A Formula Sheet for Financial EconomicsWilliam Benedict McCartneyApril 2012AbstractThis document is meant to be used solely as a formula sheet. It contains very little in the way of explanation and is not meant to be usedas a substitute for a financial economics text. It is aimed specifically atthose students preparing for exam MFE offered by the Society of Actuaries, but it should be of some use to everyone studying financial economics.It covers the important formulas and methods used in put-call parity, option pricing using binomial trees, Brownian motions, stochastic calculus,stock price dynamics, the Sharpe ratio, the Black-Scholes equation, theBlack-Scholes formula, option greeks, risk management techniques, estimations of volatilities and rates of appreciation, exotic options (asian,barrier, compound, gap, and exchange), simulation, interest rate trees,the Black model, and several interest rate models (Rendleman-Bartter,Vasicek, and Cox-Ingersoll-Ross)1

1Forwards, Puts, and Calls1.1ForwardsA forward contract is an agreement in which the buyer agrees at time t to paythe seller at time T and receive the asset at time T.Ft,T (S) St er(T t) St er(T t) F Vt,T (Dividends) St e(r δ)(T t)(1)A prepaid forward contract is an agreement in which the buyer agrees at timet to pay the seller at time t and receive the asset at time T.PFt,T(S) St or St P Vt,T (Dividends) or St e δ(T t)1.2(2)Put-Call ParityCall options give the owner the right, but not the obligation, to buy an asset atsome time in the future for a predetermined strike price. Put options give theowner the right to sell. The price of calls and puts is compared in the followingput-call parity formula for European options.Pc(St , K, t, T ) p(St , K, t, T ) Ft,T(S) Ke r(T t)1.3(3)Calls and Puts with Different StrikesFor European calls and puts, with strike prices K 1 and K 2 where K 1 K 2 , weknow the following0 c(K1 ) c(K2 ) (K2 K1 )e rT(4) rT(5)0 p(K2 ) p(K1 ) (K2 K1 )eFor American options, we cannot be so strict. Delete the discount factor on the(K2 K1 ) term and then you’re okay. Another important result arises for threedifferent options with strike prices K 1 K 2 K 3c(K2 ) c(K3 )c(K1 ) c(K2 ) K2 K1K3 K2(6)p(K2 ) p(K1 )p(K3 ) p(K2 ) K2 K1K3 K2(7)Exam MFE loves arbitrage questions. An important formula for determiningarbitrage opportunities comes from the following equations.K2 λK1 (1 λ)K3λ K3 K2K3 K12(8)(9)

The coefficients in front of each strike price in equation 8 represent the numberof options of each strike price to buy for two equivalent portfolios in an arbitragefree market.1.4Call and Put Price BoundsThe following equations give the bounds on the prices of European calls andputs. Note that the lower bounds are no less than zero.PP(Ft,T(S) Ke r(T t) ) c(St , K, t, T ) Ft,T(S)(10)P(Ke r(T t) Ft,T(S)) p(St , K, t, T ) Ke r(T t)(11)We can also compare the prices of European and American options using thefollowing inequalities.1.5c(St , K, t, T ) C(St , K, t, T ) St(12)p(St , K, t, T ) P (St , K, t, T ) K(13)Varying Times to ExpirationFor American options only, when T 2 T 11.6C(St , K, t, T2 ) C(St , K, t, T1 ) St(14)P (St , K, t, T2 ) P (St , K, t, T1 ) St(15)Early Exercise for American OptionsIn the following inequality the two sides can be thought of the pros and consof exercising the call early. The pros of exercising early are getting the stock’sdividend payments. The cons are that we have to pay the strike earlier andtherefore miss the interest on that money and we lose the put protection if thestock price should fall. So we exercise the call option if the pros are greaterthan the cons, specifically, we exercise ifP Vt,T (dividends) p(St , K) K(1 e r(T t) )(16)For puts, the situation is slightly different. The pros are the interest earned onthe strike. The cons are the lost dividends on owning the stock and the callprotection should the stock price rise. Explicitly, we exercise the put optionearly ifK(1 e r(T t) ) c(St , K) P Vt,T (dividends)3(17)

