HJM Model for Interest Rates and Credit

HJM Model for Interest Rates and Credit
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Published Date:26-07-2017
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Lecture 24 HJM Model for Interest Rates and Credit Denis Gorokhov (Executive Director, Morgan Stanley) Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Denis Gorokhov Denis Gorokhov is an Executive Director at Morgan Stanley. During his 9 years at the firm Mr. Gorokhov worked on pricing exotic derivatives with emphasis on credit, counterparty risk, asset-backed securities, inflation, and longevity. Mr. Gorokhov obtained a number of original analytic results for fixed income pricing problems which are implemented in Morgan Stanley risk management systems. He created a flexible modeling framework used for pricing over 400 different types of exotic derivatives including key transactions in the recent Morgan Stanley history: TARP transaction with US Treasury, purchase of 20% stake in Morgan Stanley by Mitsubishi UFJ Securities, and credit valuation adjustment with monoline insurer MBIA. Prior to joining Morgan Stanley Mr. Gorokhov worked in the fields of superconductivity and mesoscopic physics and authored 20 papers in leading physics journals. Mr. Gorokhov holds a Ph.D. degree in theoretical physics from ETH-Zurich and spent several years as a post-doctoral researcher at Harvard and Cornell. Only Mr. Gorokhov’s comments today are his own, and do not necessarily Source / Footnotes represent the views of Morgan Stanley or its affiliates, and are not a product of below this line Morgan Stanley Research. 2 2 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Introduction Guide 2.68 Subtitle Guide 2.64  HJM (Heath-Jarrow-Morton) model is a very general framework used for pricing interest rates and credit derivatives. Guide 1.95  Big banks trade hundreds, sometimes even thousands, of different types of Guide 1.64 derivatives and need to have a modeling/technological framework which can quickly accommodate new payoffs.  Compare this problem to that in physics. It is relatively straightforward to solve Schrodinger equation for the hydrogen atom and find energy levels. Guide 0.22 But what about energy spectra of complicated molecules or crystals? Physicists use advanced computational methods in this case, e.g. LDA (local density approximation).  Similarly, in the world of financial derivatives there is a very general framework, Monte Carlo simulation, which in principle can be used for pricing any financial contract.  The HJM model naturally fits into this concept. Only Only Source / Source / Footnotes Footnotes  Before discussing the HJM model it is very important to understand how the below this below this line line Monte Carlo method appears in finance. Stock options are the best Guide 2.80 example. 3 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Dynamic Hedging Guide 2.68 Subtitle Guide 2.64  Dealers are trying to match buy and sell orders from clients. It is not always possible and they have to hedge the residual positions. Guide 1.95 Guide  In the example below a dealer sold a call option on a stock, the loss may be 1.64 unlimited. To hedge the exposure the dealer takes a position in an underlying and adjusts it dynamically. Guide Payoff 0.22 Premium Stock price 0 Only Only Source / Source / Strike Footnotes Footnotes below this below this line line Guide 2.80 4 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Stock Price Dynamics Guide 2.68 Subtitle Guide 2.64 Guide 1.95 Guide 1.64 Guide 0.22 Only Only Source / Source / Footnotes Footnotes below this below this line line Guide S&P-500 index, 1990 -2012 2.80 5 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Lognormal Stochastic Process Guide 2.68 Subtitle Guide  Usually stock dynamics is assumed to be a sum of drift and diffusion and a 2.64 lognormal stochastic process is a good first approximation. Guide 1.95  For this process one can calculate the probability distribution function Guide 1.64 exactly. This is why the price of the call option in the Black-Scholes model can be calculated analytically.  The probability distribution function is Gaussian in the log coordinates. dSSdtSdB Guide t 0.22  1 S 1 1 2 2 t P(S ,S ;Tt) exp (ln ( )(Tt)) T t 2 S 2 2 (Tt) 2S  T t Only Only Source / Source / Footnotes Footnotes below this below this line line Guide 2.80 6 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Black-Scholes Formalism Guide 2.68 Subtitle Guide 2.64  Usually derivation of the Black-Scholes equation is based on the result from Guide stochastic calculus known as Ito’s lemma. 1.95 Guide  Roughly speaking, Ito’s lemma says that the derivative of a stochastic 1.64 function with respect to time has an additional deterministic term.  