Financial Markets and Derivative Pricing PDF Autumn 2024

Financial Markets and Derivative Pricing Dorje C. Brody School of Mathematics and Physics University of Surrey, Guildford GU2 7XH (MSc Financial Data Science, University of Surrey) Financial Markets and Derivative Pricing Autumn Term 2024 References These lecture notes are meant to be self-sufficient. However, you might find the books below helpful. “Investment Science” David G. Luenberger Oxford University Press, 1998 “Financial Calculus: An Introduction to Derivative Pricing” Martin Baxter and Andrew Rennie Cambridge, 1996 MATM068: Mathematics, University of Surrey -2- c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 1. Basic theory of interest Principal and interest The basic idea of interest is rather familiar. You invest £1.00 in a bank account that pays 5% interest per year. One year later, you have the principal of £1.00 plus interest of £0.05. In general, if the annualised interest rate is r, then the initial investment would be multiplied by (1 + r) after one year. (Note that one usually quotes the interest rates in percent. For example, one might say that ‘the interest rate is 5%’, which means that r = 0.05.) Under simple interest rule, if the initial principal has a notional amount N , then the value V of the investment after t years is V = (1 + rt)N. (1) In particularly, the account grows linearly in time. MATM068: Mathematics, University of Surrey -3- c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Most banks, however, accumulate compounding interest. Starting from a unit principal, after one year the amount becomes (1 + r). After second year, the amount (1 + r) is multiplied by another factor of (1 + r), making it (1 + r)2 in total. Thus, under compound interest, the the amount grows geometrically. In particular, under compound interest, the initial principal N will have the value V after t years, given by V = (1 + r)tN. (2) Note that most banks calculate and pay interest more frequently—quarterly, monthly, or even daily. In this case it is the convention to quote the interest rate on a yearly basis, but apply the appropriate proportion of that interest rate over each compounding period. For example, if the interest is paid quarterly, then after a quarter the growth is given by the factor (1 + r/4). MATM068: Mathematics, University of Surrey -4- c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Figure 1: Simple (1 + rt) vs compound ((1 + r)t) interest. The rates are set to be fixed r = 5%. MATM068: Mathematics, University of Surrey -5- c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Thus, in this case after one year a unit principal will be multiplied by the factor (1 + r/4)4. The so-called effective yearly interest rate r′ can then be calculated by setting 1 + r′ = (1 + r/4)4. If the year is divided into m equally spaced periods for which interest is paid on each period, then we have the growth (1 + r/m)m, and the effective yearly interest rate r′ is defined by 1 + r′ = (1 + r/m)m. It is worth remarking that for any r > 0 and m > 1 we have ′ r m 1+r = 1+ > 1 + r. (3) m A continuously compounded interest is then defined by the limit r m lim 1 + = er. (4) m→∞ m Exercise: Verify this limit. Hint: You may use the fact that (1 + r/x)x = exp[x ln(1 + r/x)]. Changing the variable x → y = 1/x allows the use of l’Hospital’s rule. MATM068: Mathematics, University of Surrey -6- c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 What about the growth after an arbitrary period of time? In this case we let t denote time variable such that t = 1 means one year. If the compounding is via m periods, then after t years we have r mt h r mit 1+ = 1+ , (5) m m which in the limit approaches ert. In this case the account grows exponentially. Remark: in real banks, rates are often readjusted so that the geometric compounding deviates away from the constant-rate compounding. In this manner banks are able to superficially offer attractive figures. Present and future values Interest rate positivity implies that money invested today leads to increased value in the future. This concept can be reversed in time to calculate the value that should be MATM068: Mathematics, University of Surrey -7- c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Figure 2: Continuous (ert ) vs geometric ((1 + r)t) compounding, with r = 5%. MATM068: Mathematics, University of Surrey -8- c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Geometric compounding, with declining rates starting from r = 5%, vs geometric compounding Figure 3: with fixed rate r = 5%. MATM068: Mathematics, University of Surrey -9- c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 invested today to yield some fixed value in the future. In other words, given the value in the future, we can determine its present value. This is the effect of the discounting. Thus the value of £1 in one year from now, in the case of a simple interest, has the present value £1/(1 + r). The process of evaluating future obligations as an equivalent present value is called discounting. The factor used to determine the amount of discounting is called discount factor D. For compound interest, the present value of a unit amount of cash is determined by the discount factor; r −mt D = 1+. (6) m Note that in real market banks offer different rates for lending and for borrowing. An ideal bank offers the same rate of interest to both deposits and loans, and MATM068: Mathematics, University of Surrey - 10 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 charges no transaction. Real banks are, of course, never ideal, but for the purpose of simplifying the analysis we often assume the banks being ideal. Given a string of cash flow, we can speak also about its future value. Suppose that we fix a time cycle for compounding (e.g., yearly) and let r denotes the interest rate for that period. Consider a cash flow given by {x0, x1, · · · , xn}, where xk denotes the amount of cash flow paid at the beginning of the k-th period. Then the future value of the cash flow, when it is paid into an ideal bank account with simple rate, is F V = x0(1 + r)n + x1(1 + r)n−1 + · · · + xn−1(1 + r) + xn. (7) Similarly, the present value of such a cash flow is given by x1 x2 xn−1 xn P V = x0 + + + ··· + +. (8) 1 + r (1 + r)2 (1 + r)n−1 (1 + r)n MATM068: Mathematics, University of Surrey - 11 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Internal rate of return and net present value A slightly more complicated concept, which is however important in practical situations, is the concept of internal rate of return. Given a cash flow stream, we write the expression for present value and then find the value of r̄ that renders this present value equal to zero. Thus we have x1 x2 xn−1 xn 0 = x0 ++ + · · · + + (9) 1 + r̄ (1 + r̄)2 (1 + r̄)n−1 (1 + r̄)n for some choice of r̄. Note that the internal rate of return r̄ is defined without reference to a prevailing interest rate. It is determined entirely by the cash flows of the stream. It is not a priori clear whether a value for r̄ exists such that (9) is satisfied. To see this, we proceed as follows. MATM068: Mathematics, University of Surrey - 12 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Putting z = (1 + r̄)−1, (9) can be written 0 = x0 + x1z + x2z 2 + · · · + xn−1z n−1 + xnz n. (10) Before we study the solution to this equation, let us look at a simple example, where the cash flows are: {x0, x1, x2, x3} = {−2, 1, 1, 1}. Then we have 0 = −2 + z + z 2 + z 3. (11) In figure 4 we plot the function f (z) = z 3 + z 2 + z − 2. We see that f (z) = 0 has a single root at z ≈ 0.81. Hence in this case we find that the internal rate is r̄ ≈ 0.23. (12) The question now is whether a unique root of (10) always exists. Fortunately, we have the following result. Suppose the cash flow stream has x0 < 0 and xk ≥ 0 for all k = 1, 2,... , n. MATM068: Mathematics, University