Interest-Sensitive Financial Instruments

Interest-Sensitive Financial Instruments Valuing fixed cash flows Two basic rules: - Value additivity: Find the portfolio of zero-coupon bonds which replicates the cash flows of the security, the price of the security is the price of the replicating portfolio. - Valuation using interest rates: T CF With spot rates V 0 = t ( 1+ s t ) t t =1 T CF With forward rates V 0 = t t t =1 ( 1+ f n ) n =1 With short rates: The value of a security is the average value that is produced by simulating multiple interest rate paths and then discounting the cash flows back along each path Using yield-to-maturity (YTM): The value of a security can be reproduced by solving for its YTM, and then discounting all of its future cash flows at the YTM. Not really a method. Valuing interest-sensitive cash flows The only source of uncertainty for now are the level of interest rates; default risk, and liquidity risk. Projecting future interest rate paths and the corresponding cash flows along those paths, and then discounting back to get an average PVCF, is the method for valuing instruments whose cash flows are interest-rate sensitive. There may be differences in the prices obtained by using interest rate paths and from using the other traditional valuation approaches as: - Traditional approaches give the present value of expected cash flows, while - Interest rate path valuation gives the expected value of present values of cash flows.

The uniqueness of Yield To Maturity (YTM) Some observations: - YTM cannot be calculated unless you already know the price. Of course, valuation with spot rates is just as easy if they are known. - There is a unique YTM if a security has only positive future cash flows. The Relation between YTM and Spot Rates of Interest Dollar Duration = MD V MD = Modified duration V = PV based on the YTM of the portfolio Example: An investor purchases a 1-year zero for $1,000 that has a maturity value of $1,100 and a 10-year zero for $1,000 that has a maturity value of $3,707.. The YTM of this portfolio is found iteratively by solving the equation 000 = 1100 1+ y Now let us approximate the YTM: ( ) + 3707. y = 13.641% ( 1+ y) 10 The 1-year spot rate is 1100 1 = 10% and the 10-year spot rate is 1000 3707. 1000 1/10 1 = 14% Recall that Modified Duration = Macaulay Duration 1+ YTM For the 1-year, MD = 1 1.13641 = 0.879964097

For the 10-year, MD = 10 1.13641 = 8.799640975 For the 1-year, V = 1100 1.13641 = 0.967.96 For the 10-year, V = 3707. 1.13641 10 = 103.04 For the 1-year, DD = MD V = 0.879964097 967.96 = 851.77 For the 10-year, DD = MD V = 8.79964097 103.04 = 9081.58 The YTM estimate is produced like this: 851.77 851.77 + 9081.58 ( 0.10) 9081.58 851.77 + 9081.58 ( 0.14) = 13.657%

Interest rate contingent securities have payoffs that are dependent on the evolution of the term structure of interest rates. The key issue in valuing interest rate contingent securities is the model of the term structure of interest rates. This chapter considers one-factor models. Example At each node, short rate moves up to a factor of 1+ with probability p and down to a factor of 1 1+ ( ) p r t ( 1+ ) with probability 1 p. For one periods: r t 1 p r t 1+ The short rate is assumed to follow a Markovian Process, meaning that successive moves are independent (both the probability and the magnitude). This lets the tree to recombine under certain conditions, but it would not necessarily recombine if the probability and/or magnitude changed from one period to the next, which is allowed in this model. As the number of time periods in the model, N, becomes large, the distribution of terminal short rates approaches a lognormal distribution with parameters = 0, = N ln( 1+ )