22.1Binomial TreesThe One-Period Replicating PortfolioThe main idea is to replicate the payoffs of the derivative with the stock and arisk-free bond. eδh S0 u Berh Cuδh e S0 d Berh Cd(18)(19)To replicate the derivative we buy shares and invest in B dollars.Cu Cd(20)S0 (u d)uCd dCu(21)B e rhu dSince we designed the portfolio to replicate the option they must, since there isno arbitrage, have the same time-0 price. e δhC0 S0 B2.2(22)Risk-Neutral ProbabilitiesWe define the risk-neutral probability of the stock price going up as followsp e(r δ)h du d(23)Then the price of the option isC0 e rh [p Cu (1 p )Cd ](24)A key result of the risk-neutral world is that the expected price of the stock atfuture time t isE [St ] p S0 u (1 p )S0 d S0 ert2.3(25)Multi-Period TreesThe single period binomial trees formulas can be used to go back one step at atime on the tree. Also, note that for a European option we can use this shortcutformula.C0 e 2rh [(p )2 Cuu 2p (1 p )Cud (1 p )2 Cdd ](26)For American options, however, it’s important to check the price of the optionat each node of the tree. If the price of the option is less than the payout, thenthe option would be exercised and the price at that node should be the payoffat that point.4

2.4Trees from VolatilitiesAssuming a forward treeu e(r δ)h σd e(r δ)h σ1p 1 eσAssuming a Cox-Ross-Rubinstein treeu eσ d e σ h(27)h(28)(29)hh (30)h(31)Assuming a Jarrow-Rudd (lognormal) forward tree1u e(r δ 2 σd e2.52 )h σ h(r δ 21 σ 2 )h σ h(32)(33)Options on CurrenciesThe easiest way to deal with these options is to treat the exchange rate as astock where x(t) is the underlying, rf is the dividend rate and r is the risk-freerate. It is helpful to understand these following equation. 1.00 x(t)(34)Remember that if you treat x(t) as the underlying asset than r is the risk-freerate for dollars, rf is the risk-free rate for pounds, and the option is considereddollar-denominated.c(K, T ) p(K, T ) x0 e rf T Ke rT2.6(35)Options on FuturesThe main difference between futures and the other previously discussed assetsis that futures don’t initially require any assets to change hands. The formulasare therefore adjusted as followsCu CdF0 (u d) 1 du 1 rhB C0 eCu Cdu du d (36)(37)Note that the previous equation becomes clearer when we define p , which,because it is sometimes possible to think of a forward as a stock with δ r, as5

1 d(38)u dThe other formulas all work the same way. Notably, the put-call parity formulabecomesp c(K, T ) p(K, T ) F0 e rT Ke rT2.7(39)True Probability PricingWe’ve been assuming a risk-free world in the previous formulas as it makesdealing with some problems nicer. But it’s important to examine the followingreal-world or true probability formulas.E(St ) pS0 u (1 p)S0 d S0 e(α δ)t(40)(α δ)t d(41)u dIt is possible to price options using real world probabilities. But r can no longerbe used. Instead γ is the appropriate discount ratep eS0 Beαt ertS0 BS0 BC0 e γt [Cu (1 p)Cd ]eγt 2.8(42)(43)State PricesState prices are so called because it’s the cost of a security that pays one dollar upon reaching a particular state. Remember these following formulas fordetermining state pricesQH QL e rtSH QH SL QL PF0,t(S)C0 CH QH CL QL(44)(45)(46)The above equations can easily be adapted for trinomial and higher order trees.The economic concept of utility also enters the stage in the following equations.Understand that, for example, UH is the utility value in today’s dollars attachedto one dollar received in the up state.QH pUH(47)QL (1 p)UL(48)This leads to the following result.p pUHQH pUH (1 p)ULQH QL6(49)