One can create a portfolio consisting of an option and a position in the underlying (hedge) which is riskless (thanks to Ito’s lemma). Guide 0.22 2 CC 1 C 2 2 dC dt dS S dt, CC(S,t) 2 tS 2S C CS, choose  S ddCdSr(CS)dt, portfolio is riskless 2 C 1 CC 2 2 Only Only  SrSrC 0, r interest rate Source / Source / 2 Footnotes Footnotes t 2SS below this below this line line Guide 2.80 7 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Black-Scholes Miracle Guide 2.68 Subtitle Guide 2.64  There are 2 striking facts about the Black-Scholes equation. Guide 1.95  First, the drift of the stock does not show up in the BS equation. It happens Guide because we hedge the option with the stock. 1.64  Second, the risk is eliminated completely, i.e. by holding a certain position in underlying we can fully replicate the option. This is closely related to the deterministic second-order term in Ito’s lemma. It is worth analyzing it in detail. Guide 0.22 2 C 1 CC 2 2  SrSrC 0 2 t 2SS Only Only Source / Source / Footnotes Footnotes below this below this line line Guide 2.80 8 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Ito’s Lemma under Microscope Guide 2.68 Subtitle Guide 2.64 ' ' SSS dtS dt ,  N(0,1) i1 i i i i i 2 CC 1 C 2 ' Guide C(S ,t )C(S ,t ) dtSSSS , i 0...N1, 1.64 i1 i1 i i i1 i i1 i 2 tS 2S 2 CC 1 C 2 ' 2 2 ' C(S ,t )C(S ,t ) dtSS S dt i1 i1 i i i1 i i 2 tS 2S 2 CC 1 C 2 2 2 ' Guide C(S ,t )C(S ,t ) dt (S ,t )SS S dt 0.22 N N 0 0 i i i1 i i 2 tS 2S i i • In the limit N-infinity the sum in the last term of the last equation becomes deterministic and we obtain Ito’s lemma dt dt’ Only Only Source / Source / Footnotes Footnotes below this below this line line Guide … i i-1 i+1 N-1 N 2.80 0 1 9 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Black-Scholes Miracle - 2 Guide 2.68 Subtitle Guide 2.64 N Guide 2 1.95  Exercise 1: Prove that  ,  N(0,1) is deterministic in the limit N-infinity.  i i Guide i1 1.64 • Exercise 2: Look up a “proof” of Ito’s lemma in Hull’s book (J.C. Hull, Options, th Futures, and Other Derivatives) and find an error (pages 232-233 in 5 edition).  We see that complete risk elimination in the Black-Scholes model is due to Guide the existence of 2 different time scales dt and dt’. One can derive the BS 0.22 equation in the limit dt-0, dt’-0, and dt/dt’-infinity.  From the business point of view this means that we hedge on a small time scale dt’ and the profit/loss noise is finite, however as long as the time interval increases to dt the profit/loss noise disappears Only Only Source / Source / Footnotes Footnotes below this below this line line Guide 2.80 10 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Solving Black-Scholes Equation Guide 2.68 Subtitle Guide 2.64 2 C 1 CC 2 2  SrSrC 0 Guide 2 1.95 t 2SS Guide ' 1.64 dS S 1 1 2 2 ' C(S,t) expr(Tt) exp (ln (r )(Tt)) Payoff(S )  ' 2  ' S 2 2 (Tt) 2S   BS equation is similar to the heat equation and can be solved using Guide 0.22 standard methods.  The Green function in the solution above is strikingly similar to the probability distribution function of the lognormal distribution shown above.  The 2 expressions differ only by the discount factor as well as the stock drift substituted by interest rate r. Only Only Source / Source / Footnotes Footnotes below this below this line line Guide 2.80 11 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Interpretation: Monte Carlo Simulation Concept Guide 2.68 Subtitle Guide 2.64 Stock price Guide 1.95 Guide 1.64 dSrSdtSdB t r(Tt) C(S,t)e EPayoff (S ) T Guide 0.22 time maturity  One needs to simulate different stock paths in the risk-neutral world, calculate the average of the payoff, and discount.  It turns out that this approach works not just for equity derivatives but can Only Only Source / Source / Footnotes Footnotes be generalized for interest and credit cases as well. below this below this line line Guide  Every security in the risk-neutral world grows on average with the risk free 2.80 rate. 12 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Interest Rates Derivatives: Basic Concepts Guide 2.68 Subtitle Guide 2.64  Businesses borrow money to finance their activity. Guide  As a compensation lenders charge borrowers certain interest. 1.95 Guide 1.64  Interest rates fluctuate with time and, similar to the equity case, there exists a market of derivatives linked to the level of interest rates.  Time value of money: 1 to be paid in 1 year form now is worth less than 1 paid in 2 years form now. For example, if 1- and 2-year interest rates are both Guide equal to 5%, then one needs to invest 1/(1+0.05)= 0.95 today to obtain 1 0.22 in 1 year form now and 1/(1+0.05)2=0.90 to obtain 1 in 2 years form now.  