of Surrey - 13 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Figure 4: The function f (z) = z 3 + z 2 + z − 2. MATM068: Mathematics, University of Surrey - 14 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Then there is a unique root to the equation (10). Exercise: Verify this. Hint: Note that the polynomial f (z) = x0 + x1z + x2z 2 + · · · + xn−1z n−1 + xnz n has the properties that f (0) < 0 and f ′(z) > 0. When the various cash flows are present valued, we can consider the difference between present value of benefits and present value of loss. This difference is usually called net present value. To be worthy of any consideration, the cash flow stream associated with an investment must have a positive net present value. Net present value vs internal rate of return It is worth remarking that depending on the choice for the optimality criteria, the investment decision can be different. MATM068: Mathematics, University of Surrey - 15 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 The example below illustrates this clearly. Suppose that you have the opportunity to plant trees that later can be used for lumber. This project requires an initial outlay of money in order to purchase and plant the trees. Assume that no other cash flow occurs until the trees are harvested. However, you have a choice as to when to harvest: after 1 year or after 2 years. If you harvest after 1 year, you get return quickly; but if you wait an additional year, the trees will have grown further to yield higher return. We assume that the cash flow streams associated with these two alternatives are: {−1, 2} or {−1, 0, 3} Suppose, in addition, the prevailing interest rate is 10%. We can now calculate the net present values of the two cash flows: 2 3 −1 + = 0.82 or −1+ = 1.48 (1 + 0.1)1 (1 + 0.1)2 MATM068: Mathematics, University of Surrey - 16 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Hence under this consideration it is best to cut the trees later. Exercise: What is the value of the interest rate r that would have made the two strategies indifferent? Hint: See figure 5 below. Alternatively, we can calculate the internal rate of return. Thus we must solve the two equations: −1 + 2z = 0 or − 1 + 3z 2 = 0. Bearing in mind z = 1/(1 + r) we this find √ 1 3 √ z = ⇒ r̄ = 100% or z= ⇒ r̄ = 3 − 1 ≈ 73% 2 3 Hence under this consideration it is best to cut the trees sooner. Exercise: Which strategy would you choose? MATM068: Mathematics, University of Surrey - 17 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 2 3 Figure 5: The functions −1 + 1+r and −1 + (1+r) 2. MATM068: Mathematics, University of Surrey - 18 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 2. Fixed-income securities Markets for future cash An interest rate is a price, or rent, for the most popular of all traded commodities—namely, money. The one-year interest rate, for example, is just the price that must be paid for borrowing money for one year. Note, however, that the overall market associated with interest rates is more complex than simple bank accounts. There are numerous products, or financial instruments, related to various rates, that are traded in the market. If there is a well-developed market for an instrument so that it can be traded ‘freely’ in the market, then the instrument is called a security. An obscure product (e.g., mortgage-backed instruments) can also be made popular—this is the idea of ‘securitisation’. MATM068: Mathematics, University of Surrey - 19 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Fixed-income securities are financial instruments that are traded in well-developed markets and ‘promise’ a fixed (i.e. definite) income to the holder over a span of time. Hence they represent the ownership of a definite cash flow stream. Note that the classification of a fixed-income security is somewhat vague. Originally, this classification meant that the security pays a fixed cash flow stream to the owner. The only uncertainties about the promised stream were associated with whether the issuer of the security might default. Nowadays, however, many ‘fixed-income’ securities promise cash flows whose magnitudes are tied to various contingencies. Thus the notion of a ‘fixed-income’ security must be interpreted in the broad sense. The most familiar fixed-income instrument is an interest-bearing bank deposit: the deposit account. MATM068: Mathematics, University of Surrey - 20 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 The term money market refers to the market for short-term (1 year or less) loans by corporations. Within this market commercial paper is the term used to describe unsecured loans to corporations. In the import/export industry, banker’s acceptance is often used. This is a promise by one company to another of a delayed payment. Eurodollar deposits are deposits denominated in dollars but held in a bank outside the US. There are many other fixed-income securities. Some of these include the following: US Treasury bills are issued in denomination of $10,000 or more, with fixed terms to maturity of 13, 26, and 52 weeks. US Treasury notes have maturities of 1 to 10 years and are sold in denomination as small as $1,000. MATM068: Mathematics, University of Surrey - 21 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 US Treasury bonds are issued with maturities more than 10 years, and make coupon payments. Gilts are bonds issued by the United Kingdom, South Africa, or Ireland. Corporate bonds are issued by corporations for the purpose of raising capital for operations and new ventures. Callable bonds gives the issuer the right to repurchase the bond at a specified price. Note that some Treasury bonds are callable. A discount bond (or a zero-coupon bond) is a promise to deliver one unit of currency on the termination (maturity) day. Most bonds entail coupon payments, but coupon payments can be decomposed into a series of discount bonds. In this sense, a discount bond can be regarded as the most fundamental instrument in the fixed-income market, from a modelling point of view. Three other important concepts in the fixed-income market are mortgage, MATM068: Mathematics, University of Surrey - 22 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 annuity, and inflation. A future homeowner usually will ‘sell’ a home mortgage to generate immediate cash to pay for home. An annuity is a contract that pays the holder money periodically, according to a predetermined schedule, over a period of time. Pension benefits often take the form of annuities. Inflation products are linked to real rate of interest. In this case the cash flows are associated with inflation-linked price index such as the consumer price index (CPI). The main index in the Euro zone is the HICPxT. In the UK we have the retail price index (RPI), published by National Statistics. Valuation formulae Many fixed-income instruments include an obligation to pay a stream of equal MATM068: Mathematics, University of Surrey - 23 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 periodic cash flows. This is characteristic of standard coupon bonds that pay the holder a fixed sum on a regular basis. A natural question that arises in this context is how to determine the price of such instruments. Our first step towards this direction is to examine the price of a perpetual annuity. This product pays a fixed amount of money each year forever. In financial markets it is not always possible to purchase a perpetual annuity. In the UK such a products used to be common, and are called consoles (consolidated annuities). Consoles still exist today, but form only a small part of the UK government’s debt portfolio. From what we have learned, the present value of a perpetual annuity can easily be calculated. MATM068: Mathematics, University of Surrey - 24 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Suppose that an amount N