Ho and Lee Model of the Term Structure of Interest Rates Tahirivonizaka Fanirisoa Zazaravaka Rahantamialisoa ([email protected]) African Institute for Mathematical Sciences (AIMS) Supervised by: Dr. Raouf Ghomrasni AIMS, South Africa

20 May 2010 Submitted in partial fulfillment of a postgraduate diploma at AIMS

Abstract Modelling interest rate is of great importance in finance as it is useful to price derivatives and in risk management. In 1986, Ho and Lee presented the first arbitrage free model term structure of interest rate which relates the bond to yields curves. This model takes the term structure of interest rate as exogenously given. The aim of this essay is to derive the theoretical basis of the Ho and Lee model and investigate the suitability of the model for the market. Moreover, the pricing of interest rate derivatives under this model will be investigated. We present the implementation of the model and the algorithms behind it.

Famintinana Ny fanaovana modely ny zana-bola dia manandanja lehibe eo amin’ny fitatanam-bola satria ilaina izy amin’ny sampan’ny vidiny sy aminny fitantanana ny risika. Tamin’ny 1986, Ho sy Lee dia naneho ny fitsarana malalaka voalohany ny rafitra modelinny zana-bola izay mikasika ny biloka sy ny toetran’ny zana bola. Io modely io dia mandray ny rafitrin’ny zana-bola ho toy ny nomena exogenously. Ny tanjon’ny fanandramana dia ny hiderivena ny fototry ara-theorikan’ny modelin’i Ho sy Lee ary hanadihadiana (hitrandraka, hikaroka) ny mety fampiharana ilay modely ho anny tsena. Na izany aza, ny fanamboarana ny sampan’ny zana-bola eo ambanin’ilay modely dia ho voahadihady. Izahay dia maneho ny fampiharana an’ilay modely sy ny aligoritima izay mifehy azy.

Declaration I, the undersigned, hereby declare that the work contained in this essay is my original work, and that any work done by others or by myself previously has been acknowledged and referenced accordingly.

Tahirivonizaka Fanirisoa Zazaravaka Rahantamialisoa, 20 May 2010 i

Contents Abstract

i

Introduction

1

1 Bond Markets and The Arbitrage-free Pricing

2

1.1

1.2

Bond Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.1

Zero-Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.2

Zero-Coupon Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Arbitrage-Free Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.1

No-Arbitrage Opportunity Principle . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.2

First Fundamental Theorem of Asset Pricing . . . . . . . . . . . . . . . . . . . .

4

2 The Ho and Lee Model in Discrete Time 2.1

2.2

6

Model for Zero-Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1.1

One Period Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1.2

Multi-Period Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Models for Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1

Short-Rate of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2

Forward-Rate Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.3

Spot Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Pricing Interest Rate Derivative 3.1

3.2

23

Derivatives Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1.1

Bond Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.2

Interest Rate Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Numerical Approach of the Ho and Lee Model . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.1

Algorithm of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.2

Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

A The Code of the Program for the Ho and Lee Model

31

References

39

ii

Introduction The modern theory of option pricing began in 1973 with the now classical Black and Scholes models where the risk free asset is assumed to follow a geometric Brownian motion with constant volatility and drift. In this model, the risk free is deterministic with a constant interest rate. Black and Scholes obtained a closed form formula for call and put European options. However, in practice, the fact that the interest rate is constant is far being realistic as we know there are different economic perturbations, such as inflation and deflation, which might affect future prices of financial derivatives and make them uncertain. Moreover, one should take into account the existence of interest rate risk which plays a central role in the financial theory. Several approaches have been proposed to model the time evolution of stochastic interest rates. In this essay, we present the model introduced in 1986 by Thomas S.Y.Ho and Sang-bin Lee, published in the Journal of Finance [HL86] and considered as the first model of term structure of interest rate. The Ho and Lee model is constructed under the assumption of no arbitrage opportunity and in a binomial setting (i.e discrete time and discrete space). The main idea of Ho and Lee is to model the stochastic behaviour of the structure as a whole and it becomes the most commonly used type of dynamical stochastic model for interest rates. This model can be seen as an analogue of the Cox-Ross-Rubinstein (1979) model for stock options applied to the valuation of interest rate contingent claims. The main drawbacks of Ho and Lee model are: first the interest rates can take a negative value with positive probability and the volatility of interest rates is constant for all periods. In this work, we proceed as follow: Chapter 1 gives some important definitions related to interest rates and the general theory of pricing in mathematical financial modelling. In Chapter 2, we present the Ho and Lee model in its whole in the discrete time binomial approach. We introduce the unique no-arbitrage price for a bond which show by construction that, the model is arbitrage-free. To finish the Chapter, we derive the models for interest rate from those of bonds. In the last chapter, by means of the First Fundamental Theorem of Asset Pricing for interest rate derivatives we present the interest rate related derivative pricing and give a numerical simulation of the prices under the Ho and Lee model.

1

1. Bond Markets and The Arbitrage-free Pricing 1.1

Bond Markets

Since the dynamics and the behaviour of the term structure of interest rates depend on those of bonds, it is convenient to give a brief definition of bonds. The bond is defined as securitized form of a loan which generates the interest rate. A bond market is a financial market where we buy and sell debt securities or bonds which are traded before their maturity. Usually the bond market is referred to the government bond market where the bond is traded as a stock, because of its liquidity, lack of credit risk and, therefore, sensitivity to interest rates.

1.1.1

Zero-Coupon Bonds

A zero-coupon bond with maturity date T , is a claim or a contract which guarantees the holder 1 unity of cash (1 dollar or 1 Euro) to be paid at fixed time T . In general, we call it also T -bond and its price at time t is denoted by P (t, T ). By convention, to ensure the regularity and the consistency of the bond market, the price of the zerocoupon must satisfy the following assumptions: • There exists a constraint on the zero-coupon bond price, namely P (t, t) = 1 for all t. • For all 0 6 t 6 T , 0 < P (t, T ) 6 1. Definition 1.1. The term structure of a zero-coupon bond is defined as the graph of the function P (t, T ) versus T for a fixed value of t. At fixed time t, it determines the evolution of the prices for all zero-coupon bonds of all possible maturities. The term structure of the zero-coupon bond at time t is also known as zero-coupon bond price curve at t. Usually, the term structure of zero-coupon bond is a smooth graph. Remark 1.2. If the maturity T is fixed, the price P (t, T ) becomes a stochastic process because it determines all prices of the bond at different time t with fixed maturity T . Proposition 1.3. In the deterministic case, the zero-coupon bond price satisfies the relation of rollover, for all s 6 t 6 u: P (s, u) = P (s, t)P (t, u) In this relation, the right side is interpreted as the amount to invest at time s in order to have at t, the precise amount that needs to be invested at this time to have 1 unity at time u. This equality allows us to predict and estimate the value of P (t, u) at time s because at this time the value of P (s, t) and P (s, u) are known. Definition 1.4. A cash account is interpreted as the amount of cash, denoted by B(t), which is accumulated up to time t starting with one at time 0 and continually reinvesting at the short rate of interest r(t) such that it is given by the recurrent relation: B(t + 1) = B(t)er(t) where B(0) = 1. 2

