6.1.1 Positive part function

We define the positive part function¹ by

The plot of the function is shown in the top-left part of Figure 6.1. One may recognize the notation ( · )₊ as it was introduced in Chapter 5 to shorten the writing of call and put options’ payoffs.

6.1.2 Maximum function

We can generalize the positive part function as follows. For a fixed real number a, we consider the maximum function x↦max {x, a}, defined by

With this notation, a is a parameter of the function while x is the variable. The maximum function is illustrated in the top-right plot of Figure 6.1. Note that if we take a = 0, then we recover the positive part function.

When working with call and put options, one very important property of the maximum function is its behavior with respect to translations. For any real number b, we have

or, equivalently,

These properties are easily deduced by looking at the plot of max {x, a}, as given in Figure 6.1, and shifting it by ± b.

6.1.3 Stop-loss function

We can also generalize the positive part function in another direction. For a fixed real number a, we will call the function x↦(x − a)₊, the stop-loss function at a. From the definition of the positive part function, we can write

Again, with this notation, a is a parameter of the function while x is the variable.

One should easily recognize from the stop-loss function the structure of a call option payoff. Moreover, the stop-loss function is well known in actuarial science as (x − a)₊ determines the amount paid by the insurer when a claim x is subject to a deductible a.

It is also possible to look at the reversed version of the stop-loss function. For a fixed real number a, we have

Again, with this notation, a is a parameter while x is the variable. Both the stop-loss and reversed stop-loss functions are illustrated in the bottom panels of Figure 6.1. One should recognize from the reversed stop-loss function the payoff structure of a put option.

Using the above properties of the maximum function, we get that, for any other real number b,

(6.1.1)

and, in particular,

(6.1.2)

Finally, if we take the difference of a stop-loss function with its reserved version, we get the following simple relationship:

(6.1.3)

Indeed, since the functions (x − a)₊ and (a − x)₊ are never different from zero at the same time (for the same value of the variable x), we have

The identities in equations (6.1.2) and (6.1.3) are at the core of the upcoming parity relationships between vanilla options and the underlying asset.

6.1.4 Indicator function

For a fixed real number a, we define the indicator function of {x > a} by

With this notation, a is a parameter of the function while x is the variable. The function takes the value 1 if and only if the variable x is greater than a; otherwise it takes the value 0. The same function can also be written as

In fact, in general, for any interval (a, b), we can define

In other words, if the variable x is in the interval (a, b) then the indicator function is equal to 1, otherwise it is equal to 0. For example, the graph of is given in Figure 6.2. Indicator functions will be used mainly in Section 6.3.

6.2.1 Simple payoff design

Let us consider the following four assets:

• a risky asset not generating any income (for example, a non-dividend paying stock) and whose final payout at time T is S_T;
• a call and a put option issued on the above (underlying) asset, both with maturity T and strike price K;
• a zero-coupon bond with maturity time T and face value K or, equivalently, a risk-free bank account whose accumulated balance with capital and interest is K at time T.

Using the maximum function, we will illustrate how to design two simple payoffs:

1. S_T − K, which is a payoff similar to that of a forward contract with maturity time T and delivery price K;
2. max (S_T, K), i.e. the payoff of an investment guarantee with maturity time T.

First, using equation (6.1.3) with a = K and x = S_T, we can write

(6.2.1)

Therefore, the payoff of a strategy which consists in being long a call option and short a put option is equal to the payoff of a strategy consisting in being long one unit of the underlying asset and having a loan worth K, principal and interests included. No matter the scenario, i.e. the value of the random variable S_T, the two positions will have the same payoff.

Reorganizing equation (6.2.1), we can write

from which we deduce that a long call option can be replicated by a long put option together with a long position in a contract with payoff S_T − K (or, said differently, one unit of the underlying asset and a loan worth K, principal and interests included). Again, no matter the value of S_T, the two positions will have the same payoff. This is a synthetic call. We use the word synthetic because legally we do not own a call. Of course, we can also reorganize equation (6.2.1) to create a synthetic put.

Second, an investment guarantee with payoff max (S_T, K), i.e. a product providing the largest value between the asset price S_T and the guarantee K at maturity, also plays an important role in life insurance as it forms the basis of many equity-linked insurance and annuity policies (see Chapter 8).

