Derivative Securities

A contingent claim is a contract between a seller and a buyer where the seller agrees to deliver a certain payoff at a future time. A contingent claim is called a derivative if this contract is written as a payoff depending on some underlying, such as stocks, bonds, indices, portfolios, or funds. Options are specific derivatives characterized by parameters like strike price and expiration date. While definitions may vary, the mathematical interpretation remains consistent.

Examples of Derivatives

In the presentation of the different derivatives we consider a financial market with a generic asset \(S\).

Forward/Future Contract

A forward contract is an agreement between two parties to buy or sell an asset \(S\) at a future time for a price \(K\) fixed today. The payoff for the contract owner is determined by the difference between the asset price and the agreed price:

\[ C^{fw} = S - K \]
Forward Payoff

European Put/Call Option

A European Call or Put option grants the right, but not the obligation, to buy or sell an asset \(S\) at a future time for a strike price \(K\) fixed today.
The payoff is

\[ \begin{align*} C^{call} & = (S - K)^+ = \begin{cases} S - K & \text{if } S \geq K, \\ 0 & \text{otherwise}. \end{cases}\\ C^{put} & = (K - S)^+ = \begin{cases} K - S & \text{if } K \geq S, \\ 0 & \text{otherwise}. \end{cases} \end{align*} \]

European call and put options are related by the formula

\[ C^{call} - C^{put} = S - K \]
European Call/Put Payoff

Straddle Option

The goal of a straddle option is to profit from significant price movements of the underlying asset, regardless of the direction. It grants the right to get the price deviation (positive or negative) of the underlying asset with respect to a strike \(K\).

The payoff is:

\[ C^{straddle} = |S - K| = \begin{cases} S-K & \text{ if } S\geq K\\ K-S & \text{ if } S <K \end{cases} \]

Note that a straddle option is equivalent to holding a call and a put option with the same strike.

\[ C^{straddle}(K) = C^{call}(K) + C^{put}(K) \]
Straddle Option Payoff

Butterfly Option

A butterfly option has somehow the counter effect to a straddle option in the sense that it is designed to profit from price movement around a given target \(K\) within an interval \(\underline{K} < K <\overline{K}\).

The payoff is:

\[ C^{butterfly} = \begin{cases} 0 & \text{if }S<\underline{K} \text{ or }S>\overline{K}\\ S-\underline{K} & \text{if } \underline{K}\leq S\leq K\\ \overline{K} - S & \text{if } K<S \leq \overline{K} \end{cases} \]

Note that such a straddle option can also be written as a combination of put and call options strikes

\[ C^{butterfly}(\underline{K}, K, \overline{K}) = C^{call}(\underline{K}) + C^{call}(\overline{K}) - 2 C^{call}(K) \]

Usually, the strike \(K\) is at the mid point between \(\underline{K}\) and \(\overline{K}\).
Butterfly Option Payoff

Note that each of these options can be expressed as \(f(S)\), where \(S\) is the underlying asset, and \(f: \mathbb{R} \to \mathbb{R}\) is some continuous function.

Definition: Contingent Claim

A contingent claim \(C\) is a positive random variable.(1)
1. In a strictly more general sense it does not need to be strictly positive but usually bounded from below. See for instance forward contracts.
A derivative \(C\) is a contingent claim that depends only on the information provided by the underlying on which it is written. In other terms, \(C = f(\boldsymbol{S}_1)\) for some continuous function \(f:\mathbb{R}^d \to [0, \infty)\).(1)
1. In a more general fashion, it means that \(C\) is \(\sigma(\boldsymbol{S}_1)\)-measurable, which with some proof work can be shown to be equivalent to \(C = f(\boldsymbol{S}_1)\) for some measurable function \(f\).

