1 Introduction

1.1 What is Mathematical Finance?

Mathematical Finance can be broadly viewed as the study of the mathematics used to model and analyse financial markets, in order to gain a better understanding of the behaviour of these markets in reality, and therefore make well-judged investment decisions. Of course, modern financial markets are extremely complex, but, under some simplifying assumptions, the mathematical theory becomes rather elegant and provides robust methods for the pricing and hedging of a broad range of financial products known as derivatives.

In this course, we’ll focus on the mathematics behind this pricing of derivatives. In Michaelmas term we’ll look at discrete-time models, where we assume that any trading must happen at prescribed times. Here the minimal mathematics required is little more than some basic linear algebra; however, the full power of our approach is realised from a probabilistic viewpoint. Some of the nice theory we’ll see include filtrations on a probability space, conditional expectation and martingales. In the discrete-time setting we will be able to construct these objects explicitly (essentially, because the probability space we work with is finite) and use them to calculate the prices of a range of complex financial products. This will also provide a solid ground on which to develop the theory for continuous-time models (see Epiphany term) where the probability theory is rather more subtle and typically non-explicit; for example, even the existence of the general version of conditional expectation requires some sophisticated measure theory. Later in this term, we’ll use our discrete-time model to approximate a continuous-time model. This will ultimately lead to the famous Black–Scholes pricing formula. We’ll see the formula this term, but to fully appreciate it we would need to develop a rigorous continuous-time model—that will be the focus of Epiphany term.

Although our main focus is on the mathematics rather than the economics of finance, we will need some financial terminology so that we have a language in which to frame the mathematical theory. To see a flavour of the kinds of problem that we’ll be interested in, let’s consider the following reasonably realistic situation.

A Swedish company C&H, today (denoted by $t=0$) signs a contract with an American counterpart ACME. The contract stipulates that ACME will deliver 1000 units of computer terminals to C&H exactly six months from now (denoted by $t=T$). Furthermore it is stipulated that C&H will pay 1000 US dollars per unit to ACME at the time of delivery (i.e., at $t=T$). Let us assume that the present currency rate between SEK (Swedish krona) and the US dollar is 9.00 SEK/$.

One of the problems for C&H with this contract is that it involves a considerable currency risk. Since C&H does not know the currency rate prevailing six months from now, it does not know how many SEK it will have to pay at $t=T$. If the currency rate at $t=T$ is still 9.00 SEK/$ it will have to pay 9,000,000 SEK, but if the rate rises to, say, 9.50 it will face a cost of 9,500,000 SEK. Thus C&H faces the problem of how to guard itself against this currency risk. Below are some strategies that C&H can take:

C&H can buy $1,000,000 today at the price of 9,000,000 SEK, and keep this money (in a Eurodollar account) for six months.
- Advantages: Currency risk is completely eliminated.
- Drawbacks: C&H has to tie up a substantial amount of money for a long period of time, or maybe C&H does not have access to 9,000,000 SEK today.
C&H can go to forward market and buy a forward contract for $1,000,000 with delivery six months from now. Such a contract may, for example, be negotiated with a commercial bank, and in the contract two things will be stipulated:
- The bank will, at $t=T$, deliver $1,000,000 to C&H.
- C&H will, at $t=T$, pay for this delivery at the rate $K$ SEK/$.
  - The exchange rate $K$, which is called the forward price at $t=0$, for delivery at $t=T$, is determined at $t=0$.
  - By the definition of the forward contract, the cost of entering the contract equals zero, and the forward rate $K$ is thus determined by supply and demand on the forward market. (Note, however, that even if the price of entering a forward contract (at $t=0$) is zero, the contract may very well fetch a nonzero price during the time interval $(0, T]$).

Let us assume that the forward rate today for delivery in six months equals 9.10 SEK/$. If C&H enters the forward contract this simply means that there are no outlays today, and that in six months it will get $1,000,000 at the predetermined total price of 9,100,000 SEK. Since the forward rate is determined today, C&H has again completely eliminated the currency risk.

However, the forward contract also has some drawbacks, which are related to the fact that a forward contract is a binding contract. To see this let us look at two scenarios.

Suppose that the currency rate at $t=T$ turns out to be 9.20. Then C&H can congratulate itself, because it can now buy dollars at the rate 9.10 despite the fact that the market rate is 9.20. In terms of the million dollars at stake C&H has thereby made an indirect profit of $9,200,000-9,100,000=100,000$ SEK.
Suppose on the other hand that the exchange rate at $t=T$ turns out to be 8.90. Because of the forward contract this means that C&H is forced to buy dollars at the rate 9.10 despite the fact that the market rate is 8.90, which implies an indirect loss of $9,100,100-8,900,000=200,000$ SEK.