is paid at the end of each period, and that the per-period interest rate is r. Then the present value of the perpetual annuity is ∞ X A A P = =. (13) (1 + r)k r k=1 Exercise: Verify this. In the case of a continuously compounding interest rate, we can calculate the price of a perpetual annuity as follows: Z ∞ −rt A P = Ae dt =. (14) 0 r Thus both calculations lead to the same expression. Exercise: Explain why the price of a perpetual annuity should be a decreasing function of the rate r. Of more practical importance is the case where the payment stream has a finite lifetime. MATM068: Mathematics, University of Surrey - 25 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 The present value in this case, for a fixed interest rate r per period, is n X A A 1 P (n) = k = 1 − n. (15) (1 + r) r (1 + r) k=1 Exercise: Verify this. In the case of a continuously compounding interest rate, we have Z n −rt A −nr P (n) = Ae dt = 1−e. (16) 0 r One important consideration to make in the study of annuity prices is the effect of interest rate dynamics. This is most easily studied in the context of continuously compounding rate. Suppose we let r(t) denote the value of the interest rate at time t ≥ 0. Then the price of an annuity that has lifetime of T years is calculated according to the formula Z T − 0t r(s)ds R P (T ) = Ae dt. (17) 0 We can compare the prices of annuities for different interest rate dynamics. MATM068: Mathematics, University of Surrey - 26 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 In one extreme the interest rate remains constant: r(s) = r. In this case the expression (17) reduces to (16). Another, more realistic situation to consider is a varying interest rate. As an example, take a periodically varying interest rate r(t) = 0.05(1 + cos2(0.2t)) (18) The dynamics of the two systems of interest rates are sketched in figure 6. If we now calculate the price of a T -maturity annuity in these two examples, we find that there are notable differences. The results are shown in figure 7. We see that for T ≫ 1 the price of the annuity for varying rate is just slightly cheaper than that of a fixed rate in this example. Ultimately, one needs a dynamical approach to understand the prices of financial instruments. MATM068: Mathematics, University of Surrey - 27 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Figure 6: Varying interest rate r(t) = 0.05(1 + cos2(0.2t)) vs fixed rate r(t) = 0.075. MATM068: Mathematics, University of Surrey - 28 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 T -year maturity annuity price in the case of a varying interest rate r(t) = 0.05(1 + cos2(0.2t)) Figure 7: and a fixed rate r(t) = 0.075. MATM068: Mathematics, University of Surrey - 29 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Yield and duration A bond in detail is defined to be an obligation by the bond issuer to pay money to the bond holder, according to rules specified at the time bond is issued. The idea is that the issuer of the bond initially sells the bonds to raise capital immediately, and then is obliged to make the prescribed payments. Usually bonds are issued with coupon rates close to the prevailing interest rate so that they will sell at close to their face value. However, each coupon payment can be viewed in the form of a discount bond, whose face value is the value of the coupon. Hence all the analysis ‘boils down’ to the analysis of discount bonds. Given a bond, we can speak of its yield. A bond’s yield is the interest rate implied by the payment structure. Specifically, it is the interest rate at which the present value of the cash flow stream is exactly equal the current price. MATM068: Mathematics, University of Surrey - 30 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 This is called the yield to maturity. It is just the internal rate of return of the bond at the current price. Suppose that a bond with face value F makes m coupon payments of c/m each year and there are n periods remaining. Hence coupon payments sum to c each year. Suppose also that the current price of the bond is P. The price of a bond, having yield R, is then given by −n R c 1 P =F 1+ + 1−. (19) m R (1 + R/m)n Exercise: Verify this. Hint: From the definition we have −n Xn R c/m P =F 1+ +. (20) m [1 + R/m]k k=1 It should be evident that the price of long-dated bonds are more sensitive to interest rate changes than the price of short-dated bonds. MATM068: Mathematics, University of Surrey - 31 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Foe example, in the simplest case of a continuously compounding rate with zero coupon payment, the bond price is P = e−rT. In this case, upon differentiation we have dP = −T P, (21) dr hence the sensitivity of the bond price on interest rate increases linearly in maturity T. One measure of time length, called duration, gives a direct measure of interest rate sensitivity. The duration of a fixed-income instrument is a weighted average of the times that cash flows are made. The weighting coefficients are the present values of the individual cash flows. The definition of the duration associated with a cash flow is as follows. t0P V (t0) + t1P V (t1) + t2P V (t2) + · · · + tnP V (tn) D=. (22) PV Here, P V (tk ) denotes the present value of the cash flow that occurs at time tk , MATM068: Mathematics, University of Surrey - 32 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 and n X PV = P V (tk ). (23) k=0 Thus D is a weighted average of the cash flow times. For a zero-coupon bond, the duration equals its maturity. A coupon-paying bond have a duration strictly less than the maturity. Thus duration can be interpreted as a generalised maturity measure: it represents the average of the maturities of all the individual payments. MATM068: Mathematics, University of Surrey - 33 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 3. Term structure of interest rates Yield curve We have looked at specific bonds with specific maturities. A natural question to address now is: how is a bond with one maturity related to a bond with another maturity? What we have in the bond market is a family of bonds, one for each maturity. The yield to maturity of a bond is tied to the general condition of the fixed income market. However, bond yields are not exactly the same. The variation in yields across bonds is partly due to their ratings. It should be clear, for example, that a triple-A rated bond is likely to cost more, and hence yield less, than bonds with lower ratings. Another factor affecting the bond yield is the maturity. MATM068: Mathematics, University of Surrey - 34 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 It is often, although not always, the case that long-dated bonds have higher yields than short-dated ones. The dependence of the bond yield on maturity is captured in the yield curve. In figure 8 examples of yield curves are shown. An yield curve that is increasing in maturity has a ‘normal’ shape. Otherwise, it is said to be ‘inverted’. The inverted curve is seen when the short-term rates increase rapidly, and investors believe that the rise is temporary. This means that, to understand the maturity dependence, one has to have in mind how the short-term interest rate changes in time. Term structure Term structure theory puts aside the notion of yield and instead focuses on pure interest rate. MATM068: Mathematics, University of Surrey - 35 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 0.1 0.08 0.06 0.04 0.02 0 5 10 15 20 25 30 T Yield curves in the Vasicek model. Both curves have the same asymptotic yield of 4%, but with Figure 8: different initial interest rate of 2% and 10%. MATM068: Mathematics, University of Surrey - 36 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 The idea that underlies term structure theory is that interest rate charged for money depends on the length of time that the money is held. There are