Pricing a Contingent Claim

Given a contingent claim \(C\), the goal is to determine a fair price \(\pi(C)\) at which it can be sold. To do this, we analyze the situation from the seller's perspective:

Time 0: The seller receives \(\pi(C)\) and deposits it into their bank account. This amount is used to invest in a strategy \(\boldsymbol{\eta}\) in \(\mathbb{R}^d\). The holdings in the bank account become \(\pi(C) - \boldsymbol{\eta} \cdot \boldsymbol{S}_0\) and the golding in assets becomes \(\boldsymbol{\eta}\cdot \boldsymbol{S}_0\). The initial portfolio value is \(\bar{V}_0(\boldsymbol{\eta}) = \pi(C)\).
Time 1: The seller delivers the payoff \(C\) to the buyer while benefiting from their investment strategy \(\boldsymbol{\eta}\). The discounted portfolio value minus the discounted payoff is:

\[ V_1(\boldsymbol{\eta}) - \frac{C}{1 + r} = \pi(C) + \boldsymbol{\eta} \cdot \Delta\boldsymbol{X}_1 - \frac{C}{1 + r} \]

In an ideal situation, the seller of the option would like to get out with gains without downside risk, that is, fully hedge the position. Hence, the seller aims to ensure:

\[ \pi(C) + \boldsymbol{\eta} \cdot \Delta\boldsymbol{X}_1 \geq \frac{C}{1 + r} \]

Such a price \(\pi(C)\) together with the smart strategy \(\boldsymbol{\eta}\) means that the seller super hedge his position. Considering all the possible prices and smart strategies that super hedge the position allows to define the lowest price at which the seller is willing to sell the option without taking risk, the super-hedging price

\[ \bar{\pi}(C) = \inf \left\{ x \in \mathbb{R} : x + \boldsymbol{\eta} \cdot \Delta\boldsymbol{X}_1 \geq \frac{C}{1 + r}, \; \text{for some } \boldsymbol{\eta} \in \mathbb{R}^d \right\}. \]

Conversely, the buyer's perspective leads to the sub-hedging price, \(\underline{\pi}(C)\):

\[ \underline{\pi}(C) = \sup \left\{ y \in \mathbb{R} : y + \boldsymbol{\nu} \cdot \Delta\boldsymbol{X}_1 \leq \frac{C}{1 + r}, \; \text{for some } \boldsymbol{\nu} \in \mathbb{R}^d \right\}. \]

It seems economically reasonable that prices \(y\) from the buyer sub-hedging their position should be lower than the prices \(x\) of the seller super-hedging their positions. This is however true if the market is arbitrage free as seen in the subsequent proposition.

To simplify our exposition, let us provide the notation for pricing measures

\[ \mathcal{P}^\ast := \left\{ P^\ast \sim P \colon E^{P^\ast}\left[ \Delta \boldsymbol{X}_1 \right] = 0 \text{ and } dP^\ast/dP \text{ is bounded}\right\} \]

In particular, the FTAP can be formulated as "The market is arbitrage free if and only if \(\mathcal{P}^\ast\) is not empty".

Remark: Geometric Interpretation I

Define \(I\) and \(J\) as the set of super- and sub-hedging prices, respectively:

\[ \begin{align*} I & := \left\{ x \in \mathbb{R}\colon x+\boldsymbol{\eta}\cdot \Delta \boldsymbol{X}_1 \geq \frac{C}{1+r}\text{ for some smart strategy }\boldsymbol{\eta} \in \mathbb{R}^d \right\}\\ J & := \left\{ y \in \mathbb{R}\colon y+\boldsymbol{\nu}\cdot \Delta \boldsymbol{X}_1 \leq \frac{C}{1+r}\text{ for some smart strategy }\boldsymbol{\nu} \in \mathbb{R}^d \right\}\\ \end{align*} \]

For which holds \(\bar{\pi}(C) = \inf I\) and \(\underline{\pi}(C) = \sup J\).

These two sets, \(I\) and \(J\), are eventually intervals

Lemma

\(I\) is either an interval of the form \([\bar{\pi}(C), \infty)\) or \((\bar{\pi}(C), \infty)\)
\(J\) is either an interval of the form \((-\infty, \underline{\pi}(C)]\) or \((-\infty, \underline{\pi}(C))\)

Proof

We show only the first assertion, the second follows the same argumentation. Clearly \(\bar{\pi}(C)\) is the lower bound of the set \(I\), whether or not it is a minimum or an infimum. We just have to show that for any \(x\) in \(I\), if \(m>0\), then \(x+m\) is also in \(I\). This follows however from the definition, as for a smart strategy \(\boldsymbol{\eta}\) such that

\[x + \boldsymbol{\eta}\cdot \Delta \boldsymbol{X}_1 \geq \frac{C}{1+r}\]

It follows immediately that

\[x +m + \boldsymbol{\eta}\cdot \Delta \boldsymbol{X}_1 \geq m +\frac{C}{1+r} \geq \frac{C}{1+r}\]

showing that \(x+m\) is in \(I\).

