WIXLIB

I.4.b – Deposits and LoansWhen a corporate client has an excess of treasury and won’tneed it until a certain period of time, it can make a simpledeposit to the bank. Then bank receives 1M to thecustomer and pays him the LIBOR rate plus a spread. Tomake the spread bigger, the client can still sell a product tothe client.In this case the product to be sold to the bank would be acap on LIBOR, which price would be embedded in thespread the bank add to LIBOR. The corporate client wouldhence receive bigger interests on his deposit, but if LIBORtends to be bigger than the cap’s strike, the bank would onlypay him interest based on the strike level and not LIBOR.The client can also buy a floor from the bank, which wouldmake the spread lower, but would protect him from to lowLIBOR rates.I.4.c – Extendable SwapsA corporate client could also need to enter a receivingswap. Imagine that a client wants to enter a 3 yearsreceiving swap, and also thinks that the rates will increasein 3 years. He could enter a 3 year swap with the bank andsell it a receiver swaption expiring in 3 years, offering thebank the opportunity to enter a payer swap at the samefixed rate than the one it enters today against the client. Tomatch the price of the swaption, the fixed rate received bythe client from the bank would be higher.If the client’s view of rates increasing in 3 years werecorrect, then the bank would not exercise the payerswaption., and he would have receive a higher rate than the3 year swap rate. In exchange, it offers the banks thepossibility to extend the swap for another two years if rateswould stay low. This kind of product might especially bepopular when rates are low, with a lot of uncertainty on thefuture, giving the options (which basically are insuranceagainst the future) more value.I.5 – Discounting and Credit RiskWe see that to discount cashflows with respect to time , wemultiply them by bond prices (which we also call discountfactors). This is due to the no-arbitrage assumption. Inreality, this is not true : markets do a distiction betwen therates on which the option is, and the rates which will be15

used to discount the cashflows. To make the the thesisreadible, we will consider that this is not the case here. Buthere are some reasons about the differences.The discount curve might vary with respect to products :for example, GBP european swaptions are discounted onSONIA (« Sterling OverNight Index Average »), but theinterests are paid on LIBOR (« London Interbank OfferedRate ») ; on the opposit, GBP bermudean swaptions keepsbeing discounting on LIBOR.The discount curve might also change as the clientchange :we can include the risk of credit in the discountingcurve, by adding a spread related to his default risk. Forexample, a top tier bank with good credit rating and a majorcompany with a high default risk would not pay the sameprice for a similar product. This is commonly taken into thediscounting method.Moreover, it is more and more common to use a collateralsto reduce the risk of default between two counterparties.Let us take another example. Bank A sells a swaptionexpiring in 10 years to Bank B. Bank A will receive 10,000,000 from bank B, and bank B will handle an optionwhich allows him to receive money from bank B in 10years. Let us suppose Bank A goes bankrupt : it won’t beable to honor the contract in 10 years. Hence the optiondoes not exist anymore and bank B has lost 10 millions. Toavoid these problems, bank A can post a collateral : It willmake a deposit in bank B of the previous 10 millions, andbank B will pay interest to A on these 10 millions. Hence, ifbank A defaults, the swaption dissapears, but bank B has itsmoney back. Bank A can also chose to post a collateral inEUR or in SEK instead of GBP, hence B will have to payinterests with respect to these currencies. The swaptionprice should be discounted at the rates corresponding tothe currency of the collateral ! That is why we see more andmore products based on a market but discounted withrespect to another market. The discounting model itself isnowadays complicated enough to be a complete subject ofanother thesis.16

