1、Lectures on Financial MathematicsHarald Langc Harald Lang, KTH Mathematics 2010PrefacePrefaceMy main goal with this text is to present the mathematical modellingof flnancial markets in a mathematically rigorous way, yet avoiding math-ematical technicalities that tends to deter people from trying to
2、accessit.Trade takes place in discrete time; the continuous case is consideredas the limiting case when the length of the time intervals tend to zero.However, the dynamics of asset values are modelled in continuous time asin the usual Black-Scholes model. This avoids some mathematical techni-calitie
3、s that seem irrelevant to the reality we are modelling.The text focuses on the price dynamics of forward (or futures) pricesrather than spot prices, which is more traditional. The rationale for this isthat forward and futures prices for any good|also consumption goods|exhibit a Martingale property o
4、n an arbitrage free market, whereas this isnot true in general for spot prices (other than for pure investment assets.)It also simplifles computations when derivatives on investment assets thatpay dividends are studied.Another departure from more traditional texts is that I avoid the no-tion of obje
5、ctive“ probabilities or probability distributions. I think theyare suspect constructs in this context. We can in a meaningful way assignprobabilities to outcomes of experiments that can be repeated under simi-lar circumstances, or where there are strong symmetries between possibleoutcomes. But it is
6、 unclear to me what the objective“ probability distri-bution for the price of crude oil, say, at some future point in time wouldbe. In fact, I dont think this is a well deflned concept.The text presents the mathematical modelling of flnancial markets. Inorder to get familiar with the workings of the
7、se markets in practice, thereader is encouraged to supplement this text with some text on flnancialeconomics. A good such text book is John C. Hulls: Options, Futures, 1;:;T, so that day 0 is today, and day T is the day of delivery. Each dayn a futures price Fn of cofiee beans with delivery day T is
8、 noted, and FTequals the prevailing spot price of cofiee beans day T. The futures priceFj is not known until day j; it will depend on how the cofiee bean crop isdoing, how the weather has been up to that day and the weather prospectsup till day T, the expected demand for cofiee, and so on. One can a
9、t anyday enter a futures contract, and there is no charge for doing so. The longholder of the contract will each day j receive the amount FjFj1 (whichmay be negative, in which he has to pay the corresponding amount,) so ifI enter a futures contract at day 0, I will day one receive F1F0, day twoF2 F1
10、 and so on, and day T, the day of delivery, FT FT1. The totalamount I receive is thus FT F0. There is no actual delivery of cofieebeans, but if I at day T buy the beans at the spot price FT, I pay FT, getmy cofiee beans and cash the amount FT F0 from the futures contract.In total, I receive my beans
11、, and pay F0, and since F0 is known alreadyday zero, the futures contract works somewhat like a forward contract.The difierence is that the value FT F0 is paid out successively duringthe time up to delivery rather than at the time of delivery.Here are timelines showing the cash ows for pay now“, for
12、wardsand futures contracts:Pay nowday 0 1 2 3 T1 Tcash ow P 0 0 0 0 XForward contractday 0 1 2 3 T1 Tcash ow 0 0 0 0 0 X G1I: Present-, Forward and Futures PricesFutures contractday 0 1 2 3 T1 Tcash ow 0 F1F0 F2F1 F3F2 FT1FT2 XFT1The simplest of these three contracts is the one when we pay in advanc
13、e,at least if the good that is delivered is non-pecuniary, since in that casethe interest does not play a part. For futures contracts, the interest rateclearly plays a part, since the return of the contract is spread out overtime.We will derive some book-keeping relations between the present prices,
14、forward prices and futures prices, but flrst we need some interest ratesecurities.Zero Coupon BondsA zero coupon bond with maturity T and face value V is a contractwhere the long holder pays ZTV in some currency day 0 and receives Vin the same currency day T. The ratio ZT of the price today and thea
15、mount received at time T is the discount factor converting currency attime T to currency today. One can take both long and short positions onzero coupon bonds.We need a notation for the currency, and we use a dollar sign $, eventhough the currency may be Euro or any other currency.Money Market Accou
16、ntA money market account (MMA) is like a series of zero coupon bonds,maturing after only one day (or whatever periods we have in our futurescontracts|it might be for example a week.) If I deposit the amount $1day 0, the balance of my account day 1 is $er1, where r1 is the shortinterest rate from day