Section 1.2. Arbitrage-Free Pricing

1.1.2

Page 3

Zero-Coupon Rates

Definition 1.5 (Forward Rates). The forward rate at time t is the rate that appears in a contract guaranteeing a riskless rate of interest over the future interval [T, S], where T, S are two fixed points in the future time satisfying t 6 T < S. Generally, the forward rate at time t between the period T and S is given by: log P (t, S) − log P (t, T ) f (t, T, S) = − S−T Definition 1.6 (Spot Rates). A spot rate at fixed time t is a family of yield curve or graph which is also derived from the zero-coupon bond price P (t, T ) with various maturity T and gives the average return of the bonds after elimination of the distorting effects of maturity. Then the spot rate at time t of T -bond is given by: log P (t, T ) y(t, T ) = − T −t Definition 1.7 (Short Rates). In general, we define the short rate as the rate of interest on a bank account or a cash account between a small variation of time, it can be change on a daily basis by the bank with no control on part of the bank account holder. The short rate is also defined as a special case of the forward rate at time t over the interval [t, t + ∆t], more precisely the rate of return on a zero-coupon bond maturing at the next time. Then the short rate is given by: r(t) = f (t, t, t + ∆t)

1.2 1.2.1

Arbitrage-Free Pricing No-Arbitrage Opportunity Principle

In our work, we consider that the financial market is a bond market constituted by zero-coupon bond associated and a bank account with interest rate r(t) which is a function of time. Before starting, it is convenient to introduce the notion contingent claim or derivative security. For a given a price process X = (Xt )t>0 , the derivative security on X is any financial instrument such that its value is contingent to X. More precisely, the contingent claim on X is a stochastic variable whose its value is a function of the price process X. Throughout the remainder, we assume that the state space Ω is a finite and discrete. Definition 1.8 (Trading strategies). A trading strategy is a stochastic process {(x(t), y(t)) ; t = 1, . . . , T } such that (x(t), y(t)) is interpreted as the number of units of assets held by the investors between the time t − 1 and t in the bond market. Sometimes, we call also the trading strategy as a portfolio strategy. Definition 1.9 (Self-financing). For a given two assets T -bond and a cash account from the market, a portfolio strategy is called self-financing if the following condition is satisfied for all t: x(t)B(t) + y(t)P (t, T ) = x(t + 1)B(t) + y(t + 1)P (t, T )

Section 1.2. Arbitrage-Free Pricing

Page 4

Proposition 1.10 (No-Arbitrage Principle). The market model is arbitrage free or viable if and only if no arbitrage possibility can appear on a financial market, which means, there does not exist any self-financing portfolio strategy h with initial value V0 (h) = 0 such that the final value VT (h) > 0 with strictly positive probability (P (VT (h) > 0) > 0) or P -almost surely. More precisely, no investor can start with nothing and make a profit without risk at the end. Proposition 1.11 (Law of One Price). In financial market, a security must have a single price, no matter how that security is created or else an arbitrage opportunity would exist. In other words, we say that for a security market model the law of one price is hold for two portfolio strategies h1 and h2 if for T > 0, VT (h1 ) = VT (h2 ) then V0 (h1 ) = V0 (h2 ). Otherwise an arbitrage opportunity can appear in the market, since from this definition the arbitrage free involves the law of one price equivalently if the law of one price is not hold then there is not an arbitrage free. Definition 1.12 (Replication strategy). A contingent claim X = (Xt ) is said to be replicable, if there exists a portfolio h such that if Vt (h) designs its initial value then the contingent claim might have the same value as the portfolio after one time interval. Which means at the next time t + ∆t: Vt+∆t (h) = Xt+∆t In general, the portfolio is created with assets of risk-free and of risky security. Definition 1.13 (Complete Market). A market is said to be complete if all claims on the market can be replicated.

1.2.2

First Fundamental Theorem of Asset Pricing

Definition 1.14 (Filtration). Given a non empty discrete and finite set Ω, a filtration F = (Ft )t>0 on Ω is a sequence of σ-algebra F0 , F1 , . . . , Ft , . . . , such that for all s < t, F0 ⊂ Fs ⊂ Ft . In finance, Ft represents the information given by the market up to time t and the more time increases the more informations we get about the market. Definition 1.15 (Martingale). Given a filtered probability space (Ω, F, (Ft ), P), the stochastic process X = (Xt )t>0 is said to be martingale with respect to (Ω, F, P) or simply F-martingale if and only if: 1. For all t > 0, Xt is Ft -measurable 2. For all t > 0, E(|Xt |) < +∞ 3. For all 0 6 t 6 s, E(Xs |Ft ) = Xt Definition 1.16 (Risk Neutral Probability Measure). Given a filtration (Ft ) on a state space Ω, and ˜ t = Xt /B(t), a probability a price process X = (Xt )t>0 , where the discounted prices are given by X measure Q is a risk neutral probability measure on Ω or an equivalent martingale measure if it satisfies: • For all ω ∈ Ω, we have: Q(ω) > 0 ˜ t+1 − X ˜ t |Ft ] = 0 • For t=0,. . . ,T -1, EQ [X

Section 1.2. Arbitrage-Free Pricing

Page 5

where EQ is called a risk-neutral expectation. Theorem 1.17 (First Fundamental Theorem of Asset Pricing). Given a bond price P (t, T ) and a finite probability space (Ω, F, P ) where F define a filtration on Ω. Bond prices evolve in way that is arbitrage free if and only if there exists a measure Q, equivalent to P, under which, for each T, the discount price process P˜ (t, T ) is a martingale for all t, T such that 0 6 t < T. The proof of the First Fundamental Theorem of Asset Pricing can be found in [EK98]. Corollary 1.18. If Q defines an equivalent martingale measure then Q is unique if and only if the market is complete. The reader may refer to [Pli97] for the proof of this corollary.

2. The Ho and Lee Model in Discrete Time In this chapter, we are going to describe the Ho and Lee Model by construction. In the first section, we will show the Ho and Lee model as the term structures of interest rate satisfying the no-arbitrage principle with its recombining binomial property. All the theory that we present here has been generalised based on the papers [Cai04], [HL86] and [Pli97].

2.1 2.1.1

Model for Zero-Coupon Bonds One Period Binomial Model

A one period model is characterised by its single trading period which means that we only have two trading dates. Consider P (t, T ) the price at time t of a zero-coupon bond with maturity time T , with t = 1, 2, 3, . . . , T and T = t, t + 1, . . . In the paper [HL86], Ho and Lee assume in their model that for one period market the zero-coupon bond price is defined as a stochastic price process which follows the binomial model. That is to say, if t designs the present time, at the time t + 1 the price can go up or down relative to an additional factors related to the economics problems, which is uncertain and no previsible. This shows the existence of a perturbation f actor which is a non deterministic function of the expiration time T − t, denoted by η(T − t), and defines a stochastic behaviour. For the Ho and Lee model the perturbation factor is defined by two functions u and d where u(t, T − t) represents the factor of the movement up whereas d(t, T − t) corresponds to the movement down between the period t and T . Depending on the movement of the prices in the future times, the function η(T − t) is given by:   u(t, T − t) if the price goes up η(T − t) =   d(t, T − t) if the price goes down With this perturbation factor, the relation of rollover in proposition 1.3 will be transformed into a stochastic recurrent: P (t, T ) P (t + 1, T ) = η(T − t − 1) P (t, t + 1) More precisely, the stochastic price of the zero-coupon bond can be expressed:  P (t, T )   u(t, T − t − 1) if the price goes up   P (t, t + 1)  P (t + 1, T ) =   P (t, T )   d(t, T − t − 1) if the price goes down P (t, t + 1)

(2.1)

Definition 2.1 (Risk-Free Rate of Interest). In discrete time, for all t, the risk-rate of interest between the time t and t + 1 is determined by: r(t) = − log P (t, t + 1) 6

