MTH6120 Further Topics in Mathematical Finance Lesson 2

MTH6120 Further Topics in Mathematical Finance Lesson 2 Contents 1.2.3 Non-constant interest rates....................... 15 1.3 Arbitrage and Black-Scholes Theory....................... 16 1.3.1 Informal definition and basic example.................. 17 1.3.2 Toy example............................... 18 1.2.3 Non-constant interest rates The previous sections assumed that interest rates are constant in time. This is implicit in the assumption that we can compound interest rates as in P (1 + 112 ) 12 r where r is the monthly interest rate. In the real world interest rates typically vary due to supply and demand, and monetary policy so that after one month the one month interest will no longer be r and hence the formula above is not valid. There are two more advanced mathematical modelling techniques to handle the case in which interest rates are not constant: Deterministic 1 interest rates This is the case where even though the rates vary with time we presume that the future values are known now. This is what is covered in [IMF, 3.4]. Stochastic interest rates Here we allow interest rates to take values in the future unknown today. This uncertainty is modelled by making all interest rates random variables. We will cover this in a later chapter of this module. We will now review the first case as discussed in [IMF]: Deterministic interest rates We will consider two functions and show how they are related. Definition 1.26 ([IMF, 3.4], see also [Ross, 4.4]). 1. P (t) stands for the amount of cash obtained by placing 1 unit of cash in a deposit from time 0 to time t. 2. The variable r(t) indicates the instantaneous rate for a deposit placed at time t. 1 The word deterministic is sometimes used to emphasize that there is nothing random here. It can be understood as the opposite of stochastic or random. 15

So intuitively r(t) is the interest rate I get for a nano-second deposit placed at time t, and P (t) is the money I have in the bank at time t if I started with 1 unit of cash at time 0. The deterministic rates assumption in this section is that these are both known functions (i.e. P (t) and r(t) are just normal real functions, there is nothing stochastic here We have the following result: Theorem 1.27 (compare with [IMF, Theorem 3.18]). We have the following relation between r(t) and P (t) P (t) = e t 0 r(s)ds. Proof. If at time t we have P (t) units of cash in the bank then, one nanosecond later, at time t + t, we should have something like: P (t) (1 + r(t) t) but this must be equal to P (t + t), so we have the equality: P (t + t) = P (t) (1 + r(t) t). If we isolate r(t) from the equation above we get P (t + t) P (t) P (t) t = r(t) If we take lim x 0 we get P (t) P (t) = r(t) This is an ordinary differential equation whose solutions are P (t) = Ce t 0 r(s)ds. Since P (0) = C = 1 is the cash we have at time 0 the statement follows. Remark 1.28. Note this theorem is plausible as we can rewrite t e t 0 r(s)ds = e 1 t t 0 r(s)ds t. And the quantity r(t) = 1 r(s)ds is some kind of average instantaneous rate from time 0 t 0 to time t, and P (t) = exp (r(t)t). Definition 1.29 ([IMF, 3.4.2], [Ross, 4.4]). The function r(t) = 1 t yield curve. t r(s)ds is called the 0 1.3 Arbitrage and Black-Scholes Theory [Ross, Chapter 6] 16

1.3.1 Informal definition and basic example In this section we will revise the notion of arbitrage and how it leads to the Black-Scholes formulæ. Recall the basic imprecise but intuitive definition : Definition 1.30 ([IMF, Definition 4.1]). An arbitrage is an opportunity to make money with no risk. One expects not to find such opportunities in the market as the buying and selling activity of market makers will eliminate them. An interesting fact of modern finance is that often we can determine prices or link different prices by using the assumption that there is no arbitrage. This is sometimes called the noarbitrage principle or sometimes no-free-lunch principle. Example 1.31 (baby example of reasoning by no-arbitrage). Assume that bank A offers us a monthly compounded interest rate, r 1M, and bank B offers a yearly compounded interest rate of r 1Y. Then we will see that we must have ( 1 + r 1Y = 1 + 1 12 1M) 12 r, or otherwise there is arbitrage in the market. Assume that instead we have ( 1 + r 1Y < 1 + 1 12 1M) 12 r, (1) (the reverse inequality is treated similarly). Then we could ask for a loan of $1,000,000,000 in bank B that would require payment of $1,000,000,000(1 + r 1Y ) in one year time. We would invest this money in a one month deposit in bank A which would pay $1,000,000,000 ( 1 + 1 r ) 12 1M at the end of the first month. At the end of the second month our bank account (in bank A) would hold $1,000,000,000 ( 2. 1 + 1 r 12 1M) And at the end of the year we would have $1,000,000,000 ( 12. 1 + 1 r 12 1M) We can use this money to pay back our loan and would be left with ( ( $1,000,000,000 1 + 1 ) 12 12 r 1M (1 + r 1Y )) which by assumption is a strictly positive number. Thus we make a profit with no risk. This is an arbitrage. If we assume that arbitrages do not exist then inequality (1) cannot hold Exercise 1.32. Explain why, in the example above, the reverse inequality, ( 1 + r 1Y > 1 + 1 12 1M) 12 r, also leads to arbitrage. An important result in the theory of no-arbitrage pricing relates non existence of arbitrage with the possibility of defining a probability with certain features. What follows is an imprecise version of this result which we will make rigorous in the following section 17