17、 0 to day 1. The rate r1 is known already day 0.The next day, day 2, the balance has grown to $er1+r2, where r2 is theshort 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 balancehas grown to $er1+r2+r3, where r3 is the
18、 short interest rate from day 2to day 3; it is random as seen from day 0 and day 1, and its outcomeoccurs day 2, and so on. Day T the balance is thus $er1+rT which isa random variable. In order to simplify the notation, we introduce thesymbol R(t;T) = rt+1 +rT.2I: Present-, Forward and Futures Price
19、sRelations between Present-, Forward- and Futures PricesLet P(T)0 X be the present price today of a contract that delivers therandom value X (which may take negative values) at time T. Likewise,let G(T)0 X denote the forward price today of the value X delivered attime T and F(T)t X the futures price
20、 as of time t of the value X deliveredat time T. We then have the following theorem:TheoremThe following relations hold:a) P(T)0 , G(T)0 and F(T)0 are linear functions, i.e., if X and Y are randompayments made at time T, then for any constants a and bP(T)0 aX +bY = aP(T)0 X+bP(T)0 Y,and similarly fo
21、r G(T)0 and F(T)0 .b) For any deterministic (i.e., known today) value V, G(T)0 V = V,F(T)0 V = V and P(T)0 V = ZTVc) P(T)0 X = ZTG(T)0 Xd) P(T)0 XeR(0;T) = F(T)0 Xe) P(T)0 X = F(T)0 XeR(0;T)ProofThe proof relies on an assumption of the model: the law of one price.It means that there can not be two c
22、ontracts that both yield the samepayofi X at time T, but have difierent prices today. Indeed, if there weretwo such contracts, we would buy the cheaper and sell the more expensive,and make a proflt today, and have no further cash ows in the future. Butso would everyone else, and this is inconsistent
23、 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 contracton X and at the same time a long position on a zero coupon bond withface value G(T)0 X, then we have a portfolio which costs ZTG(T)0 X today,and yie
24、lds the income X at time T. By the law of one price, it hence mustbe that c) holds.To prove d), consider the following strategy: Deposit F0 on the moneymarket account, and take er1 long positions on the futures contract on Xfor delivery at time T.The next day the total balance is then F0er1 +er1(F1
25、F0) = F1er1.Deposit this on the money market account, and increase the futures posi-tion to er1+r2 contracts.3I: Present-, Forward and Futures PricesThe next day, day 2, the total balance is then F1er1+r2 +er1+r2(F2F1) = F2er1+r2. Deposit this on the money market account, and increasethe futures pos
26、ition to er1+r2+r3 contracts.And so on, up to day T when the total balance is FTer1+rT =Xer1+rT. In this way we have a strategy which is equivalent to acontract where we pay F0 day zero, and receive the value FTer1+rT =Xer1+rT day T. This proves d).Since relation d) is true for any random variable X
27、 whose outcomeis known day T, we may replace X by XeR(0;T) in that relation. Thisproves e).It is now easy to prove b). The fact that P(T)0 V = ZTV is simplythe deflnition of ZT. The relation G(T)0 V = V now follows from c) withX = V. In order to prove that F(T)0 V = V, note that by the deflnitionof
28、money market account, the price needed to be paid day zero in orderto receive VeR(0;T) day T is V; hence V = P(T)0 VeR(0;T). The relationF(T)0 V = V now follows from d) with X = V.Finally, toprovea), notethatifwebuyacontractswhichcostP(T)0 Xday zero and gives the payofi X day T, and b contracts that
29、 gives payofi Y,then we have a portfolio that costs aP(T)0 X+bP(T)0 Y day zero and yieldsthe payofi aX +bY day T; hence P(T)0 aX +bY = aP(T)0 X+bP(T)0 Y.The other two relations now follow immediately employing c) and d). Thiscompletes the proof.Comparison of Forward- and Futures PricesAssume flrst t
30、hat the short interest rates ri are deterministic, i.e.,that their values are known already day zero. This means that eR(0;T) isa constant (non-random,) so$1 = P(T)0 $eR(0;T) = P(T)0 $1eR(0;T) = $ZTeR(0;T);henceZT = eR(0;T):ThereforeZTG(T)0 X = P(T)0 X = F(T)0 XeR(0;T)= F(T)0 XeR(0;T) = F(T)0 XZT;an
31、d henceG(T)0 X = F(T)0 X:We write this down as a corollary:4I: Present-, Forward and Futures PricesCorollaryIf interest rates are deterministic, the forward price and the futuresprice coincide: G(T)0 X = F(T)0 XThe equality of forward- and futures prices does not in general hold ifinterest rates are
32、 random, though. To see this, we show as an examplethat if eR(0;T) is random, then F(T)0 $eR(0;T) G(T)0 $eR(0;T).Indeed, note that the function y = 1x is convex for x 0. This impliesthat its graph lies over its tangent. Let y = 1m +k(xm) be the tangentline through the point (m; 1m). Then 1x 1m + k(x
33、 m) with equalityonly for x = m (we consider only positive values of x.) Now use this withx = eR(0;T) and m = ZT. We then haveeR(0;T) Z1T +k(eR(0;T) ZT)where the equality holds only for one particular value of R(0;T). In theabsence of arbitrage (we will come back to this in Ch. IV,) the futuresprice
