FMP 13: Properties of Options
Six quantities drive what a stock option is worth: the strike price, the price of the stock beneath it, the risk-free rate, how volatile that stock is, how long the contract has left to run, and any dividends payable while the option is alive. Valuation work under the Black-Scholes-Merton assumptions blends all six into one number. The results below rest on arbitrage alone, so they say what an option price cannot be. The expected return on the underlying stock, notably, plays no part in either exercise.
The direction of each effect
Raising the stock price helps a call and hurts a put; raising the strike price does the reverse, since those two numbers fix what changes hands on exercise. Volatility widens the spread of outcomes, and an option keeps the good tail while cutting the bad one, so calls and puts both gain. A higher risk-free rate lifts a call, whose holder pays the strike later, and depresses a put, whose holder waits longer to receive it. Dividends drag the stock down on each ex-dividend date, against a call and in favour of a put.
Time to maturity is the awkward one. An American option with longer to run holds every exercise opportunity the shorter one has and adds more, so extra time can never hurt it. A European option is tied to a single date, and pushing that date out can destroy value: a deep in the money European put almost certain to be exercised is worth more when the strike arrives sooner.
| Factor increases | European call | European put | American call | American put | Same for all four? |
|---|---|---|---|---|---|
| Stock price | Up | Down | Up | Down | No |
| Strike price | Down | Up | Down | Up | No |
| Risk-free rate | Up | Down | Up | Down | No |
| Volatility | Up | Up | Up | Up | Yes |
| Time to maturity | Ambiguous | Ambiguous | Up | Up | No |
| Dividends | Down | Up | Down | Up | No |
Source: the six factors named in the chapter; signs and the final column are set out for study.
Options listed on exchanges, including those on individual stocks, are mostly American, so the right to exercise ahead of maturity is genuinely available. Whether using it pays is another matter, and for a call on a stock paying no dividends the answer is no.
Take an American call struck at USD 30 on a stock trading at USD 50, two months from maturity and with no dividends along the way. Buying at USD 30 and selling at USD 50 hands over USD 20 per share, and since a contract covers 100 shares that is USD 2,000 per contract. The money is real, yet collecting it early is wrong once interest rates are positive.
Two situations for the holder
In Situation 1 the investor wants to finish up owning the shares; in Situation 2 the investor does not.
Work through Situation 1, where shares acquired on exercise would be kept. Finish above USD 30 and early exercise was wasteful, because the USD 30 could have sat at the risk-free rate for two more months. Finish below USD 30 and early exercise locks in a loss of USD 30 minus whatever the stock is worth at expiry, while an untouched option lapses and costs nothing more. That second ending is insurance: holding the call shields the investor against the stock sinking under USD 30, and the shield vanishes on exercise.
The holder who does not want the shares
In Situation 2 the option should be sold rather than exercised. Exercising and then selling the shares captures the intrinsic value, USD 20 here, and nothing else. Selling the option captures intrinsic value plus time value, which is the worth of the optionality just described. A live American call on a non-dividend-paying stock therefore trades above its intrinsic value whenever interest rates are positive.
The formal version of that argument throws off a lower bound as a by-product. Portfolio A holds one call together with cash worth PV(K), the strike price discounted back from maturity at the risk-free rate. Portfolio B holds the stock alone.
By maturity the cash in Portfolio A has grown into exactly K. Should the stock finish above K, the call is exercised and the cash buys the share; should it finish below K, the call lapses and Portfolio A keeps K. So Portfolio A ends up worth the greater of the two, while Portfolio B is worth the stock whatever happens: Portfolio A matches it in some states and beats it in others. An inequality holding at maturity must hold today too, or an arbitrageur could buy Portfolio A, sell Portfolio B short, and sit on a position that can never lose and sometimes wins.
Positive interest rates make K bigger than PV(K), so this bound is stronger than the intrinsic value S minus K. That settles the early exercise question, since the call is worth more alive than the S minus K a holder would pocket by exercising. The ceiling is easier: a right to buy one share cannot be worth more than the share.
Dividends break the clean result. A stock falls when it goes ex-dividend and a call holder collects nothing, so the option loses value across that date. Exercising in the final moments before an ex-dividend date, and so owning the share in time to collect, can be the better choice, depending on the size of the dividend and on how deep in the money the option is.
The comparison that decides it
Write K for the strike price and K* for the present value of K, discounted back to the current ex-dividend date from whichever comes first, the next ex-dividend date or option maturity, at the risk-free rate. Exercise on an ex-dividend date is never optimal, however high the stock has climbed, when the dividend comes in under K minus K*. Once the dividend clears K minus K*, exercise turns optimal for a stock price high enough.
