Lecture Notes 5, Option Values (Ch 9) & Trading Strategies (Ch 10)

K Foster, CCNY, Spring 2010

Learning Outcomes (from CFA exam)

Students will be able to:

§ determine the minimum and maximum values of European options and American options;

§ calculate and interpret the lowest prices of European and American calls and puts based on the rules for minimum values and lower bounds;

§ explain how option prices are affected by the exercise price and the time to expiration;

§ explain putcall parity for European options, and relate putcall parity to arbitrage and the construction of synthetic options;

§ contrast American options with European options in terms of the lower bounds on option prices and the possibility of early exercise;

§ explain how cash flows on the underlying asset affect putcall parity and the lower bounds of option prices;

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

From Hull Chapter 9

Notation:

C value of American call option (sometimes distinguish between C₀ and C_T, the value now and value at maturity)

P value of American put option

c value of European call option

p value of European put option

Six important factors affecting option prices:

Current stock price, S₀
Strike, K
Time to expiration/maturity, T
Volatility, σ
risk-free interest rate, r
dividends of stock over T

Look at each:

First, the value of an option depends on the spread between S and K a call is worth max(S_T K,0) while a put is worth max(K S_T,0). So if the stock price, S₀, rises while K is constant, then this increases the value of a call but reduces the value of a put; if the stock price falls while K is constant then the call price falls while the put price rises. It works vice versa for K. Generally higher stock prices result in higher call prices but lower put prices.

Time to expiration is generally a positive factor, certainly for stocks without dividends. (If substantial dividends are paid, which reduce the value of a stock on the day after, then the dividend schedule can affect the option value.) American options, which can be exercised anytime, are certainly always worth more when there is a longer time to expiration more choice can never be bad!

Volatility (or uncertainty) is a positive factor. This might seem odd unless you remember that options are basically providing insurance, and it certainly seems sensible that more uncertainty raises the value of insurance.

The risk-free interest rate affects the present value of the payoffs to options (since the payoff will be received in the future, a higher rate means a lower present value). However since the expected future price is influenced by the risk-free rate, a call would be worth more while a put would be worth less.

Since dividends reduce the value of a stock on the day after, higher dividends raise the value of a put and lower the value of a call.

Upper Bounds for Option Prices

A call gives the right to buy the stock so it can never be worth more than the stock price; c ≤ S₀ and C ≤ S₀

A put gives the right to sell at K so it can never be worth more than K, p ≤ K and P ≤ K; in fact p ≤ Ke^-rT.

Lower Bounds for Option Prices

These are trickier. For a call, consider 2 portfolios: portfolio A buys a call, c, and invests an amount of cash Ke^-rT. Portfolio B just buys a share at S0. The value of portfolio A will be, at date T,

max(S_T K,0) + Ke^rte^-rT = max(S_T K,0) + K = either S_T or K, whichever is bigger, so = max(S_T,K).

Portfolio B will evidently be worth S_T at the end of the period so portfolio A will always be worth more than portfolio B (or equal in value to it), or

c + Ke^-rT ≥ S₀ so c ≥ S₀ Ke^-rT

For a put, on the other hand, consider 2 more portfolios: portfolio C buys a put and the stock, so p + S₀, while portfolio D invests the cash Ke^-rT. The value of portfolio C will be, at date T,

max(K S_T,0) + S_T = either K or S_T, whichever is bigger, so = max(S_T,K).

Portfolio D will be worth K at the end of the period, so again portfolio C is always worth more than D (or is worth the same amount), so

p + S₀ ≥ Ke^-rT thus p ≥ Ke^-rT - S₀

Put-Call Parity

We can also note that portfolios A and C have the exact same value, max(S_T,K). So their values must be equal, so

c + Ke^-rT = p + S₀

c p = S₀ Ke^-rT.

This connects the values of the call and put options a connection that seems reasonable since both depend on the properties of the very same underlying stock. The excess of the value of a call over a put depends on the excess of what we can interpret as the expected present value of the "intrinsic" value, S K.