Using the translation property of the maximum function as given in equation (6.1.1), we get

(6.2.2)

or, equivalently,

(6.2.3)

Therefore, being long an investment guarantee can be mimicked by one of the two following strategies:

• a long call option together with a risk-free investment;
• a long protective put, i.e. a long put option together with one unit of the underlying asset.

Those strategies are summarized in the next table.

Again, reorganizing equation (6.2.3), we can write

which says that a long put option can be replicated by buying an investment guarantee and shorting the underlying asset. This is another synthetic put. Of course, we can also reorganize equation (6.2.2) to create another synthetic call.

6.2.2 Put-call parity

Using some of the results from the previous section, we will now derive the so-called put-call parity.

Using equation (6.2.1) (or combining equations (6.2.2) and (6.2.3)), we can write

In other words, the payoff of a call option and a risk-free investment (with final value K) is equal to the payoff of a put and one unit of the underlying asset.

By the no-arbitrage principle, since none of these securities generates any cash flows between inception time 0 and maturity time T, their prices at any previous times must also satisfy this relationship: for any 0 ⩽ t ⩽ T,

(6.2.4)

where C_t and P_t are the call and put prices at time t. In particular, at inception, we have

(6.2.5)

This is the classical put-call parity relationship.

In other words, at time 0, buying a call and investing Ke^{− rT} at the risk-free rate r will generate the same payoff as buying a put and buying the underlying asset.

Example 6.2.1 Call and put prices

A stock currently trades at $25 and a 3-month at-the-money European call option on that stock sells for $2. If the annual interest rate is 2.7% (continuously compounded), what is the current (no-arbitrage) price of an otherwise equivalent put option?

We have C₀ = 2, S₀ = K = 25, r = 0.027 and T = 0.25. To avoid arbitrage opportunities, the put-call parity in (6.2.5) must hold. Therefore, we must have

from which we deduce that P₀ = 1.8318.

 ◼ 

Example 6.2.2 Risk-free rates

A 2-month European call currently sells for $2 whereas a European put option sells for $3. Their common strike price is $53 and the stock trades for $51. Let us determine the risk-free rate you can lock in by trading in the stock, the call and the put.

Using the put-call parity of equation (6.2.5), we can replicate the payoff of a long bond with capital K (i.e. a risk-free investment) using a long put, a long stock and a short call:

where K = 53 and T = 1/6. Therefore, the annual risk-free rate (continuously compounded) that we can lock in by trading in the other assets is

 ◼ 

Example 6.2.3 Arbitrage opportunities arising from the put-call parity

With a strike price of $50, a 1-month European call currently trades for $3 whereas the corresponding European put option sells for $2.50. The stock currently trades for $51. Assume the 1-month rate is 0%. Let us determine whether there are arbitrage opportunities between the four assets and, if so, let us describe how to exploit this opportunity.

First of all, we verify whether the put-call parity of equation (6.2.5) holds or not. Since we have

and

the put-call parity is violated. Therefore, there are arbitrage opportunities. Indeed, in the current situation, a long put with a long stock is overpriced compared with a long call and an investment at the risk-free rate. In order to benefit from this mismatch, we can buy the cheapest portfolio (call and bond) and sell the costliest one (put and stock). Hence, we short-sell/write a put and short-sell the stock, and we buy the call and invest at the risk-free rate. This strategy generates an immediate cash amount of 0.50 and, as we saw above, at maturity all positions will offset each other. We have thus identified one arbitrage opportunity arising from the (violated) put-call parity.

 ◼ 

In conclusion, the put-call parity in equation (6.2.4) links four financial assets through a no-arbitrage relationship: a call, a put, a stock and a zero-coupon bond (risk-free investment). Therefore, it is always possible to replicate one of these assets using the other three: to replicate a

1. long call: we need a long put, a long stock and a short bond (i.e. a loan);
2. long put: we need a long call, a long bond and a short stock;
3. long stock: we need a long call, a long bond and a short put;
4. long bond: we need a long put, a long stock and a short call.