We are now in position to show the first proposition regarding the super- and sub-hedging price

Proposition

If the market is arbitrage free, then for any super-hedging price \(x\), sub-hedging price \(y\) and any pricing measure \(P^\ast\) it holds

\[ y \leq E^{P^\ast}\left[ \frac{C}{1+r} \right] \leq x \]

In particular, we get

\[ \underline{\pi}(C) \leq \inf_{P^\ast \in \mathcal{P}^\ast} E^{P^\ast}\left[ \frac{C}{1+r} \right]\leq \sup_{P^\ast \in \mathcal{P}^\ast} E^{P^\ast}\left[ \frac{C}{1+r} \right]\leq \bar{\pi}(C) \]

Proof

Let \(x\) be a super-hedging price and \(y\) be a sub-hedging price. By definition, there exists smart strategies \(\boldsymbol{\eta}\) and \(\boldsymbol{\nu}\) in \(\mathbb{R}^d\) such that

\[ y + \boldsymbol{\nu}\cdot \Delta \boldsymbol{X}_1 \leq \frac{C}{1+r} \leq x + \boldsymbol{\eta}\cdot \Delta \boldsymbol{X}_1 \]

Now, since the market is arbitrage free, according to the FTAP, for any pricing measure \(P^\ast\) in \(\mathcal{P}^\ast\) which is not empty, by taking expectation of this inequality, it holds

\[ y + \underbrace{E^{P^\ast}\left[\boldsymbol{\nu}\cdot \Delta \boldsymbol{X}_1\right]}_{=0} \leq E^{P^\ast}\left[\frac{C}{1+r} \right]\leq x + \underbrace{E^{P^\ast}\left[\boldsymbol{\eta}\cdot \Delta \boldsymbol{X}_1\right]}_{=0} \]

showing the first assertion.

Since this inequality holds for any super hedging price \(x\), any sub hedging price \(y\) and any pricing measure \(P^\ast\), by the definition of \(\bar{\pi}(C)\) and \(\underline{\pi}(C)\) the second assertion follows.

We however still need to define the notion of of a fair price for a contingent claim and how it relates to the super-/sub-hedging price.

Definition: Fair Price of Contingent Claims

Given a contingent claim \(C\), a price \(\pi(C)\) is called a fair price if the original financial market extended with the new financial instrument \(S^{d+1}\) given by

\[ S^{d+1}_0 = \pi(C) \quad \text{and}\quad S^{d+1}_1 = C \]

is arbitrage free.

We denote by \(\Pi(C)\) the set of all possible fair prices for the contingent claim \(C\).

In other terms, if the financial market considers the new financial instrument \(C\) traded at price \(\pi(C)\) it remains arbitrage free. As pendant to the super- and sub-hedging price, with the help of the FTAP we can connect arbitrage free prices to pricing measures as follows.

Proposition

Let \(C\) be a contingent claim on an arbitrage-free financial market. Then the set of fair prices for the contingent claim \(C\) is non-empty and given by:

\[ \Pi(C) := \left\{ E^{P^\ast}\left[ \frac{C}{1+r} \right] : P^\ast \sim P \text{ pricing measure with bounded density} \right\}. \]

Proof

A price \(\pi(C)\) for \(C\) is fair if the extended financial market is arbitrage-free. By the FTAP, it follows that there exists a pricing measure \(\hat{P}\) equivalent to \(P\) with bounded density such that:

\[ E^{\hat{P}}\left[\frac{\boldsymbol{S}_1}{1+r}\right] = \boldsymbol{S}_0, \quad \text{and} \quad E^{\hat{P}}\left[\frac{C}{1+r}\right] = \pi(C). \]

In particular, \(\hat{P}\) is a pricing measure equivalent to \(P\) for the smaller market \(\boldsymbol{S}\), that is, an element of \(\mathcal{P}^\ast\), showing that:

\[ \Pi(C) \subseteq \{ E^{P^\ast}[C/(1+r)] : P^\ast \in \mathcal{P}^\ast \}. \]