What C&H would like to have is a contract which guards it against a high rate at $t=T$, while still allowing it to take advantage of a low rate at $t=T$. Such contracts do in fact exist, and they are called European call options. We’ll see a full definition later, for now we note that the contract is written at time $t=0$ and specifies a strike price (or exercise price) $K$ and an expiry date $T$, with the following properties:
- The holder of the contract has, exactly at the time $T$, the right to buy $X$ US dollars at the price $K$ SEK/$.
- The holder of the option has no obligation to buy the dollars.

Concerning the nomenclature, the contract is called an option precisely because it gives the holder the option (as opposed to the obligation) of buying some underlying asset (in this case US dollars). A call option gives the holder the right to buy, whereas a put option gives the holder the right to sell the underlying object at a prespecified price. The prefix European means that the option can only be exercised at exactly the date of expiration. There also exist American options, which give the holder the right to exercise the option at any time before the date of expiration.

Options of the type above (and with many variations) are traded on options market all over the world, and the underlying objects can be anything from foreign currencies to stocks, oranges, timber or pig stomachs. For a given underlying object there are typically a large number of options with different dates of expiration and different strike prices.

We now see that C&H can insure itself against the currency risk very elegantly by buying a European call option, expiring six months from now, on a million dollars with a strike price of, for example, 9.00 SEK/$. If the exchange rate at $t=T$ exceeds the strike price $K$, say that it is 9.20, then C&H exercises the option and buys at 9.00 SEK/$. Should the exchange rate at $t=T$ fall bellow the strike price, it simply abstains from exercising the option.

Note, however, that in contrast to a forward contract, which by definition has the price zero at the time at which it is entered, an option will always have a positive price, which is determined on the existing options market. This means that our friends in C&H will have the rather delicate problem of determining exactly which option they wish to buy, since a higher strike price (for a call option) will reduce the price of the option.

One of the main problems in this course is to see what can be said from a theoretical point of view about the market price of an option like the one above. In this context, it is worth noting that the European call has some properties which turn out to be fundamental.

Since the value of the option (at $t=T$) depends on the future level of the exchange rate, the holding of the option is equivalent to a future stochastic claim.
The option is a derivative asset in the sense that it is defined in terms of some underlying financial asset.

Since the value of the option is contingent on the evolution of the exchange rate, the option is often called a contingent claim. Later on we will give a precise mathematical definition of this concept, but for the moment the informal definition above will do. An option is just one example of a financial derivative; some other commonly traded derivatives include: European calls and puts; American options; Forward rate agreements; Convertibles; Futures; Bonds and bond options; Caps and floors; Interest rate swaps.

This list is far from complete but the main point is the fact that financial derivatives exist in a great variety and are traded in huge volumes. We’ll study some of these in more detail later in the course. We can now formulate the main two problems which concern us in this course.

Main Problems: Take a fixed derivative as given.

What is a “fair” price for the contract?
Suppose that we have sold a derivative, such as a call option. Then we have exposed ourselves to a certain amount of financial risk at the date of expiration. How do we protect (“hedge”) ourselves against this risk?

Let us look more closely at the pricing question above. There exist two natural and mutually contradictory answers.

Answer 1: “Using standard principles of operations research, a reasonable price for the derivative is obtained by computing the expected value of the discounted future stochastic payoff.”

Answer 2: “Using standard economic reasoning, the price of the contingent claim, like the price of any other commodity, will be determined by market forces. In particular, it will be determined by the supply and demand curves for the market for the derivatives. Supply and demand will in their turn be influenced by such factors as aggregate risk aversion, liquidity preferences, etc., so it is impossible to say anything concrete about the theoretical price of derivatives.”

Main Result: Both answers above are incorrect! It is possible to talk about the “correct” price of a derivative, and this price is not computed by the method given in Answer 1.

Main Ideas

A financial derivative is defined in terms of some underlying asset which already exists on the market.
The derivative cannot therefore be priced arbitrarily in relation to the underlying prices if we want to avoid mispricing between the derivative and the underlying asset.
We thus want to price the derivative in a way that is consistent with the underlying prices given by the market.
We are not trying to compute the price of the derivative in some “absolute” sense. The idea instead is to determine the price of the derivative in terms of the market prices of the underlying assets.

Example 1.1 Suppose that we are market makers who buy and sell stocks and options. For a particular stock the current price is 100 and we can buy or sell any number of units at this price. After one time period the value of this stock will be 200 with chance $p > 0$ or $50$ with chance $1-p > 0$. This is common knowledge, though the value of $p$ can only be estimated.

