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Lecture Notes in Financial Mathematics

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Lectures on Financial Mathematics Harald Lang  c Harald Lang, KTH Mathematics 2012Preface Preface My main goal with this text is to present the mathematical modelling of financial markets in a mathematically rigorous way, yet avoiding math- ematical technicalities that tends to deter people from trying to access it. Trade takes place in discrete time; the continuous case is considered as the limiting case when the length of the time intervals tend to zero. However, the dynamics of asset values are modelled in continuous time as in the usual Black-Scholes model. This avoids some mathematical techni- calities that seem irrelevant to the reality we are modelling. The text focuses on the price dynamics of forward (or futures) prices rather than spot prices, which is more traditional. The rationale for this is that forward and futures prices for any good—also consumption goods— exhibit a Martingale property on an arbitrage free market, whereas this is not true in general for spot prices (other than for pure investment assets.) It also simplifies computations when derivatives on investment assets that pay dividends are studied. Another departure from more traditional texts is that I avoid the no- tion of “objective” probabilities or probability distributions. I think they are suspect constructs in this context. We can in a meaningful way assign probabilities to outcomes of experiments that can be repeated under simi- lar circumstances, or where there are strong symmetries between possible outcomes. But it is unclear to me what the “objective” probability distri- bution for the price of crude oil, say, at some future point in time would be. In fact, I don’t think this is a well defined concept. The text presents the mathematical modelling of financial markets. In order to get familiar with the workings of these markets in practice, the reader is encouraged to supplement this text with some text on financial economics. A good such text book is John C. Hull’s: Options, Futures, & Other Derivatives (Prentice Hall,) which I will refer to in some places. 13/9–2012 Harald LangContents Contents I: Introduction to Present-, Forward- and Futures Prices . . . 1 Zero Coupon Bonds ............................ 2 Money Market Account .......................... 2 Relations between Present-, Forward- and Futures Prices ..... 3 Comparison of Forward- and Futures Prices . ............ 4 Spot Prices, Storage Cost and Dividends . .............. 6 Comments . ................................. 6 II: Forwards, FRA:s and Swaps . . . . . . . . . . . . . . . . . . . 8 Forward Prices . .............................. 8 Forward Rate Agreements . . . ..................... 10 Plain Vanilla Interest Rate Swap .................... 11 Exercises and Examples ......................... 12 Solutions . . . ................................ 14 III: Optimal Hedge Ratio . . . . . . . . . . . . . . . . . . . . . . 17 Exercises and Examples.......................... 19 Solutions . . . ................................ 19 IV: Conditions for No Arbitrage . . . . . . . . . . . . . . . . . 21 Theorem (The No Arbitrage Theorem) . . .............. 22 The No Arbitrage Assumption...................... 23 V: Pricing European Derivatives . . . . . . . . . . . . . . . . . 25 Black’s Model ................................ 25 The Black-Scholes Pricing Formula . .................. 26 Put and Call Options ........................... 26 The Interpretation of σ and the Market Price of Risk . . . .... 26 Comments . ................................. 27 Exercises and Examples ......................... 28 Solutions . . . ................................ 30 VI: Yield and Duration . . . . . . . . . . . . . . . . . . . . . . . 32 Forward Yield and Forward Duration ................. 34 Black’s Model for Bond Options .................... 36 Portfolio Immunising . . . ........................ 37 Exercises and Examples.......................... 37 Solutions . . . ................................ 39 VII: Risk Adjusted Probability Distributions . . . . . . . . . 42 An Example ................................. 43 Forward Distributions for Different Maturities . . . ......... 44 Exercises and Examples.......................... 46 Solutions . . . ................................ 