# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] Option bounds

Put-Call Parity

We introduced these new products called Call and Put options by unbundling a forward and we had the following relationship:

$\mbox{Long Forward = Long Call + Short Put}$

or alternatively,

$F_T = S_T - K = \mbox{max}(S_T - K, 0) - \mbox{max}(K - S_T, 0)$

$\mbox{Value of Forward at T = Call Payoff at T - Put Payoff at T}$

Since this is a relationship about a product being ‘broken’ into its constituents, if this holds at time $T$ (i.e. in future value terms), this must also hold today (i.e. in PV terms), and we have the following result:

$\mbox{Value of Forward Today = Value of Call Today - Value of Put Today}$

$\mbox{PV}(S_T - K) = \mbox{Call Price Today (C) - Put Price Today (P)}$

or more succinctly in PV terms as:

$\boxed{S - PV(K) = C - P}$

This is referred to as the Put-Call Parity in options markets.

Having introduced the machinery of payoff diagrams we may also see this graphically. Consider the following payoff. It is known by alternative names of a Call + Bond or a Protective Put.

(Call + Bond / Protective Put; Click to zoom)

The reason they are are known by multiple names is that one can break the same payoff down alternatively as a portfolio of a Call Option (C) and a Zero-Coupon Bond (PV(K)), or a portfolio of a Put Option (P) and Stock (S) as:

(Call + Bond; Click to zoom)

or

(Protective Put; Click to zoom)

Since both Call + Bond and Protective Put represent the same payoff at expiration (T), they must be valued at the same price. That is, we must have:

\begin{aligned} \mbox{Call Option + Zero Coupon Bond} = \mbox{Protective Put} &= \mbox{Put Option + Stock} \\ \Rightarrow \mbox{PV(Call Option) + PV(Zero Coupon Bond)} &= \mbox{PV(Put Option) + PV(Stock)} \end{aligned}

Writing the present values of a Call and Put Option as $\mbox{C}$ and $\mbox{P}$ respectively, we again get the Put-Call Parity as:

$\boxed{C + PV(K) = P + S}$

So if one knows the price of a Put Option on a stock with strike $K$ and time to maturity $T$, one can find the price of the Call Option on the same stock for same strike $K$ and time to maturity $T$.

Option Bounds

Call Option: Lower Bound

Continuing with the same examples of a Call + Bond and Protective Put we now compare the payoff from a Call + Bond with that from holding a Stock, i.e. geometrically we compare the following two payoffs:

(Comparing Call + Bond and Stock; Click to zoom)

It’s clear that when the stock price at expiration $S_T$ is more than $K$, both a Call + Bond and a Stock have the same payoff, but whenever the stock price at expiration, $S_T$ is less than $K$, a Call  + Bond offers $K \ge S_T$, but a Stock offers only $S_T$.

That is, in many ‘states of the world’ a Call + Bond offers the same payoff as holding a stock, but in some states it offers more than what holding a stock offers. This suggests that the present value of Call + Bond must always be more than or equal to that of holding a simple stock. That is, we must have:

\begin{aligned} \mbox{PV(Call + Zero Coupon Bond)} &\ge \mbox{Stock} \\ \Rightarrow \mbox{PV(Call Option)} &\ge \mbox{PV(Stock - Zero Coupon Bond)} \end{aligned}

Writing the present values of a Call as $\mbox{C}$ gives us the Lower Bound for the present value (price) of a Call Option as:

$\boxed{C \ge S - PV(K)}$

A more trivial lower bound for a Call Option, is, of course $C \ge 0$. If the strike price $K$ is ‘high enough’ (so as to make $S - PV(K) < 0$), then, technically, the lower bound for a Call Option must be:

$\boxed{C \ge\mbox{max}\big[ S - PV(K), 0 \big]}$

Call Option: Intrinsic Value

The fact that $PV(K) \le K$ gives us another result that:

\begin{aligned} C &\ge \mbox{max}\big[S - PV(K), 0 \big] \\ \Rightarrow C &\ge \mbox{max}\big[ S - K, 0 \big] \end{aligned}

The term $\mbox{max}\big( S - K, 0 \big)$ is referred to as the Intrinsic Value of the Call Option, and a European Call Option is always worth more than its Intrinsic Value.