Example: The initial short rate is 5%, = 0.5, and p = 0.5. Then the evolution of interest rates is as follows (all moves up and down have the same probability of 0.50): 6.5% 7.81% 5.00% 5.00% 4.00% 3.0% Let us compute the price of a one-year zero-coupon bond with par value $100. Since the initial short rate is 5%, its price is just 100 1.05 = 95.4 Now for the price a two-year zero-coupon bond with face value $100. Then the prices are computed recursively as follows: 0.5 100 + 1 0.5 1.065 ( ) 100 1.065 = 94.1 (time 1 up ) 0.5 100 + 1 0.5 1.04 ( ) 100 1.04 = 96.15 (time 1 down ) And finally, 0.5 94.1 1.05 + ( 1 0.5 ) 96.15 1.05 = 90.61 (time 0 value) You work right-to-left and calculate the price of the security recursively, discounting at the short rate at each node. Now as we know the prices of the one- and two-year zero-coupon bonds, we can calculate the spot and forward rates of interest. We already know that s 1 = f 0 = 5%. Then 90.61( 1 + s ) = 100 s = 5.054% And

( 1+ s ) = ( 1+ f 0 )( 1+ f 1 ) f 1 = 5.108% We can calculate the price of successive zero-coupon bonds and generate the entire implied spot and forward yield curves. Now we would like to price a security whose cash flows are interest rate contingent. Consider an interest rate cap that pays: Days in Period Max (Index ( -Strike,0) Notional 360 Let s use our previous interest rate model to price a two-year interest rate cap with a strike rat of 6% and a notional of $100. The interest rate tree is 6.5% 7.81% 5.00% 5.00% 4.00% 3.0% The cap payoffs are: 1.81 0.5 0.00 0.00 0.00 0.00 and the cap price is calculated as follows:

0.5 1.81 + 1 0.5 1.065 ( ) 0 + 0.5 = 1.10 (value in up node at time 1, 1.065 don t forget to add the cash flow at the current node) 0.5 0 + 1 0.5 1.04 ( ) 0 1.04 0.5 1.10 + 1 0.5 1.05 ( ) 0 1.05 + 0.00 = 0.00 (value in down node at time 1) = 0.5 (value at time 0) Arbitrage-free pricing Key point: In an arbitrage-free market, if two portfolios of securities produce the same payoffs in all scenarios, then they must have the same price. Suppose you wanted to price a one-year option that paid 0.5 if interest rates went up and 0.00 if rates go down, in the above given model. We will try to replicate the payoffs of this option using the risk-free security and the two-year zero-coupon bond. Note that a risk-free security goes from 1.00 to 1.05 in the first time period, while the two-year zero coupon bond, now worth 90.61, will be worth 94.1 in the up state, and 96.15 in the down state. We want to create a portfolio from risk-free securities and two-year zero-coupon bonds that will replicate the option s payoffs in all scenarios. If we can do this, then in a market free of arbitrage, the price of the option must equal the price of the replicating portfolio. Let D be the number of dollars invested in the risk-free security and be the number of two-year zero-coupon bonds bought. Then the replicating portfolio must satisfy the system of equations: 1.05D + 94.1 = 0.5 1.05D + 96.15 = 0 Solving, we get D = 11.773 and = - 0.1315, meaning that we should invest 11.773 in the risk-free security and short 0.1315 zero-coupon bonds. The price of the option must equal the price of the replicating portfolio: D + 90.61 = 11.773+ 90.61( 0.1315) = 0.119.

Some comments on this methodology: - Shorter time periods will produce a better price estimate, - The replicating strategy is dynamic, the portfolio will have to be rebalanced at each node to maintain replication. Restrictions on the term structure imposed by the absence of arbitrage For any interest rate contingent security S, = r s, s where r is the risk-free rate, is the security s expected return, and is s s its standard deviation. This quantity is called the market price of risk. It is a measure of how much additional expected return investors are demanding for a given level of risk. In an arbitrage-free market this value must be the same for all securities. If there are no free lunches, then no securities are better than others. Sometimes the market price of risk is written as ( r,t) to indicate that the market price of risk can change from one node to the next. Another important concept, the martingale probability,, is defined as: = S ( 1+ r) d, S d S u where S d is the price of the security if rates go down and S u is the price of the security if rates go up. For most fixed income securities S d > S u. The martingale is probability is the probability such that S = S + ( 1 )S u d 1+ r The only time the martingale probabilities will equal the true probabilities is when the market price of risk is zero. This is called a risk-neutral world. This is why is often called a risk-neutral probability. Just as the market price of risk, the martingale probability at a given node is the same for all securities. It is sometimes written as ( r,t) to reflect the fact that it may vary by node.