We can get some idea of the rationale behind this by thinking of the value of a call and put, just moments before the exercise date. A call is worth the excess of the stock price over the strike while a put is worth the excess of the strike over the stock price. So just a moment before exercise, c = max(S_T K,0) and p = max(K S_T,0), or to write in greater detail,

and

so that c p = S_T K in either case. Put-call parity is just the present value of that relationship.

For American options, a similar result (for non-dividend stocks) can be derived:

S₀ K ≤ C P ≤ S₀ Ke^-rT.

Early Exercise of American Options

For non-dividend stocks, an American call is never optimally exercised before the expiration date. There are two reasons for this: that the insurance value should be kept for the duration of the period (you would never want to pay for a year of insurance but then cancel after 6 months!) and the fact that the strike price must be paid, so the time value of money indicates that we want to defer that payment.

However for non-dividend stocks, an American put might be optimally exercised early, because there is a maximum of the put value: if the stock price goes to zero then the put is worth K.

Dividends make the arguments above a bit more Byzantine but don't change the basic intuition.

Trading Strategies

From Hull Chapter 10

Combining a stock position with an option allows investor to put a cap or floor on payoff. But this has a more subtle implication it can allow us to take another view of put-call parity.

Consider the profit to buying a stock and buying a put with strike K. The profit function of the stock, S₀, is

and the profit function for the put, p, purchased at some cost (so that the investor loses some money if the stock price rises and the put is not exercised) is

so the profit to the combined function is the purple line:

The profit to this combination, which I've labeled "Z", looks remarkably like the profit to buying a call, c, at some cost:

What could cost be? Recall our formula for put-call parity, that for a stock that does not pay dividends, p + S₀ = c + Ke^-rT. What is the payoff to a portfolio with one put and one share of stock? That's the " p + S₀" part. What is it equal to? A call, "c", plus some amount of cash, " Ke^-rT ". So we could also derive our formula on put-call parity from the equivalence of the payoff functions.

This is a taste of a more general result: if two portfolios give the same payoff functions, then if they do not have the same current market value then there are arbitrage opportunities (which, in a perfectly functioning market, would be absent). Put another way, if markets are efficient then two portfolios that give the same payoffs should have the same value. We'll do some work to show that any payoff pattern can be replicated with combinations of puts and calls in order to show that, once we've gotten valuations for puts and calls, we've done all the work that's necessary any other portfolio can be valued!

We can go through to re-order the put-call parity equation and figure other payoffs. Rearrange so that S₀ c = Ke^-rT p and this says that a short put ("-p") with some amount of cash has the same value as a long stock ("S₀") and short call ("-c") position. This diagram is:

where now "Z" looks like the payoff function to a short put. We can keep on rearranging to show that the long put and short call can also be replicated.

Market participants have named a variety of different combinations of options. Among them are:

Bull Spread: buy a call at K₁ and sell a call at K₂ (K₂ > K₁)

Money-ness determines the cost

if both K₁ and K₂ are out of the money then this is low-cost
if K₁ is initially in the money but K₂ is out
if both are in-the-money

Can also be replicated with puts: buy a put at K₁ and sell at K₂

Bear Spread: buy a put at K₄ and sell a put at K₃ (K₃ < K₄)

Can be replicated with calls as well

Box Spread

combines bull spread and bear spread, but K₁=K₃ and K₂=K₄ so the payoff is always (K₂
K₁).
if cost is different from (present value of) payoff then arbitrage

Butterfly Spread

options with strike prices K₅, K₆, K₇ (K₅ < K₆ < K₇)
buy call at K₅
sell two calls at K₆
buy call at K₇

Straddle

Buy a call and buy a put, both at K

Profit if large moves in either direction
straddle write or top straddle has opposite payoffs
profit if stock price moves little but loss if stock goes up or down

Strip buys one call and two puts; strap buys two calls and one put (all with same strike) will skew the payoffs
Strangle buys a put with strike K₈ and a call with strike K₉ (K₈ < K₉)

All of these were assumed to use options with same expiration date but "calendar spreads" use different expirations to further complicate the position payoffs

If options are available at any given strike price, then we can replicate ANY payoff function using calls and puts.