3Continuous-Time FinanceMore specifically, this section is going to cover Brownian motions, stochasticcalculus and the lognormality of stock prices and introduce the Black-Scholesequation.3.1Standard Brownian MotionThe important properties of an SBM are as follows. One, Z(t) N(0, t). Two,{Z(t)} has independent increments. And three, {Z(t)} has stationary increments such that Z(t s) Z(t) N(0, s). Also useful is the fact that given aZ(u) : 0 u t, Z(t s) N(Z(t), s).3.2Arithmetic Brownian MotionWe define X(t) to be an arithmetic Brownian motion with drift coefficient µand volatility σ if X(t) µt σZ(t). Note that an arithmetic Brownian motionwith µ 0 is called a driftless ABM. Finally, X(t) N(µt, σ 2 t).3.3Geometric Brownian MotionArithmetic Brownian motions can be zero, though, and have a mean and variance that don’t depend on the level of stock making them a poor model for stockprices. To solve these problems we consider a geometric Brownian motion.Y (t) Y (0)eX(t) Y (0)e[µt σZ(t)](50)The following equations can be used to find the moments of a GBM. In equation51, let U be any normal random variable.1E(ekU ) ekE(U ) 2 k2V ar(U )(51)So specifically for geometric Brownian motions1E[Y k (t)] Y k (0)e(kµ 2 k2σ 2 )t(52)Also note, then, that Y (t) is lognormally distributed as followslnY (t) N(lnY (0) µt, σ 2 t)3.4(53)Ito’s LemmaFirst define X as a diffusion and present the following stochastic differentialequation.dX(t) a(t, X(t))dt b(t, X(t))dZ(t)Then for7(54)

Y (t) f (t, X(t))dt(55)We have1dY (t) ft (t, X(t)) fx (t, X(t))dX(t) fxx (t, X(t))[dX(t)]22(56)where[dX(t)]2 b2 (t, X(t))dt3.5(57)Stochastic IntegralsRecall the fundamental theorem of calculus.Zd ta(s, X(s))ds a(t, X(t))dt 0The rule for stochastic integrals looks very similar.Z tdb(s, X(s))dZ(s) b(t, X(t))dZ(t)(58)(59)03.6Solutions to Some Common SDEsFor arithmetic Brownian motions, we can say the followingdY (t) αdt σdZ(t)(60)Y (t) Y (0) αt σZ(t)(61)For geometric Brownian motions, there are several equivalent statements.dY (t) µY (t)dt σY (t)dZ(t) σ2dt σdZ(t)d[lnY (t)] µ 2 2µ σ2 t σZ(t)Y (t) Y (0)e(62)(63)(64)And for Ornstein-Uhlenback processes, which will become very useful when weget to interest rate models, we knowdY (t) λ[α Y (t)]dt σdZ(t)Z tY (t) α [Y (0) α]e λt σe λ(t s) dZ(s)08(65)(66)

3.7Brownian Motion VariationFor any function Y(t), the kth-order variation isZb dY (t) k(67)aFor standard Brownian motions we can say that the total variation is infinity,the quadratic variation is (b a) and higher-order variations are zero. And forarithmetic Brownian motions we know that the total variation is infinity, thequadratic variation is (b a)σ 2 and higher-order variations are zero.3.8Stock Prices as a GBMA stock price that pays constant dividends at a rate δ, such that the rate ofappreciation is α δ, follows the following stochastic differential equation andsolutiondS(t) (α δ)dt σdZ(t)S(t))t σZ(t)(69)σ2)dt σdZ(t)2(70)σ2)t, σ 2 t)2(71)S(t) S(0)e(α δ d[lnS(t)] (α δ σ22(68)Therefore we can say thatS(t) LN(lnS(0) (α δ 3.9Since Stock Prices are LognormalStock prices being lognormal gives many convenient formulas. First, we can determine the percentiles of different values of the future stock price and thereforeconfidence intervals as follows100p th percentile of S(t) S(0)e(α δ σ22 )t σ tN 1 (p)100(1 - β)% lognormal CI for S(t) is S(0)e(α δ σ22 )t zβ/2 σ t(72)(73)Similarly we know the probability that the future stock price will be above orbelow some value. We find this probability using the following two equations.P (S(t) K) N ( dˆ2 )ln S(0)K (α δ dˆ2 σ t9σ22 )t(74)(75)