various rates that appear in the consideration of term structures. Spot rates are the basic interest rates defining the term structure. The spot rate st is the rate of interest, expressed in yearly terms, charged for money held from the present time (t = 0) to time t. Thus s2, for example, denotes the rate that is paid for money held for 2 years, expressed on an annualised basis. In this case, if you deposit an amount N in the account, then after 2 years it will become N (1 + s2 )2. The definition of spot rates assumes a compounding convention. If we have yearly compounding, then the (1 + st)t is the factor multiplied to the notional amount after t years. If we have the convention of compounding m periods per year, then the spot rate st is defined such that (1 + st/m)mt is the corresponding factor. MATM068: Mathematics, University of Surrey - 37 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 For continuous compounding, the spot rate st is defined such that estt is the corresponding factor. Once the spot rates are determined, we can proceed to determine the corresponding discount factor Dt. For yearly compounding, we have 1 Dt =. (24) (1 + st)t For m periods per year, we have 1 Dt = mt. (25) (1 + st/m) For a continuous compounding, we have Dt = e−stt. (26) Discount factors are important because they transform future cash flows directly into an equivalent present value. Hence for a cash flow {x0, x1,... , xn} the present value, relative to the MATM068: Mathematics, University of Surrey - 38 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 prevailing spot rates, is P V = x 0 + D1 x 1 + D2 x 2 + · · · + Dn x n. (27) The spot rates can be determined by the prices of coupon bonds. First, the one-year spot rate s1 can be determined by the 1-year Treasury bill rate: 1 1 1 + s1 = ⇒ s1 = − 1. (28) P1 P1 Now consider a 2-year bond, with coupon c and face value F. Then the price of the bond is c c+F P = + 2 , (29) 1 + s1 (1 + s2) from which we find 1/2 (1 + s1)(c + F ) s2 = − 1. (30) (1 + s1 )P − c Exercise: Verify this. By iterating this procedure, we obtain the spot rates {s1, s2, s3,...}. MATM068: Mathematics, University of Surrey - 39 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Another useful and important concept is that of forward rates. They are interest rates for money to be borrowed between two dates in the future, but under terms agreed today. To understand the concept, consider the following example of a 2-year setup. Suppose that s1 and s2 are known. Starting from £1 today we will have £(1 + s2)2 after 2 years. Alternatively, we can place £1 in a 1-year account today to obtain £(1 + s1) a year from now. Simultaneously, we agree to place the amount £(1 + s1) in another 1-year account, but in 1 year’s time from now, at an agreed rate f12. The rate f12 is the forward rate for money to be lent in this way. The final amount of money we receive at the end of 2 years under this compound plan is £(1 + s1)(1 + f12). We now observe that, if the two plans were to result in a different amount of MATM068: Mathematics, University of Surrey - 40 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 money after two years, then no one would agree to enter into the plan offering lower return. Therefore, the value of the forward rate f12 over the period 1 ∼ 2 year has to be such that (1 + s2)2 = (1 + s1)(1 + f12). (31) It follows that (1 + s2)2 f12 = − 1. (32) 1 + s1 Hence the forward rate is determined by the two spot rates. The argument employed here to determine the value of the forward rate is a so-called arbitrage argument. The point is that if f12 were to take any value other than (32), and if the three rates s1, s2, and f12 are offered in the market, then a clever market participant can make a risk-free profit. Exercise: Establish explicit strategies that guarantees a profit higher than the spot rates offered in the market, if the forward rate is (i) higher, and (ii) lower, MATM068: Mathematics, University of Surrey - 41 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 than the arbitrage-free rate (32). The forward rates spanning a single time period are called short rates: rk = fk,k+1. (33) The spot rate sk is obtained from the short rates because interest earned from time zero to time k is identical to the interest that would be earned by rolling over an investment each year. Hence we have (1 + sk )k = (1 + r0)(1 + r1)(1 + r2) · · · (1 + rk−1). (34) This relation generalises because all forward rates can be found from the short rates in a similar manner: (1 + fij )j−i = (1 + ri)(1 + ri+1)(1 + ri+2) · · · (1 + rj−1). (35) Notation: forward rate between the two times t1 and t2 > t1 is denoted ft1t2 , where both times are expressed in units of year. In general, forward rates are expressed on an annualised basis. What we see here is that forward rates are implied by the values of the underlying spot rates. MATM068: Mathematics, University of Surrey - 42 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 More generally, the implied forward rates are determined according to the formula for i < j (1 + sj )j = (1 + si)i(1 + fij )j−i. (36) Here the left side is the growth from an investment for j years via spot rate sj. The right side is the growth of an investment for i years via spot rate si, and then a further j − i-year investment via the forward rate fij. It follows that 1 j j−i (1 + sj ) fij = i − 1. (37) (1 + si) In the case of a continuously compounding rate, an analogous line of argument shows that esj j = esiiefij (j−i), (38) from which it follows that sj j − sii fij =. (39) j−i MATM068: Mathematics, University of Surrey - 43 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Explaining term structures As remarked above, the yield curve is usually sloped gradually upwards as maturity increases. The spot rate curve has similar characteristics. It is natural to ask if there is a simple explanation for these typical shapes. One explanation that people give is based on an expectation argument. The argument is that the 2-year spot rate, for example, is higher than the 1-year spot rate because the market ‘expects’ that the 1-year rate will rise next year. Suppose that we have s1 = 5% and s2 = 5.5% for the two spot rates. Then according to (32) we have f12 = 6%. The expectation argument asserts that the market expectation of the 1-year rate next year is 6%. Exercise: Does the expectation argument make sense? Reason? The liquidity preference argument asserts that investors usually prefer MATM068: Mathematics, University of Surrey - 44 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 short-term fixed income securities over long-term securities. This is because they do not wish to tie up their capital in a long term investment. It should be remarked, however, that the bond market is highly liquid so that if the holder of a long-term bond requires cash, he/she can sell the bond to raise capital. Exercise: What is your view on the liquidity preference argument? Running present value The present value of a cash flow is obtained by summing the discounted cash flows of all future cash flows. There is an alternative way of expressing this by use of the discount factor. This leads to the concept of a running present value. This allows us to calculate the present value in a recursive manner. MATM068: Mathematics, University of Surrey - 45 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 As before, suppose the cash flow stream is given by {x0, x1, x2,... , xn}. We let the present value at time zero, given in (27), denoted by P V (0). Now imagine that k time periods have passed and we are anticipating the remainder of the cash flow stream {xk , xk+1, xk+2,... , xn}. We can calculate the ‘present’ value P V (k), viewed at time k, using the discount factor that would be applicable then. Note that the present value (27) at time zero can be expressed in the form P V (0) = x0 + D1 (x1 + D12x2 + · · · + D1nxn) , (40) where D1k = Dk /D1. Analogous argument shows that the running present values satisfy the recursion relation P V (k) = xk + Dk,k+1P V (k + 1), (41) where Dk,k+1 = 1/(1 + fk,k+1) is the discount factor for the short rate at time k. Exercise: Verify this. MATM068: Mathematics, University of Surrey - 46 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 4. Elements of probability theory Probability space and random variables Historically, the development of probability theory was based more on intuition than on mathematical axioms. As a consequence, the resulting theory was rather intricate to the extent that one might well doubt predictions based upon such theories. It was Kolmogorov, in 1933, who finally provided an axiomatic formulation which is now universally accepted as the basis for modern probability theory. Since then probability theory has been no more ambiguous than other branches of mathematics. The intuition motivating probability theory is the notion of randomness that we encounter in experiments, whose results are not precisely predictable a priori and can only be determined after observing the outcomes. Familiar examples are the tossing of a coin and the throwing of a die. MATM068: Mathematics, University of Surrey - 47 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Even though the fairness of, say, a coin can only be determined after observing a large number of outcomes, the knowledge of the fairness of the coin does not allow us to predict the outcome of an individual trial. Abstractly, one considers a space Ω of all possible outcomes of an experiment, each individual outcome being represented as a point ω ∈ Ω. The space Ω is called the sample space, and subsets of Ω are called events. Each subset corresponds to a collection of outcomes. If the outcome ω is in the subset A ⊂ Ω, then the event A is said to have occurred. For example, in the case of a die the set A = {2, 4, 6} ⊂ Ω corresponds to the event “the outcome is an even number.” If in a particular throw the outcome happens to be 5, then we know that the event A did not occur. Thus, one might expect that a probability can be associated with each outcome, and that there should be a probability function p(ω) which expresses the MATM068: Mathematics, University of Surrey - 48 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 likelihood that ω occurs. In the case of a balanced die, this implies that 1 p(1) = p(2) = · · · = p(6) =. 6 More generally, we expect to have X p(ω) = 1. (42) ω∈Ω If, however, Ω contains uncountably many elements, this is problematic. There is no reasonable way of adding up an uncountable set of numbers, each of which equals 0, and obtaining a nonzero sum. This indicates that it may not be possible to start from probabilities associated with individual outcomes and build a meaningful theory. Thus, we start with the notion that probabilities are already defined for events. In such a case, P(A) is defined for a class B of subsets A ⊂ Ω. The question that arises is what should B be and what properties should be possessed by P(−), defined on B. MATM068: Mathematics, University of Surrey - 49 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Probability measures It seems natural to demand the following: (i) The entire space Ω and the empty set ∅ are in B. (ii) If A ∈ B, then the complement Ac should also be in B. Any class of sets satisfying these properties is called a field. Definition 1. A probability measure is a nonnegative function P(−), defined for sets A ∈ B, that possesses the following properties: P(A) ≥ 0 for all A ∈ B, (43) P(Ω) = 1 and P(∅) = 0. (44) If A, B ∈ B are disjoint, then P(A ∪ B) = P(A) + P(B). (45) In particular, P(Ac) = 1 − P(A) (46) for all A ∈ B. MATM068: Mathematics, University of Surrey - 50 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Note that P(∅) = 0 is not an independent condition in the sense that it can be derived from (45) and (46), along with P(Ω) = 1. A condition that is somewhat more technical, but important from a mathematical point of view, is that of countable additivity. The class B, in addition to being a field, is assumed to be closed under countable unions, i.e. if An ∈ B for every n, then A = ∪nAn ∈ B. In other words, we insist that the union of a countable collection of measurable sets is also measurable. Such a class is called a σ-field, and the probability itself is assumed to be defined on a σ-field B of Ω. Thus, a σ-field is closed under countably many iterations of the usual set operations. Definition 2. A function P defined on a σ-field is called a countably additive probability measure if in addition to equations (43), (44) and (45), it satisfies the following condition: for any sequence of pairwise disjoint sets An with MATM068: Mathematics, University of Surrey - 51 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 A = ∪nAn we have X P(A) = P(An). (47) n We now consider the construction of countably additive probability measures on the real line. The natural σ-field for the definition of a probability measure on the real line R is the Borel σ-field, the most important of all σ-fields. This is defined as the smallest σ-field containing all intervals (and therefore containing all open sets). Specifically, we consider the class Ia,b of subsets of the real numbers, where Ia,b = {x : a ≤ x < b} i.e. the collection of intervals that are left-closed and right-open. Now suppose we are given a function F (x) on the real line which is nondecreasing and satisfies lim F (x) = 0 and lim F (x) = 1. x→−∞ x→∞ MATM068: Mathematics, University of Surrey - 52 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Then we can define an additive probability measure P on the Borel σ-field by setting P(Ia,b ) = F (b) − F (a) for every interval. A theorem of Lebesgue states that P is countably additive if and only if F (x) is a right continuous function of x, i.e. F (x + ǫ) → F (x) as ǫ ↓ 0. Conversely, for each such F (x), there is a unique probability measure P on the Borel subsets of the line such that F (x) = P(I−∞,x), and the correspondence between P and F (x) is one-to-one. Random variables A pair (Ω, F), consisting of a set Ω and σ-field F of subsets of Ω, is called a measurable space. If, in addition, we are given a probability measure P, then the triple (Ω, F, P) is MATM068: Mathematics, University of Surrey - 53 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 called a probability space. Such triples (Ω, F, P) model experiments involving random outcomes. Given a probability space (Ω, F, P), we can then introduce the concept of a random variable. Definition 3. A random variable or a measurable function is a map X : Ω → R, i.e. a real-valued function X(ω) on Ω such that for every Borel set B ⊂ R, X −1(B) = {ω : X(ω) ∈ B} is a measurable subset of Ω, i.e. X −1(B) ∈ F. Alternatively stated, X : Ω → R is a map with the property that {ω : X(ω) ≤ x} ∈ F for all x ∈ R. This is natural inasmuch as we can consider the probability P(X ≤ x). Note that sums and products of measurable functions are also measurable. Two random variables X and Y are said to be independent if P(X ≤ x; Y ≤ y) = P(X ≤ x)P(Y ≤ y). MATM068: Mathematics, University of Surrey - 54 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 A random variable satisfying |X(ω)| < ∞ for all ω ∈ Ω is said to be a bounded measurable function. As an example, consider the indicator function of A ∈ F given by 1 if ω ∈ A 1A(ω) = (48) 0 if ω ∈ / A. Clearly, 1A(ω) is bounded and measurable. Consider another example. If {Ak : 1 ≤ k ≤ n} is a finite disjoint partition of Ω into measurable sets, then the function n X θ(ω) = cj 1Aj (ω) (49) j=1 is a measurable function, and is referred to as a simple function. R If θ is simple, then the integral θ(ω)dP is defined to be Z Xn θ(ω)dP(ω) = cj P(Aj ). (50) j=1 MATM068: Mathematics, University of Surrey - 55 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 In general, the integral is defined for all bounded measurable functions. In particular, if X is a bounded measurable function and A is a measurable set, we define Z Z X(ω)dP(ω) = 1A(ω)X(ω)dP(ω). (51) A Ω Distributions and expectations We let (Ω, F, P) denote a probability space. Recall that a random variable X is a real-valued measurable function on (Ω, F). Any such function induces a probability distribution µ on the Borel subsets B of the line. More explicitly, this is given by µ(B) = P(X −1(B)). (52) The distribution function F (x) corresponding to µ is determined via F (x) = µ((−∞, x]) = P(ω : X(ω) ≤ x). (53) MATM068: Mathematics, University of Surrey - 56 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 The measure µ is called the distribution of the random variable X and F (x) is called the distribution function of X. The expectation of a random variable, assumed integrable, is defined by Z EP[X] = X(ω)dP. (54) When the probability space is understood as given, then without ambiguity we can simplify the notation and write E[X] = EP[X]. In particular, in terms of the measure µ we have Z E[X] = xdµ(x). (55) In general, we can also consider the expectation of a function f (X) of a random variable. In this case, we can write, similarly, Z Z E[f (X)] = f (X(ω))dP = f (x)dµ(x). MATM068: Mathematics, University of Surrey - 57 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Of particular interest is the variance of X, given by Var[X] = E[X 2] − (E[X])2. (56) More generally, we can consider a vector-valued random variable given by X = (X1, X2,... , Xn) on a probability space (Ω, F, P). Such a random variable induces a distribution µ = P(X −1) on the Borel subsets of Rn , called the joint distribution of (X1, X2,... , Xn). Given a pair of random variables X and Y , their covariance is defined by Cov[X, Y ] = E [(X − E[X])(Y − E[Y ])] = E[XY ] − E[X]E[Y ], (57) and their correlation by Cov[X, Y ] ρ(X, Y ) = p. (58) Var[X]Var[Y ] Therefore, if X and Y are independent random variables, we have ρ(X, Y ) = 0, and we say that the two variables are uncorrelated. MATM068: Mathematics, University of Surrey - 58 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Note, however, that the converse does not hold. Thus, ρ(X, Y ) = 0 does not imply the independence of X and Y. Characteristic functions* If µ is a probability distribution on the real line, its characteristic function is defined by Z φ(t) = exp(itx)dµ(x). (59) The characteristic function of any given probability distribution µ is a continuous function of t and is nonnegative definite, i.e. X φ(tj − tk )zj z̄k ≥ 0 j,k for all real t1, t2,... , tn and complex z1, z2,... , zn. Given a characteristic function φ(t), we would like to recover the distribution function F (x). MATM068: Mathematics, University of Surrey - 59 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Using the Fourier inversion formula, we conjecture that Z ∞ dF (x) 1 = exp(−itx)φ(t)dt, (60) dx 2π −∞ provided that F (x) is differentiable. In particular, we have b ∞ 1 Z Z F (b) − F (a) = dx exp(−itx)φ(t)dt, 2π a −∞ hence, in this case, the distribution function F (x), and consequently µ, is determined uniquely by the characteristic function. Conversely, a theorem of Bochner states that, if φ(t) is positive definite, continuous at t = 0, and is normalised so that φ(0) = 1, then φ(t) is the characteristic function of a unique probability distribution on R. If µ is a probability distribution on R, then the moment mk of µ is defined by Z ∞ mk = xk dµ(x). (61) −∞ MATM068: Mathematics, University of Surrey - 60 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 In terms of the characteristic function, we have the formula k −k d φ(t) mk = i (62) dtk t=0 for the moments of the distribution. Let us now consider Gaussian random variables. Let N (m, σ) denote the family of normal probability distributions with mean m and variance σ. The density of a Gaussian random variable is given by 2 1 (x − m) ρ(x) = √ exp − 2. (63) 2πσ 2σ Hence, one easily calculates that the characteristic function is itX 1 2 2 E e = exp itm − 2 σ t , (64) where the random variable X ∼ N (m, σ). Alternatively, a Gaussian random variable X = X(ω) can be defined as one with a characteristic function of the form (64). MATM068: Mathematics, University of Surrey - 61 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 This follows from the observation above that the distribution function is determined uniquely by the characteristic function. Conditional probability* Conditional probability and conditional expectation are among the most fundamental concepts of probability theory. Suppose that we have a probability space (Ω, F, P), and a set A ∈ F of positive measure. Then, conditioning with respect to A means that we restrict our considerations to the set A, and likewise restrict the σ-field by the σ-field FA of subsets of A that are in F. For a given B ⊂ A we define P(B) PA (B) = , P(A) MATM068: Mathematics, University of Surrey - 62 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 which is equivalent to defining, for arbitrary B ∈ F, P(A ∩ B) PA(B) = , P(A) since B ⊂ A. In the latter case, PA(−) is a measure defined on F as well, but is concentrated on A and assigns null probability to the complement Ac. The definition of conditional probability relative to A is P(A ∩ B) P(B|A) =. (65) P(A) Similarly, the conditional expectation of an integrable function X(ω), given a set A ∈ F of positive measure, is defined by R A X(ω)dP E[X|A] =. (66) P(A) In general, if we have a partition of Ω into a finite or a countable number of MATM068: Mathematics, University of Surrey - 63 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 disjoint sets A1, A2, A3 ,... with Aj ∩ Ak = ∅ for j 6= k, then X P(B) = P(Aj )P(B|Aj ). (67) j In the case where X is a random variable taking the value a on a measurable set A, we can write P(B ∩ {X = a}) P(B|{X = a}) = , P(X = a) provided P(X = a) 6= 0. We would like to define conditional probability in a manner that makes sense even when P(X = a) = 0. This involves dividing 0 by 0, and should thus involve differentiation of some kind. In the countable case where X takes values {aj } on {Aj }, we may regard the probability P(B|{X = aj }) as a function fB (X) that is equal to P(B|Aj ) on X = aj. MATM068: Mathematics, University of Surrey - 64 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Thus, fB (aj ) = P(B|{X = aj }) can be written as Z fB (X)dP = P(B ∩ {X = aj }) X=aj for each j. Summing over any collection E of indices j, we have Z fB (X)dP = P(B ∩ {X ∈ E}). X∈E Now, the sets of the form X ∈ E generate a sub σ-field Σ ⊂ F, and we can write the definition as Z fB (X)dP = P(B ∩ A) (68) A for all A ⊂ Σ. Similar considerations apply to the conditional expectation of a random variable G given X: Z Z g(X)dP = G(ω)dP, A A MATM068: Mathematics, University of Surrey - 65 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 which can be rewritten as Z Z g(ω)dP = G(ω)dP A A for all A ⊂ Σ, and instead of demanding that g be a function of X, we demand that g be Σ-measurable, which, after all, is the same thing. The key point to note here is that the random variable X is now out of the picture. What is important is the information we have if we know X, and that is the same, if we replace X by a one-to-one function of it. The σ-field Σ abstracts just this very information. In other words, the proper notion of conditioning involves a sub σ-field Σ ⊂ F. If G is an integrable function and Σ ⊂ F is given, we seek another integrable function g that is Σ-measurable and satisfies Z Z g(ω)dP = G(ω)dP (69) A A for all A ⊂ Σ. MATM068: Mathematics, University of Surrey - 66 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 We call such a g the conditional expectation of G given Σ, and denote it by g = E[G|Σ]. Theorem 1 (Conditional expectation). The conditional expectation has the following properties: 1. If g = E[f |Σ], then E[g] = E[f ] (law of total probability). 