Chapter 1 Introduction

These lecture notes are based on the content we will cover in Mathematical Finance in Michaelmas term, 2022 - 23. They are based on previous sets of notes put together by Nic Georgiou, Andrew Wade, and others.

1.1 What is Mathematical Finance?

Mathematical Finance is the study of the mathematics used to model and analyse financial markets. These models are constructed to try to better understand how markets behave in reality, and to inform decisions about investments. In reality, these markets are incredibly complex, but under some simplifying assumptions, the mathematics becomes quite elegant, and allows us to develop methods for pricing and valuing portfolios based on a wide range of financial derivatives.

In Michaelmas term, we’ll focus on discrete-time versions of these models, where we assume that trades can only happen at specified moments. We’ll build up a theory using probabilistic concepts like filtrations, conditional expectation, and martingales. Because discrete-time models lead to countable probability spaces, we’ll be able to do a lot of this work in very concrete settings, and calculate the prices of some quite complex financial products.

In Epiphany term, you’ll see the continuous-time versions of the models. Here, the continuous (uncountable!) probability spaces mean that the theory becomes much more complex and subtle, and some of the generalisations to continuous time require some pretty sophisticated measure theory. The understanding of concrete fundamental concepts you build up in Michaelmas term will put you on a solid footing to start working with the more abstract theory to come in Epiphany.

1.1.1 Finance or probability?

I consider myself a probabilist, and that’s the approach I’m bringing to this course. I see it as being about the mathematics of finance, rather than the economics, and we’ll be coming at a lot of the material from a probability perspective. That said, we’ll need to use some financial terminology to describe the concepts we’re using, and many of the examples in the course will be from a financial context.

1.2 Financial derivatives

The main focus of this course is on financial derivatives, and how they should be priced and can be hedged. In this chapter, we set up the framework under which we work, including a lot of the definitions we’ll need. In Chapters 2 and 3 we’ll build up our theory using an example of a discrete-time market; then in chapter 4 and 5 we’ll look at extensions to this theory. To illustrate the types of problems we’ll be discussing, we start with an example.

Example 1.1 A Norwegian company, Elg Inc, wants to import 1000 moose harnesses from a Scottish company, McMoose.

Each harness costs £1,000 (GBP), and Elg Inc will be required to pay McMoose when the harnesses are delivered, in six months’ time.

We will assume that the current exchange rate between Norwegian krone and pounds is 11.5 NOK : 1 GBP.

One of the issues for Elg Inc is the currency risk associated with the transaction. Since they cannot know what the exchange rate will be in six months, it is impossible to know today (at \(t=0\)) what the price in NOK will be at \(t = T\). If the exchange rate is still 11.5, then they will have to pay 11,500,000 NOK, but if the rate increases to, say, 12, then the cost will be 500,000 NOK higher.

Elg Inc can avoid this currency risk in several different ways, including:

  1. Elg Inc could buy £1,000,000 today, at a price of 11,500,000 NOK, and keep this money in a bank account for six months.

  • Pros: The currency risk is completely eliminated.

  • Cons: This is a lot of money to tie up for six months, and Elg Inc might not have access to this much today.

  1. Elg Inc could buy a forward contract for £1,000,000 with delivery six months from now. This is an agreement with (e.g.) a bank that Elg Inc will buy £1,000,000 from them, at an exchange rate which is agreed at time \(t = 0\). The rate is called the forward price, and usually denoted \(K\); here we might have \(K = 12\). Typically forward contracts involve no upfront cost.

  • Pros: There is nothing to pay now, and the currency risk is completely eliminated; Elg Inc know exactly how much they will have to pay. If the exchange rate at \(t=T\) is higher than \(K\), then Elg Inc will have saved some money in buying at the lower rate of \(K\) NOK : 1 GBP - but…

  • Cons: …if the exchange rate at \(t=T\) is lower than \(K\), then Elg Inc will still have committed to pay the higher rate, and will lose out compared to paying the market rate for their £1,000,000.

  1. Elg Inc could buy a contract allowing them the option to buy at an agreed price, but not the obligation. In this scenario, Elg Inc wants a European call option: they agree on a strike price of \(K\) NOK : 1 GBP and an expiry time \(T\) with the broker, and pay a (hopefully fair!) upfront cost for the agreement. If the exchange rate at time \(T\) is higher than \(K\), then they can exercise the option and pay the lower price; if the exchange rate at time \(T\) is lower than \(K\), they can ignore the option and simply buy £1,000,000 at the market price.