And we can determine all the moments of the future stock price.1E[S k (t)] S k (0)ek(α δ)t 2 k(k 1)(σ2t)(76)Also very useful is the conditional expected price formulas.E[S(t) S(t) K] E[S(t)]N ( dˆ1 )N ( dˆ2 )(77)N (dˆ1 )N (dˆ2 )(78)E[S(t) S(t) K] E[S(t)]where dˆ2 is defined as before and dˆ1 isln S(0)K (α δ dˆ1 σ t3.10σ22 )t(79)The Sharpe Ratio and HedgingX(t) mdt sdZ(t) and a continuouslyFor any asset that has the dynamics dX(t)compounded dividend rate δ, the Sharpe ratio is defined asm δ r(80)sRecalling that for a stock, m α δ the Sharpe ratio of any asset written ona GBM isφ α r(81)σThe key thing to remember is that for any two assets with the same dynamicsthe Sharpe ratio must be the same. Using this fact we can derive the followinghedging formulas. We can hedge a position of long one unit of X by buying Nunits of Y and investing W dollars.φ N sX X(t)sY Y (t)W X(t) N Y (t)3.11(82)(83)The Black-Scholes EquationIf we look at any derivative with value V (S(t), t), use Itô’s lemma to finddV (S(t), t) , and put this into the Sharpe ratio formula we can derive the BlackScholes equation.1rV Vt (r δ)SVs σ 2 S 2 Vss210(84)

3.12Risk-Neutral Valuation and Power ContractsRecall that to switch from the real world to the risk-neutral world we exchangeα for r which leads to the following risk-neutral dynamics.dS(t) (r δ)dt σd[Z̃(t)]S(t)(85)Z̃(t) Z(t) φt(86)Using these risk-neutral equations we can show thatV (S(t), t) e r(T t) E [V (S(T ), T ) S(T )](87)This equation can be used to derive the following time-t price of a power contract. The payoff of a power contract is S a (T ) at time T and the price is1pFt,T(S a ) S a (t)e( r a(r δ) 2 a(a 1)σ112)(T t)(88)

4The Black-Scholes Formula4.1Binary OptionsBinary in this case is a fancy way of saying all or nothing.Binary OptionCash-or-Nothing CallCash-or-Nothing PutAsset-or-Nothing CallAsset-or-Nothing PutPricee r(T t) N (d2 )e r(T t) N ( d2 )S(t)e r(T t) N (d1 )S(t)e r(T t) N ( d1 )Where d1 and d2 are defined as before except that α is replaced with r.4.2The Black-Scholes FormulaWith the binary option formulas in hand it’s just a hop, skip, and a jump tothe Black Scholes Formulas.c(S(t), K, t) S(t)e δ(T t) N (d1 ) Ke r(T t) N (d2 ) r(T t)p(S(t), K, t) KeN ( d2 ) S(t)e δ(T t)N ( d1 )(89)(90)It’s helpful to note that, just like with the binomial formulas, for options oncurrencies we can replace δ with rf and for options on futures we can replaceδ with r. Another option is to use the more general prepaid forward version ofthe Black-Scholes option pricing formulas.4.3The Prepaid Forward VersionPPc(S(t), K, t) Ft,T(S)N (d1 ) Ft,T(K)N (d2 )(91)PPp(S(t), K, t) Ft,T(K)N ( d2 ) Ft,T(S)N ( d1 )(92)whereF P (S)1 2ln F t,TP (K) 2 σ (T t)t,T d1 σ T t(93)F P (S)1 2ln F t,TP (K) 2 σ (T t) t,T d1 σ T td2 σ T t(94)So far an option on any asset, just plug in the correct prepaid forwardprice. As a reminder, for currencies this is x(t)e rf (T t) and for futures this isF (t)e r(T t) . In other words, we are simply replacing δ with the appropriatevariable.12

4.4Greeks and Their UsesGreeks are what are used to measure the risk of a derivative. The followingthree are the most important. VS , Γ VSS S , θ Vt(95)The formula for the delta of a call and the delta of a put are useful to havememorized. c e δ(T t) N (d1 )(96) p e δ(T t) N ( d1 )(97)From the Black-Scholes equation we get the following relationship.1θ (r δ)S σ 2 S 2 Γ rV(98)2The following are the most important properties of the greeks. is positivefor calls and negative for puts. Γ is always positive. θ is negative for calls andputs generally, but can be positive for very in-the-money puts. Also, ν is alwayspositive while ψ is negative for a call and positive for a put and ρ is positive fora call and negative for a put. Note that the signs of these greeks is assumingwe are long.We can approximate the price of a new derivative if we know the price of aslightly different derivative using the Delta-Gamma approximation.1V (S ε, t) V (S, t) (S, t)ε Γ(S, t)ε2(99)2This assumes an instantaneous change in the underlying price. A more usefulformula is the Delta-Gamma-Theta Approximation.1V (S(t h), t h) V (S(t), t) (S(t), t)ε Γ(S(t), t)ε2 θ(S(t), t)h (100)2ε S(t h) S(t)(101)To find the Delta, Gamma, or Theta for a portfolio, as opposed to a singlesecurity, simply add up the greeks of the options in the portfolio.4.5ElasticityWe can write the expected return and the volatility of a derivative.mV Ωα (1 Ω)rsV Ωσ 13S σV(102)(103)