46 VIII: Conditional Expectations and Martingales . . . . . . . 48 Martingale Prices . ............................. 49Contents IX: Asset Price Dynamics and Binomial Trees . . . . . . . . . 51 Black-Scholes Dynamics . . ........................ 51 Binomial Approximation . . . ...................... 51 The Binomial Model . . .......................... 52 Pricing an American Futures Option . . . ............... 53 American Call Option on a Share of a Stock ............. 54 Options on Assets Paying Dividends .................. 54 Comments .................................. 55 Exercises and Examples . ......................... 56 Solutions ................................... 58 X: Random Interest Rates: The Futures Distribution . . . . 66 XI: A Model of the Short Interest Rate: Ho-Lee . . . . . . . 69 The Price of a Zero Coupon Bond . . ................. 70 Forward and Futures on a Zero Coupon Bond . . . ......... 71 The Forward Distribution . ....................... 71 Pricing a European Option on a Zero Coupon Bond . ....... 72 XII: Ho-Lee’s Binomial Interest Rate Model . . . . . . . . . . 73 Comments .................................. 74 Exercises and Examples . ......................... 75 Solutions ................................... 77I: Present-, Forward and Futures Prices I: Introduction to Present-, Forward- and Futures Prices Assume that we want to buy a quantity of coffee beans with delivery in nine months. However, we are concerned about what the (spot) price of coffee beans might be then, so we draw up a contract where we agree on the price today. There are now at least three ways in which we can arrange the payment: 1) we pay now, in advance. We call this price the present price of coffee beans with delivery in nine months time, and denote it by P. Note that this is completely different from the spot price of coffee beans, i.e., the price of coffee for immediate delivery. 2) we pay when the coffee is delivered, i.e., in nine months time. This price is the forward price, which we denote by G. 3) we might enter a futures contract with delivery in nine months time. A futures contract works as follows: Let us denote the days from today to delivery by the numbers 0, 1,... ...,T, so that day 0 is today, and day T is the day of delivery. Each day n a futures price F of coffee beans with delivery day T is noted, and F n T equals the prevailing spot price of coffee beans day T. The futures price F is not known until day j; it will depend on how the coffee bean crop is j doing, how the weather has been up to that day and the weather prospects up till day T, the expected demand for coffee, and so on. One can at any day enter a futures contract, and there is no charge for doing so. The long holder of the contract will each day j receive the amount F −F (which j j−1 may be negative, in which he has to pay the corresponding amount,) so if I enter a futures contract at day 0, I will day one receive F −F , day two 1 0 F − F and so on, and day T, the day of delivery, F − F . The total 2 1 T T−1 amount I receive is thus F − F . There is no actual delivery of coffee T 0 beans, but if I at day T buy the beans at the spot price F , I pay F , get T T my coffee beans and cash the amount F − F from the futures contract. T 0 In total, I receive my beans, and pay F , and since F is known already 0 0 day zero, the futures contract works somewhat like a forward contract. The difference is that the value F − F is paid out successively during T 0 the time up to delivery rather than at the time of delivery. Here are timelines showing the cash flows for “pay now”, forwards and futures contracts: Pay now day 0123 ··· T−1 T cash flow −P 000 ··· 0 X Forward contract day 0123 ··· T−1 T cash flow 0 0 0 0 ··· 0 X− G 1I: Present-, Forward and Futures Prices Futures contract day 0 123 ··· T−1 T cash flow 0 F −F F −F F −F ··· F −F X−F 1 0 2 1 3 2 T−1 T−2 T−1 The simplest of these three contracts is the one when we pay in advance, at least if the good that is delivered is non-pecuniary, since in that case the interest does not play a part. For futures contracts, the interest rate clearly plays a part, since the return of the contract is spread out over time. We will derive some book-keeping relations between the present prices, forward prices and futures prices, but first we need some interest rate securities. Zero Coupon Bonds A zero coupon bond with maturity T and face value V is a contract where