  • Pros: the currency risk is completely eliminated; Elg Inc will either pay \(K\) NOK : 1 GPB or the exchange rate at time \(T\), whichever is lower.

  • Cons: there is an upfront cost for the contract; how much should Elg Inc be willing to pay?

Vocabulary

Both the forward contract and the option are examples of financial derivatives or derivative assets: they are products which can be bought and sold on the financial market, but whose values are based on a separate underlying asset. In this example, the underlying asset is the currency itself.

Holding the option is equivalent to a future stochastic claim: its value depends on the exchange rate at a time in the future, which is unknown. Options are a type of contingent claim, because their value is contingent on an external factor, in this case the exchange rate. We will see contingent claims in more detail later in the term.

There are two main categories of options: call and put. Call options, as in this example, give the holder the right (but not the obligation) to buy the underlying asset at an agreed price; put options give the holder the right to sell it. The prefix European means that the option can only be exercised at exactly time \(T\); American options can be exercised at any time \(t \in [0,T]\).

The value of European call and put options The value of a European call option at time \(T\) depends on the value of the underlying asset it’s based on. We write \(S_t\) for the price of the underlying asset at time \(t\): this could be the current exchange rate, or the price of a stock.

A call option is only valuable if the strike price, \(K\), is less than the price of the asset; that is, if \(S_T < K\). In this case, the holder can exercise the option, buy at price \(K\), and immediately sell at price \(S_T\), making a profit of \(S_T - K\). On the other hand, if \(S_T \leq K\), the option is worthless: rather than make a loss of \(S_T - K\), we can simply choose not to exercise the option.

As a result, the value of the option at time \(T\) is \[ \begin{cases} S_T - K & \mathrm{if\ } S_T > K \\ 0 & \mathrm{if\ } S_T \leq K. \end{cases} \]

We can also view this as \(\max (S_T - K, 0)\), which we write as \((S_T - K)^+\) (as it’s the positive part of this value).

Similarly, the return at time T of a put option is \(\max (K - S_T, 0) = (K - S_T)^+\).

Remark. Notice that selling call options is not the same thing as buying put options! Mathematically, this is because \[ (K - S_T)^+ \not= -(S_T - K)^+\] (one side is always positive or zero, and the other is always negative or zero). In terms of the transactions, the difference arises from the fact that the choice always lies with the buyer of the option; the seller is obligated to go along with the buyer’s decision).

Exercise 1.1 Draw the graphs of the return values of the following combinations of options, as a function of the asset price:

  • a call option with strike price \(K\)
  • a put option with strike price \(K\)
  • buying one call plus one put option with the same strike price \(K\) (this is known as a straddle)
  • buying one call option and selling one put option with the same strike price \(K\)
  • buying one call option with strike price \(K_1\), and selling another call option with strike price \(K_2\) (this is known as a bull spread; you’ll want \(K_2 > K_1\))

1.3 Pricing derivatives

Example 1.2 Elg Inc shares are floated on the stock market. The value of one share is £100 today (at \(t = 0\)), and we know that tomorrow (at \(t=1\)) the value will either increase to £200, with some unknown probability \(0 < p < 1\), or decrease to £50 with probability \(1-p\).

We are a broker trading in stocks and options, and we’re considering the following (European call) option: the holder has the right (but not the obligation) to buy one share for £150, at time \(t=1\). At time \(t=0\), what is the fair price \(C\) for this option? We have to be willing to both buy and sell options at this price, and we will assume that there’s a wealthy trader “Agent A” who will take advantage and run us out of business if we mis-price the option and create an arbitrage.

One approach is to think about the value of the option at time \(t=1\): if the share price is £50, then the option is worthless and we throw it away. If the share price is £200, then by exercising the option and immediately selling the share, the holder will make a profit of £50. So the expected value of the option is \(50 p\) – to decide what \(C\) should be, we’ll need an estimate for \(p\). Let’s see what happens if we estimate that \(p=0.2\), so that \(C =\) 10.

Agent A comes along, and asks to sell us a share and buy 3 options at time \(t=0\). With this transaction, we initially hand over \[ 100 - 3 \times 10 = 70.\]

An important note here is that Agent A can sell us as many shares as he likes, through short selling. He borrows the shares from some third party at time 0, and he will have to return them at time 1. In practice, this means that for every share he sells us now, he will have to buy one back later.

At time 1, if the share price has gone down, the option is worthless. In this case, he can buy one share of the stock for £50, return it to the third party, and leave with \[ 70 - 50 = 20\] in profit.