Reciprocally, let \(\pi(C)\) be an element of \(\{ E^{P^\ast}[C/(1+r)] : P^\ast \in \mathcal{P}^\ast \}\). It follows that \(\pi(C) = E^{P^\ast}[C/(1+r)]\) for some \(P^\ast \sim P\) with bounded density. Hence, \(P^\ast\) is a pricing measure equivalent to \(P\) for the extended market, showing by the FTAP that the extended market is arbitrage-free. Hence, \(\pi(C) \in \Pi(C)\), proving the reverse inclusion.(1)

The proof is not fully complete unless we can show that we can show that \(\Pi(C)\) is non-empty. As done previously, we pick a probability measure \(\tilde{P}\) equivalent to \(P\) such that \(E^{\tilde{P}}[C] < \infty\). Under \(\tilde{P}\), the market is arbitrage-free. The FTAP guarantees the existence of a pricing measure \(P^\ast\) equivalent to \(P\) with bounded density. In particular, \(E^{P^\ast}[C] < \infty\), and therefore:

\[ \pi(C) = E^{P^\ast}[C/(1+r)] \in \Pi(C). \]

We are now in place to show the relation ship between super- sub-hedging prices, pricing measures and fair prices. In a nuttshell that the following illustration holds

Super- Sub-Hedging

Theorem: Super/Sub Hedging and Fair Prices

Let \(C\) be a contingent claim. For

\[ \begin{align*} J & = \left\{ y\in \mathbb{R}\colon y + \boldsymbol{\eta}\cdot \Delta \boldsymbol{X}_1 \leq \frac{C}{1+r} \text{ for some }\boldsymbol{\eta}\in \mathbb{R}^d \right\} && \text{Risk free subhedging prices}\\ \underline{\pi}(C) & = \sup J && \text{Sub-hedging price}\\ I & = \left\{ x\in \mathbb{R}\colon x + \boldsymbol{\nu}\cdot \Delta \boldsymbol{X}_1 \leq \frac{C}{1+r} \text{ for some }\boldsymbol{\nu}\in \mathbb{R}^d \right\} && \text{Risk free superhedging prices}\\ \underline{\pi}(C) & = \inf I && \text{Super-hedging price}\\ \Pi(C) & = \left\{ \pi(C)\colon \text{fair prices} \right\} && \text{Fair prices}\\ \mathcal{P}^\ast & = \left\{ P^\ast \sim P \colon P^\ast \text{ is a pricing measure} \right\} && \text{Pricing measures} \end{align*} \]

If the market is arbitrage free, then it holds that \(J\), \(I\), and \(\Pi(C)\) are intervals such that \(J\leq \Pi(C) \leq I\). Furthermore

\[ \begin{align*} J & = (-\infty, \underline{\pi}(C)] & \Pi(C) & = \left\{ E^{P^\ast}\left[ \frac{C}{1+r} \right] \colon P^\ast \in \mathcal{P}^\ast \right\} & I & = [\overline{\pi}(C), \infty ) \end{align*} \]

and

\[ \begin{align*} \underline{\pi}(C) & = \inf \Pi(C) & \overline{\pi}(C) & = \sup \Pi(C) \end{align*} \]

Proof

We already saw that \(J\) and \(I\) are intervals (see geometric interpretation remark above). We also say that \(\Pi(C) = \{E^{P^\ast}[C/(1+r)]\colon P^\ast \in \mathcal{P}^\ast\}\) and that \(J \leq \Pi(C) \leq I\). The fact that \(\Pi(C)\) is also an interval follows directly from \(\mathcal{P}^\ast\) is a convex set, same argumentation as for \(\mathcal{C}\) in the proof of the FTAP.

We are left to show that \(\underline{\pi}(C) \in J\) and \(\overline{\pi}(C) \in I\) and that \(\underline{\pi}(C) = \inf \Pi(C)\) as well as \(\overline{\pi}(C) = \sup \Pi(C)\).