Section 2.1. Model for Zero-Coupon Bonds

Page 7

Proposition 2.2. Knowing P (t, t + 1), we note that the cash account B(t) is linearly proportional to the return on the one year time zero-coupon bond P (t, t + 1), namely, for all t: B(t + 1) =

B(t) P (t, t + 1)

Therefore, the cash account is predictable as we know at time t its value at time t + 1. Corollary 2.3. From definition 2.1 and proposition 2.2, for a given time t, the cash account can be expressed uniquely by: " t−1 # t−1 Y X 1 B(t) = = exp r(k) P (k, k + 1) k=0

k=0

Proof. From proposition 2.2, we proceed by iteration starting at time t until the time 0 to get: B(t) =

B(t − 1) B(t − 2) B(0) = = ··· = P (t − 1, t) P (t − 1, t)P (t − 2, t − 1) P (t − 1, t)P (t − 2, t − 1) · · · P (0, 1)

Since we know that B(0) = 1, we obtain: B(t) =

t−1 Y k=0

1 P (k, k + 1)

Replacing P (k, k + 1) with its expression given in definition 2.1, it follows that for all t: " t−1 # t−1 Y X 1 B(t) = r(k) = exp P (k, k + 1) k=0

k=0

First it is necessary to precise that by convenience, we have to consider the following assumptions: • To ensure that the price of going up is greater than the price of going down and prices are positives, we assume, for all 0 6 s 6 t: u(t, s) > d(t, s) > 0 • There is no dependence upon prices or upon t of u(t, s) and d(t, s), and can then be write simply as u(s) and d(s). • From the expression of P (t, T ), it is necessary to assume u(0) = d(0) = 1 to ensure that P (t, t) = 1 for all t. After introduced the zero-coupon bond, let us see now for which condition the model has a property of arbitrage free if all prices change between time t and time t + 1. Theorem 2.4 (The No-Arbitrage Principle). Let (Ω, (Ft , Ft+1 ), P ) be a probability space, where (Ft , Ft+1 ) define a filtration. Consider the equivalent measure Q on the filtration, then the price process P (t, T ) is arbitrage free under Q if and only if the following properties are satisfied:

Section 2.1. Model for Zero-Coupon Bonds

Page 8

1. For all t and T > t + 1, we have: u(T − t) > 1 > d(T − t) 1 − d(T − t) u(T − t) − d(T − t) q defines the equivalent martingale measure Q which represent the probability of the price going up. (Derived from the First Fundamental Theorem of Asset Pricing)

2. There exists q , with 0 < q < 1, such that for all t and T > t + 1:

q=

Proof. → First let us prove that if the model does not contain any arbitrage opportunity then we have the two properties. 1. We are going to show by contradiction the first property which is, for all t and T > t + 1: u(T − t) > 1 > d(T − t) or equivalently, for all t and T > t + 2: u(T − t − 1) > 1 > d(T − t − 1) First suppose that u(T − t − 1) > d(T − t − 1) > 1, then: u(T − t − 1) d(T − t − 1) 1 > > P (t, t + 1) P (t, t + 1) P (t, t + 1) u(T − t − 1)P (t, T ) 1 d(T − t − 1)P (t, T ) 1 1 × > × > P (t, t + 1) P (t, T ) P (t, t + 1) P (t, T ) P (t, t + 1) Here two possibilities can happen, the price can either go up or down. But we notice that using the relation 2.1, for both cases the following inequality is always satisfied: P (t + 1, T ) 1 > P (t, T ) P (t, t + 1)

(2.2)

Moreover, let us construct at time t a portfolio with initial value V (t) = 0, such that: • Make a short position with on P (t, T ) with interest rate r(t). • Cash balance: V (t) = P (t, T ) − P (t, T ) = 0. At time t + 1 , we sell the bond at price P (t + 1, T ) and pay the loan with the interest rate. The cash balance of these transactions gives: V (t + 1) = P (t + 1, T ) − P (t, T )er(t) . According to definition 2.1, the value of this portfolio can be written in the form: V (t + 1) = P (t + 1, T ) −

P (t, T ) P (t, t + 1)

and we deduce from inequality 2.2 that V (t + 1) > 0. Thus realising an arbitrage opportunity.

Section 2.1. Model for Zero-Coupon Bonds

Page 9

Now consider the second case where we suppose that u(T − t − 1) < d(T − t − 1) < 1 and proceeding the same way as in the first case, we have: u(T − t − 1)P (t, T ) 1 d(T − t − 1)P (t, T ) 1 1 × < × < P (t, t + 1) P (t, T ) P (t, t + 1) P (t, T ) P (t, t + 1) and here also two possibilities can happen, the price can go up or go down. In both cases, we always have: P (t + 1, T ) 1 < P (t, T ) P (t, t + 1)

(2.3)

Using the same strategy as before, i.e we borrowing the amount P (t, T ) at time t and investing it with the interest rate r(t) such that the initial value of the portfolio is zero (V (t) = 0). At time t + 1, the value of the portfolio becomes: V (t + 1) = P (t, T )er(t) − P (t + 1, T ) =

P (t, T ) − P (t + 1, T ) P (t, t + 1)

According to inequality 2.3, we deduce that V (t + 1) > 0 and since the initial value of this portfolio is zero, an arbitrage profit can appear in the market. We have then a contradiction with the No-Arbitrage Principle, hence for all t and T > t + 2: u(T − t − 1) > 1 > d(T − t − 1). We conclude that to avoid an arbitrage opportunity, it is necessary to have for all t and T > t + 1: u(T − t) > 1 > d(T − t)

(2.4)

2. Let us use the strategy of replicating to show that there exists a probability q which defines the equivalent martingale measure Q, that is P r(P rice goes up) = q and P r(P rice goes down) = 1 − q under Q. In fact, we replicate P (t + 1, t + 2) with T-bond (with T > t + 2) and cash such that at time t we invest x units of cash and y units of P (t, T ) from the bond market, so the value of the portfolio is: xB(t) + yP (t, T ) Then at time t + 1 the value of the portfolio will be: xB(t + 1) + yP (t + 1, T ) According to the law of one price this portfolio must be equal to P (t + 1, t + 2), i.e:  P (t, t + 2) P (t, T )   xB(t + 1) + yu(T − t − 1) = u(1)   P (t, t + 1) P (t, t + 1)    P (t, T ) P (t, t + 2)   xB(t + 1) + yd(T − t − 1) = d(1) P (t, t + 1) P (t, t + 1) By definition, B(t) = B(t + 1)P (t, t + 1) with B(t) > 1 for all t, so we can write:  P (t, T ) P (t, t + 2)   x + yu(T − t − 1) = u(1)   B(t) B(t)    P (t, t + 2) P (t, T )   x + yd(T − t − 1) = d(1) B(t) B(t)

Section 2.1. Model for Zero-Coupon Bonds

Page 10

Hence we eliminate x first, and we get the value of y, namely: y =

[u(1) − d(1)]P (t, t + 2) [u(T − t − 1) − d(T − t − 1)]P (t, T )

and after using this expression we can compute x: x = d(1)

P (t, t + 2) P (t, T ) [d(1)u(T − t − 1) − u(1)d(T − t − 1)]P (t, t + 2) − yd(T − t − 1) = B(t) B(t) [u(T − t − 1) − d(T − t − 1)] B(t)