Exercising early buys the dividend and costs the interest on the strike price over the waiting period, so a large dividend and a short wait favour exercise. A call struck at USD 40 on a stock trading at USD 45, with one week to maturity and a USD 2 dividend just declared, is the clear case: one week of interest on USD 40 is negligible beside USD 2.
An American call runs seven months on a stock paying USD 0.5 at the three-month point and USD 0.5 again at the six-month point. The strike price is USD 40 and the risk-free rate is 8 percent per annum with annual compounding.
Why employees exercise early
Options granted to staff on stocks that pay nothing are exercised ahead of maturity all the time. That is not irrational: a grant of this kind cannot be traded, so an employee who does not want the shares reaches the cash only by exercising and selling. The constraint, not poor judgement, explains why employee stock options go earlier than exchange-traded options.
Dividends also weaken the lower bound, and the two-portfolio argument shows by how much. Redefine Portfolio A as a European call together with cash worth PV(K) plus PV(Divs), where PV(Divs) discounts each dividend from its own ex-dividend date back to today at the risk-free rate. Portfolio B stays the stock alone.
By maturity the PV(K) piece has become K and the PV(Divs) piece has become FV(Divs), the future value of those dividends once reinvested at the risk-free rate. Where the stock finishes above the strike price, K is swapped for the share, so Portfolio A finishes worth the greater of the terminal stock price and K, plus FV(Divs). Portfolio B finishes at the terminal stock price plus FV(Divs), so Portfolio A again dominates and the ranking carries back to today.
A European call has one year left. The stock changes hands at USD 64 against a strike price of USD 60. Dividends of USD 1 each are due at the three-month, six-month and nine-month points, and the risk-free rate is 4 percent per annum with annual compounding.
Once dividends enter, the bound built from the two portfolios belongs to the European contract alone, because the right to exercise immediately puts a floor at the intrinsic value.
Put options behave differently. American puts do get exercised ahead of maturity, and this happens even where the stock pays nothing at all.
Picture a put with two months to run on a stock paying no dividends, where the shares have collapsed to USD 1 against a strike price of USD 20. The contract lets the holder sell for USD 20 an asset now worth almost nothing, so exercising straight away is close to certainly right.
Where the asymmetry comes from
Cash flows point in opposite directions for the two contract types. A call holder pays the strike price, so delay is a benefit and interest on that money accrues to the holder while the option lives. A put holder receives the strike price, so delay is a cost, and exercising early puts USD 20 in hand two months sooner, where it can earn the risk-free rate.
Something is still surrendered. Some small chance remains that the stock climbs back over USD 20 inside the two months, and that chance is the optionality destroyed on exercise. Deciding whether to exercise an American put is therefore a contest between collecting the strike price early to earn interest and keeping the optionality for the cases where the stock recovers.
What tilts the decision
Exercise grows less attractive to the holder of a put as the stock price increases, as the interest rate decreases, as the time to maturity increases, and as the dividends expected during the life of the option increase. A higher stock price shrinks the payoff from selling at the strike, a lower interest rate shrinks the reward for taking the cash early, and a longer remaining life leaves more room for the stock to climb past the strike, so the optionality given up is worth more.
Dividends work the opposite way round from the call case. Exercising a put means selling the shares and forgoing dividends still to come, so a put holder who knows dividends are on the way is more inclined to wait.
Two more portfolios pin down the put. Portfolio C holds a European put plus one share, and Portfolio D holds cash worth PV(K).
Where the strike price beats the stock at maturity, the put is exercised, the share goes, and Portfolio C is worth K; otherwise the put lapses and Portfolio C is worth the stock. So Portfolio C finishes worth the greater of the terminal stock price and K, while Portfolio D grows into exactly K. Portfolio C is never behind, and the ranking carries back to today.
Exercise is available at any moment on the American contract, which supplies a tighter floor of its own at the intrinsic value K minus S. The ceilings follow from the most each contract can deliver. A European put pays at most K, and only on the maturity date, so it cannot be worth more than PV(K) today. An American put converts into K at any time, so its ceiling is K.
Adding dividends
With dividends in the picture Portfolio D is enlarged to PV(K) plus PV(Divs), while Portfolio C keeps the reinvested dividends alongside the share. Portfolio C then finishes at the greater of the terminal stock price and K, plus FV(Divs), and Portfolio D at K plus FV(Divs). The comparison survives and the floor is pushed up.
A three-month European put is struck at USD 25 while the underlying pays no dividends and trades at USD 22. Take the risk-free rate as 6 percent per year with annual compounding.