6.3.1 Decomposing call and put options

Using indicator functions we can rewrite the payoff of a call as

(6.3.1)

and the payoff of a put as

(6.3.2)

Therefore, a call option is the combination of two simple products (a similar decomposition can also be obtained for the put option):

• a long position in a derivative that pays S_T, if S_T > K, and 0 otherwise;
• a short position in a derivative that pays K, if S_T > K, and 0 otherwise.

It turns out that these two derivatives exist and are collectively known as binary (or digital) options.

6.3.2 Binary or digital options

A binary or digital option is a basic derivative in which, at maturity, the holder receives an amount of money or the underlying asset, if the underlying asset price at maturity is small/large enough, or nothing otherwise. Two elements define binary options: the condition under which money is received and what is paid in that case (cash or asset).

A binary call option is based upon the condition that S_T > K, i.e. the option is activated if and only if S_T > K (at maturity), while a binary put option is based on the condition S_T < K, i.e. the option is activated if and only if S_T < K.

Then, for both binary calls and binary puts, the amount received at maturity is either fixed or calculated with the underlying asset price. For an asset-or-nothing option, the buyer receives the cash equivalent of the stock price at maturity when the condition is activated and for a cash-or-nothing option, the buyer receives $1. Therefore, most binary options are settled in cash, but it is possible to opt for a physical delivery of the asset in an asset-for-nothing binary option.

To summarize, the payoffs of binary options are given by

• asset-or-nothing call: ;
• asset-or-nothing put: ;
• cash-or-nothing call: ;
• cash-or-nothing put: .

They are illustrated in Figure 6.3.

Example 6.3.1 Payoff of binary options

Three-month binary options are issued on the stock of ABC inc. For a strike price of $50, compute the payoffs of the four binary options in the scenario where the final stock price is equal to $52 (after 3 months).

In the scenario where S_T = 52, we get:

• asset-or-nothing call: the condition S_T > K is met because 52 > 50. Therefore, the holder receives $52;
• asset-or-nothing put: the condition S_T < K is not met because 52 > 50. Therefore, the holder receives nothing;
• cash-or-nothing call: the condition S_T > K is met because 52 > 50. Therefore, the holder receives $1;
• cash-or-nothing put: the condition S_T < K is not met because 52 > 50. Therefore, the holder receives nothing.

 ◼ 

Finally, from equation (6.3.1), the payoff of a standard call option is equal to being long one unit of an asset-or-nothing call and short K units of a cash-or-nothing call, all options having the same strike price K. Similarly, from equation (6.3.2), the payoff of a standard put option is equal to being long K units of a cash-or-nothing put and short one unit of an asset-or-nothing call, all options having the same strike price K.

Example 6.3.2 Financial engineering with binary options

Consider the following assets:

• a non-dividend paying stock whose current price is $45;
• a standard put option on that stock with a strike price of $41 currently sells for $2;
• a cash-or-nothing put option on that stock with a strike price of $41 sells for $0.27.

Let us find the current price of an otherwise-equivalent asset-or-nothing put.

As a standard put option with strike price 41 is equivalent to being long 41 units of a cash-or-nothing put also with strike price 41 and short one unit of an asset-or-nothing call with strike price 41, at time 0, we must have

where x is the initial price for an asset-or-nothing put. We find that x = 9.07.

 ◼ 

6.3.3 Gap options

A gap call option (resp. gap put option) is an option to buy (resp. to sell) the underlying asset for K if the stock price at maturity S_T is greater than (resp. less than) a predetermined trigger level H. Here, K is known as the strike price whereas H is the trigger price.

Therefore, the payoff of a gap call option is equal to

whereas the payoff of a gap put option is equal to

Figure 6.4 illustrates the payoff of both the gap call and put options. Clearly, if H = K, then a gap option is equivalent to its otherwise-equivalent standard vanilla option.

 It is important to note that, depending on the relationship between K and H, it is possible that the payoff of a gap option takes on negative values. Indeed, for a gap call, if H < K and if at maturity the final underlying price is such that H < S_T < K, then in this scenario . There is a similar possibility for a gap put.

Note that we can engineer the payoffs of gap options as follows:

Therefore, we deduce that a gap call is equivalent to being long one unit of an asset-or-nothing call (with strike price H) and being short K units of a cash-or-nothing call (also with strike price H). Similarly, a gap put is equivalent to being long K units of a cash-or-nothing put (with strike price H) and being short one unit of an asset-or-nothing put (also with strike price H).