Theorem 1.33 (Arbitrage Theorem). One and only one of the following two statements is true: 1. There is arbitrage in the market. 2. There exists a probability measure in which the expected value of the Profit&Loss of any transaction is zero. Remarks 1.34. 1. This result is, in some, sense intuitive as the condition (b) expresses that we do not expect, on average, to make (or lose) money which is what no-arbitrage means. 2. The theorem is equivalent to the following statement: The following are equivalent: (a) The market is free of arbitrage. (b) There exists a probability measure in which the expected value of the Profit&Loss of any transaction is zero. 1.3.2 Toy example In order to appreciate the details of no arbitrage arguments we will develop a very simple toy model that is rich enough to capture interesting features of a betting market. In this model we only have two times: now and T, and n possible outcomes of a game 1,..., m. Example 1.35. An example of such model would be the game of tossing a dice. At time T we toss the dice and the possible results are 1, 2, 3, 4, 5, 6. We might sometimes use the symbols,,,,, and instead to prevent confusions with other sub-indices that will arise. Definition 1.36. A bet on our game is a function r defined on each of the possible outcomes. I.e. a list of numbers r(j) for j = 1,..., m. The number r(j) indicates the profit or loss of the outcome j. Example 1.37. 1. In a game of dice a bet would be a list of numbers r( ), r( ), r( ), r( ), r( ), and r( ). Each of them would indicate how much money we will make for each possible outcome. 2. Imagine a game has three possible outcomes denoted 1,2 and 3 (for example a game of football can result in team A win, team B win or draw). A possible bet would be r(1) = 4, r(2) = 1, r(3) = 100. This means that if the outcome is 1 then we make 4, if it is 2 we lose 1, and if the outcome is 3 then we lose 100. 3. In a game of dice we could have r( ) = 100, r( ) = 2, r( ) = 3, r( ) = 0, r( ) = 20, and r( ) = 100. 18

4. Note that we could have bets that are inconsistent. For example, bets r 1 and r 2 described by r 1 ( ) = 10 r 2 ( ) = 5 are inconsistent (arbitrageable) in that we can create an arbitrage by purchasing bet r 1 and selling bet r 2. The resulting position will make zero Profit&Loss if the outcome is,,,, or, and makes 5 if the outcome is. In order to describe what an arbitrage is in this context we need to define what a portfolio of bets is Definition 1.38. If we have a collection of bets r 1,... r n, then a portfolio of bets (also called a betting strategy) consists of the specification of n numbers called the notionals: x 1,..., x n. The payoff or return of the portfolio of bets is defined to be R = x i r i. Note that this is an equality of functions; you can write it on each individual outcome as: R(j) = x i r i (j) j = 1,..., m. Remark 1.39. In the case of dice, the payout of a portfolio of bets can be illustrated as: R( ) = x i r i ( ), R( ) = x i r i ( ), R( ) = R( ) = x i r i ( ), R( ) = x i r i ( ), R( ) = x i r i ( ), x i r i ( ). In this framework we can give a rigorous definition of what arbitrage means. Definition 1.40. Given a collection of bets r 1,... r n, an arbitrage is a portfolio of bets, x 1,..., x n such that its return is 0 for all possible outcomes (never loses money), and is > 0 for some outcomes (makes money on some occasions). 19