On the other hand, if the share price has gone up, each option represents a profit of £50. In this case, Agent A will spend £450 to exercise the options; return one share to the third party; and sell the two remaining shares for £400. This time, he leaves with \[ 70 - 450 + 400 = 20 \] in profit.

In either case, Agent A has returned his borrowed share and made a £20 profit from us: there is no risk! This is an example of an arbitrage opportunity. Since Agent A has effectively limitless money, he can instead sell a million shares, buy three million options, and make £20 million profit - or even more - and ruin us.

In order to calculate the fair price for the option, we should think about Agent A’s profit in general terms. Let’s consider a portfolio consisting of \(x\) units of the stock and \(y\) units of the option; here, negative values for \(x\) and \(y\) represent short selling. The initial cost to Agent A is \[100 x + C y.\]

At time 1, this portfolio is either worth \(50x\) (if the share price goes down), or \(200 x + 50y\) (if the share price goes up). To eliminate the risk and make the two values equal to each other, we set \(y = -3x\).

Overall, this portfolio has a risk-free profit of \[ 50 x - (100 x - 3 Cx) = (3C - 50)x. \]

If this is non-zero, Agent A can ruin us! If it’s positive, he takes a hugely positive value for \(x\) (this is buying the stock and short selling the option). If it’s negative, he can take a hugely negative value for \(x\) (this corresponds to short selling the stock and buying the option, as in the first part of the example). The only way to avoid arbitrage is to set \(C= 50/3\).

Note that the price \(C\) does not depend on \(p\) at all!

Vocabulary

The portfolio selected by Agent A in this example took the form “sell one share and buy 3 options, at time 0.” In general, a portfolio is a description of the shares, options, and cash we hold at any given time. The values of any of these things can be negative if we engage in short selling: borrowing units of an asset at one time, to be returned later.

When it is possible to assemble a portfolio that with certainty performs better than the risk-free interest rate, we say that there is an arbitrage opportunity in the market.

1.4 Risk-free interest

We will assume that any market we study contains an opportunity for risk-free investment. By this, we mean that there is a (unique) fixed interest rate \(r\), and that any money we borrow or invest will accrue interest at this rate.

To compute the amount of interest, we need to know the nominal rate \(r\), plus the rule for compounding. Suppose we have an initial investment of \(B(0)\) at \(t=0\).

If the interest is compounded once per year at nominal rate \(r\), then we will have \[ B(t + 1) = B(t)\ (1+r), \qquad\qquad\mbox{so that} \qquad B(t) = B(0)\ (1+r)^t. \]

If the interest is compounded \(n\) times per year (here \(n\) is often 2, 4, 12, 52, 365, etc), then we multiply by \((1 + r/n)\) at each of the \(n\) equally-spaced intervals. Now we have \[ B(t + 1/n) = B(t)\ (1 + r/n) \qquad\qquad\mbox{so that } \qquad B(m/n) = B(0) \ (1+r/n)^m. \]

Finally, interest may be compounded continuously. We can think of this in terms of the previous case by looking at the limit as \(n \to \infty\); if we write \(m = tn\) then \[ B(t) = B(0) \ (1 + r/n)^m \] becomes \[ B(t) = B(0) \ (1 + rt/m)^m\] and so in the continuous case, \[ B(t) = \lim_{n \to \infty} B(0) \ (1+ rt/m)^m = B(0) \ e^{rt}.\]

From now on, we will work in units of “one compounding period” (day, week, month, etc) and will suppose that we have chosen the “correct” value of \(r\) so that we can always work in the setting \(n=1\).

Remark. In textbooks or the wider literature, you may come across references to the effective interest rate, \(r_{\mathrm{eff}}\). This is the rate which, when compounded once per year, gives the same amount of interest as the nominal rate compounded \(n\) times; in other words, \[ 1 + r_{\mathrm{eff}} = (1+r/n)^n.\]

Present value analysis

Now that we have introduced interest rates into the model, the timings of cash deposits matter: £10 today is worth more to us than £10 next week, because of the extra interest. In order to compare different cash flows, we write everything in terms of the present value.

In order to calculate the present value of any amount of cash we will receive in the future, we use the discount factor \(\alpha = 1/(1+r)\). If we will receive (or pay out) an amount \(x\) at time \(t\), its present value at time 0 is given by \(x \ \alpha^t\). A cash flow \((x,t)\) in which we receive values \(x_i\) at times \(t_i\) for \(i=1,2, \dots, n\) has present value \[ P(x,t) = \sum_i x_i \alpha^{t_i}.\]

Example 1.3 Which of these cash flows has the highest (and the lowest) present value?