We arrive at these two separate approaches when valuing a security, both of which will should the same price: - Use the martingale probabilities and discount at the risk-free rate: S = S + ( 1 )S u d 1+ r - Use the true probabilities and discount at the risk-adjusted rate: S = ps + ( 1 p)s u d 1+ S Assuming S d > S u, we have = p + p( 1 p) or = p ( ). p 1 p Arbitrage-free models of interest rates These are two fundamental approaches to modeling the term structure of interest rates (single-factor): - Model short rate and get the yield (term structure) from short rate (calibrated to market data), - Arbitrage-free models which exactly fit the term structure of interest rates, and maybe the volatilities of discount bond yields. Generating the interest rate tree for a constant-volatility version of the extended Vasicek model (an arbitrage-free model with an exact fit to the term structure). This is a single-factor model, projecting only the continuously compounded yield on a discount bond maturing in t time periods. This model is trinomial and additive, since changes in the rate at each node are determined by adding a multiple of r and there are three branches emanating from each node (increases by a multiple of some amount = additive, goes to the upper, lower, or middle node = trinomial). This type of model is in contrast to the other models we have seen, which have all been multiplicative and binomial. The node i, j ( ) refers to the node at time t = i t (after i periods t have passed), with r = r 0 + j r (after the rate had a total of j shifts of size r). These are the shifts possible: ( i +1, j +1) ( i.j) ( i +1, j) ( i +1, j 1)

( i +1, j +1) ( i +1, j) ( i.j) ( i +1, j 1) ( i.j) ( i +1, j +1) ( i +1, j) ( i +1, j 1) Steps in the Vasicek model: Set initial short rate at (0,0) equal to r 0 (current market rate) For other nodes: 1. Compute value of ( n,t) from ( n t) = 1 t ( n + )R( n + ) + t + 1 t ln Q( n, j) e r j t +ar j t j ( ) is the value of a security that pays one dollar if node (n,j) is where Q n, j reached and nothing otherwise,. Compute = ( i t) ar i, j j (this is the drift from the node), 3. Choose the middle branch for which the calculated short rate is closes to r j + i, j t. This will tell you if your trinomial tree will resemble (A), (B), or (C) in the above picture. Noting that r = 3 t, compute from = i, j t + ( j k) r, where k is 1 if the upper branch is chosen, 0 if the middle branch is chose, and 1 if the lower branch is chosen 4. Compute probabilities for the three paths from: p 1 p p 3 ( i, j) = ( i, j) = 1 ( i, j) = t r + r t r r t r + r + r r

Note that probabilities must add up to one, and probability of moving up should be slightly higher than probability of moving down, as the yield curve usually slopes up. 5. Repeat for various nodes as necessary until entire model is developed Example in the book for the first node You are given that: = 0.014,a = 0.1, t = 1 So r 0 Maturity 1 3 4 5 Interest rate 10.0% 10.5% 11.0% 11.5% 11.5% = 10%. Furthermore: ( n t) = 1 t ( n + )R( n + ) + t When n = 0 and t = 1, ( 0) = R( ) + = ( 0.105) + 0.014 + ( a )r 0 = + 1 t ln Q( n, j) e r j t +ar j t j + ( 0.1 ) ( 0.10) = 0.001. Then = ( 0) ar = 0.01 ( 0.1 )( 0.10) = 0.0101 0,0 0 So r 0 + 0,0 t = 0.10 + ( 0.0101) ( 1) = 0.1101 r = 3 t = 0.014 3( 1) = 0.045 Now you must choose which of the ones below is r 0 + 0,0 t closest to: r 0 + r = 0.145 (the middle branch would move up) r 0 = 0.10 (the middle branch stays the same) r 0 r = 0.07575 (the middle branch moves down) Here, r 0 + 0,0 t = 0.1101, so this is closest to the second one and the middle branch will stay level. Now compute: = 0,0 t + ( j k) r = 0.0101( 1) + ( 0 0) r = 0.0101 Copyright 06 by Krzysztof Ostaszewski

