bilaterally negotiated or offered “over-the-counter” by e.g. a bank
Futures markets are organized exchanges where standardized contracts are traded…and the final settlement is in cash for the difference between spot and forward