Let us proove that \(\overline{\pi}(C) = \sup \Pi(C)\). We already know that \(\overline{\pi}(C) \geq \sup \Pi(C)\) Suppose \(\sup \Pi(C) < \infty\), otherwise the equality is trivial. Let \(m > \sup \Pi(C)\). By definition, the market extended with \((m, C)\) admits an arbitrage opportunity. That is, there exist \(\boldsymbol{\eta} \in \mathbb{R}^d\) and \(\mu \in \mathbb{R}\) such that:

\[ \begin{cases} P\left[\boldsymbol{\eta} \cdot \Delta \boldsymbol{X}_1 + \mu \left(\frac{C}{1+r} - m\right) \geq 0\right] = 1, \\ P\left[\boldsymbol{\eta} \cdot \Delta \boldsymbol{X}_1 + \mu \left(\frac{C}{1+r} - m\right) > 0\right] > 0. \end{cases} \]

Since the original market is arbitrage-free, it follows that \(\mu \neq 0\). Taking the expectation of the positive random variable \(\boldsymbol{\eta} \cdot \Delta \boldsymbol{X}_1 + \mu \left(\frac{C}{1+r} - m\right)\) with respect to \(P^\ast \in \mathcal{P}^\ast\) yields:

\[ \mu E^{P^\ast}\left[\frac{C}{1+r} - m\right] \geq 0. \]

By definition of \(m\), this implies \(\mu < 0\). Defining \(\boldsymbol{\nu} = -\boldsymbol{\eta}/\mu \in \mathbb{R}^d\) yields:

\[ m + \boldsymbol{\nu} \cdot \Delta \boldsymbol{X}_1 \geq \frac{C}{1+r}, \]

showing that \(m\) is in \(I\), and therefore \(m \geq \bar{\pi}(C)\). Consequently:

\[ \sup \Pi(C) \geq \bar{\pi}(C). \]

The same argumentation shows that \(\overline{\pi}(C) = \inf \Pi(C)\).

Warning, the last part of the assertion calls for compactness arguments

We are left to show that \(\overline{\pi}(C)\) is in \(I\). Without loss of generality, due to the law of one price, we may assume that there is no redundancy; that is, \(\boldsymbol{\eta} \cdot \Delta \boldsymbol{X}_1 = 0\) implies \(\boldsymbol{\eta} = 0\).

Pick a sequence \((m_n)\) of elements in \(I\) such that \(m_n \downarrow \overline{\pi}(C)\) and denote by \(\boldsymbol{\eta}^n\) the corresponding sequence of strategies such that:

\[ m_n + \boldsymbol{\eta}^n \cdot \Delta \boldsymbol{X}_1 \geq \frac{C}{1 + r}. \]

If \((\boldsymbol{\eta}^n)\) is bounded, up to a subsequence, we may assume that \(\boldsymbol{\eta}^n \to \boldsymbol{\eta} \in \mathbb{R}^d\), for which it holds:

\[ \frac{C}{1 + r} \leq m_n + \boldsymbol{\eta}^n \cdot \Delta \boldsymbol{X}_1 \to \bar{\pi}(C) + \boldsymbol{\eta} \cdot \Delta \boldsymbol{X}_1, \]

showing that \(\bar{\pi}(C)\) is in I.

If \(\lim \|\boldsymbol{\eta}^n\| = \infty\), up to a subsequence, it follows that \(\boldsymbol{\eta}^n / \|\boldsymbol{\eta}^n\| \to \boldsymbol{\mu}\) with \(\|\boldsymbol{\mu}\| = 1\).

However, it follows that:

\[ 0 \leq \lim \frac{C}{\|\boldsymbol{\eta}^n\|(1 + r)} = \lim \frac{\boldsymbol{\eta}^n}{\|\boldsymbol{\eta}^n\|} \cdot \Delta \boldsymbol{X}_1 + \frac{m_n}{\|\boldsymbol{\eta}^n\|} = \boldsymbol{\mu} \cdot \Delta \boldsymbol{X}_1 \]

Since the market is arbitrage-free, it must follow that \(\boldsymbol{\mu} \cdot \Delta \boldsymbol{X}_1 = 0\), which by the non-redundancy assumption implies \(\boldsymbol{\mu} = 0\), leading to a contradiction since \(\|\boldsymbol{\mu}\| = 1\).

Definition

A contingent claim \(C\) is called replicable — attainable, hedgeable, or redundant — if there exists a portfolio with start value \(\bar{V}_0\) and strategy \(\boldsymbol{\eta} \in \mathbb{R}^d\) such that:

\[ C = \bar{V}_1 = \bar{V}_0 + \boldsymbol{\eta} \cdot \left(\boldsymbol{S}_1 - \boldsymbol{S}_0(1+r)\right). \]

In other term, a contingent claim is replicable if its outcome can be fully attained by a self financing portfolio.