II – The Hull-White Short Rate ModelsII.1 – The One-Factor Hull White ModelWe assume that our short rate follows the dynamics :, , ,Z with Z a Wiener process, and a deterministicfunction of time.This is a more general dynamics than the Vasicek model :, , ,Z with a positive constant. The solution of the equation ofthe previous SDE is a a a b a ,Z c 1 1 1The short rate has a normal distribution, with meanand variance " a 1 1 a d Y 1 Ya 2we see that when , we have" d Y2and the distribution of tends to h Ga , Ya H.i jkFrom the expression of " r t and Var r t , we see thatthe bigger the value of a, the « faster » r t tends to its limitdistribution. We call a the mean reversion : it defines howthe short rates dynamics tend to a limit mean.Negatives RatesSince the Hull-White model implies that the short rate has anormal distribution, this short rate could technically take17

every value of ℝ, and a fortiori negatives values. In fact, wecan even compute the probability of it relatively easily: 0 8pd q " 09 sq r" upd twith q a random variable such that q h 0,1 . Hence wehave: 0 w s r" upd tWhat would happen if the short rate were near 0? Whenrates on the market are very low, volatility tends to be alsovery low. In the One-Factor Hull-White model, this wouldbe equivalent to have a bigger mean reversion and asmaller \theta(t). From the expression of r(t), we see thatthe bigger the mean reversion, the lower the variance orr(t).Note: One would think that negative rates would beimpossible in real life ; however, we have seen recently thatbanks trading CHF (Swiss Franc) exchange a negativeovernight rate.II.2 - The Two-Factor Hull-White ModelII.2.a – The motivation for multiple factor modelsThe short rate and its distributional properties suffice tocharacterize the yield curve, as we have the relation :B , " xexp b , {and the fact that with all the bond prices, we canreconstruct the yield curve.However, a poor short rate model would lead to a poorrepresentation of the yield curve and its evolution. One-18

factor models such as the classic Hull-White gives 100%correlated LIBOR rates. We see that in reality this is not thecase, as we often see the yield curve steepening (short termLIBOR rates get lower, long term LIBOR rates get higher). Aone-factor model only allows us to parallel moves of theyield curve.II.2.b – DefinitionWe have seen that the One-Factor Hull-White model is amodel where the rates tends to reach a limit mean given by at a certain pace, given by the mean reversion . Thefunction is deterministic, but an intuitive way wouldbe to add it a stochastic component c , in fact to give itthe structure of the One-Factor Hull-White model, with amean reversion , lower than . Hence, after a certain time,our short rate model would tend to a one-factor stochasticfunction which would then itself tend to a deterministicfunction. We would then introduce a correlation factor }between the Wiener processes driving the dynamics of and c .This leads us to study the Two-Factor Hull-White model :, c , 2 ,Z2 ,c c , Y ,ZY with ,Z2 ,ZY }, .II.3 – Equivalence to the Two-Additive-Factor GaussianModelThe two factor Hull-White model is defined such that itassumes the short rate evolves in the risk-adjusted measureaccording to :, c , 2 ,q2 , 0 1,c c , Y ,qY , c 0 0with q2 , qY a two dimensional Brownian motion such that,q2 ,qY }̅ , , 1 , , , 2 , Y positive constants, and 1 }̅ 1. The deterministic function is chosen to fitthe current term structure of interest rates.We define the the new stochastic process c 19

where 2. a By differentiating , we get :, c , 2 ,q2 c , Y ,qY , G c c H , 2 ,q2 Y ,qY , , ,q with : 2 Y 2}̅ 2 ,q2 Y ,qY ,q Y 2 YYYMoreover, we define another new stochastic process : c The differentiation of gives us :c Y c , ,qY , , ,qY , with : Hence we can writewhere Y , , ,q , , ,qY 1 a b a , 1Hence we see that the Hull-White Two-Factor Model isequivalent to a « Two-Additive-Factor Gaussian Model ».This equivalence is going to be very useful to us. If it is moreeasy to interpret the different parameters of the Hull-White20