the long holder pays Z V in some currency day 0 and receives V T in the same currency day T. The ratio Z of the price today and the T amount received at time T is the discount factor converting currency at time T to currency today. One can take both long and short positions on zero coupon bonds. We need a notation for the currency, and we use a dollar sign , even though the currency may be Euro or any other currency. Money Market Account A money market account (MMA) is like a series of zero coupon bonds, maturing after only one day (or whatever periods we have in our futures contracts—it might be for example a week.) If I deposit the amount 1 r 1 day 0, the balance of my account day 1 is e , where r is the short 1 interest rate from day 0 to day 1. The rate r is known already day 0. 1 r +r 1 2 The next day, day 2, the balance has grown to e , where r is the 2 short interest rate from day 1 to day 2; it is random as seen from day 0, and its outcome is determined day 1. The next day, day 3, the balance r +r +r 1 2 3 has grown to e , where r is the short interest rate from day 2 3 to day 3; it is random as seen from day 0 and day 1, and its outcome r +···+r 1 T occurs day 2, and so on. Day T the balance is thus e which is a random variable. In order to simplify the notation, we introduce the symbol R(t,T)= r +··· + r . t+1 T 2I: Present-, Forward and Futures Prices Relations between Present-, Forward- and Futures Prices (T) Let P X be the present price today of a contract that delivers the 0 random value X (which may take negative values) at time T. Likewise, (T) let G X denote the forward price today of the value X delivered at 0 (T) time T and F X the futures price as of time t of the value X delivered t at time T. We then have the following theorem: Theorem The following relations hold: (T) (T) (T) a) P , G and F are linear functions, i.e., if X and Y are random 0 0 0 payments made at time T, then for any constants a and b (T) (T) (T) P aX + bY= aP X+ bP Y , 0 0 0 (T) (T) and similarly for G and F . 0 0 (T) b) For any deterministic (i.e., known today) value V, G V= V , 0 (T) (T) F V= V and P V= Z V T 0 0 (T) (T) c) P X= Z G X T 0 0 (T) (T) R(0,T) d) P Xe = F X 0 0 (T) (T) −R(0,T) e) P X= F Xe 0 0 Proof The proof relies on an assumption of the model: the law of one price. It means that there can not be two contracts that both yield the same payoff X at time T, but have different prices today. Indeed, if there were two such contracts, we would buy the cheaper and sell the more expensive, and make a profit today, and have no further cash flows in the future. But so would everyone else, and this is inconsistent with market equilibrium. In Ch. IV we will extend this model assumption somewhat. To prove c), note that if we take a long position on a forward contract on X and at the same time a long position on a zero coupon bond with (T) (T) face value G X, then we have a portfolio which costs Z G Xtoday, T 0 0 and yields the income X at time T. By the law of one price, it hence must be that c) holds. To prove d), consider the following strategy: Deposit F on the money 0 r 1 market account, and take e long positions on the futures contract on X for delivery at time T. r r r 1 1 1 The next day the total balance is then F e + e (F − F )= F e . 0 1 0 1 Deposit this on the money market account, and increase the futures posi- r +r 1 2 tion to e contracts. 3I: Present-, Forward and Futures Prices r +r r +r 1 2 1 2 The next day, day 2, the total balance is then F e +e (F − 1 2 r +r 1 2 F )= F e . Deposit this on the money market account, and increase 1 2 r +r +r 1 2 3 the futures position to e contracts. r +···+r 1 T And so on, up to day T when the total balance is F e = T r +···+r 1 T Xe . In this way we have a strategy which is equivalent to a r +···+r 1 T contract where we pay F day zero, and receive the value F e = 0 T r +···+r 1 T Xe day T. This proves d). Since relation d) is true for any random variable X whose outcome −R(0,T) is known day T, we may replace X by Xe in that relation. This proves e). (T) It is now easy to prove b). The fact that P V= Z V is simply T 0 (T) the definition of Z . The relation G V= V now follows from c) with T 0 (T) X = V . In order to prove that F V= V , note that by the definition 0 of money market account, the price needed to be paid day zero in order (T) R(0,T) R(0,T) to receive Ve day T is V ; hence V = P Ve . The relation 0 (T) F V= V now follows from d) with X = V . 