Put-call parity ties the price of a European call to the price of a European put on the same stock, sharing a strike price and a maturity. The portfolios already built do the work, because they turn out to be worth the same at maturity rather than one dominating the other. Portfolio A is a European call together with cash worth PV(K) plus PV(Divs); Portfolio C is a European put plus one share.
Matching the two payoffs
Follow Portfolio A forward. Its PV(K) has become K and its PV(Divs) has become FV(Divs). If the terminal stock price beats the strike price, the call is exercised, K buys the share, and the portfolio holds one share plus FV(Divs); if not, the portfolio holds K plus FV(Divs). Portfolio A is therefore worth the greater of the terminal stock price and K, plus FV(Divs).
Portfolio C has reinvested its dividends into FV(Divs). Where the terminal stock price beats the strike price, the put goes unused and the portfolio is the share plus FV(Divs); where it does not, the share is sold for K and the portfolio is K plus FV(Divs). Portfolio C lands on the same expression.
| Component | If S(T) > K | If S(T) < K | |
|---|---|---|---|
| Portfolio A | Call option | S(T) minus K | 0 |
| Portfolio A | Cash PV(K) | K | K |
| Portfolio A | Cash PV(Divs) | FV(Divs) | FV(Divs) |
| Portfolio C | Put option | 0 | K minus S(T) |
| Portfolio C | Share and its dividends | S(T) plus FV(Divs) | S(T) plus FV(Divs) |
| Difference | Portfolio A minus Portfolio C | 0 | 0 |
Source: the portfolio comparison in the chapter; the closing row is added here.
Two portfolios worth the same in every state at maturity have to cost the same today, or the cheaper could be bought and the dearer sold for certain profit.
A stock paying no dividends is quoted at USD 29. A four-month call struck at USD 30 costs USD 2, the risk-free rate is 4 percent per annum with annual compounding, and no arbitrage opportunities exist.
One limit matters. Parity is an equality for European options only, and no exact relationship of this kind links the price of an American call to the price of the matching American put.
When parity fails the trade writes itself. Price both portfolios today, buy whichever is cheaper, sell whichever is dearer, and carry the package to maturity, where the payoffs cancel and the opening price gap is left behind as profit. Reverse the mispricing and every leg of the trade simply flips.
A stock paying no dividends trades at USD 52. A six-month European call struck at USD 55 is quoted at USD 3.20, and a six-month European put on identical terms is quoted at USD 6.00. The risk-free rate is 5 percent per year with annual compounding.
The same logic enforces the bounds
Bound violations are traded the same way. Take a four-month European call selling for USD 2.50 on a stock at USD 54 struck at USD 50, with a dividend of USD 1.50 due in one month and a risk-free rate of 3 percent per annum, annually compounded. The floor works out at 2.99, so the option is under it. Buying the call and selling the stock short brings in USD 51.50 today, against a USD 1.50 dividend to fund after one month and at most USD 50 to close the short position at four months, so the worst case still leaves a profit.
Everything so far assumed a stock. Options are written on currencies, indices, commodities and much besides, each paying income in its own way. Rebuilding the argument around forward prices sidesteps the problem, because a forward price already contains whatever income the asset throws off before maturity.
Redefine the portfolios once more. Portfolio A is a European call together with cash worth PV(K). Portfolio C combines a European put, a forward contract to buy the asset for F at option maturity, and cash worth PV(F), where F is the forward price for a contract maturing when the options do.
At maturity the cash in Portfolio A has become K, so it is worth the asset price if the call is exercised and K otherwise. In Portfolio C the cash has become F, exactly what the forward contract calls for, so the asset is delivered, and with the put alongside it that portfolio is worth K when the asset price sits below K and the asset price otherwise. Both finish at the greater of the strike price and the asset price. A forward contract costs nothing at inception, so Portfolio C is priced today at the put price plus PV(F).
What falls out of it
Neither option price can go under zero, and setting each in turn to zero produces general floors: a call is worth at least PV(F) minus PV(K), extending the earlier stock result, and a put is worth at least PV(K) minus PV(F).
The equation also answers a neat question. A European call and a European put on one asset, sharing a strike price and a maturity, cost the same when PV(F) equals PV(K), which happens when F equals K. Equal prices signal a strike price set at the forward price.
The practical gain is that no dividend forecast is needed. Forward prices reflect all income on the asset up to maturity, so an analyst can take quoted forwards, interpolate between the maturities that trade to reach the option maturity, and apply parity without estimating a dividend. For an option on the GBP/USD exchange rate the forward price is built from the USD risk-free rate and the GBP risk-free rate, and for an index option from the risk-free rate and the dividend yield.