For simplicity, let us exclude the possibility of earning interest on this market. This means a cash amount $R$ at time 1 is equivalent to a cash amount $R$ at time 0. (Our general model will incorporate interest rates—you might like to revisit this example later and repeat the calculations assuming a risk-free interest rate of $r$ per time period.)

The option offered on this stock is the right (but not the obligation) to buy a unit for 150 at time 1. What price $C$ should we charge per option? We will assume that there is a wealthy arbitrager Agent A who will not take any risks at all but will ruin us if we offer up a guaranteed profit by mis-pricing the option and creating an arbitrage.

The option has time 1 value of 0 or 50 respectively if the share price becomes 50 or 200 (at value 50 throw the option away, at value 200 buy the stock for 150 and sell immediately for 200). Hence the expected value of the random payoff from each unit of the option is $50p$. Naively, it seems reasonable to set this to be the price we charge for the option (i.e.. $C=50p$). But observe that this means we need to make an estimate for $p$. Let’s see what happens if we assess that $p = 0.2$ and therefore choose $C = 10$.

Along comes Agent A who sells us $10^6$ shares (which have been borrowed from a third party and will be returned at time 1) and buys $3 \cdot 10^6$ options. Agent A pockets $(100 - 3 \cdot 10)\cdot 10^6$ from us at time 0 on this trade.

At time 1 the new share price is known:

If it is 200, then Agent A exercises the options to buy $3\cdot 10^6$ shares costing a total price of $450 \cdot 10^6$, returns the borrowed $10^6$ shares and sells the rest to earn $200 \cdot 2 \cdot 10^6$ for a return of $(70 - 50)\cdot 10^6$.
If the share price is 50, then Agent A throws away the options, buys $10^6$ shares at $50$ each to return them to the third party and again pockets $(70 - 50)\cdot 10^6$.

Either way we have lost a stack of money. Agent A found an arbitrage opportunity because we mispriced the option!

In general, Agent A can buy $x$ units of the stock and $y$ units of the option from us (where $x$, $y$ can be of either sign – the finance industry uses long and short to mean positive and negative). This costs $100x + Cy$ at time 0. At time 1 the investment is worth $200x + 50y$ at stock price 200 or $50x$ at stock price 50. Agent A can make these two amounts equal by choosing $y = -3x$ which would eliminate any risk! Then, the risk-free profit of such an investment would be \[ 50x - (100x - 3Cx) = ( 3C - 50 )x. \] Hence, if we charge more than $50/3$ the investor ruins us by taking $x$ to be a hugely positive (buying the stock and short selling the option). If we charge less the investor ruins us by taking $x$ to be hugely negative (short selling the stock and buying the option). The only possibility to avoid arbitrage is $C=50/3$.

Notice that $p$ does not appear in the expression for the price! Our estimate for the value of $p$ is not important for pricing, roughly speaking because we need to account for both eventualities of the share price either increasing to 200, or decreasing to 50. (In fact, there is a unique value of $p$ for which the expected payoff of $50p$ would agree with the price $C = 50/3$, namely $p=1/3$. For this very special probability, the correct price can be viewed as an expectation of the random payoff from the option. We’ll see much more on this later in the course.)

1.2 Some terminology

Risk-free interest A cash amount $x$ (positive or negative) in the bank earns interest at rate $r$ per period so after one period the cash amount in the bank will be $x(1+r)$.

Portfolios We will describe our assets at any time as a portfolio. It consists of the shares, options and cash we have at that time and we suppose that any of these can be negative. For shares and options this can be achieved by selling short i.e. selling things you haven’t got. (In practice, this really happens!) When you do this you will be required to buy the relevant stocks or shares at some future date at the market price to honour your short sale.

Arbitrage When it is possible to assemble a portfolio that with certainty returns more than the risk free interest rate then we say that an arbitrage opportunity exists.

Options A huge and important market exists for the buying and selling of options. There are good economic reasons for this, one of which is that there is a decent theory for working out price structures for them.

Definition 1.1 A European call option gives its holder the right to buy from the writer an asset at a nominated price at a specified future time (but the holder need not exercise the option).

A European put option gives its holder the right to sell to the writer an asset at a nominated price at a specified future time (but the holder need not exercise the option).

For these options, the nominated price is often called the exercise price or strike price while the specified time is the expiry date or exercise date.

Note that there are other versions of call/put option (which we will see later in the course).

In practice, there are a range of possible assets on which options are available but the most common are stocks (shares in major companies), currencies and indices (it’s possible to buy units in the Dow Jones which is an investment based on all the companies listed in that index).