0 (T) Finally, to prove a), note that if we buy a contracts which cost P X 0 day zero and gives the payoff X day T, and b contracts that gives payoff Y , (T) (T) then we have a portfolio that costs aP X+bP Y day zero and yields 0 0 (T) (T) (T) the payoff aX + bY day T; hence P aX + bY= aP X+ bP Y . 0 0 0 The other two relations now follow immediately employing c) and d). This completes the proof. Comparison of Forward- and Futures Prices Assume first that the short interest rates r are deterministic, i.e., i R(0,T) that their values are known already day zero. This means that e is a constant (non-random,) so (T) (T) R(0,T) R(0,T) R(0,T) 1 = P e = P 1e =Z e , T 0 0 hence −R(0,T) Z = e . T Therefore (T) (T) (T) −R(0,T) Z G X= P X= F Xe T 0 0 0 (T) (T) −R(0,T) = F Xe = F XZ , T 0 0 and hence (T) (T) G X= F X. 0 0 We write this down as a corollary: 4I: Present-, Forward and Futures Prices Corollary If interest rates are deterministic, the forward price and the futures (T) (T) price coincide: G X= F X 0 0 The equality of forward- and futures prices does not in general hold if interest rates are random, though. To see this, we show as an example (T) (T) R(0,T) R(0,T) R(0,T) that if e is random, then F e G e . 0 0 1 Indeed, note that the function y = is convex forx 0. This implies x 1 that its graph lies over its tangent. Let y = + k(x− m) be the tangent m 1 1 1 line through the point (m, ). Then ≥ + k(x− m) with equality m x m only for x = m (we consider only positive values of x.) Now use this with −R(0,T) x = e and m = Z . We then have T R(0,T) −1 −R(0,T) e ≥ Z + k(e − Z ) T T where the equality holds only for one particular value of R(0,T). In the absence of arbitrage (we will come back to this in Ch. IV,) the futures price of the value of the left hand side is greater than the futures price of the value of the right hand side, i.e., (T) (T) R(0,T) −1 −R(0,T) F e Z + k(F e − Z ) T 0 T 0 (T) −R(0,T) (T) But F e = P 1 = Z , so the parenthesis following k is T 0 equal to zero, hence (T) R(0,T) −1 F e Z 0 T On the other hand, (T) (T) R(0,T) R(0,T) Z G e = P e =1 T 0 0 (T) −1 R(0,T) so G e =Z , and it follows that 0 T (T) (T) R(0,T) R(0,T) F e G e . 0 0 In general, if X is positively correlated with the interest rate, then the futures price tends to be higher than the forward price, and vice versa. 5I: Present-, Forward and Futures Prices Spot Prices, Storage Cost and Dividends Consider a forward contract on some asset to be delivered at a future time T. We have talked about the forward price, i.e., the price paid at the time of delivery for the contract, and the present price, by which we mean the price paid for the contract today, but where the underlying asset is still delivered at T. This should not be confused with the spot price today of the underlying asset. The present price should equal the spot price under the condition that the asset is an investment asset , and that there are no storage costs or dividends or other benefits of holding the asset. As an example: consider a forward contract on a share of a stock −rT to be delivered at time T. Let r be the interest rate (so that Z = e ) T rT and S the spot price of the share. Since 1 today is equivalent to e 0 (T) rT at time T, the forward price should then be G = S e . But only if 0 0 there is no dividend of the share between now and T, for if there is, then one could make an arbitrage by buying the share today, borrow for the cost and take a short position on a forward contract. There is then no net payment today, and none at T (deliver the share, collect the delivery (T) (T) rT price G of the forward and pay the S e = G for the loan.) But it 0 0 0 would give the trader the dividend of the share for free, for this dividend goes to the holder of the share, not the holder of the forward. Likewise, the holder of the share might have the possibility of taking part in the annual meeting of the company, so there might be a convenience yield. Comments You can read about forward and futures contracts John Hull’s book 1 “Options, Futures, & other Derivatives ”. He describes in detail how fu- tures contracts work, and why they are specified in the somewhat peculiar way they are. The mathematical modelling of a futures contract is a slight simplifi- cation of the real contract. We disregard the maintenance account, thus avoiding any problems with interest on the balance. Furthermore, we as- sume that the delivery date is defined as a certain day, not a whole month. We also disregard the issues on accounting and tax. We use “continuous compounding” of interest rates (use of the expo- nential function.) If you feel uncomfortable with this, you may want to read relevant chapters in Hull’s book. We will use continuous compound- ing unless otherwise explicitly stated, since it is the most convenient way to handle interest. It is of course easy to convert between continuous compounded interest and any other compounding. 