t p 1 ( 0,0) = r + r + r = 0.014 ( 1) 0.0101 0.045 0.045 ( ) + + 0.0101 ( ) ( 0.045) = 0.46 p ( 0,0) = 1 t r ( ) 0.0101 = 0.493 ( ) ( 0.045) r = 1 0.014 1 0.045 t p 3 ( 0,0) = r + r + r = 0.014 ( 1) 0.0101 0.045 0.045 ( ) + 0.0101 ( ) ( 0.045) = 0.45 Therefore a summary of the first node is as follows: 1.4% with probability 0.46 10.00% 10.00% with probability 0.493 Q i, j otherwise. Therefore: 7.58% with probability 0.045 ( ) is the value of a bond that pays $1 if node ( i, j) is reached and zero Q( 1,1 ) = 1 0.46 = 0.4 1.10 Q( 1,0) = 1 0.493 = 0.45 1.10 Q( 1, 1) = 1 0.45 = 0.41 1.10 They can be found from the iterative relationship j + Q( i, j) = Q( i 1, j *) q( j*, j)e r j* t where q j*, j j * = j ( ) is the probability of moving from ( i 1,j *) to i, j ( ).

Valuing interest rate contingent claims in this model Securities whose cash flows all arrive in the terminal nodes can be valued as Value = Q n, j F n, j j ( ) ( ) ( ) if node ( n, j) is reached. This method can where the security pays F n, j also be extended to value interest rate contingent securities with multiple cash flows if the security can be decomposed into a portfolio of Europeanstyle derivative securities (like a cap that can be decomposed into caplets).

May 003 SOA Course 6 Examination, Problem No. A-4 (9 points) You are given the following with respect to an Extended Vasicek Trinomial Lattice Model: = 0.0, t = 1 year, R( 1) = 0.08, R( ) = 0.09, R( 3) = 0.10, a = 0.4. (a) Describe the key characteristics of this model. (b) Calculate the value of ( 0) using the Hull and White approximation. (c) Calculate the value of p ( 0,0). (d) Calculate the value of a one-year cap with a notional amount of 100 and a strike interest rate of 9.5%. Solution. (a) The Extended Vasicek Trinomial Lattice Model is an arbitrage-free model (it provides an exact fit to the current term structure of spot rates). It uses a single stochastic factor driving its evolution, r t, the continuously compounded yield on a one period zero-coupon bond. The model is additive (future rates are obtained from past rates by adding values generated by the driving stochastic process). Any short rate on the interest rate tree generated by the model can be written as r 0 + j r, where r 0 is the initial short rate (also the initial spot rate, as the first spot rate is for one period), and j is an integer (can be positive, negative or zero). This model was analyzed by Hull and White, who pointed out that the values of r and t must be chosen so that r is between 1 3 t and 3 t, and they chose to set r = 3 t, where is the one-period volatility parameter. In this model: ( i, j) is the node of the tree where t = i t and r = r 0 + j r, R( i) is the initial spot rate for maturity i t, r j = r 0 + j r, = ( i t) ar i, j j is the drift rate, the expected change of r at node ( i, j), p k ( i, j) is the probability of following branch k, where k = 1(up),(middle),3(down),