It is very convenient to describe time value of money using discount factors. discount factor 1.0 Only Only Source / Source / Footnotes Footnotes below this below this line line Guide 2.80 0 time 13 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Forward Rates Guide 2.68 Subtitle Guide 2.64  It is very convenient to express discount factors using forward rates. Guide  Mathematically, the relation between discount factors and forward rates can 1.95 Guide be expresses as 1.64 T d(t,T) exp( f (t,s)ds)  t Guide forward 0.22 rate time Only Only Source / Source / Footnotes Footnotes below this below this line line Guide 2.80 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Yield of 10-year US Treasury Note Guide 2.68 Subtitle Guide 2.64 Guide 1.95 Guide 1.64 Guide 0.22 Only Only  Interest rates are extremely low at present. Source / Source / Footnotes Footnotes below this below this line line Guide 2.80 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Libor Rates Guide 2.68 Subtitle Guide  LIBOR rate is a fundamental rate in the world of interest rate derivatives. This 2.64 rate is used an important benchmark in the world of lenders/borrowers. Very Guide often borrow rates are quoted as a spread over LIBOR. 1.95 Guide 1.64  Roughly speaking, USD LIBOR (London Interbank Exchange Offer) rate is the rate at which well rated banks lend US dollars to each other in London for a short term on unsecured basis.  There exist liquid 1M, 3M, 6M, and 12M LIBOR rates in different currencies. Guide 0.22  LIBOR swap is a fundamental interest rates derivative. The first counterparty makes periodic LIBOR payments, while the second one pays predetermined fixed rate. LIBOR Standard USD 3M-LIBOR swap Only Only payments Source / Source / Footnotes Footnotes below this below this line line time fixed Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Interest Rate Derivatives Guide 2.68 Subtitle Guide 2.64  Besides standard interest rate swaps there exist numerous types of interest rate derivatives. They are used for risk management of interest rate exposure. Guide 1.95 Guide  European swaption: An option to enter a forward starting LIBOR Swap. 1.64 Guide 0.22 time today expiry  Libor caps/floors: Essentially put/call options on LIBOR rate in the future.  Cancellable swaps: One of the parties has a right to cancel the LIBOR swap. Only Only Source / Source / Footnotes Footnotes below this below this line line Guide 2.80 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 LIBOR Swap Quotes Guide 2.68 Subtitle Guide 2.64 Guide 1.95 Guide 1.64 Guide 0.22 Only Only Source / Source / Footnotes Footnotes below this below this line line Guide 2.80 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Pricing LIBOR Swaps, Discount Curve Cooking Guide 2.68 Subtitle Guide 2.64  The LIBOR swap consists of 2 streams: fixed and floating ones. They can be priced if we know the discount factor curve d=d(t). Guide 1.95  The present value of a fixed payment C paid at time t is given by Guide 1.64 PVCd(t) fixed  There is a very neat result: present value of the float leg of a LIBOR swap plus a notional payment at the end is equal to the notional. The PV of the swap receiving fixed, paying float, and maturing at time T is Guide 0.22 PV (T) cNd(t )Nd(T)N  i i  But where from do we know the discount function d(t)? The swap market is very liquid and we know swap rates corresponding to different maturities. One can reverse engineer curve d(t) so that the observed swap rates are fair rates, i.e. for all swaps PV(T)=0. Only Only  This procedure is know as discount curve bootstrapping (“cooking”). Source / Source / Footnotes Footnotes below this below this line line  Note: Here, we do not take into account OIS discounting. Guide 2.80 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.Guide 4.69 Pricing Interest Rate Derivatives Guide 2.68 Subtitle Guide 2.64  In order to price a stock option one needs to know the stock value today as well as future stock dynamics. Guide 1.95  The Monte Carlo simulation framework allows to calculate the price. Guide 1.64  It turns out that the same framework can be used for pricing interest rate derivatives. However, there is a very important distinction: while stock evolution is a point-like process, in the interest rate world we interested in dynamics of a curve, i.e. a one-dimensional object. The problem is more complicated. Guide 0.22 STOCK OPTIONS IR OPTIONS INITIAL VALUE known not known, curve needs to be cooked DYNAMICS point-like object 0ne-dimensional Only Only object Source / Source / Footnotes Footnotes below this below this line line Guide 2.80 Developed for educational use at MIT and for publication through MIT OpenCourseware. No investment decisions should be made in reliance on this material.

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