Example 6.3.3 Payoff of a gap put option

The strike price of a gap put option is $74 whereas its trigger price is $70. Compute the payoff of this gap put option in the scenario that the stock price is equal to $72 at maturity.

The gap put option’s payoff is activated if the stock price at maturity is below the trigger price, otherwise it is worthless. In this scenario, since S_T = 72 > 70 = H, the payoff is equal to 0.

 ◼ 

Example 6.3.4 Price of a gap call option

The price of an asset-or-nothing call option with a strike price of $30 is $9 and the price of a cash-or-nothing call option also with a strike price of $30 is $0.25. Find the price of a gap call option with a trigger price of $30 and a strike price of $25.

As we saw before, the payoff of this gap call can be written as

This payoff is the same as being long one unit of the asset-or-nothing call and being short 25 units of the cash-or-nothing call.

Therefore, the actual price of the gap call option is

 ◼ 

6.4.1 Bounds on European options prices

Let us establish bounds on the price of call and put options. Simply said, price bounds are values that a no-arbitrage price cannot exceed. The main focus will be on the financial arguments needed to obtain those bounds rather than their sharpness, i.e. how close they are from the true value. In fact, as we will see in the examples below, most of those bounds are not very informative. The bounds are perhaps more useful to better understand the design of call and put options and the behavior of American options, as we will see in Section 6.5.

Again, unless stated otherwise, the calls and puts are written on the same underlying asset S not generating any cash flows/dividends, have the same strike price K and maturity date T.

Let us start by looking for lower bounds. As the payoff of a call and the payoff of a put option are always (in all scenarios) non-negative, we must have C₀ ⩾ 0 and P₀ ⩾ 0, or clearly there is an arbitrage opportunity.

We can say more. From the put-call parity in equation (6.2.5), we have P₀ = C₀ + Ke^{− rT} − S₀ and, since P₀ ⩾ 0, we then have

from which we deduce that

Finally, since we must also have that C₀ ⩾ 0, we conclude with the following:

(6.4.1)

Using similar arguments, we can also obtain the following lower bound for a put option’s initial price:

(6.4.2)

Example 6.4.1 Lower bounds on call and put prices

The stock of ABC inc. currently trades at $34. If the 3-month rate is 1% (continuously compounded), identify lower bounds on at-the-money call and put options prices both maturing in 3 months.

From equation (6.4.1), we have

and from equation (6.4.2), we have

Therefore, for a strike price of $34, if we find a call trading for less than 8.49 cents, there is an arbitrage opportunity. On the other hand, the bound on the put option price is not informative.

 ◼ 

Let us now look for upper bounds. First, using arbitrage arguments, we will show that C₀ < S₀. To do so, let us assume the opposite, i.e. let us assume that S₀ ⩽ C₀. We consider the following portfolio:

• sell the call;
• buy the underlying asset;
• lend C₀ − S₀ at the risk-free rate.

This portfolio does not require any investment at time 0. Therefore, if it yields a non-negative payoff at maturity, it is an arbitrage opportunity and the inequality S₀ ⩽ C₀ cannot be true.

At maturity, the payoffs are depicted in the following table.

Selling a call and buying the underlying asset yields the strictly positive final payoff min {S_T, K}. Moreover, our investment at the risk-free rate yields (C₀ − S₀)e^rT ⩾ 0 at maturity. Overall, the total value of the portfolio at maturity is given by the strictly positive payoff

This is an arbitrage opportunity. Consequently, we must have C₀ < S₀.

For the put, using the put-call parity together with the newly obtained inequality C₀ < S₀, we get

yielding P₀ < Ke^{− rT}.

In conclusion, we have obtained the following lower and upper bounds on the initial prices of European call and put options (on an underlying that is not generating cash flows, i.e. a non-dividend-paying asset):

Example 6.4.2 Bounds on call and put options

Recall example 6.4.1. We now compute the upper bounds on those call and put prices. The present value of the strike price is . Therefore, the bounds on the call option premium are

whereas the bounds on the put option premium are

Any call or put option (with these features) traded for a price outside these bounds will generate an arbitrage opportunity.