(b) In order to calculate ( 0), the drift function parameter, we use the information we have about the spot rates, and the fact that the model is supposed to be arbitrage-free. Consider the price of a unit two-period zero coupon bond. Given that the two-period spot rate is 9%, its price is: e 1 0.09 = e 0.18 = 0.835701. On the other hand, the price is the expected value of buying a one-period zero coupon bond, and then buying a one-period zero coupon bond at the end of the first period. Thus the price is: 0.835701 = e 0.08 E e r 1 ( r0 = 0.08). From this, we conclude that E e r 1 ( r0 = 0.08) = e 0.08 0.835701 = 0.9048374. At this point use the Hull and White approximation: E e r 1 ( r0 ) = e r 0 ( t 1 0,0 ( t ) ). We have = ( 0) ar, and a = 0.4. Thus: 0,0 0 0.9048374 = e 0.08 ( 1 ( ( 0) 0.4 0.08) 1 ). We conclude that: 1.03 ( 0) = 0.9048374 e 0.08 = 0.9801967, ( 0) = 1.03 0.9801967 = 0.0518033. We will later need = ( 0) ar = 0.0518033-0.4 0.08 = 0.0198033. 0,0 0 (c ) The probabilities p k ( i, j) are in general calculated from the system of equations: p 1 ( i, j) = t r + r t + r p ( i, j) = 1 r r t p 3 ( i, j) = r + r r where = i, j t + ( j k) r. The parameter k is the middle node of the next value of the short rate, with possible values of k being j + 1, j, j - 1, chosen so that the short rate reached in the middle node, r k = r 0 + k r is as close as possible to the current value of the short rate plus its expected change, in our case r 0 + 0,0 t=0.08 + 0.0198033 = 0.0998033. Here we have j = 0, and k can be 1, 0, or - 1. Recall that

r = 3 t = 0.0 3 = 0.034641 Thus: r 1 = 0.08 + 0.034641 = 0.114641 r 0 = 0.08 r 1 = 0.08 0.034641 = 0.054641 therefore, the most logical choice of k is 0. Given that, we have = 0,0 t + ( 0 0) r = 0.0198033. Now we arrive at the three equations for the probabilities: p 1 ( 0,0) = p ( 0,0) = 1 p 3 ( 0,0) = 0.0 0.034641 + 0.0198033 + 0.0198033 ( ) ( 0.034641) = 0.61590758. 0.0 0.0198033 ( 0.034641) 0.034641 0.0 0.034641 0.034641 0.034641 ( ) = 0.33985714. + 0.0198033 0.0198033 ( ) ( ) = 0.044358. 0.034641 (d) This one year cap will pay the interest in excess of 9.5% on $100 notional principal, as long as short rate exceeds 9.5%, and nothing otherwise. The short rate in this model can be: 11.4641% with probability 0.61590758, 8% with probability 0.33985714, 5.4641% with probability 0.044358. Since these are arbitrage-free probabilities, the price of the one-year cap will be the expected present value of its cash flows: 1.9641 with probability 0.61590758, a year from now, at r 0 = 8%, 0.0000 with probability 0.384095, a year from now, at r 0 = 8%. This is equal to: 1.9641 0.61590757 = 1.1. 1.08

The Black-Scholes model Two variations: - Binomial branching model, - Continuous-time hedging approach. Discrete-time model S = Initial stock price S u = ( 1+ z u )S (stock price after an up movement) S d = ( 1+ z d )S (stock price after a down movement) p = probability of an up movement 1 p = probability of a down movement r = risk-free rate R = 1 + r C S, ;K expiration, and strike price K. Main point in valuation: if a portfolio produces the same payoffs in all scenarios (i.e. if it is riskless), then it must earn the risk-free rate. As we showed previously, the call option price is just the future call option values discounted at the risk-free rate using risk-neutral probabilities. ( ) = price of a call option on the stock with current price S, time to Assume that the investor buys m shares of the stock and shorts one call option. We would like to determine m such that the portfolio is worth the same regardless of whether the stock moves up or down in the next period. This implies Stock ( up ) Option ( up ) = Stock ( down ) Option ( down ) or ms u C( S u, 1;K) = ms d C( S d, 1; K ) This means that: m = C( S, 1; K ) C( S, 1; K ) u d S u S d Using this value of m, the investor s portfolio will cost ms C( S, ;K) initially. If the stock price moves up, then the portfolio will be