  1. \(x_i = 100\), \(t_i \in \{1,3,5,7,9\}\)
  2. \(x_i = 50\), \(t_i \in \{1, 2,3,4,5,6,7,8,9,10\}\)
  3. \(x_i = 200\), \(t_i \in \{1,3,5,7,9\}\) and \(x_i = -100, t_i \in \{2,4,6,8,10\}\)
  4. Wildcard: construct your own cash flow.

Example 1.4 You have a bank account in which the annual interest rate is 5%, compounded monthly You plan to pay in £\(D\) every month for thirty years (360 months), and then withdraw £1000 every month for the following twenty years (months 361 - 600). What is the minimum deposit \(D\) you should be making, to ensure you have enough in your account?

Here \(r = 0.05/12 = 1/240\), so \(\alpha = 241/240\). The present value of all the deposits is \[ D + D \alpha + D \alpha^2 + \dots + D \alpha^{359} = D \sum_{i=0}^{359} \alpha^i = D \ \frac{1-\alpha^{360}}{1-\alpha}. \]

Next, the present value of all the withdrawals is \[ 1000 \alpha^{360} ( 1 + \alpha + \alpha^2 + \dots + \alpha^{239}) = 1000 \alpha^{360} \frac{1 - \alpha^{240}}{1-\alpha}. \]

These are equal when \[ D = 1000 \alpha^{360} \frac{1-\alpha^{240}}{1-\alpha} \frac{1-\alpha}{1-\alpha^{360}} = 1000 \alpha^{360} \frac{1-\alpha^{240}}{1-\alpha^{360}} \approx 182.065. \]

1.5 Portfolios and arbitrage

Throughout this term, our aim is to understand the pricing and hedging of financial derivatives. We’ll primarily do this through the study of portfolios. One important idea here is that, if two portfolios have the same future payoffs (whatever happens to the values of the different assets on the market), then they should have the same current price. This means that, if we can find a portfolio that matches the payoff from a derivative asset, we can determine the price of the derivative by analysing the portfolio. In Chapter 2, we’ll see this in more detail for a particular market; for now, we set up some basic portfolio theory.

Portfolios

We start off with a market consisting of \(M\) different assets, over a time period \([0,T]\). For \(t \in [0,T]\), we write \(S_t = (S_t^1, \dots, S_t^M)\) for the vector of the prices of the \(M\) different assets. The study of share prices for many different stocks suggests that, over small intervals, the price change \[ \frac{S_{t+ \delta t} - S_t}{S_t} \] should be approximately Normally distributed, and that the price changes over two (or more) non-overlapping intervals should be modelled by independent random variables. For now, we will not concern ourselves too much with how the prices behave, and just let them take values in the positive real numbers. We will usually assume that each \(S_t^j\) is a left-continuous process, so that \(\lim_{s \uparrow t} S_s = S_t\) for every \(t \in (0,T]\).

A portfolio, or a trading strategy, represents the way our holdings evolve over time. We can do this in one of two ways: by describing our initial holdings along with a sequence of trades, or by describing how much of each asset is held at each time \(t \in [0,T]\).

In the first case, our portfolio is given by a list of vectors \(P = ((v_i, t_i), i = 1, \dots, k)\), where the times \(t_i\) are increasing, so that \(0 \leq t_1 < t_2 < \dots < t_k \leq T\), and where the vectors \(v_i \in \mathbb{R}^M\) represent the amount of each of the \(M\) assets we will buy or sell at times \(t_i\).

In the second, we write \(P = (h_t; t \in [0,T])\), where \(h_t\) is the vector representing the amount of each asset we hold at time \(t\). To move between the two representations, we can use the fact that \[ h_t = \sum_{ i : t_i \leq t} v_i.\] Note that, when a trade happens at time \(t\), the amount of each asset we held before the trade is represented by \(h_{t^-}\), and the amount we hold after the trade is represented by \(h_t\).

We can view a portfolio as a function \(t \mapsto h_t\); it is a step function which is constant on the intervals \([t_i, t_{i+1})\), \(i = 1, \dots, k-1\). In fact, any right-continuous piecewise step function \(f:[0,T] \to \mathbb{R}^m\) can represent a portfolio, and to recreate the corresponding sequence of trades, we just take all the non-zero values of \(f(t) - f(t^-)\) for \(t \in [0,T]\).