Let us calculate the initial value of the portfolio which is: [d(1)u(T − t − 1) − u(1)d(T − t − 1)]P (t, t + 2) [u(1) − d(1)]P (t, t + 2) + u(T − t − 1) − d(T − t − 1) u(T − t − 1) − d(T − t − 1)   1 − d(T − t − 1) u(T − t − 1) − 1 = u(1) P (t, t + 2) + d(1) u(T − t − 1) − d(T − t − 1) u(T − t − 1) − d(T − t − 1)

xB(t) + yP (t, T ) =

Again according to the law of one price and the principle of replicating, the initial value of the portfolio should be equal to P (t, t + 2). It follows that for all t and T > t + 2: u(1)

1 − d(T − t − 1) u(T − t − 1) − 1 + d(1) =1 u(T − t − 1) − d(T − t − 1) u(T − t − 1) − d(T − t − 1)

in other words, this previous recurrent relation is equivalent to for all t and T > t + 1: u(1)

1 − d(T − t) u(T − t) − 1 + d(1) =1 u(T − t) − d(T − t) u(T − t) − d(T − t)

(2.5)

Define for all T > t + 1, the function q(T − t) such that: q(T − t) =

1 − d(T − t) u(T − t) − d(T − t)

and 1 − q(T − t) =

u(T − t) − 1 u(T − t) − d(T − t)

(2.6)

such that q(T − t) satisfies relation 2.5, namely, for all t and T > t + 1: u(1)q(T − t) + d(1)(1 − q(T − t)) = 1

(2.7)

Thus, immediately we deduce that , for all t and T > t + 1, equation 2.7 has a root: q(T − t) =

but from inequality 2.4, we have for all T > t + 1:

1 − d(1) u(1) − d(1) u(T − t) − d(T − t) > 1 − d(T − t) > 0.

Since u(1) and q(1) does not depend on time, more precisely they are known from the initial condition, we can conclude that the function q(T − t) is a constant function compared to the variable T − t and satisfied 0 < q(T − t) < 1 for all t and T > t + 1. This prove us that we can find a risk neutral probability q, such that q = q(T − t) for all t and T > t + 1. In other words, we can define a quantity q which is unique and satisfy: q=

1 − d(1) u(1) − d(1)

with

0t+2 : P r(P rice goes up) = q(T − t − 1) = q and since P˜ (t + 1, T ) = P (t + 1, T )/B(t + 1) then, u(T − t − 1)P (t, T ) d(T − t − 1)P (t, T ) EQ [P˜ (t + 1, T )|Ft ] = q(T − t − 1) + (1 − q(T − t − 1)) B(t + 1)P (t, t + 1) B(t + 1)P (t, t + 1) From relation 2.6 and proposition 2.2, we get:   P (t, T ) (1 − d(T − t − 1))u(T − t − 1) + (u(T − t − 1) − 1)d(T − t − 1) EQ [P˜ (t + 1, T )|Ft ] = u(T − t − 1) − d(T − t − 1) B(t)   u(T − t − 1) − d(T − t − 1) ˜ = P (t, T ) u(T − t − 1) − d(T − t − 1)

Thus, we obtain for all t and T > t + 2:

EQ [P˜ (t + 1, T )|Ft ] = P˜ (t, T ).

Hence q defines an equivalent martingale measure Q under which P˜ (t, T ) is a martingale. → Conversely, let us show that if Properties 1. and 2. are satisfied then the model is arbitrage free. That is to say, if there exists an equivalent martingale measure Q defined by q such that q is a risk neutral probability which is defined in Property 2. then there is no arbitrage possibility between times t and t + 1 in one period binomial model. M

So let us take and construct a portfolio (aT ) associated with the zero-coupon bond price such that t+1 PM its value at time t is a P (t, T ) = 0. T =t+1 T Now, calculate the expectation of the portfolio under the equivalent martingale measure Q: " M #   M X X P (t + 1, T ) EQ aT P (t + 1, T ) Ft = aT B(t + 1)EQ Ft B(t + 1) T =t+1

T =t+1

= B(t + 1)

M X T =t+1

aT

P (t, T ) B(t)

=0 That is to say, there is no portfolio with zero initial value and a non-zero value in the future t + 1. Hence according to the Fundamental Theorem of Asset Pricing, we can affirm that any arbitrage opportunity can be seen in the market between the period t and t + 1. Remark 2.5. From definition of the function q(T − t) in 2.6 and the fact that q(T − t) = q, for all t and T > t + 1, then in general an arbitrage free for one period binomial model are characterized by: qu(T − t) + (1 − q)d(T − t) = 1

Section 2.1. Model for Zero-Coupon Bonds

Page 12

Remark 2.6. We have seen in theorem 2.4 that by the construction, the risk neutral probability q is unique because of the fact it is constant and does not depend of the time t and the maturity T . Therefore, according to corollary 1.18 we conclude that the market model is a complete market.

2.1.2

Multi-Period Binomial Model

Now, we are going to see the Ho and Lee model in a general form based on the theory of the single period model. Consider that the model evolves in a multi-period time with N + 1 trading dates (t = 0, δt, 2δt, . . . , T ), where N is the number of steps between the period 0 and T . In other words, for a fixed time step δt given by T = N δt such that 0 < δt 6 1 and for all running time t, we define t = nδt where n is the number of steps between the period 0 and t and this will be carried on throughout the rest of the work. Let (Ω, F, (Ft )t∈T , P ) be a probability space where (Ft )t∈T define a filtration, with F0 = {∅, Ω}, which is a sub-model describing how the information about the zero-coupon prices is revealed to the investors. The multi-period market model has two new features which is not shared with the one period model: the information sub-model and the stochastic process sub-models of prices. We assume that the perturbation factor is independent of the time t and the information Ft , so we can simplify the writing in the both cases (’up’ and ’down’), for all t and T > t + δt: u(t, T − t, Ft ) = u(T − t) and d(t, T − t, Ft ) = d(T − t) Definition 2.7. Define P (t, T, s) the price of zero coupon bond at time t with maturity T at the time-state (t, s), where s is the state of the world or the node point on the binomial lattice tree, by:  P (t − δt, T, s − 1)   u(T − t) if the price goes up   P (t − δt, t, s − 1)  P (t, T, s) =   P (t − δt, T, s)   d(T − t) if the price goes down P (t − δt, t, s) Throughout this section, we show that the Ho and Lee model satisfies both classical constraints of model of financial pricing which are: • There is no arbitrage opportunities for the model. • Independence of the path in the multiple period (Recombining Binomial Model), which means we have ”U p − down = Down − up”. That is to say, the model does not allow an arbitrage opportunity and additionally the binomial lattice tree is recombining. Our task is to look for the conditions which ensure that the Ho and Lee model follows these constraints.