worth: ( ) = C( S, 1; K ) C ( S, 1;K) u d ms u C S u, ;K From this we get ms u C( S u, ;K) = 1+ z d S u S d ( )C S u, 1; K ( ) 1 + z u S u C( S u, 1; K ) ( )C S d, 1; K ( ) z u z d Because this portfolio return the same in both scenarios we know that this must also be its value in the down scenario.the return on the portfolio must satisfy the following no-arbitrage condition: (Initial value of portfolio) ( 1 + r) = Value of portfolio in either scenario or ( ms C( S, ;K)( 1+ r) = 1+ z )C( S d u, 1; K) ( 1 + z u )C S d, 1; K z u z d We can solve this for the call price: 1 C( S, ;K) = C S 1+ r ( ( u, 1;K) + ( 1 ) C( S d, 1;K) ) ( ) ( ) where = r z d. The call price is just the expected present value of its z u z d cash flows, discounted at the risk-free rate using risk-neutral probabilities. Continuous-time model version gives the famous Black-Scholes formula..

The Black-Scholes model can be applied to value a Treasury Bill call option. Call option price is: F( B,t) = BN( d 1 ) Ke r( T t ) N( d ) ln B K + r + 1 ( T t) where d 1 = and d T t = d 1 T t. There are problems with using this model: - The volatility of the price of the T-Bill decreases as the maturity date approaches, while the Black-Scholes formula assumes constant volatility, - The Black-Scholes formula also assumes a constant risk-free rate, but this would imply that there is no volatility in the prices of risk-free securities (and we are valuing a call on one, so such call would be worthless!). The text suggests that use of the Vasicek model to correctly value a call option on a T-Bill. In that model, the call option price formula is: C( r,t,t C ) = P( r,t,t)n( b) KP( r,t,t C )N b p where P( r,t,t) = A( t,t)e B( t,t )r ( ) = 1 ( 1 e ( T t ) ) B t,t ( ) = e B t,t A t,t R( ) = b = 1 P ( ( ) ( T t )) R ( ) ( ) 4 B ( t,t ) P r,tt ln P( r,t,t C )K + = T T P ( C ) ( T C T) p ( )

In practice, finding a closed-form solution is not always possible, and instead numerical techniques are used. They are: - The finite difference method. The basic idea of this method is to approximate the derivatives in a partial differential equation with discrete difference equations. This will generate a system of linear equations that can be solved using the tools of linear algebra. Let us illustrate by valuing a zero-coupon bond paying $1 at maturity. If B r is the partial derivative of the bond price with respect to interest rates, B is the partial derivative of the bond price with respect to time, and B is the second partial derivative of the bond price with respect to interest rates, then the valuation equation is 1 ( r)b + ( r)b r rb B = 0 since the expected instantaneous return on the bond must be equal to the instantaneous risk-free rate. We would like to solve this equation but we are unable to exactly calculate the partial derivatives B, B r, B. Let B( r, ) be the price of this bond for an interest rate r and a remaining time to maturity. We can write some boundary equations: - B( r,0) = 1 at maturity, - ( 0)B r B ( r,0) = 0 by plugging in r = 0 in the valuation equation, - B(, ) = 0 as the interest rate goes to infinity. We work this by breaking up r and into increments, so that r is broken up into i increments of size b and is broken up into j increments of size k. This allows us to calculate values of B for small changes in r and. Then we can take the differences between the values we calculate and approximate the partial derivatives. We set up a grid: 0k 1k ( m 1)k mk 0h B 0,0 B 1,0 B m 1,0 B m,0 1h B 1,0 B 1,1.. B i, j. ( n 1)h B n 1,0 nh B n,0 B n,m Using derivative approximations, we get the following relationship that can be used to solve for the entire grid recursively: -