Formally, the space of all portfolios (or the space of all right-continuous piecewise step functions) is a vector space: we can take any linear combination of two portfolios to find another one. Given two portfolios \(X\) and \(Y\), and two real numbers \(\alpha\) and \(\beta\), we define \[ \alpha X + \beta Y = \{\alpha h_t^X + \beta h_t^Y; t \in [0,T]\}. \] For example, the portfolio \(X+Y\) means we perform all the trades of both \(X\) and \(Y\), while \(-X\) means we take all the trades of \(X\) “in reverse” (swapping buy for sell, and vice versa).

We can also create portfolios in which we don’t know in advance precisely when the trades will happen. For example, we may wish to create a portfolio in which we buy shares of a stock at time 0, and then sell them as soon as the share price doubles. In that case, the time of our trade is \(\inf \ \{ t : S_t \geq 2 S_0\}\). This is a permissible trade, because we will recognise the moment when it happens; on the other hand, we can’t decide to sell “whenever the price is highest between time 0 and time \(T\),” because we can’t know whether or not that moment has arrived. If you took Markov Chains last year, you might recognise that we’re talking about stopping times.

The value of a portfolio at time \(t\) is given by the total value of each of the assets held; for a portfolio \(P = (h_t; t \in [0,T])\) we write \[ V_t = h_t \cdot S_t = \sum_{i=1}^m h_t^i S_t^i. \]

The amount of cash required to initialise a portfolio is given by \(V_0\) - note that this can be negative, for example if we’re short selling. The cash received at time \(t=T\) when we close out the portfolio is \(V_T\), which can also be negative - for example to buy back shares that were sold earlier.

Example 1.5 If our portfolio \(X\) is “sell 100 shares of the stock at time 0,” then we can express \(X\) as a sequence of trades by \[ X = ((-100, 0)),\] or as a sequence of holdings by \[ X = ( h_t^X, t \in [0,T]) \qquad \qquad \mbox{where} \qquad h_t^X = (-100 ) \ \forall t \in [0,T].\] The value of \(X\) is always \(V_t = -100 S_t\).

If our portfolio \(Y\) is, as we discussed earlier, “buy 100 shares at time 0, and sell them as soon as the share price doubles,” then the sequence of trades is \[Y = ( (100, 0),\ (-100,\inf \ \{ t : S_t \geq 2 S_0\} ) ), \] and the holdings are given by \[ h_t^Y = \begin{cases} 100 & \sup_{s \in [0,t]} (S_s) < 2 S_0 \\ 0 & \sup_{s \in [0,t]} (S_s) \geq 2 S_0 \end{cases}.\] We can write this as \(h_t^Y = 100 \ \ind \ \{ \sup_{s \in [0,t]} (S_s) < 2 S_0 \}\). Similarly, the value of the portfolio is given by \(V_t = 100 S_t \ \ind \ \{ \sup_{s \in [0,t]} (S_s) < 2 S_0 \}\).

Question: what are the trades, holdings, and value of the portfolio \(X+Y\)?

Cash flows and arbitrage

As well as understanding how the holdings of shares, derivatives, and other assets evolve in a portfolio, we also need to think about the cash flow. Typically, each trade \((v,t)\) will generate a cash flow \((x,t)\), and we need to take these into account, using the present value \(P(x,t)\). To simplify the calculations, we will only focus on cash flows which have a trivial cash flow, that is, \(x_i = 0\) for all \(i\).

(The only exception to this is when \(t_1 = 0\), meaning there is a trade at time 0. In this case, we will allow \(x_1 \not = 0\).)

If we consider the value of the portfolio both immediately before and immediately after the \(i\)th trade, we have \[V_{t_i^-} = h_{t_i^-} \cdot S_{t_i^-},\] and \[ V_{t_i} = h_{t_i} \cdot S_{t_i}.\] Since we are assuming that \(S_t\) is left-continuous, \(S_{t_i^-}\) and \(S_{t_i}\) are the same; we therefore have \[ V_{t_i} -V_{t_i^-} = (h_{t_i} - h_{t_i^-}) \cdot S_{t_i} = v_i \cdot S_{t_i}. \] (Remember that \(v_i\) is the vector representing the trade at time \(t_i\).) When this value is always 0, we say that the portfolio is self-financing; in other words, it is self-financing if \(V_t = V_{t^-}\) for all \(t \in (0,T]\). Note that by excluding \(t=0\), we allow for portfolios which require or generate cash at their initialisation.