model and their influence on the price and volatilitystructures, the shape of the Gaussian model allows us toeasier calculations for the prices of bonds and derivatives.We have a perfect equivalence between the two models.If we write the HW2F as follow :, c , 2 ,q2 , 0 1,c c , Y ,qY , c 0 0such that is a deterministic function of time and,q2 ,qY }And the G2 as follow : . ,. . , ,Z2 , , ,ZY with a deterministic function of time such that 0 1, then we the coresponding equivalences betweeneach parameters of the two models : 2Y 2 Y 2}̅ Y 2 }̅ } YYY 1 a b a , 1or, put in the other way : 2 Y Y 2} Y } }̅ 2, , We will now proceed as follow : 1) We will calculate andcalibrate everything in the G2 framework, and then 2)21

analyze the equivalent parameters of the Two-Factor HullWhite model.22

III - Interest rates and product pricingsIII.1 – Zero-Coupon BondsWe denote by , the price at time of a zero-couponbond maturing at and with unit face value, so that , " # & ! %'ℱ )where " denotes the expectation under the risk-adjustedmeasure .We have seen that , % , is a gaussian process,hence it is entirely defined by its mean and variance.Bd , ℱ B d 8 b 1 a B ,Z2 c B b 1 B ,ZY c 9 Y B Y BYY Y b 1 a B ,c Y b 1 B ,cB b 1 a B 1 B ,c 2} 21 Ya B 3 Y Y a B 22 Y2 B 1 Y B 3 Y 2 2 a B 1 B 1 2} x a B 1{ and the mean" , ℱB " xb % , ℱ {B " xb . , ℱ {Because of the linearity and homogeneity of the expectationlinear application, and since is a linear process we canwrite :23

BB" , ℱ b " . ℱ , b " ℱ , B b , Hence by the definition of x, y and , the calculation ispretty straightforward, and gives :" , ℱ 1 a B 1 B . B b , We know that if q is a normal random variable with mean2 and variance Y , then " exp q exp G Y Y H . , is a normal random variable whose mean andvariance are now known. Hence2 D D& 2 D D& &a , 2 % % ¡ ,B YIn particular, we have 0, -2 % % ¡ 1,B YThis expression must fit the actual market, which meansthat for each we must have : 0, -2 % % ¡ 1,B YWhich leads us to -and for & % % % % 2 0, Y¡ 1,B & % % - % % & -& 0, 2 ¡ 1,B ¡ 1, Y 0, % % % %We can now rewrite the price of the zero coupon bond,fitting the current market, with our Hull-White parameters :24

, 0, 2 D a 0, D& 2 D D& 2 ¡ 1,B ¡ ,B ¡ 1, YWe introduce two new functions to shorten our nextequations : 0, 2 ¡ 1,B ¡ ,B ¡ 1, Y 0, 1 B * , , [ , Which gives us , [ , a, ,B , ,B ln , is linear in . and , which is the beauty ofthe Two-Additive-Factor Gaussian Model. This will be veryhelpful in both derivating the price of other products andnumerical implementation.III.2 – Zero-Coupon Bond OptionsThe price at time t of a European call option with maturity Tand strike K , written on a zero-coupon bond with unit facevalue and maturity isq*§ , , , X " # & % % , X ℱ )Under the T-forward (risk-adjusted) measure,processes . and follow the dynamics : Y 1 a B } 1 B { , ,Z2B Y, x 1 B } 1 a B { , ,ZYB ,. x . and with ,Z2B ,ZYB }, .The solution of these equations are for :25the

. . a % B , b a ,Z2 c % % B , b ,ZY c %with B , 8 Y } 9 1 a % Y Y Y a B a B Y% 2} B B a a % We get B , simply by replacing by , by , by and by in the expression of B , , thanks to thesimmetry of . and .Under the B measure, ℱ has a normal distribution,entirely defined by its mean and variance :" ℱ . a % % B , B , Y Y Ya % Y 1 Y % d ℱ Y 1 22 a % 1 2 The change of numeraire implies thatq*§ , , , X , " , X ℱWe know from previous calculations that , 0, 2 D ªDa 0, 2 D ªD 2 B B ¡ 1,« ¡ B,« ¡ 1,« Y We know that ln , conditional to ℱ is normallydistributed with mean26