Proposition

Let \(C\) be a contingent claim in an arbitrage-free market. Then:

\(C\) is replicable if and only if \(\overline{\pi}(C) = \underline{\pi}(C)\) which is the unique price of the contingent claim
If \(C\) is not replicable, then \(\overline{\pi}(C) > \underline{\pi}(C)\), and:

\[ \Pi(C) = \left(\overline{\pi}(C), \underline{\pi}(C)\right). \]

Proof

Clearly, if \(C\) is replicable, it follows that \(\underline{\pi}(C) = \overline{\pi}(C)\) by the definition of \(\underline{\pi}\) and \(\overline{\pi}\). The reciprocal follows from the second assertion. To prove the second assertion:

\(\Pi(C)\) is an interval
Non-replicability implies that \(\bar{\pi}(C) \not\in \Pi(C)\): Indeed, by the previous theorem, there exists a strategy \(\boldsymbol{\eta}\) such that:

\[ \bar{\pi}(C) + \boldsymbol{\eta} \cdot \Delta \boldsymbol{X}_1 \geq \frac{C}{1+r}. \]

Since \(C\) is not replicable, this inequality cannot hold with equality \(P\)-almost surely. This demonstrates \(\bar{\pi}(C) \not\in \Pi(C)\).

Since \(\overline{\pi}(C) = \sup \Pi(C)\), it follows that \(\Pi(C)\) forms an open interval:

\[ \Pi(C) = \left(\underline{\pi}(C), \overline{\pi}(C)\right) \]

A condition that garantees that every contingent claim is replicable and henceforth with a unique price is when \(\mathcal{P}^\ast\) contains a unique pricing measure. In such a market every contingent claim is uniquely priced and hedgable. In other terms every contingent claim in a complete financial market is redundant since they can all be achieved by a portfolio. A market with a unique pricing measure is called a complete market

On the other hand, the proposition shows that unless the claim is attainable, then fair prices of a contingent claim are not unique. The choice of a fair price (and therefore a deal) for such a contingent claim is henceforth due to an agreement between the buyer and the seller to choose a price within the interval \(\Pi(C)\). Therefore, they both have to accept some downside risk since neither will have a price in their confort risk free zone.

Example: Forward Contract

Recall that forward contract are contingent claims of the form(1)

Note first that forward contract are not necessarily positive random variable, but usually for contingent claims we can assume that they are greater than some constant.

\[ C^{fw}(K) = S_1 - K \]

Suppose that our market is arbitrage free and choose any pricing measure \(P^\ast\) equivalent to \(P\). Taking expectation of the the discounted value of the contingent claim yields

\[ E^{P^\ast}\left[ \frac{C^{fw}}{1+r} \right] = E^{P^\ast}\left[ \frac{S_1}{1+r} \right] - \frac{K}{1+r} = S_0 - \frac{K}{1+r} \]

showing that the fair price of the forward contract is unique. It is not surpising as the payoff can be immediately replicated by a portfolio.

However in financial markets, people speaks and quote the so called forward price. The definition of a forward price \(F\) is the value of the strike \(K\) such that the fair price of the forward contract is equal to \(0\). In other term, the forward price \(F\) is given by

\[ F = (1+r) S_0 \]

European Call and Put Options

The European call and put options are ubiquitous in finance as the most simple types of options.

Recall the payoff of such options

\[ \begin{align*} C^{call}(K) & = (S_1-K)^+ & C^{put}(K) & = (K-S_1)^+ \end{align*} \]

We suppose that the market is arbitrage free and denote with \(\pi^{call}(K)\) and \(\pi^{put}(K)\) the fair prices for each option.