 ◼ 

Since these relationships will hold at any other time t in between inception and maturity, we have: for all 0 ⩽ t ⩽ T,

(6.4.3)

(6.4.4)

6.4.2 Put-call parity with dividend-paying assets

We already know that holding a stock or having a long position in a forward contract is quite different when dividends are paid by the stock. This is also the case for options. When holding a dividend-paying stock, the shareholder receives the dividends (cash flows between time 0 and time T). However, when holding a call option on this stock, no dividends are received during the life of the option.

Suppose now that S_t is the ex-dividend time-t price of a stock, i.e. after payment of dividends. We know that the payoff of a call (or a put) is calculated on the ex-dividend price. From equation (6.2.1), which says that

we know that being long a call option and short a put option is equal to the payoff of a long forward contract with delivery price K (not the forward price, as it is the convention on the market). We obtained, in Chapter 3, the value of such a contract.

By the no-arbitrage principle, this relationship also holds at any prior time t, even for dividend-paying assets, because neither the call, the put nor the forward generates any cash flows between inception and maturity. Consequently, using the same notation as in Chapter 3, we have that, for any time t such that 0 ⩽ t ⩽ T:

• If dividends are paid discretely (cash dividends) and reinvested at the risk-free rate for a total accumulated amount of D_T, then
  (6.4.5)
  where we used the fact that the time-t value of a (long) forward contract on a discrete-dividend paying asset is given by S_t − Ke^{− r(T − t)} − D_t (see Chapter 3). In particular, at time 0, we have
  
  This is the put-call parity relationship when the underlying asset is paying discrete dividends.
• If dividends are paid continuously at rate γ (dividend yield) and reinvested in the asset, then
  (6.4.6)
  where we used the fact that the time-t value of a (long) forward contract on an asset paying continuous dividends is given by . In particular, at time 0, we have
  
  This is the put-call parity relationship when the underlying asset is paying continuous dividends.

Again, if one of these relationships is violated, then there is an arbitrage opportunity.

Example 6.4.3 Put option on a stock paying discrete dividends

A dividend-paying stock currently trades at $35 and a $1 dividend is expected in a month. An at-the-money 2-month call option on that stock trades at $4 and the risk-free rate is 3% (continuously compounded). To avoid arbitrage opportunities, what should be the price of a 2-month at-the-money put option?

Using the put-call parity of equation (6.4.5), we get

and deduce that P₀ = 4.8229.

 ◼ 

Example 6.4.4 Put-call parity on a stock index

The S&P 500 index is currently at 1000. A 6-month option to sell the index for 990 currently trades for 25 whereas a similar option to buy the index trades for 50. You also know that the (average) dividend yield generated by the stocks in the index is 3% (continuously compounded). What is the price of a 6-month Treasury zero-coupon bond?

Let B₀ be the initial price of a 6-month zero-coupon bond (with face value 1). Using the put-call parity of equation (6.4.6), we get

We get B₀ = 0.9698.

 ◼ 

6.5.1 Lower bounds on American options prices

We will first compare European and American options that are otherwise identical, i.e. they are written on the same underlying asset, have the same strike price and maturity. To simplify the comparison, let us denote by P^A_t and P^E_t the time-t value of an American put and a European put, respectively. Also, let us denote by C^A_t and C^E_t the time-t value of an American call and a European call, respectively. We will use this notation temporarily, until the end of this section.²

Comparing European and American option prices

Intuitively, an American option should be worth at least as much as its European counterpart because an American option holder is able to exercise any time prior to or at maturity. As it gives more flexibility, it should have more value; otherwise there would be an arbitrage opportunity because a rational investor should not lose from having the possibility to exercise early. This is the case for both call and put options.

Indeed, assume instead that the European call is worth more than its American counterpart, i.e. assume C^A₀ < C₀^E. In this case, we buy the American option, sell the European option and invest the (positive) difference C^E₀ − C₀^A at the risk-free rate. We hold on to the American call until maturity, i.e. we do not exercise early, so both options will offset each other at maturity. We will thus end up with the profits generated by the risk-free investment. The same strategy works for put options and for any other time t between inception and maturity.

In conclusion, we have obtained the following relationships:

(6.5.1)

for all 0 ⩽ t ⩽ T.