Let’s look at the value of call and put options in more detail. Write $S_t$ for the price at time $t$ of the underlying asset on which an option is based. The call options only has non-zero value at the expiry date when the asset price is greater than the strike price ($S_T > K$). In this case the holder exercises the option to buy at price $K$, sells the asset immediately for $S_T$ and receives $S_T - K$ at time $T$. If $S_T \leq K$ there is no benefit in using the option: it is not exercised and the payoff is 0. Putting this together we see that a European call option pays out \[ \max (S_T - K, 0) \equiv (S_T - K)^+\] at the expiry date $T$. In other words, the value of the European call option at time $T$ is $(S_T-K)^+$.

Our main aim this term is to explore the pricing of European call options (i.e., their value at times $t$ other than $t=T$) in terms of the strike price $K$, expiry date $T$, the interest rate and any components of the stock price model which might be relevant.

The return at time $T$ of a put option is similarly calculated—put options are only worth exercising when $S_T < K$ and they pay out \[ \max (K - S_T, 0) \equiv (K - S_T)^+ \]

Remark. Observe that selling call options is different from buying put options (because the choice is always with the holder, and the obligation always with the writer). This can be easily seen from the fact that \[ -(S_T-K)^+ \neq (K-S_T)^+. \]

(The plus signs are crucial here!)

There are also American call and put options which only differ from Europeans in the fact they can be exercised before the expiry date. Many investors buy combinations of slightly different (usually American) options to limit exposure to risk.

Spreads and straddles Some of these combinations have standard names. A combination of options of one type is called a spread e.g. in a bull spread one buys a $(K_1, T)$ call and sells a $(K_2, T)$ call (with $K_2 > K_1$) on the same stock. A straddle consists of a call and put with parameters $(K, T)$ on the same stock (which can be bought or sold of course). If the strike prices of the call and put differ then this is usually called a strangle. Butterfly spreads involve three options of the same type with the same expiry date. Calendar spreads include options with different expiry dates.

Exercise: draw graphs of the payoffs from these various combinations of options as a function of asset price.

In addition to these there are a host of more complex options, most of which will not be discussed in this course.

1.3 Risk-free interest

A feature of investor behaviour that is hugely important in markets is their risk aversion. Specifically, given two investments with the same expected return investors generally will prefer the one with smaller variation in the returns. For instance you have 10,000 to invest in either opportunity $A$ or $B$. After a year, $A$ pays out $0$ with chance 0.45 or 20,000 with chance 0.55 while $B$ pays out 10,000 with chance 0.5 or 12,000 with chance 0.5. Nearly all investors prefer $B$ to $A$ though they have the same expected payoff of 11,000.

We will suppose that any market contains opportunities for risk-free investment that returns interest at a known rate $r$ (by risk-free we mean that they only fail to pay out due to major societal breakdown). We will often assume that money can be borrowed at that same rate though in practice the cost of borrowing is generally higher than the return from investing.

To compute interest amounts we need to know the nominal rate $r$ plus the rule for compounding. Suppose you invest $D(0)$ at time $t = 0$ at a nominal interest rate of $r$ per year, where the interest is compounded $n$ times per year (where $n = 1, 2, 4, 12, 52, 365$ are typical). With amount $D(t)$ on deposit at time $t$ interest of $D(t) \times r/n$ is added to your initial amount at time $t + 1/n$ i.e. \[ D(t + 1/n) = D(t)(1 + r/n) \qquad\mbox{so that}\qquad D(m/n) = D(0)(1 + r/n)^m \] The other form of compounding is called continuous. For any fixed $t > 0$, $m = tn$ and then let $n \to \infty$. We see that \[ D(t) = \lim_{n\to \infty} D(0)(1 + r/n)^m = \lim_{m\to \infty} D(0)(1 + rt/m)^m = D(0) \e^{rt} \] Alternatively we know that amount $D(t)$ earns $rD(t)\delta t$ over a short period $\delta t$ or more formally, $D(t + \delta t) - D(t) = rD(t)\delta t + o(\delta t^2)$. Letting $\delta t \to 0$ we obtain $D'(t) = rD(t)$ and from this it follows that $D(t) = D(0)\e^{rt}$.

Sometimes the effective interest rate is referred to in the literature. This is simply the value $r_{\mathrm{eff}}$ such that $1 + r_{\mathrm{eff}} = (1+r/n)^n$ which, when compounded annually, produces the same amount of interest as the nominal rate $r$ compounded $n$ times. When expressed in percentage terms this is the APR.