Since both prices are fair, there exists a pricing measure \(P^\ast \sim P\) in the extended market where those two options are traded together with the underlying. It holds in particular that

\[ \pi^{call}(K) = E^{P^\ast}\left[ \frac{(S_1 - K)^+}{1+r} \right] \quad \text{and}\quad \pi^{put}(K) = E^{P^\ast}\left[ \frac{(K - S_1)^+}{1+r} \right] \]

Put Call Parity

Using the fact that \((S_1-K)^+ - (K-S_1)^- = S_1-K\) we can derive the so called put/call parity by taking expectation under \(P^\ast\):

\[ \pi^{call}(K) - \pi^{put}(K) = E^{P^\ast}\left[\frac{(S_1-K)^+ - (K-S_1)^+}{1+r}\right] = E^{P^\ast}\left[ \frac{S_1-K}{1+r} \right] = S_0 - \frac{K}{1+r} \]

Put Call Parity

\[ \pi^{call}(K) - \pi^{put}(K) = S_0 - \frac{K}{1+r} \]

Universal Price Bounds

In an arbitrage free market, let \(\pi^{call} = E^{P^\ast}[C^{call}/(1+r)]\) be any fair price for this call option. We are interested at providing bounds for the fair price of this call option.

On the one hand, it holds that \(S-K \leq (S-K)^+\). Taking the expectation under \(P\) of the discounted value of this inequality together with the fact that \(\pi^{call} \geq 0\) yields

\[ \left(S_0 - \frac{K}{1+r}\right)^+ \leq \pi \]

On the other hand, it holds that \((S_1-K)^+ \leq S_1\), taking expectation of the discounted value of this inequality yields

\[ \pi^{call} \leq S_0 \]

showing the universal bounds for call options \((S_0 - K/(1+r))^+ \leq \pi{call}\leq S_0\) for any fair price for the call.

Using put call parity, for any fair price for the put \(\pi^{put}\) it holds that \(K/{1+r}\geq \pi^{put}\geq (S_0-K/(1+r))^+ + K/(1+r) - S_0 = (K/(1+r) - S_0)^+\).

Universal Price Bounds

\[ \begin{equation*} \begin{cases} \left( S_0 -\frac{K}{1+r} \right)^+ \leq \underline{\pi}\left( C^{call}(K) \right) \leq \overline{\pi}\left( C^{call}(K) \right)\leq S_0\\ \\ \frac{K}{1+r} \leq \underline{\pi}\left( C^{put}(K) \right) \leq \overline{\pi}\left( C^{put}(K) \right)\leq \left( \frac{K}{1+r} - S_0 \right)^+ \end{cases} \end{equation*} \]

Jargon

A lot of jargon is connected to these options.

Intrinsic Value: The intrinsic value of the option is the value if it were executed now, that is

\[IV^{call} = (S_0-K)^+ \quad \text{and}\quad IV^{put} = (K-S_0)^+\]
In/At/Out of the Money: An option is called in the money if its intrinsic value is strictly positive, at the money if the underlying price equal the strike, out of the money if the intrinsic value is \(0\) and the underlying price is not equal to the strike.
Moneyness: Moneyness is a concept that has no rigorous definition but stems from a particular property of the call/put option, the positive homogeneity of their payoff, that it \((\lambda x)^+ = \lambda x^+\) for any \(\lambda >0\).

We can therefore normalize the payoff of options by either \(K\), \(S_0\) or \(K/(1+r)\), etc. Most of those normalizations are brought in connection with the resulting Black-Scholes-Merton formula, but let us stress some aspects of this definition. The simple version of moneyness is related to the intrinsic value of the option. In the case of a call, we can normalize by the strike where the simple spot moneyness is defined as \(S_0/K\). Indeed, it holds

\[ \pi^{call}(K) = K E^{P^\ast}\left[ \frac{1}{1+r}\left( \frac{S_1}{K} - 1 \right)^+ \right] \]

The intrinsic value of the normalize option in the inner part of the expectation is \((S_0/K-1)^+\) and therefore is in the money iff the simple (call) spot moneyness is greater than one and out of the money otherwise.

In the case of the put option we normalize by the current underlying price \(S_0\), that is, the simple spot moneyness is defined as \(K/S_0\):

\[ \pi^{put}(K) = S_0 E^{P^\ast}\left[ \frac{1}{1+r}\left(\frac{K}{S_0}- \frac{S_1}{S_0} \right)^+ \right] \]

The intrinsic value of this normalized option in the inner part of the expectation is \((K/S_0 -1)^+\) which is positive iff the simple (put) spot moneyness is greater than one and out of the money otherwise.

Since those definition is rather confusing, always rely on your mathematical knowledge about what each should mean.