2. If f is nonnegative, then g = E[f |Σ] is almost surely nonnegative. 3. The map f → g is linear in the sense that if c1 and c2 are constants, we have E[c1f1 + c2f2|Σ] = c1E[f1|Σ] + c2E[f2|Σ]. 4. If g = E[f |Σ], then Z Z |g(ω)|dµ ≤ |f (ω)|dµ. 5. If h is a bounded Σ-measurable function, then E[f h|Σ] = hE[f |Σ]. 6. If Σ2 ⊂ Σ1 ⊂ F, then (tower property of conditional expectation) E[E[f |Σ1]|Σ2] = E[f |Σ2]. MATM068: Mathematics, University of Surrey - 67 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 7. Jensen’s inequality. If φ(x) is a convex functio of x and g = E[f |Σ], then E[φ(f (ω))|Σ] ≥ φ(g(ω)), and if its expectation satisfies E[φ(f )] ≥ E[φ(g)]. MATM068: Mathematics, University of Surrey - 68 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 5. Mean-variance portfolio theory Asset return Typically, when making an investment, the initial outlay of capital is known, but the amount to be returned is uncertain. We shall be studying such scenarios here in the context of a single investment period. That is, money is invested at the initial time, and payoff is attained at the end of the period. Although this assumption is rather restrictive, such a situation does occur in many applications. An example is a zero-coupon bond that will be held to maturity. Another example is an investment in a ‘real’ project that will not provide payment until it is completed. An investment instrument that can be bought and sold is often called an asset. MATM068: Mathematics, University of Surrey - 69 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 The total return of an investment is then defined to be amount received total return =. amount invested Equivalently, if X0 and X1 are the amount invested and received, respectively, and if R is the total return, then X1 R= (70) X0 The rate of return, on the other, is defined by amount received − amount invested rate of return = , amount invested or equivalently by X1 − X0 r=. (71) X0 Clearly, we have the relation R = 1 + r, (72) as well as X1 = (1 + r)X0. (73) MATM068: Mathematics, University of Surrey - 70 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Short sales Sometime it is possible to sell an asset that you do not own, through the process of short selling. To do this, you borrow the asset from someone who owns it (e.g., a brokerage firm). You then sell the borrowed asset to someone else, receiving an amount X0. At a later date, you repay your loan by purchasing the asset for, say, X1, and return the asset to your lender. You will then make the profit X0 − X1. Short selling is thus profitable if the asset price declines. Evidently, short selling is rather risky. This is because the potential loss is, at least in theory, unlimited. As a result, short selling is sometimes prohibited, and is often purposely avoided by many institutions. MATM068: Mathematics, University of Surrey - 71 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 However, it is not universally forbidden, and has become in recent years an important investment tool for many hedge funds. When short selling a stock, one is in effect duplicating the role of the issuing corporation. One sells the stock to raise immediate capital. If the stock pays dividends during the borrowed period, one has to pay that same dividend to the person from whom the stock was borrowed. Portfolio return Consider now the case in which n different assets are available. We can then form a portfolio consisting of n assets. Suppose that the initial wealth is given by X0, which we wish to distribute over n assets, each with the amount X0i. MATM068: Mathematics, University of Surrey - 72 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Clearly, this means that n X X0 = X0i. i=1 If we exclude the possibility of short selling, then we can write X0i = wiX0, (i = 1, 2,... , n) where wi is the weight of the asset i, satisfying X n wi = 1. i=1 The numbers {wi}i=1,...,n thus determine the portfolio position. Let the initial value of the asset i be V0i. At a later time the value changes according to V0i → V1i. The total return associated with the asset i is thus V1i Ri = i. V0 Since we invest X0i into asset i whose value is V0i, the portfolio position θ i is MATM068: Mathematics, University of Surrey - 73 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 given by i i X 0 X0 θ = i = wi i. (74) V0 V0 The value invested in asset i thus changes according to X0 θ iV0i → θ iV1i = wi i V1i = wiX0Ri. (75) V0 It follows that the total wealth changes as X n Xn X0 → θ iV1i = wiX0Ri, (76) i=1 i=1 from which it follows that the total return of the portfolio is Pn i i X n θ V0 R = i=1 = wiRi. (77) X0 i=1 In terms of rate of return we thus have Xn r= w i ri , (78) i=1 where ri is the rate of return associated with asset i. MATM068: Mathematics, University of Surrey - 74 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Mean return of a portfolio In real investment, the rate of return associated with an investment position is not known for sure. In such an uncertain environment, we can nevertheless meaningfully speak about expectation values. What concerns us here, in particular, are mean and variance of the return of a portfolio. Suppose that there are n assets with random rate of return {ri}i=1,...,n. The expectation value of the random variable ri is written r̄i = E[ri]. As before, we form a portfolio of n assets with weights {wi}. The (random) rate of return associated with the portfolio position is r = w 1 r1 + w 2 r2 + · · · + w n rn. (79) Taking the expectation, we obtain the expected rate of return: E[r] = w1E[r1] + w2E[r2] + · · · + wnE[rn]. (80) MATM068: Mathematics, University of Surrey - 75 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Variance of a portfolio Similarly, we can investigate the variance of the portfolio return. Let σi2 denote the variance σi2 = E[(ri − r̄i)2] (81) of the rate of return associated with asset i. Also let σij denote the covariance: σij = E[(ri − r̄i)(rj − r̄j )]. (82) The variance σ 2 of the portfolio return is then σ 2 = E[(r − r̄)2]. (83) From n X n X r= wiri and r̄ = wir̄i (84) i=1 i=1 we have n X r − r̄ = wi(ri − r̄i). (85) i=1 MATM068: Mathematics, University of Surrey - 76 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Upon substitution we obtain   n X σ2 = E  wiwj (ri − r̄i)(rj − r̄j ) i,j=1 n X = wiwj E [(ri − r̄i)(rj − r̄j )] i,j=1 X n = wiwj σij. (86) i,j=1 This result shows that the variance of the portfolio return can be calculated from the covariances of the pairs of asset returns and the asset weights. Effect of diversification Portfolios with only a few assets may be subject to a high degree of risk, represented by a relatively large variance. As a general rule, the variance of the return of a portfolio can be reduced by including additional assets in the portfolio. MATM068: Mathematics, University of Surrey - 77 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 This is known as diversification. To see the effect of diversification, let us first consider the simplest case whereby all the assets are uncorrelated. Suppose also that the mean and the variance of the returns are all equal and are given by E[ri] = m and E[(ri − m)2 ] = σ 2. Suppose further that we allocate an equal amount of money to each asset so that wi = 1/n. The portfolio mean return this is r̄ = m, whereas the portfolio return variance is X n var(r) = wiwj E[(ri − m)(rj − m)] i,j=1 X n = wiwj σ 2δij i,j=1 X n = wi2σ 2, (87) i=1 MATM068: Mathematics, University of Surrey - 78 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 but since wi = 1/n we find σ2 var(r) =. (88) n Therefore, we see that while portfolio return mean is independent of n, the portfolio return variance scales as 1/n. Therefore, by including a number of