0, 1 A ln d 0, d , d 0, 0, 21 a « B " . ℱ1 « B " ℱ and variancedAY YY 1 a « B 1 Ya B 2 YY 1 « B 1 Y B 2 1 a « B 1 2} « B 1 a B Using the formula 2 of the appendix, we get : , , , X 2 k , x Y¡ w 8 Xw A ln X {dA A ln X dAY9dASince A ,B is a martingale under B , we haveA ,« 2 k , " , ℱ Y¡ , which leads us to the identity A ln , 1 Y d , 2 Aand we can rewrite our price as , , , X , w ² , ln X , , Xw ²27dA1 dA ³2 , ln X , dA1 dA ³2

III.3 – Caps and FloorsWe have seen that a cap/floor is a thread of simple optionson LIBOR forward rates known as caplets and floorlets.We show that a caplet (or a floorlet) is actually equivalentto a put (or a call) option on a bond. The price of a caplet atime 0, with notional N and strike K which fixes at time 2and pays at time Y is given by : µ , 2 , Y , /, R " # &k %!/ Y 2 2 , Y R 'ℱ )where 2 , Y is the LIBOR rate, defined by : , 1 , , We can thus rewrite the caplet price as : µ , 2 , Y , /, R k" x & ! % / Y µ , 2 , Y , /, R 1 2 , Y 2 8 R9 ¶ℱ { Y 2 2 , Y 1 2 , Y 2 , Y ; R Y 2 ? ℱ ¹ /" · 2 , Y µ , 2 , Y , /, R & E ! % /" &E %! µ , 2 , Y , /, R /′" # &with G1 1 R Y 2 2 , Y H ºℱ E %! X 2 , Y 'ℱ )/ ¼ / 1 R Y 2 1X 1 R Y 2 We notice that the structure of the price of a caplet isidentical to a bond put’s with strike K and notional N’. Bythe same reasonning, we can show that a floorlet isequivalent to a call on a bond. If we know the arbitrage freeprice of an european option on a bond, we will deduce fromit the prices of caplets/floorlets, and by extension, the priceof caps/floors.28

III.4 – SwaptionsThe arbitrage free price at time 0 of a european arrearsettled payer swaption is given by numerically computingthe following one-dimensional integral :IJK½I¾ 0, , ¿, /, R, À /À 0, b 2 Á k GHY j 2Ä4·w Àℎ2 . M Æ5 . ÇC w ÀℎY . ¹ ,.5N2where À 1 (À 1) for a payer (receiver) swaption,ℎ2 . È Y 1 } } . È Y 1 } YℎY . ℎ2 . * , , 5 p1 } Æ5 . 75 [ , 5 a,B, C 1Y Y É5 . * , , 5 xÈ 1 } * , , 5 2} . È { . is the unique solution of the following equation4M 75 [ , 5 a,B, C ,B, C 15N2andÈ B 0, È B 0, 1 YaB21 Y B 2 } 1 a B } We derive this expression by taking the arbitrage free priceof the european swaption:29

Ê 0, , ¿, /, R, À 4 / 0, "B Ë·À ;1 M 75 , 5 ?¹ Ì / 0, b ·À ;15N2ℝk4 M 75 [ , 5 a,B, C ,B, C ?¹ ., , ,.5N2where f is the random vector . , , i.e., ., exp ; . È È È Y. È Y18GH 2} 9? Y 2 1 } Y2Ä 1

and see how well they fit the market. I.3 - Interest rates derivatives I.3.a - Swaps An interest rate swap is a contract in which two parties agree to exchange interest rate cash flows, based on a specified notional amount from a fixed rate, known as the swap rate to a floating rate, typically a LIBOR rate (or vice versa).