Section 2.1. Model for Zero-Coupon Bonds

Page 13

• No-Arbitrage Opportunity: T

Theorem 2.8 (Equivalence of Multi-Period Model and One Period Model). If X = (Xt )t=0 is a price process, then the following propositions are equivalent: 1. X satisfies the no-arbitrage property. 2. For all 0 6 t < T , we have that the one period market (Xt , Xt+δt ) with respect to the filtration (Ft , Ft+δt ) satisfies the no-arbitrage property. The reader may refer to [DS05] for the proof of this theorem. Proposition 2.9. For the model of Ho and Lee, the arbitrage free of the zero-coupon bond price exists if and only if there exists q a risk neutral probability such that for all t and T > t + δt: qu(T − t) + (1 − q)d(T − t) = 1

(2.8)

where u(T − t) > 1 > d(T − t). Proof. → First prove that if the price of zero-coupon bond is arbitrage free then there exists a risk neutral probability which satisfies relation 2.8. In fact, according to theorem 2.8, we have directly each one period model (P (t, T ), P (t + δt, T )) satisfies the no-arbitrage property then from remark 2.5 and theorem 2.4 there exists a risk neutral probability q such that for all t and T > t + δt: qu(T − t) + (1 − q)d(T − t) = 1 → Conversely, suppose that there exists a risk neutral probability q satisfied the relation 2.8 and let us see why this relation involves the no-arbitrage opportunity for the zero-coupon bond price. Now let us check if Q designs the equivalent martingale measure defined n by q then for each one period the discounted o ˜ ˜ price P (t, T ) is martingale under Q, where P (t, T ) = P˜ (t, T, s) | t = nδt and 0 6 s 6 n . In other words, show that for all t and T > t + δt: EQ [P˜ (t + δt, T )|Ft ] = P˜ (t, T ) this means for all 0 6 s 6 n: P˜ (t, T, s) = q P˜ (t + δt, T, s + 1) + (1 − q)P˜ (t + δt, T, s) Indeed, the theorem 2.4 give us at any time-state (t, s) of the tree, the arbitrage free is defined, for all T > t, by: qu(T − t) + (1 − q)d(T − t) = 1 where (q, 1 − q) describes a risk neutral probability which defines the equivalent martingale measure Q. For a given time-state (t, s), when multiply the precedent relation by P (t, T, s) and after dividing by P (t, t + δt, s), we get: P (t, T, s) P (t, T, s) P (t, T, s) = qu(T − t − δt) + (1 − q)d(T − t − δt) P (t, t + δt, s) P (t, t + δt, s) P (t, t + δt, s)

(2.9)

Section 2.1. Model for Zero-Coupon Bonds but we have:

Page 14

 P (t, T, s)   P (t + δt, T, s + 1) = u(T − t − δt) P (t, t + δt, s) P (t, T, s)   P (t + δt, T, s) = d(T − t − δt) P (t, t + δt, s)

hence, relation 2.9 becomes, P (t, T, s) = qP (t + δt, T, s + 1) + (1 − q)P (t + δt, T, s) P (t, t + δt, s) Putting P (t, t + δt, s) in the right side and dividing by B(t, s): P (t, T, s) P (t, t + δt, s) = [qP (t + δt, T, s + 1) + (1 − q)P (t + δt, T, s)] B(t, s) B(t, s) P (t, T, s) 1 = [qP (t + δt, T, s + 1) + (1 − q)P (t + δt, T, s)] B(t, s) B(t + δt, s) Since we discount with respect the node s, that is, if X(t) is a price process then the discounted price associated to X(t + δt, k) with respect the node s is given, for all k > s, by: ˜ + δt, k) = X(t + δt, k) X(t B(t + δt, s) therefore, we will obtain for all 0 6 s 6 n: P˜ (t, T, s) = q P˜ (t + δt, T, s + 1) + (1 − q)P˜ (t + δt, T, s) This proves that P˜ (t + δt, T ) is martingale under the equivalent martingale measure Q or more precisely each one period market in the lattice tree satisfies the no arbitrage property. Since a binomial lattice tree is a set of many single period binomial model then we can apply theorem 2.8 to show that the model does not contain any arbitrage opportunity. Remark 2.10. We note that the no-arbitrage opportunity on the binomial model for zero-coupon bonds at time t is characterized, for all s and T > t + δt, by: P (t, T, s) = P (t, t + δt, s) [qP (t + δt, T, s + 1) + (1 − q)P (t + δt, T, s)] In general, this relation allows us to compute the value of all prices of the zero-coupon bond in the binomial lattice tree using the backward induction knowing all prices at the maturity T . To determine the perturbation factors u and d, we need to use the recombining binomial model assumption. • Recombining Binomial Model: Given a time-state (t, s), consider the sub-tree where the price evolve between t and t + 2δt :

Section 2.1. Model for Zero-Coupon Bonds

Page 15 P (t+2δt,T,s+2)

mm6 mmm m m mmm mmm P (t+δt,T,s+1) QQQ pp7 QQQ ppp QQQ p p QQQ p p p Q( p P (t,T,s) P (t+2δt,T,s+1) NNN mm6 NNN mmm m m NNN mmm NN' mmm P (t+δt,T,s)

QQQ QQQ QQQ QQQ Q(

P (t+2δt,T,s)

Figure 2.1: The zero-coupon bond price P(t,T,s) at any time-state (t,s) We require that the up-down sequence is the same as the down-up sequence (see figure 2.1). In fact, by definition we can say for all t and T > t + 2δt: P (t + δt, T, s + 1) = u(T − t − δt)

P (t, T, s + 1) P (t, T, s) and P (t + δt, T, s) = d(T − t − δt) P (t, t + δt, s) P (t, t + δt, s)

Now suppose that P (t + 2δt, T, s + 1) obtained from the up-down step is equal to P (t + 2δt, T, s + 1) obtained from the down-up step, that is: d(T − t − 2δt)

P (t + δt, T, s + 1) P (t + δt, T, s) = u(T − t − 2δt) P (t + δt, t + 2δt, s + 1) P (t + δt, t + 2δt, s)

that implies, d(T − t − 2δt)u(T − t − δt) u(T − t − 2δt)d(T − t − δt) = P (t + δt, t + 2δt, s + 1) P (t + δt, t + 2δt, s) u(T − t − δt) P (t + δt, t + 2δt, s + 1) u(T − t − 2δt) = d(T − t − δt) P (t + δt, t + 2δt, s) d(T − t − 2δt) In other words, for all t and T > t + δt, this precedent relation can be written as: u(T − t) P (t, t + δt, s + 1) u(T − t − δt) = d(T − t) P (t, t + δt, s) d(T − t − δt)

(2.10)

Let us call λ the ratio P (t, t + δt, s + 1)/P (t, t + δt, s), and introduce the notion of the volatility parameter which represents this ratio. We have: u(T − t) u(T − t − δt) =λ d(T − t) d(T − t − δt)

(2.11)

Definition 2.11 (Volatility). In the Ho and Lee model, the volatility parameter is defined as the relative rate at which the price of a security moves up and down such that it is determined by geometric relation, for all t > 0 and for all 0 6 s 6 n − 1: λ=

P (t, t + δt, s + 1) P (t, t + δt, s)

and

λ>1

Section 2.1. Model for Zero-Coupon Bonds

Page 16

Remark 2.12. We note that the volatility parameter is the same in all states of the binomial lattice, since when we compute the ratio P (t, t + δt, s + 1)/P (t, t + δt, s) using definition 2.7, we find, for all t and 0 6 s 6 n: P (t, t + δt, s + 1) u(δt) = P (t, t + δt, s) d(δt) where u(δt) and d(δt) are two constants determined from the initial condition. Therefore, this prove us that there exists a necessary and a sufficient condition on the perturbation factor so that the model will be path-independent or the binomial lattice tree is recombining. Which means that from relation 2.10, the binomial lattice tree is recombining if and only if the following condition is satisfied, for all t and T > t + δt: u(T − t)d(T − t − δt)d(δt) = d(T − t)u(T − t − δt)u(δt) Now under this condition, in our multi-period model, we consider that up to time t there are j up-steps and n − j down-steps. At time t the node j is characterized by the j number up-steps and localised by the time-state (t,j). Which means that the price is independent of the order of the up and down − steps but it should only depends on the number of up − steps or down − steps. Corollary 2.13. If the binomial lattice tree is recombining then we define, for all s, t, T with s 6 t and T > t + δt, the induction hypothesis: P (t, T, s + 1) u(T − t) = = λN −n P (t, T, s) d(T − t) Proof. First from relation 2.11, we have for all t and T > t + δt:

u(T − t) u(T − t − δt) =λ d(T − t) d(T − t − δt)