Present value analysis Suppose interest is compounded discretely at rate $r$ per compounding period. Set $\alpha = 1/(1+r)$. Any cash amount $x_i$ (positive or negative) at a future time $t_i$ (in units of the compounding period) is equivalent to a value of $x_i \alpha^{t_i}$ at time 0. The term $\alpha^{t_i}$ is often called the discount factor. The standard way to compare two cash flows is to calculate their present values. A cash flow $(x, t)$ where you receive $x_i$ at $t_i$, $i = 1, 2, \dots m$ has present value \[ P(x, t) = \sum_i x_i \alpha^{t_i} \] If interest is instead compounded continuously at rate $r$, then we should set $\alpha = \e^{-r}$.

Example 1.2 You plan to retire in 30 years (360 months) and to fund this you pay amount $D$ at the start of each month into a bank that pays interest at a yearly rate of 5% compounded monthly. You plan to withdraw 1,000 per month at the start of months 361 to 600. What $D$ leads to a zero present value for this investment?

The monthly rate is $r = 0.05/12$ so let $\alpha = 1/(1+r)$. We calculate the present values of all deposits and of all withdrawals and equate them. Firstly, the present value of the deposits is \[ D + D\alpha + \cdots + D\alpha^{359} = D\sum_0^{359} \alpha^i = D\frac{1 - \alpha^{360}}{1 - \alpha} \] while the withdrawals work out to be worth \[ 1,000\alpha^{360}(1 + \alpha + \cdots + \alpha^{239}) = 1,000 \alpha^{360} \frac{1 - \alpha^{240}}{1 - \alpha} \] Hence \[ D = 1,000 \alpha^{360} \frac{1 - \alpha^{240}}{1 - \alpha} \cdot \frac{1 - \alpha}{1 - \alpha^{360}} = 1,000 \alpha^{360}\frac{1 - \alpha^{240}}{1 - \alpha^{360}} \approx 182.065 \] If it is unclear to you why this is correct then work it out by tracking the fund level in the account through to month 600. The standard method is to reduce all flows to present values and add them up.

1.4 Portfolios and arbitrage

Portfolios will play a crucial role in our pricing and hedging of financial derivatives. Roughly speaking, the idea is that if two different portfolios (perhaps formed from different amounts of cash, stocks, options, etc.) offer the same future payoffs, then they should also have the same current price. With this principle in place, then we can determine the price of a derivative if we can find another portfolio that matches the payoffs of the derivative. We will see the specifics of this for a particular market in the next chapter. For now, we set up some basic portfolio theory to make these pricing concepts concrete.

A portfolio (also called a trading strategy) represents a collection of financial products (assets) available on our market that is held over the finite time period $[0,T]$. Suppose our market consists of $M$ assets, whose prices at time $t$ are expressed by the vectors $S_t = (S^1_t, \dots, S^M_t)$ for every $t \in [0,T]$. (For now, we will not concern ourselves with questions of how to model the behaviour of these prices; for the time being we just view the prices as some positive real numbers that change over time in an uncertain but “well-behaved” way—in particular it will be convenient to assume left-continuity: $S_t = S_{t-} := \lim_{s \uparrow t} S_s$ for all $t \in (0,T]$.)

We can represent a portfolio in two ways. First, as a list of instructions telling us how much/what/when to trade during the interval $[0,T]$. Mathematically, we write $P = ( (v_i,t_i); i=1,\dots,k )$, where $0 \leq t_1 < t_2 < \dots < t_k \leq T$ are the times of trades, and $v_i \in \R^M$ represent the amounts of each asset we buy/sell at time $t_i$ (a positive number means “buy”; negative means “sell”).

Alternatively, we can represent the portfolio by a collection $P = ( h_t ; t \in [0,T] )$ of vectors in $\R^M$ which specifies how much of each asset is held at all times $t \in [0,T]$. From the sequence $( (v_i,t_i); i=1,\dots,k )$, we define \[ h_t := \sum_{i: t_i \leq t} v_i. \] (Note that at the time $t_i$ of a trade the vector $h_{t_i}$ represents the amounts of each asset held after the trade, and $h_{t_i-}$ represents the amounts held before the trade.)

Viewed as a function $t \mapsto h_t$, a portfolio is a step function which is constant on the intervals $[t_i,t_{i+1}), i=1,\dots, k-1$ (and on $[0,t_1)$ and $[t_k,T)$). Any right-continuous piecewise step function $f\colon [0,T] \to \R^M$ can represent a portfolio, and we can recreate the corresponding sequence of trades by taking all the non-zero values of $f(t) - f(t-)$ for $t \in [0,T]$ (with the convention that $f(0-) = 0$).