B i, j 1 = U i B i 1, j + V i B i, j + W i B i+1,j where U i = ( ih) k b ( ih) k h + ( ih) k b ( ih) k h + ( ih) k b V i = 1+ ihk + W i = and the values from the outside of the grid, from the boundary conditions, give us a starting point. This approach is called the implicit finite difference. Then, once the grid is calculated, note that B = B B i, j B i, j 1 k B = B r B B i +1, j i, j b B rr = B r B +1i, j B i, j + B i 1, j b And therefore we can substitute approximations for these derivatives into the valuation equation and solve for the price of the security. The trinomial lattice method can be applied to the finite difference method to produce the explicit finite difference method. It is similar to the implicit finite difference method discussed above, except that now we relate three values of f (derivative valued) at = j to one value of f at = j +1; it is therefore seen as an explicit computation of the f i, j values. The derivative approximations are as follows: f a f ax f i, j +1 f i, j k f i+1,j f i, j + f i 1, j b, f ax f f i +1, j i 1, j b

The transformed valuation equation is therefore where a i f i 1, j + b i f i, j + c i f i+1,j = f i, j +1 ( r,t) k a i = b r,t 1+ rk 1 b 1 = 1+ rk 1 ( r,t) k b c i = If you define ( r,t) k [ ( ) ( r,t) ( r,t) ] k b [ ( ) ( r,t) ( r,t) ] k b b r,t p i,i 1 = ( r,t) 1+ rk p i,i 1 = 1 ( r,t) k b k b r,t [ ( ) ( r,t) ( r,t) ] k b k p i,i +1 = ( r,t) b + [ ( r,t) ( r,t) ( r,t) ] k b Then you can rewrite the explicit finite difference method as: f i, j +1 = 1 1+ rk p i, j 1 f i 1,j + p i, j f i, j + p i,i +1 f i +1, j [ ] Notice that the p s sum up to one, and can therefore be viewed as trinomial probabilities (Vasicek s was a trinomial model). The equation just multiplies each of the values by the probability of moving to that node and then discounts for interest (standard approach).

Comparison of models for term structure Many single-factor term structure models are of the form dr = ( r,t)dt + r dz The two primary models: the Vasicek model for which Ingersoll, Ross ( CIR ) model for which = 0.5 = 0 and the Cox, Advantages and disadvantages of Vasicek s approach: Advantages: - Time-dependent function in drift term which is capable of providing an exact fit to the term structure, - Constant volatility is easy to implement. Disadvantage: - Admits the possibility of negative interest rates.

Continuous-time models are great in theory, but not in practice, and one cannot uses them directly without a closed-form solution. In practice, simulation models (a.k.a., Monte Carlo methods) are commonly used. Structure of simulation models We will consider a general two-factor model of the short rate and volatility. The assumed processes for these factors are: dr = ( u r)dt + vdz 1 dv = ( v)dt + vdz Z 1 and Z are Weiner processes with zero mean and variance dt, is the long-term average of the short rate r, and is the long-term average of the variance v. The instantaneous variance of r is v, and the instantaneous variance of v is v. These can be converted from continuous-time models to discrete-time models, as follows: r 1 r t 1 = ( r t 1 ) t + v t 1 1 t v t v t 1 = ( v t 1 ) t + v t 1 t The s are random samples from a bivariate normal distribution that can be generated as follows: get random values and from a standard 1 normal distribution N(0,1), then compute and as: 1 1 = 1, = 1 + 1 Steps in the simulation model: - Generate N individual univariate normal samples, and use them to create N joint samples as described above, - Use random samples to generate N economic scenarios, - Model cash flows along each scenario (may be path-dependent), - Discount cash flows by the short rate paths that gave rise to them, - Compute the average present value.