So by iteration following the value of t starting with δt until (N − n)δt, we will obtain: u(T − t) u(T − t − δt) u(T − t − 2δt) u(T − t − (N − n)δt) =λ = λ2 = · · · = λN −n = λN −n d(T − t) d(T − t − δt) d(T − t − 2δt) d(T − t − (N − n)δt) Then we deduce that this equality is equivalent to: P (t − δt, T, s) P (t − δt, t, s) = λN −n P (t − δt, T, s) d(T − t) P (t − δt, t, s)

u(T − t)

Hence, using definition 2.7 we get the result, for all t and T > t + δt: P (t, T, s + 1) = λN −n P (t, T, s)

Proposition 2.14. If (q,1-q) defines a risk neutral probability in the Ho and Lee model then the perturbations functions are uniquely determined, for all t and T > t + δt, as: d(T − t) =

1 1 − q + qλN −n

and

u(T − t) =

λN −n 1 − q + qλN −n

Section 2.1. Model for Zero-Coupon Bonds

Page 17

Proof. From corollary 2.13 and proposition 2.9, we get for all t and T > t + δt: qλN −n d(T − t) + (1 − q)d(T − t) = 1 then we deduce:

d(T − t) =

1 1 − q + qλN −n

and u(T − t) =

λN −n 1 − q + qλN −n

Theorem 2.15 (The uniqueness of the arbitrage free). The bond price in the Ho and Lee model is unique and his expression is given by: "n−1 # Y 1 − q + qλn−(k+1) P (0, T ) s(N −n) P (t, T, s) = λ 1 − q + qλN −(k+1) P (0, t) k=0 where the quantity P (0, T )/P (0, t) represents the forward price for the zero-coupon bond price at time t and with maturity T . Proof. Consider q as the risk neutral probability which defines the arbitrage free and λ the volatility parameter. According to corollary 2.13 and remark 2.10, there exist two recurrent relations for the zero-coupon bond price, that is: • For all t and T > t + δt, • For all 0 6 s 6 n,

P (t + δt, T, s + 1) = λN −n−1 P (t + δt, T, s)

P (t, T, s) = P (t, t + δt, s)[qP (t + δt, T, s + 1) + (1 − q)P (t + δt, T, s)]

When we combine the two relations, we have: P (t, T, s) = [qλN −(n+1) + (1 − q)]P (t + δt, T, s)P (t, t + δt, s) Since (q, 1 − q) and λ are independent of the time and the time-state, we can iterate and substitute one by one the bond price from the time t until the maturity T , namely: h i P (t, T, s) = 1 − q + qλN −(n+1) P (t + δt, T, s)P (t, t + δt, s) h ih i = 1 − q + qλN −(n+1) 1 − q + qλN −(n+2) P (t, t + δt, s)P (t + δt, t + 2δt, s)P (t + 2δt, T, s) i h i h = 1 − q + qλN −(n+1) · · · 1 − q + qλN −(n+3) P (t, t + δt, s) · · · P (t + 2δt, t + 3δt, s)P (t + 3δt, T, s) at the end, we get: h i h i P (t, T, s) = 1 − q + qλN −(n+1) · · · 1 − q + qλN −(N −1) P (t, t + δt, s) · · · P (T − 2δt, T − δt, s)P (T − δt, T, s) h i h ih i = 1 − q + qλN −(n+1) · · · 1 − q + qλ (1 − q + q) P (t, t + δt, s) · · · P (T − δt, T, s)P (T, T, s) h h i i = 1 − q + qλN −(n+1) · · · 1 − q + qλ P (t, t + δt, s) · · · P (T − δt, T, s) Therefore, we obtain the following relation: P (t, T, s) =

N −1 h Y

i 1 − q + qλN −(k+1) P (kδt, (k + 1)δt, s)

k=n

(2.12)

Section 2.2. Models for Interest Rates

Page 18

According to definition 2.11, we have for all k 6 s: P (kδt, (k+1)δt, s) = λP (kδt, (k+1)δt, s−1) = λ2 P (kδt, (k+1)δt, s−2) = · · · = λs P (kδt, (k+1)δt, 0) Then relation 2.12 becomes: P (t, T, s) = λs(N −n)

N −1 h Y

i 1 − q + qλN −(k+1) P (kδt, (k + 1)δt, 0)

(2.13)

k=n

From relation 2.13, let us compute and find the expression of the forward price:  QN −1  N −(k+1) P (kδt, (k + 1)δt, 0) 1 − q + qλ P (0, T ) = Qk=0  n−1  n−(k+1) P (kδt, (k + 1)δt, 0) P (0, t) k=0 1 − q + qλ It follows that: # −1 "Q  n−1  i N −(k+1) P (kδt, (k + 1)δt, 0) N Yh P (0, T ) k=0 1 − q + qλ = Qn−1  1 − q + qλN −(k+1) P (kδt, (k + 1)δt, 0)  n−(k+1) P (kδt, (k + 1)δt, 0) P (0, t) k=0 1 − q + qλ k=n # N −1 "n−1 h i N −(k+1) Y Y 1 − q + qλ = 1 − q + qλN −(k+1) P (kδt, (k + 1)δt, 0) n−(k+1) 1 − q + qλ k=n k=0 hence, "n−1 # N −1 i Yh P (0, T ) Y 1 − q + qλn−(k+1) N −(k+1) = 1 − q + qλ P (kδt, (k + 1)δt, 0) P (0, t) 1 − q + qλN −(k+1) k=0 k=n Using the relation 2.13, we conclude that the final expression of the zero-coupon bond price at any time-state (t, s) is given by: "n−1 # Y 1 − q + qλn−(k+1) P (0, T ) P (t, T, s) = λs(N −n) 1 − q + qλN −(k+1) P (0, t) k=0 This shows that the zero-coupon bond price is unique because the quantity q, λ, P (0, T ) and P (0, t) are given and fixed in initial time.

2.2

Models for Interest Rates

Knowing the behaviour of the zero-coupon bond price model following the value of the two parameters, let us see the dynamics of interest rate model using the relationship between the bond price and the interest rate. Since we have seen in the previous section that the bond price is parametrized by the volatility parameter and the risk neutral probability so necessary the yields curves must be characterized by the same parameter.

2.2.1

Short-Rate of Interest

At the beginning, we have introduced a basic notion of the risk-free rate of interest such that the value at the time-state (t, s) of the short-rate associated with the zero coupon bond is given, for all s 6 n, by the following relation: log P (t, t + δt, s) (2.14) r(t, s) = − δt

Section 2.2. Models for Interest Rates

2.2.2

Page 19

Forward-Rate Curves

In the binomial model the instantaneous forward rate with maturity T , contracted at time t, is defined, for all 0 6 s 6 n, by: f (t, T, T + δt, s) = −

log P (t, T + δt, s) − log P (t, T, s) δt

(2.15)

where at the initial time 0 there are N forward rates f (0, 0), f (0, δt), . . . , f (0, T − δt).