Ω is defined to be the elasticity. It’s interpretation is the same as the economicsinterpretation. Also, we can conclude that if Ω 1 than the option is riskierthan the underlying. Unlike the greeks, we cannot just sum the elasticities ofthe options in the portfolio to find the portfolio’s elasticity.Ω Σwi ViΩiP(104)Where wi is how many units of the ith derivative are in the portfolio.4.6Greeks From Trees e δhCu CdSu Sd (Su, h) (Sd, h)Su Sd11θ(S, 0) [C(Sud, 2h) ε (S, 0) ε2 Γ(S, 0) C(S, 0)]2h2Γ(Sh , h) 4.7(105)(106)(107)Delta-HedgingTo Delta-Hedge a portfolio we want to make it such that the delta of the portfoliois zero. So if we are long one unit of an option written on X we purchase Nunits of the underlying and invest W dollars.S σVsX X(t) V sY Y (t)σS(108)W X(t) N Y (t) V S (109)N It is also possible to hedge on any other greek, simply enter into other positionsuntil the value of the greek for the portfolio is zero. It is easy to determine thehedge profit and variance of the expected hedge profit. Γ 0 S(t)(1 σ h) S(t h) S(t)(1 σ h)(110)V ar(Profit) 4.81{[Γ(S(t), t)σ 2 S 2 (t)]h}22(111)Estimation of Volatilities & Appreciation RatesGiven a set of stock prices at different times, it is possible to estimate the volatilSi,ity of the stock using the following methodology. First, define ui as ln S(i 1)then find the average of the ui ’s, then the find the sample variance, and withthat estimate σ.ū 11 SnΣui lnnn S014(112)

s2u 1Σ(ui ū)2n 1suσ̂ h(113)(114)It’s also possible to calculate the expected rate of appreciation.ūσ̂ 2lnS(T ) lnS(0)σ̂ 2 δ δ (115)h2T2To test for normality, you can draw a normal probability plot. To do this arrangei 1the data in ascending order, give the ith order statistic the quantile q n 2 ,convert the q’s to z’s where z N 1 (q), and plot the data. The straighter theplot, the more normal the data is.α̂ 15

55.1Exotic Options & SimulationsAsian OptionsAsian options are options that are based on averages in place of either the priceor the strike. The average can be either an arithmetic average or a geometricaverage.A(T ) 1ΣS(ih)n1G(T ) [ΠS(ih)] n(116)(117)Then to price the option replace either the strike or the price with the appropriate path-dependent average, calculate the payoffs, and then discount them.5.2Barrier OptionsBarrier options are options that become activated (knocked-in) or deactivated(knocked-out) if the stock price passes above or below a pre-determined barrier.The price of these options can either be calculated with a binomial model or witha parity equation. Knock-in option Knock-out option Ordinary Option.5.3Compound OptionsCompound options are a pain in the butt to price and because of this theactuarial exam only asks that compound options be priced using the parityformula. Call on call Put on call Big Call - Ke rt . And similarly for puts.5.4Gap OptionsGap options are options whose strike for determining if the option is exercisedare different than the strike used to determine the payoff of the option. Theformulas don’t need to be written as long as the original Black-Scholes formulasare understood. The strike used to calculate the value of d1 and d2 is the strikethat determines if the option is exercised and the strike used to calculate theprice of the option is the strike that determines the payoff.5.5Exchange OptionsExchange options can be priced using the following parity and duality equations.1, t, T ]K1p[S(t), Q(t), K, t, T ] Kc[S(t), Q(t), , t, T ]Kc[S(t), Q(t), K, t, T ] Kp[S(t), Q(t),PPc[S(t), Q(t), K, t, T ] p[S(t), Q(t), K, t, T ] Ft,T(S) KFt,T(Q)16(118)(119)(120)