From this viewpoint it is easy to see that portfolios can be combined to make new ones (formally, the space of all portfolios is a vector space). Given two portfolios $A = (h^A_t; t \in [0,T])$ and $B = (h^B_t; t \in [0,T])$, and two reals $\alpha,\beta$ we define \[ \alpha A + \beta B = (\alpha h^A_t + \beta h^B_t; t \in [0,T]). \] For example, the portfolio $A+B$ means perform all the trades of both $A$ and $B$, and $-A$ means perform all the trades of $A$ but “in reverse” (buy becomes sell, and sell becomes buy).

So far, we have given the impression that the trades of a portfolio must happen at fixed times, but this is not always the case. For example, the instruction “buy 100 shares of stock at time $t=0$ and sell 100 shares of stock at the first time $t$ that the share price $S_t$ is at least double the initial price $S_0$” defines a valid portfolio, which has a potential trade at time $\inf\{ t: S_t \geq 2S_0 \}$ which depends on what happens to the share price over $[0,T]$. In general, we will allow what and when we trade to depend on the share prices on the market, but loosely speaking we cannot look to the future when deciding what/when to trade. Without giving too much detail of the theory here, we simply insist that the vector $h_t$ can only depend on the share prices up to time $t$, for all $t \in [0,T]$.

The value $V_t$ at time $t$ of a portfolio $P = (h_t; t \in [0,T])$ is defined as \[ V_t := h_t \cdot S_t = \sum_{i=1}^M h^i_t S^i_t. \] The value $V_0$ represents the cash amount required at time $t=0$ to initialise the portfolio, and the value $V_T$ represents the cash received at time $t=T$ from closing out the portfolio. Note that both of these values could be negative: $V_0 < 0$ means that the owner gains a cash amount when initialising the portfolio (e.g., if short selling shares); $V_T<0$ means the owner loses a cash amount when closing out (e.g., to buy back any shares that were sold earlier).

Example 1.3 Consider the simple portfolio $A$: “sell 100 shares of stock at time 0.” We can write this as either the sequence of one trade: $A= ( (-100\text{ shares}, 0) )$, or as the collection $A= (h^A_t, t \in [0,T])$, where $h^A_t = (-100\text{ shares})$ for all $t \in [0,T]$. The value of $A$ at time $t$ is $V^A_t = -100 S_t$ for all $t \in [0,T]$.

Now consider the more complex portfolio $B$, as defined earlier: “buy 100 shares of stock at time 0, and sell 100 shares of stock at the first time $t$ that the share price $S_t$ is at least double the initial price $S_0$.” Here the sequence of trades is $B = ((100 \text{ shares}, 0), (-100 \text{ shares}, \inf\{ t: S_t \geq 2S_0\})$ and the holdings are $h^B_t = (100 \1_{ \{ \sup_{s \in [0,t]}(S_s) < 2S_0 \} })$ for all $t \in [0,T]$. The value of $B$ at time $t$ is $V^B_t = 100 S_t \1_{ \{ \sup_{s \in [0,t]}(S_s) < 2S_0 \} }$. Observe that $h^B_t$ depends on the share price upto time $t$. (Here, $\1_E$ is the indicator of the event $E$, i.e., $\1_E = 1$ if $E$ occurs; $\1_E=0$ otherwise.)

Self-financing portfolios Recall that, if $t_i$ is the time of a trade of a portfolio $P = (h_t; t \in [0,T])$, then $h_{t_i}$ represents the holdings after the trade, and therefore $V_{t_i} = h_{t_i} \cdot S_{t_i}$ is the value of the portfolio after the trade at time $t_i$. In contrast, $V_{t_i-} = h_{t_i-} \cdot S_{t_i-}$ is the value of the portfolio just before the trade at time $t_i$. Since we assumed that $S_{t-} = S_t$ for all $t\in (0,T]$, we see that \[ V_{t_i} - V_{t_i-} = (h_{t_i} - h_{t_i-})\cdot S_{t_i} = v_i \cdot S_{t_i}, \] where $v_i$ is the vector representing the amounts traded at time $t_i$, and this difference represents the cash injection to the portfolio required to rebalance the amounts held of each asset (in other words, the amount of cash required to carry out the trade $v_i$ at time $t_i$).