2.2.3

Spot Rates

We define also the spot rate at the time-state (t, s) with maturity T as the yield to maturity of the T -bond, that is for all s 6 n: log P (t, T, s) (2.16) y(t, T, s) = − T −t Proposition 2.16 (Uniqueness of the interest rate). In the Ho and Lee model if q designs the risk neutral probability and λ is the volatility parameter then for a given time-state (t, s), each yield curves with maturity T are uniquely determined: 1. The instantaneous forward rate is given by:   1 1 − q + qλN s log λ f (t, T, T + δt, s) = f (0, T, T + δt) + log − N −n δt 1 − q + qλ δt 2. The short rate of interest: r(t, s) = f (0, t, t + δt) +

s log λ 1 log [1 − q + qλn ] − δt δt

3. The compounded spot rate is given by: " # n−1 1 X 1 − q + qλN −(k+1) s log λ y(t, T, s) = f (0, t, T ) + log − n−(k+1) T −t δt 1 − q + qλ k=0 Proof.

1. From theorem 2.15, we have, for all s, t, T with s 6 n and T > t + δt: " #  n−(k+1) P (0, T + δt) Q  1 − q + qλ n−1  P (t, T + δt, s) = λs(N −n+1)  k=0  P (0, t) 1 − q + qλN −k   " #   n−(k+1)  Q  1 − q + qλ P (0, T ) n−1  s(N −n)  P (t, T, s) = λ k=0 N −(k+1) P (0, t) 1 − q + qλ

Section 2.2. Models for Interest Rates

Page 20

Let us compute the following ratio of these prices of T + δt-bond and T -bond: # "n−1 Y 1 − q + qλN −(k+1) P (0, T + δt) P (t, T + δt, s) = λs P (t, T, s) P (0, T ) 1 − q + qλN −k k=0 " # N −1 ) · · · (1 − q + qλN −(n+1) )(1 − q + qλN −n ) P (0, T + δt) (1 − q + qλ = λs P (0, T ) (1 − q + qλN )(1 − q + qλN −1 ) · · · (1 − q + qλN −(n+1) )   1 − q + qλN −n P (0, T + δt) = λs 1 − q + qλN P (0, T )

Thus, from relation 2.15 we deduce the unique expression of the instantaneous forward rate at any time-state (t, s):   1 − q + qλN s log λ 1 − f (t, T, T + δt, s) = f (0, T, T + δt) + log δt 1 − q + qλN −n δt 2. From theorem 2.15, we have also: "n−1 ! # Y 1 − q + qλn−(k+1) P (0, t + δt) log P (t, t + δt, s) = log λs P (0, t) 1 − q + qλn−k k=0     (1 − q + qλn−1 ) · · · (1 − q + qλ)(1 − q + q) P (0, t + δt) = log λs (1 − q + qλn )(1 − q + qλn−1 ) · · · (1 − q + qλ) P (0, t)   s λ P (0, t + δt) = log n 1 − q + qλ P (0, t)

Finally, from relation 2.14 we get the unique expression of the short rate at the time-state (t, s): r(t, s) = f (0, t, t + δt) +

1 s log λ log [1 − q + qλn ] − δt δt

3. According to theorem 2.15, we have after computing, for all s,t,T with s 6 n and T > t + δt: "n−1 # ! Y 1 − q + qλn−(k+1) log P (t, T, s) s log λ 1 1 P (0, T ) − =− + log + log T −t δt T −t T −t P (0, t) 1 − q + qλN −(k+1) k=0 Therefore, from relation 2.16 the value of the compounded sport rate is: " # n−1 1 X 1 − q + qλN −(k+1) s log λ log − y(t, T, s) = f (0, t, T ) + n−(k+1) T −t δt 1 − q + qλ k=0

Remark 2.17. We notice that from proposition 2.16, for some case the interest rate can take a negative value and in this case the arbitrage opportunity does not necessary hold. Thus, it is necessary to fix some constraints on the volatility λ to ensure that the interest rate is always positive. More precisely, to avoid the arbitrage opportunity we must have for all s > t: r(t, s) > 0 or P (t, t + δ, s) 6 1.

Section 2.2. Models for Interest Rates

Page 21

For this purpose, let us use P (t, t + δt, n) because this correspond the highest value of price at time t whereas r(t, n) represents the lowest value of interest rate. Then for a given q, we have: 

λn

P (0, t + δt) 61 1 − q + qλn P (0, t)

=⇒

1

n

1−q

λ 6  P (0,t+δt) P (0,t)

−q



Since we want that the condition of the non negative interest rate is always satisfied between the period 0 and T , we must have:   δt T −δt 1 − q  1 < λ 6  P (0,T ) − q P (0,T −δt) Remark 2.18. From proposition 2.16, we deduce that all yields curves can be considered as a random walk with random component s log λ where actually the quantity σ = log λ represents the classic volatility or variance of the interest rate. Indeed, from proposition 2.16, for all interest rate we conclude that σ is the distance between the two consequential node for a fixed time t and despite the independence the time-state and with the time t and the maturity T then all zero-coupon rates are governed by this quantity and have the same volatility. More precisely, we get the geometric relation: σ r(t, s + 1) − r(t, s) = f (t, T, T + 1, s + 1) − f (t, T, T + 1, s) = y(t, T, s + 1) − y(t, T, s) = − δt Example 2.19. To illustrate the model, let us see the following example where δt = 1. The data used in the example are those used in [Cai04]. Given the family of initial zero-coupon bond price P (0, T ), that is: P (0, T ) = {0.94, 0.9, 0.87, 0.84} , where 1 6 T 6 4, is given by the market. In addition, we give P (1, 2, 0) = 0.94 and P (1, 2, 1) = 0.965. First definition 2.7 allows us to compute u(1) and d(1) as follows:   P (0, 2) P (0, 1)     = 0.981778 P (1, 2, 0) = d(1) d(1) = P (1, 2, 0) P (0, 1) P (0, 2) =⇒ P (0, 2) P (0, 1)     = 1.007889 P (1, 2, 1) = u(1) u(1) = P (1, 2, 1) P (0, 1) P (0, 2) This result gives us the value of the two parameter the volatility measure and the risk neutral probability: λ=

u(1) = 1.026595 d(1)

and

q=

1 − d(1) = 0.697872 u(1) − d(1)

Then by proposition 2.14, we can compute all value of each perturbation factor following the value of the maturity and we obtain: τ d(τ ) u(τ )

0 1.000000 1.000000

1 0.981778 1.007889

2 0.963749 1.015694

3 0.945917 1.023414

Now known all value of each perturbation factor, we are able to see the behaviour and the movement of all term structures of interest rate for the Ho and Lee model in the binomial lattice.

Section 2.2. Models for Interest Rates

P(t,4) s 4 3 2 1 0

Page 22

t 0 − − − − 0.84

1 − − − 0.91454 0.84529

2 − − 0.96258 0.91336 0.86665

3 − 0.98812 0.96252 0.93759 0.91330

4 1 1 1 1 1

Figure 2.2: Binomial lattices tree for zero-coupon bond price P (t, 4) for the Ho and Lee model In figure 2.2, we note that the binomial lattice tree show us that if the time goes to maturity then the price of zero-coupon bond converges to 1 for any time-state (t, s).

r(t) s 3 2 1 0

t 0 − − − 0.06188

1 − − 0.06188 0.03563

2 − 0.07083 0.04458 0.01833

3 0.09069 0.06444 0.03819 0.01194

Figure 2.3: Binomial lattices tree for the short rate r(t) for the Ho and Lee model We observe that from figure 2.3, the short rate of interest move in way that at each time the distance between two nodes is still constant, because for a given time t analytically the difference r(t, s)−r(t, s−1) for all node s, we have:       r(t, s)−r(t, s−1) = f (0, t, t+1)+log 1 − q + qλt −s log λ− f (0, t, t + 1) + log 1 − q + qλt − (s − 1) log λ This gives us: r(t, s) − r(t, s − 1) = − log λ ≈ −0.02625

3. Pricing Interest Rate Derivative In this last chapter, we are intended to evaluate derivative securities on bonds. More precisely, we evaluate derivatives based on interest rates under the Ho and Lee model. Afterwards, we give a numerical approach of the model. The essay should be self contained [Cai04], [Hul02] and [HL86].