For American Options, the difference between the Call Option Price and its Intrinsic Value is sometimes referred to as in the market as the ‘Time Value of the Call Option’ (not to be confused with the time value of money. It is an unfortunate choice of words, but that’s how it is.) One can think of the difference between the price of the Call Option and its Intrinsic Value as that part of the Call Option price that is the ‘reward for waiting for the potential upsid’ and possibly exercising the option when the stock price is even higher. This is another way to see why American Call is no more valuable than a European call, because there is always a positive ‘reward for waiting’ for a Call on a non-dividend paying stock.

Call Option: Upper Bound

Finding the Upper Bound for the Call Option price is a lot easier, and is just equal to the stock price. Think of it this way – no one is going to pay anyone to get the right to buy a stock, when one can just purchase it directly from the market, i.e. we must have:

$\boxed{C \le S}$

Put Option: Lower Bound

Again, similar to the way we proceeded in the case of a Call Option, we now compare the payoff from a Protective Put with that from holding a Zero-coupon bond, i.e. now geometrically we compare the following two payoffs:

(Comparing Protected Put and a Zero Coupon Bond; Click to zoom)

This time is the other way around, and we say that when the stock price at expiration $S_T$ is less than $K$, both a Protected Put and a Zero Coupon Bond have the same payoff $K$, but whenever the stock price at expiration, $S_T$ is more than $K$, a Protected Put offers $S_T \ge K$, but a Zero Coupon Bond, of course, always offers only $K$.

That is, in few ‘states of the world’ a Protected Put offers the same payoff as holding a Zero Coupon Bond, but in many states it offers more than what holding a Zero Coupon Bond offers. Again, this suggests that the present value of a Protected Put must always be more than or equal to the price of a Zero Coupon Bond. That is, we must have:

\begin{aligned} \mbox{PV(Protected Put)} &\ge \mbox{PV(Zero Coupon Bond)} \\ \mbox{or } \mbox{PV(Put Option + Stock)} &\ge \mbox{PV(Zero Coupon Bond)} \\ \Rightarrow \mbox{PV(Put Option)} &\ge \mbox{PV(Zero Coupon Bond) - Stock} \end{aligned}

Similarly, writing the present value of a Put as $\mbox{P}$ gives us the Lower Bound for the present value (price) of a Put Option as:

$\boxed{P \ge PV(K) - S}$

Just like in the case of a Call Option, a more trivial lower bound for a Put Option, is, of course $P \ge 0$. If the strike price $K$ is ‘low enough’ (so as to make $PV(K) - S \le 0$), then, technically, the lower bound for a European Put Option must be:

$\boxed{P \ge \mbox{max}\big[ PV(K) - S, 0 \big]}$

Put Option: Intrinsic Value

The fact that $PV(K) \le K$ doesn’t help us much in the case of a Put Option (why? Hint: Compare $K - S$ with $PV(K) - S$ – which is larger?). The term $latex\mbox{max}(K – S, 0)$, however, is still referred to as the Intrinsic Value of the Put Option, though now one can’t say that a European Put Option is always worth more than its Intrinsic Value. The same we can say for an American Put Option, however, because an American Put can always be exercised prior to maturity and therefore must be worth at least as much as its Intrinsic Value, i.e. for an American put:

$\boxed{P_{\mbox{American}} \ge K - S}$

Put Option: Upper Bound

Upper bound for the Put Option price is similarly argued. Because one can always put money in the bank, a Put Option must never be sold for more than the value of a Zero Coupon Bond with principal $K$, i.e. we must have:

$\boxed{P \le PV(K)}$

Alternatively, one could argue that the maximum payoff at maturity from a European Put Option is just $K$, so the maximum value of a Put Option today must be just the present value of $K$, i.e. $PV(K)$.

Because it may be optimal for an American put to be exercised early, the corresponding upper bound for an American put is:

$\boxed{P_{\mbox{American}} \le K}$