Cash flows In general, a portfolio consisting of trades $((v_i,t_i),i=1,\dots,k)$ will generate a cash flow $(x,t)$ at the same sequence of times $t_i, i=1,\dots,k$ and where $x_i = -v_i \cdot S_{t_i}$. When calculating the profit of a portfolio, we need to take this cash flow into account using the present value $P(x,t)$ of the cash flow. We can define the profit in general (see the remark below), but for calculation purposes it greatly simplifies matters to restrict our attention to portfolios that have a trivial cash flow, i.e., $x_i = 0$ for all $i$ (except for possibly $x_1$, in the case that $t_1 = 0$).

Definition 1.2 A portfolio $(h_t; t \in [0,T])$ is self-financing if $V_{t} = V_{t-}$ for all $t \in (0,T]$. (In other words, the cash flow associated with the portfolio is trivial.)

Remark. Notice that $t=0$ is not included in the condition; we allow trades at time 0 that require/generate a cash amount as part of the initialisation of the portfolio.

For a self-financing portfolio the profit can be simply defined, as follows.

Definition 1.3 Let $P$ be a self-financing portfolio with value $V_t$ at time $t$. Then the present value (value at $t=0$) profit of $P$ is $\alpha^T V_T - V_0$. Here, $\alpha$ is the discount factor per time period, meaning $\alpha = 1/(1+r)$ if interest is compounded discretely at rate $r$ per time period (and $T$ should be interpreted as an integer, the number of periods); or $\alpha = \e^{-r}$ if interest is compounded continuously at rate $r$ per unit time (and $T$ should be interpreted as a real number).

Remark. Restricting consideration to self-financing portfolios is no real loss of generality, provided that there is always access to risk-free interest on our market, and it is more a choice of accounting style. (For the interested reader, we remark that we can define the profit of an arbitrary portfolio to be $\alpha^T V_T + P(x,t)$, where $(x,t)$ is the cash flow generated by the portfolio, and moreover, that given any portfolio we can construct a self-financed version whose profit as defined by Definition 1.3 agrees with the general definition—see the problems sheet for more details.)

We are now in a position to give a reasonably formal definition of arbitrage, the fundamental concept underlying our approach to derivative pricing.

Definition 1.4 A portfolio $P$ is an arbitrage portfolio if the present value profit of $P$ is strictly positive, whatever happens to the share prices.

Lemma 1.1 An equivalent definition of an arbitrage portfolio, is that (i) $P$ is self-financing; (ii) $V^P_0 = 0$; and (iii) $V^P_T > 0$ whatever happens to the share prices.

Proof. See Exercise 1.14 on the Problem Sheet.

Cash versus Bonds The approach we have taken here is to not allow cash as a component of a portfolio. Instead, to gain access to risk-free interest, we assume that the market always contains a risk-free asset with a deterministic price $B_t, t \in [0,T]$. Real examples of such assets include bonds issued by financially secure and trusted institutions (e.g. Bank of England, U.S. Department of the Treasury). In reality cash and bonds do have some differences (typically cash is more “liquid,” meaning it is easier to deposit/withdraw cash from a bank account than to buy/sell bonds; typically bank interest rates change more frequently than rates of return of fixed-term bonds). But under our modelling assumptions, mathematically the two are equivalent: both offer risk-free profits at the same rate of return.

Example 1.4 Suppose the interest rate is $r$ per period (compounded discretely) and suppose $B_t$ is the price of a risk-free asset. If $B_T \neq (1+r)^T B_0$ then there is an arbitrage opportunity.

We construct an arbitrage portfolio $P$ by buying $x$ units of the risk-free asset at time 0 and holding over the whole time interval $[0,T]$, where $x \in \R$ is chosen to be positive if $B_T > (1+r)^T B_0$ and negative if $B_T < (1+r)^T B_0$. Importantly, since the value of $B_T$ is known at time 0, the value for $x$ is also determined at time 0 when the portfolio needs to be initialised. Then, the present value of the profit at time $T$ equals \[ (1+r)^{-T}V_T - V_0 = (1+r)^{-T} xB_T - x B_0 = x ((1+r)^{-T}B_T-B_0) \] which is always positive, since it is the product of two non-zero real numbers with the same sign.

The above example is easily modified to show that $B_t = (1+r)^t B_0$ for all $t=1,2,\dots,T$ (just consider the present value of the profit of $P$ at time $t$; or equivalently sell the $x$ units back at time $t$). Similar calculations work for continuously compounded interest.

Aside: Portfolios with cash

The approach we have taken here (and which we will develop further in later chapters) is to introduce a risk-free asset on our market which represents access to risk-free interest. For completeness, in this aside, we briefly describe the alternative approach which requires us to include cash as a possible component of a portfolio. We will not make use of this representation further, but it is included in case you find the comparison useful.