34、 of the value of the left hand side is greater than the futures price ofthe value of the right hand side, i.e.,F(T)0 $eR(0;T) $Z1T +k(F(T)0 $eR(0;T)$ZT)But F(T)0 $eR(0;T) = P(T)$1 = $ZT, so the parenthesis following k isequal to zero, henceF(T)0 $eR(0;T) $Z1TOn the other hand,ZTG(T)0 $eR(0;T) = P(T)
35、0 $eR(0;T) = $1so G(T)0 $eR(0;T) = $Z1T , and it follows thatF(T)0 $eR(0;T) G(T)0 $eR(0;T):In general, if X is positively correlated with the interest rate, then thefutures price tends to be higher than the forward price, and vice versa.5I: Present-, Forward and Futures PricesSpot Prices, Storage Co
36、st and DividendsConsider a forward contract on some asset to be delivered at a futuretime T. We have talked about the forward price, i.e., the price paid atthe time of delivery for the contract, and the present price, by which wemean the price paid for the contract today, but where the underlying as
37、setis still delivered at T. This should not be confused with the spot pricetoday of the underlying asset. The present price should equal the spotprice under the condition that the asset is an investment asset , and thatthere are no storage costs or dividends or other beneflts of holding theasset. As
38、 an example: consider a forward contract on a share of a stockto be delivered at time T. Let r be the interest rate (so that ZT = erT)and S0 the spot price of the share. Since $1 today is equivalent to $erTat time T, the forward price should then be G(T)0 = S0erT. But only ifthere is no dividend of
39、the share between now and T, for if there is, thenone could make an arbitrage by buying the share today, borrow for thecost and take a short position on a forward contract. There is then nonet payment today, and none at T (deliver the share, collect the deliveryprice G(T)0 of the forward and pay the
40、 S0erT = G(T)0 for the loan.) But itwould give the trader the dividend of the share for free, for this dividendgoes 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 theannual meeting of the company, so there migh
41、t be a convenience yield.CommentsYou can read about forward and futures contracts John Hulls bookOptions, Futures, T)G(T)0 X is invalid and nonsense! Indeed, R(0;T) is a randomvariable; its outcome is not known until time T1, whereas G(T)0 and P(T)0are known prices today.7II: Forwards, FRA:s and Swa
42、psII: Forwards, FRA:s and SwapsForward PricesIn many cases the theorem of Ch. I can be used to calculate forwardprices. Aswewillseelater, inordertocalculateoptionprices, itisessentialto flrst calculate the forward price of the underlying asset.Example 1.Consider a share of a stock which costs S0 tod
43、ay, and which givesa known dividend amount d in t years, and whose (random) spot priceat time T t is ST. Assume that there are no other dividends or otherconvenience yield during the time up to T. What is the forward price Gon this stock for delivery at time T?Assume that we buy the stock today, and
44、 sell it at time T. The cashow isday 0 t Tcash ow S0 d STThe present value of the dividend is Ztd and the present value of theincome ST at time T is ZtG: Hence we have the relationS0 = Zt d+ZT Gfrom which we can solve for GExample 2.Consider a share of a stock which costs S0 now, and which gives akn
45、own dividend yield dSt i t years, where St is the spot price immediatelybefore the dividend is paid out. Let the (random) spot price at time T tbe ST. Assume that there are no other dividends or other convenienceyield during the time up to T. What is the forward price G on this stockfor delivery at
46、time T?Consider the strategy of buying the stock now, and sell it at time timmediately before the dividend is paid out.day 0 tcash ow S0 StWith the notation of Ch. I, we have the relationP(t)0 St = S0 (1)Consider now the strategy of buying the stock now, cash the dividend attime t, and eventually se
47、ll the stock at time T.8II: Forwards, FRA:s and Swapsday 0 t Tcash ow S0 dSt STThe present value of the dividend is dP(t)St and the present value of theincome ST at time T is ZT G. Hence we have the relationS0 = dP(t)0 St+ZT GIf we combine with (1) we get(1d)S0 = ZT Gfrom which we can solve for GExa
48、mple 3.With the same setting as in example 2, assume that there are dividendyield payments at several points in time t1 t. The cashow for this party is thus L at time t and Lef(Tt) at time T. Since thiscontract costs nothing now, we have the relation10II: Forwards, FRA:s and Swaps0 = ZtLZTLef(Tt):Fr
49、om this we can solve for f. The interest rate f is the forward rate fromt to T.Plain Vanilla Interest Rate SwapThe simplest form of an interest swap is where one party, say A, paysparty B:the oating interest on a principal L1 from time t0 to t1 at time t1the oating interest on a principal L2 from time t1 to t2 at time t2the oating interest on a principal L3 from time t2