1 Prentice Hall 6I: Present-, Forward and Futures Prices It is important not to confuse the present price with the spot price of the same type of good. The spot price is the price of the good for immediate delivery. It is also extremely important to distinguish between constants (val- ues that are currently known) and random variables. For instance, as- −rT sume that Z = e (where r hence is a number.) Then it is true that T (T) (T) (T) −rT P X= e G X (Theorem c,) however, the relation P X= 0 0 0 (T) −R(0,T) e G X is invalid and nonsense Indeed, R(0,T) is a random 0 (T) (T) variable; its outcome is not known until time T−1, whereas G and P 0 0 are known prices today. 7II: Forwards, FRA:s and Swaps II: Forwards, FRA:s and Swaps Forward Prices In many cases the theorem of Ch. I can be used to calculate forward prices. As we will see later, in order to calculate option prices, it is essential to first calculate the forward price of the underlying asset. Example 1. Consider a share of a stock which costs S today, and which gives 0 a known dividend amount d in t years, and whose (random) spot price at timeTt is S . Assume that there are no other dividends or other T convenience yield during the time up to T. What is the forward price G on this stock for delivery at time T? Assume that we buy the stock today, and sell it at time T. The cash flow is day 0 tT cash flow −S dS 0 T The present value of the dividend is Z d and the present value of the t income S at time T is Z G. Hence we have the relation T t S = Z d + Z G 0 t T from which we can solve for G Example 2. Consider a share of a stock which costs S now, and which gives a 0 known dividend yield dS i t years, where S is the spot price immediately t t before the dividend is paid out. Let the (random) spot price at timeTt be S . Assume that there are no other dividends or other convenience T yield during the time up to T. What is the forward price G on this stock for delivery at time T? Consider the strategy of buying the stock now, and sell it at time t immediately before the dividend is paid out. day 0 t cash flow −S S 0 t With the notation of Ch. I, we have the relation (t) P S = S (1) t 0 0 Consider now the strategy of buying the stock now, cash the dividend at time t, and eventually sell the stock at time T. 8II: Forwards, FRA:s and Swaps day 0 tT cash flow −S dS S 0 t T (t) The present value of the dividend is dP S and the present value of the t income S at time T is Z G. Hence we have the relation T T (t) S = dP S + Z G 0 t T 0 If we combine with (1) we get (1− d)S = Z G 0 T from which we can solve for G Example 3. With the same setting as in example 2, assume that there are dividend yield payments at several points in time t ... t , where t T, each 1 n n time with the amount dS . As in the above example, we can buy the t j stock today and sell it just before the first dividend is paid out, so the relation (t ) 1 P S = S (2) t 0 0 1 holds. For any k =2, 3,...,n we can buy the stock at time t imme- k−1 diately before the payment of the dividend, collect the dividend, and sell the stock immediately before the dividend is paid out at time t . k day 0 t t k−1 k cash flow 0 −S + dS S t t t k−1 k−1 k The price of this strategy today is zero, so we have (t ) (t ) k−1 k−1 (t ) k 0=−P S +dP S + P S , i.e., t t t 0 k−1 0 k−1 0 k (t ) (t ) k k−1 P S =(1− d)P S t t 0 k 0 k−1 (T) (t ) n and repeated use of this relation and P S =(1− d)P S gives T t 0 0 n (T) (t1) n n P S =(1− d) P S =(1− d) S T t 0 0 0 1 where we have used (2) to obtain the last equality. Hence, by the theorem of Ch. I, we have the relation n Z G=(1− d) S (3) T 0 9II: Forwards, FRA:s and Swaps Example 4. We now consider the setting in example 3, but with a continuous dividend yield ρ, i.e., for any small interval in time (t,t + δt) we get the dividend ρS δt. If we divide the time interval (0,T) into a large