Formalizing Arbitrage
I provide a mathematical formulation for the concept of arbitrage in finance to derive insights on various properties of option contracts.
In a previous post, I tried to provide a gentle introduction on the concept of arbitrage along with the no-arbitrage principle in finance by focusing on some simple examples based on forward type contracts such as forwards and futures. There we saw the power of no arbitrage principle in the context of deriving the fair forward(delivery) price of such derivative contracts. In this post, I would like to discuss the mathematical foundations behind arbitrage and how these formulations can be extended to cover pricing of vanilla options, as well as their relation with other derivative contracts. The most of the material in this post is based on the excellent book by Mark S. Joshi, The concepts and practice of Mathematical Finance, 2nd edition.
Mathematical definition of arbitrage
In order to set the stage for what comes next, let me introduce a more formal definition of the “no-free lunch” principle.
Definition. A portfolio is said to be an arbitrage portfolio, if today it is non-positive value and in the future it has zero probability of being of negative value, and a non-zero probability of being of positive value.
By this very definition, such a portfolio provides a framework for making risk-free profit. Before we dive further, I would like to digress a bit to elaborate more on a portfolio with non-positive value. Such a situation arise when the current value of all positions, e.g. assets minus liabilities is negative. This typically happens when your short positions or obligations exceed your current long positions. In this case, typically there is a net cash inflow required to construct the portfolio, corresponding to the cost of establishing the portfolio. To give a simple example, we can consider short selling 10 shares of a stock from 50 USD each and using its proceeds to buy a zero coupon bond from 400 USD that matures at 500 USD in the future to protect ourselves against the upward movement of the stock price. In this portfolio, we sell the stocks we don’t have by borrowing 500 USD and used this money to buy the bond from 400 USD, so the total inflow of cash to build the portfolio is 500-400 = 100 USD and the net portfolio value at inception is -500 + 400 = -100 USD as we have the short position liability plus our long position in bonds. Notice that such a portfolio can be always combined with cash to render a zero cost portfolio. For example by adding 100 USD cash deposit, we end up with a portfolio that cost 0 USD with a net value of 0. In the same manner, we can combine an arbitrage portfolio (as defined above) with enough cash to build a zero-cost/value portfolio at its inception. By creating such a portfolio, we have a non-vanishing probability of receiving money in the future at no cost. The no arbitrage principle prevents the existence of such portfolios that are guaranteed to be non-negative in the future as they would lead to risk-free profit.
An important consequence of the no-arbitrage principle is the monotonicity theorem,
Monotonicity theorem. If portfolios A and B are such that portfolio A is worth at least as much as portfolio B in every possible state of the market at time $T$, then at any time $t < T$, portfolio A is worth at least as much as B ($ V_A \geq V_B$). If in addition portfolio A is worth more than portfolio B in some states of the market, then at any time $t < T$, portfolio A is worth more than portfolio B.
Proof. To prove this theorem, we can apply no arbitrage considerations to a portfolio $C$ that consist of a long portfolio $A$ and short portfolio $B$:
\[C = A - B.\]Such a portfolio has non-negative value at time $T$ independent of the state of the market at that time and must have a non-negative value for $t < T$ otherwise we would have portfolio of negative value that will always be of non-negative value in the future. If $A$ is worth more than $B$, then $C$ have a positive value at time $T$ and must have a positive value at time $t < T$, otherwise there would be a possibility of making risk-free profit by building a portfolio at zero cost at time $t$.
The no-arbitrage principle, along with the monotonicity theorem above is sufficient to obtain useful insights on the price and value of option contracts. This is what I would like to turn next.
Options in an arbitrage free market
Before we discuss the implications of no-arbitrage principle on the various properties of options, let’s start by first defining the latter.
Options can be categorized as the conditional derivative contracts that gives the buyer the right but not obligation to buy or sell an underlying asset at a fixed price, called the strike price, on a fixed date, called the exercise/maturity date. This defines a vanilla European style option. An American option differs in that it can be exercised any time up to and including the exercise date. In this post, when I use the word “option”, I will always refer to a European style option unless I state otherwise. A call option gives the buyer the right to buy the underlying asset, while a put option gives the buyer the right to sell (short) the underlying asset. Finally, the parity value (intrinsic value) of an option is the value of the option when exercised, while the premium over parity (time value) is the price of the option minus its parity value (i.e. the market price for optionality).
A rational investor who owns a call option contract will only exercise the option iff the price of the underlying $S$ is more than the strike price $K$ at the expiry time $T$. Since she can immediately sell the asset in the spot market from $S$, she pockets $S_T - K > 0$ otherwise the option expires worthless. Therefore at the expiry the value of the option (or its parity value) is worth precisely
\[V^{c} = (S_T - K)_{+} = \textrm{max}(S_T-K,0).\]We can therefore think of a call option as an asset that pays the sum $(S_T - K)_{+}$ at time $T$. Therefore, $V^c$ is also called the option’s payoff. Similarly, a put option will have the following pay-off profile
\[V^{p} = (K - S_T)_{+} = \textrm{max}(K - S_T,0).\]These properties of the options at the time of expiry notwithstanding, what can we say about the value of the options before expiry?
Theorem. A vanilla call or put option always has positive value before expiry.
- Proof: To prove the theorem, consider a portfolio $A$ as an option and an empty portfolio $B$. At the expiry, portfolio $A$ has a positive value if it is advantegous to exercise the option, otherwise it has zero value. Therefore, $A$ is worth at least as much as $B$ in all possible market states and it is worth more than $B$ in some market states (e.g states that drive the underlying price $S$ above the strike $K$). Then using the monotonicity theorem, we can conclude that $A$ is worth more than $B$ for all time prior to expiry $t < T$, and hence option have a positive value prior to expiry.