Example 1.41. If we take the two trades, r 1 and r 2, in the example 1.37.(4) then the portfolio defined by x 1 = +1 and x 2 = 1 (buy r 1 and sell r 2 ) is an arbitrage as its payoff function is R( ) = +1 r 1 ( ) 1 r 2 ( ) = 10 5 = 5 which cannot lose money and will make money in some cases. This is an arbitrage. The main theorem is the following Theorem 1.42 (Arbitrage Theorem, see [IMF, Theorem 5.4] and [Ross, Theorem 6.1.1]). Given a set of bets r 1,..., r n one and only one of the following statements holds: 1. There exists a set of probabilities for each of the outcomes p 1,..., p m such that the expectation of the bets is zero, i.e. for each i = 1,..., n 2. There is an arbitrage. E(r i ) = p 1 r i (1) + p 2 r i (2) + p m r i (m) = 0. Remark 1.43. Note that the probabilities in the theorem have nothing to do with anything physical. They are not real world probabilities. They are just some numbers that a theorem says exist and that can be useful. There are some special bets that bookmakers in the real world trade Definition 1.44 ([IMF, Example 5.3] and [Ross, Example 6.1a]). The basic trade, r i, associated to the outcome, i, is a trade of the form: { 1 if i j r i (j) = o i if i = j The number o i is called the odds of event i. Remark 1.45. 1. The function r i just codifies the bet where we win o i if we correctly guessed outcome i and lose 1 otherwise. 2. Sometimes instead of describing the whole of these bets we just say that the bookmaker is quoting outcome i with o i odds. 3. For example the bet r( ) = 1, r( ) = 1, r( ) = 1, r( ) = 1, r( ) = 1, r( ) = o 6. codifies the contract that will pay us o 6 if tossing the dice results in a 6 and will cost us 1 otherwise. 20

4. Note that if o 6 is very large then this would suggest that this event is very unlikely to happen. So it seems that o i is inversely related to probability in some sense. Assume that in a game with n outcomes, the n basic bets r 1,..., r n are quoted with odds o 1,..., o n. What can we say about the sanity of these different bets? Exercise 1.46. There will be an Arsenal vs Chelsea match on Sunday. The bookmaker quotes the following odds Outcome Odds CFC 1.8 Draw 3.5 AFC 4.33 Remember this just means that the bookmaker is offering the following three trades r AF C (AF C) = 4.33 r Draw (AF C) = 1 r CF C (AF C) = 1 r AF C (Draw) = 1 r Draw (Draw) = 3.5 r CF C (Draw) = 1 r AF C (CF C) = 1 r Draw (CF C) = 1 r CF C (CF C) = 1.8 Establish whether these trades are consistent (i.e. arbitrage-free). Hint: Use the next result. The following result analyses when a set of odds o 1,..., o n gives rise to consistent (i.e. arbitrage-free) prices. Proposition 1.47. In a game with n possible outcomes, if the odds are quoted as o 1,..., o n, define the numbers p i = 1/(1 + o i ). Then 1. If the numbers p i are a probability on the set of outcomes (i.e. they are all 0 and add up to one) the set of bets is arbitrage-free. 2. If the numbers p i are not a probability on the set of outcomes (i.e. there is one that is < 0 or the sum is not 1) then there exists an arbitrage. Proof. Let us assume there is no arbitrage amongst the basic trades r 1,..., r n. Then by Theorem 1.42 there exists probabilities p 1,..., p n such that the bets r i have expectation zero, but 0 = E (r i ) = r i (j)p j = j=1 = 1 + p i (o i + 1) j=1,j i ( 1)p j + o i p i = ( 1)p j + o i p i + p i from which we infer p i = 1/(1 + o i ) as is claimed in 1. If these numbers are not probability numbers then this leads to contradiction which proves 2. Solution (to previous exercise). We calculate the probability numbers p i = 1/(1 + o i ): p CF C = 1 1 + 1.8 = 0.3571, p Draw = j=1 1 1 + 3.5 = 0.2222, p AF C = 1 1 + 4.33 = 0.1876. The sum of these numbers is 0.7670. Therefore these numbers are not a probability and the trades are arbitrageable. 21