Advantages of simulation: - More efficient than lattice methods if modeling more than one factor, - Handles path-dependent cash flows, - Allows for more realistic jumps in interest rates (lattice tend to produce patterns of similar jumps), - Eliminates concerns about nodes recombining. Disadvantages of simulation: - Difficult for exact pricing of American options since would need an entire new set of simulations at each node. Simulation can be done with lattices Lattice methods can price American-style options, but simulation methods are needed to incorporate path-dependency. Many securities (MBSs) require both elements for accurate valuation. Simulation through lattices combines these two methods, enabling the valuation of MBSs and other securities that require both path-dependency and the accurate valuation of American-style options. Steps in simulating through a lattice: - Initialize lattice parameters and calibrate to observed prices, - Simulate by using a random number generator to determine the move at each node. - Cash flows can now be assigned to the nodes accurately, - These cash flows are discounted at the sequence of short rates which gave rise to them, - Average present value = price of the security. Implementation issues - How many paths? There is a tradeoff between accuracy and speed. Without a sound variance reduction technique, at least 1,000 paths are recommended. - How many periods? How long? Most simulations use monthly time periods, since it has been shown to produce accurate pricing and many securities pay cash flows monthly. Quarterly time periods may be used for securities and liabilities with very long maturities (30 years or more). Since early cash flows typically matter more, can use fine time intervals in the beginning and larger time intervals in the end.

Number of factors and parameter estimation An increase in factors leads to an increase in the number of parameters that must be estimated. Many models that are correctly calibrated initially to sample market securities cannot reliably price out-of-sample securities because of to the instability of the price relationships created by the presence of too many factors. The use of orthogonal (statistically independent) factors helps, but this is not easy to create. Credit, liquidity, and prepayment risk - Prepayments are usually programmed as a function of interest rates levels and their evolution (logistic function, or art tangent are common), - OAS is computed and added to the spot rate at all nodes to compensate for cash flow uncertainty and lack of liquidity. Some analysts adjust downwards the volatility when calculating the OAS, but this distorts the model. Better approach: adjust the promised cash flows to reflect expected credit losses. Initializing the parameters Models are generally calibrated to Treasuries and maybe other securities, e.g., a series of caps (various maturities). These securities are chosen since they are very sensitive to interest rate volatility over time. Options on Treasury futures are used to produce estimates of short-term volatility parameters. Alternative valuation techniques - Low-discrepancy methods. Monte Carlo methods have a tendency to cluster scenarios, leaving gaps. Low-discrepancy methods attempt to produce scenarios in a uniform fashion across the sample space, eliminating clusters and leading to more rapid convergence. - Multivariate Density Estimation (MDE). Non-parametric, model-free approach. Develops multivariate densities for security prices as functions of factors such as interest rate levels, the slope of the term structure, etc. It requires substantial data.

Fair valuation of liabilities can be performed using many of these techniques. Imperfect information and suboptimal exercise of policy options suggest that valuation of bank, pension, and insurance liabilities should proceed like the valuation of MBS securities. Liquidity risk should be incorporated in the OAS pricing. Defaults should not be factored into the valuation of liabilities that are guaranteed by the government or state insolvency programs. Two well-chosen factors should be sufficient to value most financial institution liabilities. Simulation is probably a better approach than a lattice method since: - Pricing of American options is becoming possible in simulation through new innovations, - Most financial intermediaries liabilities are non-traded and illiquid, - Parties rarely exercise options efficiently, - Path-dependency of cash flows is important. Concluding remarks: Basic principles of valuing assets and liabilities: - Behavior of interest rates and other economic factors is modeled, - Cash flows associated with assets/liabilities are mapped over time, across economic states/paths, - Adjusted or unadjusted cash flows are discounted back using the sequence of short rates that gave rise to them, - Average PV = market value.