number t n of intervals of length δt = T/n, we see from (3) that n Z G=(1−ρδt) S T 0 and when n→∞ we get −ρT Z G = e S T 0    n Derivation of the limit: ln (1− ρδt) = n ln(1− ρδt)= n − ρδt +    2 2 O(δt ) = n − ρT/n +O(1/n ) →−ρT. Taking exponential gives the limit. Example 5. Assume we want to buy a foreign currency in t years time at an exchange rate, the forward rate, agreed upon today. Assume that the interest on the foreign currency is ρ and the domestic rate is r per year. Let X be the exchange rate now (one unit of foreign currency costs X in 0 0 domestic currency,) and let X be the (random) exchange rate as of time t t. Let G be the forward exchange rate. Consider now the strategy: buy one unit of foreign currency today, ρt buy foreign zero coupon bonds for the amount, so that we have e worth of bonds in foreign currency at time t when we sell the bonds. day 0 t ρt cash flow −X e X 0 t ρt Since the exchange rate at that time is X , we get X e in domestic t t currency. Since the price we have paid today is X we have the relation 0   (t) ρt −rt ρt (ρ−r)t X = P X e = e Ge = Ge 0 t 0 i.e., (r−ρ)t G = X e 0 Forward Rate Agreements A forward rate agreement is a forward contract where the parties agree that a certain interest rate will be applied to a certain principal during a future time period. Let us say that one party is to borrow an amount L at f(T−t) time t and later pay back the amount Le at timeTt. The cash f(T−t) flow for this party is thus L at time t and−Le at time T. Since this contract costs nothing now, we have the relation 10II: Forwards, FRA:s and Swaps f(T−t) 0= Z L− Z Le . t T From this we can solve for f. The interest rate f is the forward rate from t to T. Plain Vanilla Interest Rate Swap The simplest form of an interest swap is where one party, say A,pays party B: the floating interest on a principal L from time t to t at time t 1 0 1 1 the floating interest on a principal L from time t to t at time t 2 1 2 2 the floating interest on a principal L from time t to t at time t 3 2 3 3 ··· the floating interest on a principal L from time t to t at time n n−1 n t . n The floating rate between t and t is the zero coupon rate that prevails j k between these two points in time. The amount that A pays at time t is k thus L ·(1/Z(t ,t )−1), where Z(t ,t ) of course is the price of the k k−1 k k−1 k zero coupon bond at time t that matures at t . Note that this price k−1 k is unknown today but known at time t . The total amount that B will k−1 receive, and A will pay is thus random. On the other hand, party B pays A a fixed amount c at each of the times t ,...,t . The question is what c ought to be in order to make this 1 n deal “fair”. The notation with continuous compounding is here bit inconvenient. Let us introduce the one period floating rate r ˆ : the interest from time j t to time t , i.e., if I deposit an amount a on a bank account at time j−1 j t the balance at time t is a + ar ˆ . This is the same as to say that a j−1 j j zero coupon bond issued at time t with maturity at t is 1/(1 + r ˆ ). j−1 j j Note that r ˆ is random whose value becomes known at time t . The j j−1 cash flow that A pays to B is then day t t t t ··· t 0 1 2 3 n cash flow 0 r ˆ L r ˆ L r ˆ L ··· r ˆ L 1 1 2 2 3 3 n n In order to calculate the present value of this cash flow, we first determine (t ) k the present value P (ˆ r L ). k k 0 Consider the strategy: buy L worth of zero coupon bonds at time k t with maturity at t and face value (1 + r ˆ )L . This costs nothing k−1 k k k today, so the cash flow is day t ··· t t 0 k−1 k cash flow 0 ··· −L (1 + r ˆ )L k k k 11II: Forwards, FRA:s and Swaps (t ) (t ) (t ) k−1 k k hence 0 =−P L + P L + P ˆ r L =−Z L + Z L + k k k k t k t k 0 0 0 k−1 k (t ) k P ˆ r L , so we get k k 0 (t ) k P ˆ r L =(Z − Z )L . k k t t k 0 k−1 k It is now easy to calculate the present value of the cash flow from A to B: it is n  P = (Z − Z )L AB t t k k−1 k 1 The present value of B:s payments to A is, on the other hand, n  P = c Z BA t k 1 so we can calculate the fair value of c by solving the equation we get by setting P = P . BA AB Exercises and Examples Interest rates always refer to continuous compounding. Answers are given in parenthesis; solutions to some problems are given in the next section 1. A share is valued at present at 80 dollars. In nine months it will give a dividend of 3 dollars. Determine the forward price for delivery in