uncorrelated assets into the portfolio, the variance of the return can be reduced significantly without affecting the return of the portfolio. The situation differs if the asset returns are correlated. As an example, suppose that for any pair (i, j) the returns ri and rj have constant correlation ρ > 0. Then we see that the variance of the return of the portfolio is increased. Exercise: Work out the variance of the return of the portfolio in this case. Portfolio diagram MATM068: Mathematics, University of Surrey - 79 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Suppose that two assets are represented on a mean-standard deviation diagram. We can combine these assets to form a portfolio. The mean and the standard deviation of the rate of return associated with the portfolio can be calculated from the mean, variances, and covariances of the returns of the original assets. However, because the covariance is not shown on the diagram, the exact location of the portfolio cannot be determined from the locations of the original returns. Because we only consider two assets here, we may put w1 = p and w2 = 1 − p. We then find that the portfolio return is given by r = pr1 + (1 − p)r2, and hence the average is r̄ = pr̄1 + (1 − p)r̄2. (89) What about the variance of the return of the portfolio? For this, let us write σ12 = E[(r1 − r̄1)2] and σ22 = E[(r2 − r̄2)2]. MATM068: Mathematics, University of Surrey - 80 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Figure 9: Portfolio diagram. MATM068: Mathematics, University of Surrey - 81 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 We also write ρ for the correlation so that E[(r1 − r̄1)(r2 − r̄2)] = ρσ1σ2. (90) It follows that σ 2 = E[(r − r̄)2] = p2σ12 + (1 − p)2σ22 + 2p(1 − p)ρσ1σ2. (91) The limiting case ρ = ±1 then forms the boundary of the accessible region in the (r̄, σ) plane. In particular, from (89) we have r̄ − r̄2 p=. (92) r̄1 − r̄2 Substitution of this in (91) gives σ = σ(r̄), as shown in figure 10. Specifically, for ρ = +1 we have σ = pσ1 + (1 − p)σ2, (93) which is equivalent to the straight line joining the two points (r1, σ1) and (r2, σ2). MATM068: Mathematics, University of Surrey - 82 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 The standard deviation σ of the portfolio return, as a function of r̄. The parameters are Figure 10: (r̄1, σ1) = (5%, 20%), (r̄2, σ2) = (7.5%, 30%), with correlation ρ = +0.9 (violet), ρ = +0.6 (blue), ρ = −0.4 (green), and ρ = −0.99 (red). MATM068: Mathematics, University of Surrey - 83 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Similarly, for ρ = −1 we have σ = |pσ1 − (1 − p)σ2|, (94) which corresponds to a pair of straight lines joining (r1, σ1) and (r∗, 0); and (r∗, 0) and (r2, σ2), where r∗ corresponds to the point at which σ2 σ1 (p, 1 − p) = ,. (95) σ1 + σ2 σ1 + σ2 A calculation making use of (89) then shows that ∗ σ2r̄1 + σ1r̄2 r =. (96) σ1 + σ2 Exercise: Explain why it is possible to reduce the standard deviation of the portfolio return to zero, when the correlation satisfies ρ = −1. For intermediate values of ρ ∈ (−1, 1) the mean-variance location for the portfolio will be somewhere inside the region enveloped by these three lines. In the general case we return to (91) and consider the function σ 2(p) = (σ12 − 2ρσ1σ2 + σ22)p2 − 2(σ2 − ρσ1)σ2p + σ22. (97) We plot the function σ(p) in figure 10. MATM068: Mathematics, University of Surrey - 84 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Since the coefficient of p2 is positive, we see that there is a critical value p∗ for the portfolio weight, given by σ2(σ2 − ρσ1) p∗ = 2 2, (98) σ1 − 2ρσ1σ2 + σ2 such that the variance is minimised. Feasible set Suppose now that there are n basic assets. We can plot them as points on the mean-standard deviation diagram. We form portfolios from these n P assets, using all possible weighting scheme {wi} such that wi ≥ 0 and that ni=1 wi = 1. The totality of points then constitutes the feasible set. The feasible set satisfies two important properties. 1. If there are at least three assets, then the feasible set is a solid (simply connected) region. MATM068: Mathematics, University of Surrey - 85 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 The standard deviation (97) of the portfolio return, with parameters σ1 = 0.2, σ2 = 0.3, and Figure 11: ρ = 0.15. MATM068: Mathematics, University of Surrey - 86 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 This can be seen by first considering the case for which n = 3. We have asset A, asset B, and asset C. First we consider all possible combinations of the two assets B and C. This forms a solid curve in the (r̄, σ) plane joining B and C. For each B-C combination we obtain a point X. For each X we then consider all possible combinations of X with A. Then this is also represented by a curve joining X and A. By traversing across all possible X-points, the curves will wipe a solid connected region in the (r̄, σ) plane. This line of arguments extends to the case in which there are more than three assets. 2. The feasibility region is convex to the left. What this means is that given an arbitrary pair of points in the region, the join MATM068: Mathematics, University of Surrey - 87 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 of these points never crosses the left boundary of the feasible set. This follows form the fact that all portfolio combinations of the two assets necessarily lie on the left side of the line joining the two. Minimum variance point The left boundary of a feasible set is called the minimum variance set. The minimum variance set is the boundary of the convex region. On the minimum variance set lies the minimum variance point. Quite often, for a fixed mean return r̄, investors tend to prefer a portfolio position that minimises the associated variance. Such investors are said to be risk averse. An investor who chooses not to take this position is a risk seeker. Our analysis will be focused on risk averse investors. Now if the standard deviation of the portfolio return is fixed, then investors MATM068: Mathematics, University of Surrey - 88 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 prefer to take a position that yields the highest return. Hence, within the feasible region, the ‘interesting’ part is the upper-left portion; this part is referred to as the efficient frontier of the feasible set. Our investigation can thus be limited to this frontier. The Markowitz model We are now in a position to formulate a mathematical problem that leads to minimum-variance portfolios. We continue using the same notation as previously. Specifically, we write r̄i for the mean return of asset i; σij for the covariance of ri and rj , where σii = σi2; and wi for the weights satisfying ni=1 wi = 1. P Here we also allow short selling so that {wi} can assume negative values. To find a minimum-variance portfolio, we fix the mean of the portfolio return at a level r̄. MATM068: Mathematics, University of Surrey - 89 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 Then the objective is to minimise n X 1 2 wiwj σij i,j=1 subject to the constraints Xn n X wir̄i = r̄ and wi = 1. (99) i=1 i=1 1 The factor 2 appearing here is purely for convenience. The Markowitz problem provides the foundation for single-period investment theory. The problem addresses the trade-off between expected rate of return and variance of the rate of return in a portfolio. As we will see in later lectures, if we include into our portfolio a risk-free asset, then the problem simplifies considerably. For solving the problem of optimisation under constraint, one approach is to introduce the Lagrange multipliers. MATM068: Mathematics, University of Surrey - 90 - c DC Brody 2024 Financial Markets and Derivative Pricing Autumn Term 2024 We thus consider the Lagrangian n n n ! ! X X X 1 L=2 wiwj σij −

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