Remark. It is not too restrictive to only consider self-financing portfolios. Because our market includes an opportunity for risk-free investment in the form of bonds, we can find a self-financing version of any portfolio by buying or selling bonds in the appropriate quantities to create a trivial cash flow. Although bonds are not the same as cash in practice, they play the same role in our mathematical model.

The present value profit of a self-financing portfolio \(P\) with value \(V_t, t \in [0,T]\) is given by \(\alpha^T V_T - V_0\). Remember that \(\alpha\) is the discount factor per time period, so that \(\alpha = (1+r)^{-1}\) if interest is compounded discretely at rate \(r\) per time period, or \(\alpha = e^{-r}\) if interest is compounded continuously.

We say that a portfolio \(P\) is an arbitrage portfolio if the present value profit of \(P\) is positive, whatever happens to \(S_t\).

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.

The proof of this Lemma is Exercise 1.14 on the problem sheet.

Example 1.6 If the interest rate is \(r\) per period, compounded discretely, and \(B_t\) is the price of a risk-free asset, then we must have \[ B_t = (1+r)^t B_0, \] or else there will be an arbitrage opportunity.

To see that this is true, consider the portfolio \(P\) which consists of buying \(x\) units of the risk-free asset at time 0.

The present value profit of \(P\) is given by \[ (1+r)^{-T} V_T - V_0 = (1+r)^{-T} x B_T - x B_0 = x \left( (1+r)^{-T} B_T - B_0 \right). \]

If \((1+r)^{-T} B_T - B_0\) is non zero, we can choose \(x\) to have the same sign as \((1+r)^{-T} B_T - B_0\) and ensure that the present value of the profit is always strictly positive; so \(P\) is an arbitrage portfolio.

We can modify this calculation for \(t = 1, 2, \dots, T\) to see that \(B_t\) must be equal to \((1+r)^t B_0\) for all times \(t\).

1.6 Basic option pricing

Even before we start working with a specific model, we can deduce some of the properties of option pricing. The first important principle is that option prices should not allow arbitrage opportunities.

Theorem 1.1 (The Law of One Price) Consider two self-financing portfolios \(P_1\) and \(P_2\), with costs \(C_1\) and \(C_2\), and present value payoffs \(V_1\) and \(V_2\). If \(V_1 \geq V_2\) is true whatever happens to the share price, then we must have \(C_1 \geq C_2\) or an arbitrage opportunity exists.

(In other words: if \(P_1\) is a better bet than \(P_2\), it can’t cost less.)

Proof. Consider the portfolio \(P_1 - P_2\). The present value profit is \[ \alpha^T ( V_T^1 - V_T^2) - (C_1 - C_2) \geq C_2 - C_1, \] whatever happens to the share prices; so if \(C_2 - C_1\) is positive, then an arbitrage opportunity exists.

In particular, two portfolios with identical payoffs must have the same costs to avoid creating arbitrage opportunities.

This principle, applied to European call and put options with the same strike price, gives the following theorem.

Theorem 1.2 (Put-Call Parity) Consider a European call option with cost \(C\), and a European put option with cost \(P\), on the same stock. If the initial share price is \(S_0\), interest is compounded discretely at rate \(r\), and both \(C\) and \(P\) have strike price \(K\) and expiry date \(T\), then we must have \[ P + S_0 = C + K(1+r)^{-T}, \] or else an arbitrage opportunity exists.

Proof. Remember that we have a risk-free asset \(B_t\), and that to avoid an arbitrage opportunity we must have \(\alpha^T B_T = B_0\).

Consider two portfolios, \(X\) and \(Y\). In \(X\), we buy one call option and put \(K \alpha^T\) into bonds, buying \(K \alpha^T/B_0\) units, at time 0. The price at time 0 is \(C + K \alpha^T\), and at time \(T\) the portfolio will be worth \[ V_T^X = (S_T - K)^+ + K \alpha^T B_T / B_0 = (S_T - K)^+ + K = \max(S_T, K). \]

In \(Y\), we buy one put option and one share of stock at time 0. The price at time 0 is \(P + S_0\), and at time \(T\) portfolio \(Y\) will be worth \[ V_T^Y = (K - S_T)^+ + S_T = \max(K, S_T).\]

Since both portfolios have the same payoff, they must have the same initial cost by the Law of One Price to avoid an arbitrage opportunity; so \[ C + K \alpha^T = P + S_0. \]