In the above equations, S is the underlying asset and Q is the strike asset.Exchange options can also be priced using a formula very similar to the prepaidforward version of the Black-Scholes pricing equation. For example, the priceof the option to get one unit of S in exchange for K units of Q can be writtenPPFt,T(S)N (d1 ) KFt,T(Q)N (d2 )d1 F P (S)(Q)t,Tln KFt,TP 12 σ 2 (T t) σ T t d2 d1 σ T t2σ σS2 2σQ 2ρσS σQ(121)(122)(123)(124)Exchange options can be trivially applied to options that depend on a maximumor minimum of S or Q. The parity relationship is useful to know.PPmax[S(T ), KQ(T )] min[S(T ), KQ(T )] Ft,T(S) KFt,T(Q)5.6(125)Monte-Carlo SimulationsThe way we’re going to simulate stocks is by taking advantage of the lognormality of stock prices. We’ll use the following method: start with iid uniform numbers u1 , u2 , ., un , calculate z’s where zi N 1 (u1 ), convert these to N (µ, σ 2 )random variables by letting r1 µ σz1 .To simulate a single stock price the following formula can be used.S(T ) S(0)e(α δ σ22 )T σ T z(126)Or simulate the the stock price at some time T given the stock price at a closerfuture time t.σ2S(T ) S(t)e(α δ 2 )(T t) σ[Z(T ) Z(t)](127)Monte-Carlo simulation goes like this: simulate stock prices, calculate the payoffof the option for each of those simulated prices, find the average payoff, and thendiscount the average payoff. The variance of the Monte-Carlo estimate, where2gi is the ith simulated payoff, can be calculated. The variance is e 2rT sn wheres2 5.71Σ[g(Si ) ḡ]2n 1(128)Variance ReductionThe first method of variance reduction is stratified sampling. To do this takethe iid uniform numbers and instead of taking them as is make sure the correctnumber go into each group. As an example, given 20 variables re-define the firstfive to be distributed between 0 and .25 and then next five to be distributedbetween .25 and .5 and so on. Then proceed as before.17

The other common technique involves finding an antithetic variate. To find thisestimate use the normal distribution numbers as before to find V1 and then flipthe signs on all of them and do the process again to find V2 . Calculate the2.antithetic variate to be V3 V1 V218

6Interest Rate Trees6.1Bonds and Interest RatesRecall that the price of an s-year zero isP (0, S) 1or e r(0,s)s[1 r(0, s)]s(129)After much algebra abuse we arrive at the following forward bond price formula.Ft,T [P (T, T s)] P (t, T s)P (t, T )(130)Also recall that we can calculate the non-continuous annualized rate.P (t, T )[1 rt (T, T s)] s P (t, T s)6.2(131)Caplets and CapsJust as with stock prices, trees for interest rates can be made. Interest ratecaplets and caps (which are the sums of the appropriate caplets) are most easilyunderstood by examples. The only thing to memorize is that the payoff of thecaplet is [r(t, T ) K] . To calculate the price of the caplet make a table withthe following columns.PathProbabilityTime t1 PayoffTime t2 PayoffContribution of PathThen for each path find the total contribution and sum them to find the price ofthe caplet. Remembe rto either discount the contributions or the total to findthe price.6.3Black-Derman-Toy ModelThe BDT model is a commonly usedinterest rate model with the following features: rd Rh and ru Rh e2σ1 h . And the next time step uses σ2 . Theinterest rates are all annual effective. Also, the way the rates are defined meansthat the following equality holds rruu rrud.udddThe yield volatility for period-3 is1y ln u2 h yd 11111P (ru , h, 3h) 1 ru 2 1 ruu2 1 rud1yu [P (ru , h, 3h)] 2h 119(132)(133)(134)