Comparing with the exercise value

The (early-)exercise value or intrinsic value of an American option is the amount received upon exercise. Take, for example, an American put struck at K with expiration date T. The holder can decide at any time 0 ⩽ t ⩽ T to receive K in exchange for delivery of an asset worth S_t: the (early-)exercise value of this put is then K − S_t.

From equation (6.4.3) and equation (6.5.1), we have that

(6.5.2)

for any 0 ⩽ t ⩽ T. In particular, we have C^A_t ⩾ S_t − K. In other words, at any time t during its life, the price of the American call option is larger than its exercise value S_t − K.

This is also true for American put options: for any 0 ⩽ t ⩽ T, we have P^A_t ⩾ K − S_t. If, at some time t, we observe P^A_t < K − S_t, then there is an arbitrage opportunity: we buy the American put and exercise it right away, making an immediate risk-less profit of K − S_t − P^A_t > 0.

Note that we have not said that American option prices are strictly greater than their early-exercise values, as they could be equal. Those (random) times when the American option price equals its early-exercise value play an important role in the timing of early exercise. As the latter question will only be answered later, we now focus our attention on whether a rational investor should early-exercise an American call or put.

6.5.2 Early exercise of American calls

In equation (6.5.1), we have verified that an American call is always worth at least as much as its European counterpart, i.e. C^A_t ⩾ C_t^E. It would be natural to think/believe that this last inequality is in fact a strict inequality. The idea behind this belief is that the underlying stock S may reach its maximum value at some point in time during the life of the option, so that early exercise would be optimal at that time. In fact, this is never the case for American calls written on non-dividend-paying stocks:

It is never optimal to early-exercise an American call written on a non-dividend-paying stock.

Indeed, even if an investor believes the stock price has reached its peak value at some time t, then the American call price C^A_t still is larger than its early-exercise value, as obtained above. Mathematically, for all t, we have

(see also equation (6.5.2)) no matter how large S_t is. This means that if the underlying stock S reaches a lifetime maximum, then the American call option price C^A will also contain this information and be worth more. Consequently, early-exercising an American call option at any time t is sub-optimal in the sense that we replace a financial position worth C^A_t by another position worth less, that is S_t − K.

To summarize, getting rid of an American call (on a non-dividend-paying asset) by exercising it before maturity is never a good idea. This would be equivalent to exchanging an asset that is worth more (the American call) with something of a lesser value (the exercise value). As shown by the bounds, selling the American call yields a greater income than exercising it.

Finally, note that if dividends are paid by the underlying asset S, then it could be optimal to early-exercise an American call option. For example, if dividends are discrete, then early exercise would occur right before the dividend payment.

6.5.3 Early exercise of American puts

Whether we should exercise an American put option early or not is more complicated. Using the put-call parity in equation (6.2.4), we get easily that

for all 0 ⩽ t ⩽ T. The objective is to compare P^E_t with the exercise value K − S_t.

In this case, since C^E_t ⩾ 0 and K(1 − e^{− r(T − t)}) ⩾ 0, we cannot deduce that C^E_t − K(1 − e^{− r(T − t)}) is always positive or always negative. In fact, it is sometimes positive, sometimes negative. Therefore, we cannot say if the European put is always worth more than its exercise value as we did for the European call.

If we were to exercise the American put option at time t, we would receive K − S_t. However, the European put option price P^E_t might be worth more or less than the exercise value. The same can be said about the American put option price P^A_t. In conclusion: it is not always optimal to early-exercise (or not) an American put option on a non-dividend-paying stock.

Early exercise of an American put on a non-dividend-paying stock

With the above analysis, we see that it will be optimal to early-exercise an American put at (random) time τ as soon as

because, in this scenario, we will have

which is a strict inequality. Holding on to the contract (which is here given by the European put value) is worth less than the exercise value K − S_τ, which at that time is equal to the American put value P^A_τ. Note that time τ is a random time, as we do not know if and when such a scenario will occur.

If the stock price drops dramatically (approaches 0), then the call option price will also get closer to zero. If this is the case at another (random) time τ, then the European put option is worth

since then C^E_τ ≈ 0 whereas the exercise value of the corresponding American put is approximately K.