As before suppose there are $M$ assets that can be traded and also cash which can be deposited/withdrawn. A sequence of trades is now represented by $( ((u_i,v_i), t_i); i=1, \dots,k)$ where $0 \leq t_1 < t_2 < \dots < t_k \leq T$ are the times of the trades, and $v_i \in \R^M$ the amounts of each asset traded as before, but also $u_i \in \R$ represents the amount of cash we deposit/withdraw at time $t_i$ (positive means deposit, negative means withdraw). The amounts held in the portfolio are represented by the collection $( (c_t,h_t); t \in [0,T])$, where $c_t \in R$ is the cash held in the portfolio, and $h_t \in \R^M$ the amounts of each asset ($c_t > 0$ means a cash amount is invested, $c_t < 0$ means cash amount is borrowed). The definition of $h_t$ in terms of $v_i$ is the same as before, and $c_t := \sum_{i : t_i \leq t} (1+r)^{t-t_i} u_i$ if interest is compounded discretely, or $c_t := \sum_{i : t_i \leq t} \e^{r(t-t_i)} u_i$ if it is compounded continuously. Finally, the value $\widehat{V}_t$ is defined as $\widehat{V}_t := c_t + h_t \cdot S_t$. Using these definitions, the interested reader should be able to confirm that cash here is playing the same role as a risk-free asset, for example by showing that any portfolio has a self-financed version in which any cash generated by the trades of the assets is immediately reinvested into the holdings of the portfolio.

1.5 Basic option pricing

We can deduce some properties of option prices without specifying a model for share price evolution by considering some simple portfolios. These conclusions about prices will follow from the principle that option prices should not allow arbitrage opportunities.

Theorem 1.1 (The Law of One Price) Suppose self-financing portfolio $P_i$ costs $C_i$ at time 0 for $i = 1, 2$ with $C_2 > C_1$ and suppose further that the present value of the payoffs from portfolio $P_1$ are at least as large as those from $P_2$ whatever happens to the share price. Then an arbitrage opportunity exists.

Proof. Buy portfolio $P_1$ and sell $P_2$ for a certain present value profit of at least $C_2 - C_1 > 0$. In other words, the portfolio $P_1 - P_2$ is an arbitrage portfolio, since it is self-financing, and the present value profit equals $\alpha^T(V^1_T - V^2_T) - (C_1 - C_2) \geq C_2 - C_1 > 0$

A simple consequence of this is that two portfolios with identical payoffs should have the same cost. Consideration of some simple portfolios show that the prices of European put and call options with the same strike price and expiry date are intimately related, whatever model of share prices we employ.

Theorem 1.2 (Put-Call Parity) Let $C$ and $P$ denote the prices of European call and put options on the same non-dividend paying stock with common exercise price $K$ and common expiry date $T$. Denote the initial share price by $S_0$ and suppose interest is compounded discretely at rate $r$ per time period. Then \[ P + S_0 = C + K (1+r)^{-T} \] or there is an arbitrage opportunity.

Remark. Recall that for discrete interest the time $T$ represents the number of compounding periods. If interest is instead compounded continuously at rate $r$, then replace $(1+r)^{-T}$ with $\e^{-rT}$, so that the put-call identity $P+S_0 = C + K \e^{-rT}$ holds for all real $T > 0$.

Proof. Since interest is compounded discretely, we can assume that there is a risk-free asset with deterministic price $B_t = (1+r)^t B_0$ for all $t = 0,1,\dots,T$. Setting $\alpha = 1/(1+r)$ yields the identity $\alpha^T B_T = B_0$. (For continuously compounded interest, the risk-free asset satisfies $B_t = \e^{rt}B_0$ for all $t$, so we would set $\alpha = \e^{-r}$, but otherwise the proof would proceed identically.)

Consider portfolio $A$ formed by buying one call option and $K \alpha^T/B_0$ units of the risk-free asset at time 0, and portfolio $B$ formed by buying one put option and one share of stock at time 0. At time $T$ portfolio $A$ is worth $(S_T - K)^+ + K (\alpha^T/B_0) B_T = \max(S_T-K, 0) + K = \max(S_T,K)$ while portfolio $B$ is worth $(K - S_T)^+ + S_T = \max(K,S_T)$. These are identical so, by the Law of One Price, to avoid an arbitrage opportunity the two portfolios must have the same price at time 0, i.e., $C + K \alpha^T = P + S_0$.

The obvious consequence of this result is that if we work out a formula for $C$ then one for $P$ follows immediately. Minor variations on this choice of portfolios lead to some simple bounds on $C$ and $P$ (see the problem sheet).

Mathematical Finance 2022/23 (Michaelmas)