6.4The Black FormulaThe Black formula is similar to the prepaid forward version of the Black-Scholesformula except that it uses forward prices.c(S(0), K, T ) P (0, t)[F0,T (S)N (d1 ) KN (d2 )](135)p(S(0), K, T ) P (0, t)[KN ( d2 ) F0,T (S)N ( d1 )](136)d1 lnF0,T (S)K σ2 T2σ T d2 d1 σ T6.5(137)(138)The Black Model for Bond OptionsThere is a put-call parity equation for zeros.c(K, T ) p(K, T ) P (0, T s) KP (0, T )(139)We can further specify the Black Formula for Bond Options.c(S(0), K, T ) P (0, t)[F )N (d1 ) KN (d2 )](140)p(S(0), K, T ) P (0, t)[KN ( d2 ) F N ( d1 )](141)F F0,T [P (T, T s)](142)2Fln K σ T 2σ T d2 d1 σ T(143)d1 (144)1V ar[lnP (T, T s)]TAnd we can further specify the Black Formula for Caplets as well. N ( d2 )caplet(K, T, T s) P (0, T ) F N ( d1 )1 Kσ2 F P (0, T s)P (0, T )ln[F (1 K)] d1 σ T d2 d1 σ T20(145)(146)(147)σ2 T2(148)(149)

6.6Interest Rate ModelsWe’ve been assuming that the interest rates are known. Now we will modelthem instead and see how this affects our other models.dr(t) a(r(t))dt σ(r(t))dZ(t)RT r(s)dsP (t, T ) e t(150)(151)And the dynamics of a derivative on a non-dividend paying stock are nowdV (r(t), t) α(r(t), t, T )dt q(r(t), t, T )dZ(t)V (r(t), t)(152)And so the Sharpe Ratio isφ(r(t), t)α(r(t), t, T ) r(t)q(r(t), t, T )(153)And the following hedging rules apply. If you own one interest rate derivativeV1 buy N of V2N 6.7q1 (r(t), t, T1 )V1 (r(t), t)q2 (r(t), t, T2 )V2 (r(t), t)(154)The Term Structure EquationUsing a process very similar for how we dealt with stocks and the Black-Scholesequation we derive the following equations.[σ(r)]2Vrr rV2σ(r(t))Vr (r(t), t)q(r(t), t, T ) ΩσV (r(t), t)Vt [a(r) σ(r)φ(r, t)]Vr 6.8(155)(156)Risk Neutral ValuationAgain, the thought process is similar. The formulas are adjusted appropriately,but ”look” the same.dr(t) [a(r) σ(r)φ(r(t), t)]dt σ(r)dZ̃tZ tZ̃(t) Z(t) φ(r(s), s)ds(157)(158)0dZ̃(t) dZ(t) φ(r(t), t)dt21(159)

6.9Greeks for Interest Rate DerivativesThe greeks all work the same and lead to the same approximation formulas.Some bonds do follow what’s called an ”affine” structure. A bond has an affinestructure if it has a certain pricing formula.P (r, t, T ) A(t, T )e rB(t,T )(160)The greeks for interest rate derivatives under an affine structure are easy tocalculate.6.10 B(t, T )P (r, t, T )(161)Γ [B(t, T )]2 P (r, t, T )(162)q(r, t, T ) σ(r)B(t, T )(163)Rendleman-BartterThis is just a straightforward GBM.dr(t) ar(t)dt σr(t)dZ(t)r(t) r(0)e6.112( a σ2)t σZ(t)(164)(165)VasicekThis is an Ornstein-Uhlenback process.dr(t) a[b r(t)]dt σdZ(t)Z t atr(t) b [r(0) b]e σe a(t s) dZ(s)(166)(167)0This model has some nice formulas about its bond prices.P (r, t, T ) A(T t)e rB(T t)σ2σφ 2a2ar̄ b (169)1 e a(T t)aq(r, t, T ) σB(T t)B(T t) 0dr [a(b r)] σφ]dt σdZ̃ a(b r)dt σdZ̃b0 b y(r, t, T ) σφa1ln[P (r, t, T )]T t22(168)(170)(171)(172)(173)(174)

6.12Cox-Ingersoll-RossThe short rate process and other useful formulas and equations are below.pdr(t) a[b r(t)]dt σ r(t)dZ(t)(175)P (r, t, T ) A(T t)e rB(T t)qγ (α φ̄)2 2σ 2 q(r, t, T ) σ rB(T t)yield to maturity 232aba φ̄ γ(176)(177)(178)(179)

A prepaid forward contract is an agreement in which the buyer agrees at time t to pay the seller at time t and receive the asset at time T. FP t;T(S) S tor S PV (Dividends) or Se (T t) (2) 1.2 Put-Call Parity Call options give the owner the right, but not the obligation, to buy an asset at