one year given that the rate of interest is 5% a year. (81.06 dollars) 2. A share is valued at present at 80 dollars. In nine months it will give a dividend of 4% of its value at that time. Determine the forward price for delivery in one year given that the rate of interest is 5% a year. (80.74 dollars) 3. The current forward price of a share to be delivered in one year is 110 dollars. In four months the share will give a dividend of 2 dollars and in ten months will give a dividend of 2% of its value at that time. Determine the current spot price of the share given that the rate of interest is 6% a year. (107.67 dollars) 4. The exchange rate of US dollars is today 8.50 SEK per dollar. The forward price of a dollar to be delivered in six months is 8.40 SEK. If the Swedish six month interest rate is 4% a year, determine the American six month interest rate. (6.367%) 12II: Forwards, FRA:s and Swaps 5. The forward price of a US dollar the first of August with delivery at the end of December is 0.94630 EUR. The forward price of a dollar to be delivered at the end of June next year is 0.95152 EUR. Assuming a flat term structure for both currencies and that the Euro interest rate is 4% a year—what is the American rate of interest? (2.90%) 6. Determine the forward price of a bond to be delivered in two years. 1 The bond pays out 2 EUR every 6-months during 4 years (starting 2 in six months), and 102 EUR after five years. Thus the bond is, as of today, a 5-year 4%-coupon bond with a coupon dividend every six months with a 100 EUR face value. The bond is to be delivered in two years immediately after the dividend has been paid. The present term structure is given by the following rates of interest (on a yearly basis) 6 months 5.0% 18, 24 months 5.6% 12 months 5.4% 30–60 months 5.9% (94.05 EUR) 7. A one-year forward contract of a share which pays no dividend before the contract matures is written when the share has a price of 40 dollars and the risk-free interest rate is 10% a year. a) What is the forward price? (44.207 dollars) b) If the share is worth 45 dollars six months later, what is the value of the original forward contract at this time? If another forward contract is to be written with the same date of maturity, what should the forward price be? (2.949 dollars, 47.307 dollars) 8. Determine the forward price in SEK of a German stock which costs 25 EUR today. The time of maturity is in one year, and the stock pays a dividend in nine months of 5% of the current stock price at that time. The interest rate of the Euro is 4.5% per year, and the crown’s rate of interest is 3% per year. One Euro costs 9.40 SEK today. (230.05 SEK) 13II: Forwards, FRA:s and Swaps 9. Let r be the random daily rate of interest (per day) from day i− 1 i to day i, and R(0,t)= r +··· + r . The random variable X is the 1 t t stock exchange index day t (today is day 0.) The random variables X and R(0,t) are not independent. t The forward price of a contract for delivery of the payment R(0,t) X e EUR day t is 115 EUR, a zero coupon bond which pays t 1 EUR day t costs 0.96 EUR. Determine the futures price of a contract for delivery of X EUR t day t. The stock exchange index is today X = 100. 0 10. A share of a stock currently costs 80 EUR. One year from now, it will pay a dividend of 5% of its price at the time of the payment, and the same happens two years from now. Determine the forward price of the asset for delivery in 2.5 years. The interest rate for all maturities is 6% per year. (83.88 EUR) 11. The 6 month zero rate is 5% per year and the one year rate is 5.2% per year. What is the forward rate from 6 to 12 months? (5.4% per year) 12. Show that the present value of a cash flow at time T that equals the floating interest from t to T (tT) on a principal L is the same as if the floating rate is replaced by the current forward rate. (Note that this is an easy way to value interest rate swaps: just replace any floating rate by the corresponding forward rate.) Solutions 3. (The problem is admittedly somewhat artificial, but serves as an ex- ercise.) If we buy the stock today and sell it after one year, the cash flow can be represented: month 0 4 10 12 cash flow −S 20.02S S 0 10 12 i.e.,     (10) (12) −0.06·4/12 S =2e +0.02 P S + P S . (1) 0 10 12 0 0 On the other hand, if we sell the stock after 10 months, before the dividend, then the cash flow is month 0 4 10 cash flow −S 2 S 0 10 i.e., 14