Progress Check 1. Why is it important to understand variable interest rates? 2. How can variable interest rates be modelled mathematically? 3. What does deterministic mean? What does stochastic mean? 4. What is the yield curve, r(t)? Describe the relationship between P (t) and r(t). 5. What relationship is there between the value of an account, P (t), and the instantaneous interest rate, r(t)? Prove it. 6. What is the connection between P (t) and r(t) when the interest rate r(t) is constant? What is the yield curve in this case? 7. What is the informal definition of arbitrage? Why do you think arbitrage is not typically found in the real world? 8. Explain, following example 1.31 how semi-annual rate is related to annual rate. It is suggested you discuss these questions with your colleagues to reinforce your understanding of the lesson. You should try to respond these questions without your lecture notes to check whether you have learnt the material. 9. Think of a game of coin tossing with two bets r 1 and r 2. If you take bet r 1, you will gain 1 with heads and pay 2 for tails. In bet r 2, you will lose 5 with heads and gain x for tails. Show that when x = 11 there is an arbitrage. (Hint: Experiment with the portfolio R = x 1 r 1 + x 2 r 2 for some values of x 1 and x 2 ; try numbers like ±1, ±2,... and see if you can produce an arbitrage) 10. What does odds mean? 11. What condition do the odds have to verify so that there is no arbitrage. More interesting facts 1. Interest rate modelling and trading in investment banks and hedge funds is typically referred to as Fixed Income as opposed to variable income which includes assets such as foreign exchange, equity shares, commodities etc. 2. Arbitrage theory is relatively recent. It was proposed by Stephen Ross in 1976. It is sometimes referred to as APT or Arbitrage Pricing Theory. It is at the centre of most of investment finance related to pricing and risk management of financial derivatives. This is not examinable content but will help your understanding of the course 3. Arbitrage is sometimes referred to as an arb by market participants. 4. Some people use the expressions soft arbitrage or statistical arbitrage (also stat-arb) these loosely refer to circumstances which look like an unbalance in the market but 22

fall short of being a clear-cut arbitrage. A stats-arb hedge fund is a hedge fund that uses statistical analysis to identify statistical arbitrage in the market. I have seen people lose money over what they thought was arbitrage but was not. For example, someone observed that gold in the Tokyo market was more expensive than in NY so they purchased gold in NY to sell in Tokyo in the hope to make a profit. The gold had to be transported to Tokyo to be physically delivered on a particular date, then this person discovered the size of the bullions is different so they had to melt the NY ingots... After all this the expected profit evaporated and turned into losses. 5. (material volunteered by a keen FTMF student) In the lesson we have used the odds to mean the amount, o i, of cash that you will gain in case your bet that i will be the outcome turns out right. If your bet is not right then you will pay 1 (this is what Definition 1.44 describes mathematically). We also explained that you could do the bet times x i in which case you make x i o i if the outcome i results and have to pay x i otherwise. This x i is what is called the stake in the real world. In the real world a bookmaker will also enforce payment of the 1 beforehand to prevent you from disappearing in case you do not win the bet 2. This means that if you lose the bet then the 1 has already been paid, and that if you win the bookmaker will pay you the o i plus the payment you made 1. So essentially, you pay 1 to enter the bet: if you win you get 1 + o 1 and if you lose you get nothing. This is sometimes quoted in the betting market as o i against 1 or o i /1. Which means what is described above: you pay 1, if you get it right you gain 1 + o i, if you get it wrong you gain nothing. Note that if instead of betting 1 you invest a stake of x, you will gain x + xo i if you get it right (and zero otherwise). If o i is fractional, e.g. 0.6, bookmakers, who are not specialists on decimal numbers, will prefer to quote as if you invested a higher stake that makes all numbers integers. So in the case o i = 0.6 the bookmaker, instead of quoting the odds as 0.6/1 (you make 1 + 0.6 if you invest 1 and get it right), will instead quote the odds as 6/10 (you make 10 + 6 if you invest 10 and get it right). References [IMF] Introduction to Mathematical Finance course notes, Queen Mary University, 2015/16 [Ross] Sheldon Ross, An Elementary Introduction to Mathematical Finance, 3rd Edition, Cambridge University Press, 2011 2 Incidentally this is related to something called settlement risk or credit risk: the risk that a counter-party does not pay in case a trade results in a debt for them. 23