Relationships between put and call option
We continue to explore the implications of arbitrage considerations to derive relations between put and call type option contracts. As an example of the effect an arbitrage-free market has on options, we can consider the following two portfolios:
Portfolio $A$ consists of a European call (long) with strike price $K$ and maturity $T$ , and a European put (short) also with strike price $K$ and maturity $T$ , both on the same underlying $S$.
Portfolio $B$ consists of a forward contract (long) to buy the underlying upon maturity $T$ at the delivery price $K$.
At the expiry $T$, portfolio $A$ has value $S_T - K$ independent of the market conditions, i.e irrespective of whether $S_T > K$ or $S_T < K$. In the former, we exercise the call option, while in the latter it is advantegous for us to exercise the put. Therefore
\[\begin{equation}\label{pc0} V_A(T) = \textrm{max}(S_T - K, 0) - \textrm{max}(K - S_T, 0) = S_T - K \end{equation}\]On the other hand, a long forward contract to buy the same underlying asset at time $T$ will also worth, so that $V_B(T) = S_T - K$. Then two portfolios have the same value at the expiry $V_A(T) = V_B(T)$ for every possible market conditions. According to the monotonicity theorem, we then expect the both portfolios to have the same value at all time prior to the maturity, $t < T$ because otherwise arbitrage opportunities appear. This gives the famous put-call parity relation
\[\begin{equation}\label{pc} C_t - P_t = F_t \end{equation}\]where we have labelled the value of a call/put as $C/P$ and $F$ as the value of a forward. This relation is quite useful, as with the price knowledge of a forward contract based on the same underlying, we can price a call or put if we happen to have the price information of one of the two. Another useful interpretation of put-call parity \eqref{pc} is as follows: a European vanilla call option can be considered as a forward contract on the underlying plus an insurance in the form of a put that protect the investor from the undesirable outcome of declines in the underlying price with an aim to guarantee that the underlying is worth no less than the previously agreed delivery price $K$. If an investor is certain that the price the underlying price will rise and wishes to make the most in light of this information, he/she should buy forward contracts and not calls because in this way, the investor avoids paying the unnecessary insurance premium (put price) that would guard against a falling underlying price.
Let’s continue deriving more insights on the options with respect to their underlying. Earlier, we proved that the value of a call or put option must be larger than zero before the expiry however this does not tell us much about how large its value could be. Focusing on the call options, we can derive an upper and lower bound on the call price using the monotonicity theorem.
Theorem. At time $t$, let $C_t$ be the price of a call option on a non-dividend paying stock, $S_t$, with expiry $T$ and strike $K$. Let $Z_t$ be the price of a zero-coupon bond with maturity $T$. We have
\[\begin{equation} S_0 > C_0 > S_0 - K Z_0 \end{equation}\]Proof: We start with the upper bound. At the maturity of the option, the stock is worth $S_T$ while the option is worth at most $S_T - K$. So at the expiry, the stock is always worth more than the call option. By the monotonicity theorem, the stock is always worth more than the call option for any time (e.g at $t = 0$) prior to the exercise, thus we get:
\[S_0 > C_0.\]To prove the lower bound consider a portfolio with a call option plus $K$ (unit) zero coupon bond which mature at $T$ so that at time $T$ we have one option and $K$ dollars. At this time, if the stock price is larger than strike, we exercise the option and spend $K$ dollars to buy the stock. In this case our portfolio is worth $S_T - K + K = S_T$. Otherwise, we do not exercise, and our portfolio is worth $K$ dollars. To sum up, at the expiry or portfolio is worth:
\[\textrm{max}(S_T - K, 0) + K = \textrm{max}(S_T, K).\]Therefore, it is always worth as much as the stock and in some circumstances (e.g when $S_T < K$) it is worth more. By the monotonicity theorem, our portfolio is therefore needs to worth at least as much as the stock at any other time prior to the maturity, $t < T$ as otherwise there would be an risk-free profit opportunity:
\[C_0 + K Z_0 > S_0,\quad\quad \Rightarrow \quad\quad C_0 > S_0 - K Z_0.\]Thus, we finalized the proof. If we make the assumption that the risk-free interest rates are non-negative, we can reach at an even more interesting result. In this case, the price of the unit zero coupon bond should be less than unity, $Z_0 < 1$ (with the promise of obtaining a dollar at $T$.) which ensures the following bound
\[C_0 > S_0 - K.\]This relation is a fundamental one as it tells us that before the expiry, an European call option on a (non-dividend paying) stock is always worth more than its intrinsic value, i.e the value that it would be if it were to be exercised today.
References
1. “Derivatives and Internal Models: Modern Risk Management”, Hans-Peter Deutsch and Mark W. Beinker .
2. “The concepts and practice of mathematical finance”, Second Edition, Mark S. Joshi.
Appendix A: What is a zero coupon bond?
A zero-coupon bond is a financial instrument that represents a promise to pay a fixed amount (the face value, e.g., $1) at a specific future date, known as its maturity. Unlike traditional bonds, it does not make periodic interest payments (coupons) along the way. Instead, it is sold at a discounted price today, and the difference between the purchase price and the face value represents the investor’s return.
Now consider a zero bound which I refer to as a unit zero bound that promises to pay a face value of one dollar at its maturity $T$ which we may assume to be in one year. Since we wait a year to get this money and the money has time value, this should be reflected to the fair discoundted price of the zero coupon that we buy today. For any non-negative annual interest rate, the price of this zero coupon should be less than unity as I have argued in the main text.