3.1

Derivatives Pricing

From remark 2.6, the model evolves in the complete market, that is all portfolio defined on the market satisfied the condition of self-financing and replicating. Before starting, let us give a brief definition of option. An option is a right but not obligation for the owner to trade the asset (stock or bond) for a fixed price K at a future date T where the option is called call option if the right is to buy an asset and put option if the right is to sell. Definition 3.1 (European Option). An European option with exercise price K and exercise date T on the underlying bond price is a contract characterized by: the holder of the option has the right and not the obligation, to buy or sell an asset at a precise time T . In other words, since European options are considered as derivative assets in the market, their pay-off are random variables contingent on the underlying asset. In general, the pay-off is a contract function, denoted by ϕ.

3.1.1

Bond Option

Now, we consider that the option derives from the zero-coupon bond and for a given bond price P (t, T, s) then the value of its pay-off at time t is given by:   max(P (t, T, s) − K, 0) if the option is call ϕ(P (t, T, s)) =   max(K − P (t, T, s), 0) if the option is put Proposition 3.2. If D(t, s) designs the price of the option at any time-state (t,s) for the multi-period binomial model, then the no-arbitrage price of option at time t is determined by: D(t, s) = P (t, t + δt, s)[qD(t + δt, s + 1) + (1 − q)D(t + δt, s)] where (q, 1 − q) defined as risk neutral probability in the Chapter 2. Proof. For a given time-state (t, s), consider the value of the bond option D(t, s) and let us use the replicating strategy to find its expression. Then there exists a portfolio V (t, s) constituted by bonds and cash such that: V (t, s) = x(t)P (t, T, s) + y(t)B(t, s)

and

D(t, s) = V (t, s)

At time t + δt the value of the portfolio becomes V (t + δt, s) which is defined by:   V (t + δt, s + 1) = x(t)P (t + δt, T, s + 1) + y(t)B(t + δt, s)   V (t + δt, s) = x(t)P (t + δt, T, s) + y(t)B(t + δt, s) 23

Section 3.1. Derivatives Pricing which gives:

x(t) =

Page 24

V (t + δt, s + 1) − V (t + δt, s) , P (t + δt, T, s + 1) − P (t + δt, T, s)

then we deduce: 1 [V (t + δt, s) − x(t)P (t + δt, T, s)] B(t + δt, s)   V (t + δt, s)P (t + δt, T, s + 1) − V (t + δt, s + 1)P (t + δt, T, s) 1 = B(t + δt, s) P (t + δt, T, s + 1) − P (t + δt, T, s)

y(t) =

Thus, the value of the portfolio at time t can be expressed by: P (t, T, s)[V (t + δt, s + 1) − V (t + δt, s)] P (t + δt, T, s + 1) − P (t + δt, T, s) P (t, t + δt, s)[V (t + δt, s)P (t + δt, T, s + 1) − V (t + δt, s + 1)P (t + δt, T, s)] + P (t + δt, T, s + 1) − P (t + δt, T, s) V (t + δt, s + 1) [P (t, T, s) − P (t + δt, T, s)P (t, t + δt, s)] = P (t + δt, T, s + 1) − P (t + δt, T, s) V (t + δt, s) [P (t + δt, T, s + 1)P (t, t + δt, s) − P (t, T, s)] + P (t + δt, T, s + 1) − P (t + δt, T, s)

x(t)P (t, T, s) + y(t)B(t, s) =

From definition 2.7, we have:  P (t, T, s)   P (t + δt, T, s) = d(T − t − δt)   P (t, t + δt, s)    P (t, T, s)   P (t + δt, T, s + 1) = u(T − t − δt) P (t, t + δt, s) After substituting these expression to the previous expression of V (t, s), therefore we get:   V (t + δt, s + 1)(1 − d(T − t − δt)) V (t + δt, s)(u(T − t − δt) − 1) V (t, s) = P (t, t + δt, s) + u(T − t − δt) − d(T − t − δt) u(T − t − δt) − d(T − t − δt) However, we have seen before that in the Ho and Lee model the risk neutral probability which characterized the arbitrage free is: 1 − d(T − t − δt) q= u(T − t − δt) − d(T − t − δt) More precisely, the arbitrage free of the option price at any time-state (t, s) is given determined by: D(t, s) = P (t, t + δt, s)[qD(t + δt, s + 1) + (1 − q)D(t + δt, s)]

Remark 3.3. This proposition allows to determine the price of the bond option for all periods between 0 and T using the backward induction. ˜ ˜ Corollary 3.4. If D(t) designs the discounted price of the option price at any time t then D(t) is a martingale with respect to the martingale probability q defined in the Ho and Lee model, that is, for all t: ˜ + δt)|Ft ] = D(t) ˜ EQ [D(t

Section 3.1. Derivatives Pricing

Page 25

˜ + δt)|Ft ] = D(t) ˜ Proof. By definition EQ [D(t means for all t and for a given state s, we have: ˜ + δt, s) = q D(t ˜ + δt, s + 1) + (1 − q)D(t ˜ + δt, s) D(t Indeed, from proposition 3.2, we have for all s 6 n: D(t, s) = P (t, t + δt, s)[qD(t + δt, s + 1) + (1 − q)D(t + δt, s)] B(t, s) D(t, s) = [qD(t + δt, s + 1) + (1 − q)D(t + δt, s)] B(t + δt, s) D(t, s) D(t + δt, s + 1) D(t + δt, s) =q + (1 − q) B(t, s) B(t + δt, s) B(t + δt, s) Here also, we discounted the value of option versus B(t, s) for a given state s then, ˜ s) = q D(t ˜ + δt, s + 1) + (1 − q)D(t ˜ + δt, s) D(t, Therefore, we have directly the result which means, ˜ + δt)|Ft ] = D(t) ˜ EQ [D(t

Proposition 3.5 (Uniqueness of the No-Arbitrage Option Price). In the Ho and Lee, if ϕ(P (T, S)) is the pay-off of any European option based on a bond at time T , then the free arbitrage price of this option is uniquely determined, for all t, by the following relation: ! # "  N −1  X B(t) D(t) = EQ ϕ(P (T, S)) Ft = EQ exp − r(kδt) ϕ(P (T, S)) Ft B(T ) k=n

 Proof. We want to prove that, for all t:

D(t) = EQ

 B(t) ϕ(P (T, S)) Ft B(T )

Indeed, from corollary 3.4 and using the martingale property in definition 1.15, we can write under the measure Q: ˜ )|Ft ] = D(t) ˜ EQ [D(T ˜ )|Ft ] which is equivalent to, for all t: D(t) = B(t)EQ [D(T Since the cash account B(t) is Ft -measurable, then we have:   B(t) D(t) = EQ D(T ) Ft B(T ) since D(T ) = ϕ(P (T, S)) and from corollary 2.3, then we get: " ! # N −1 X D(t) = EQ exp − r(kδt) ϕ(P (T, S)) Ft k=n

that means for all t:  D(t) = EQ

" ! # N −1  X B(t) ϕ(P (T, S)) Ft = EQ exp − r(kδt) ϕ(P (T, S)) Ft B(T ) k=n

Section 3.1. Derivatives Pricing

Page 26

Corollary 3.6. (Option Pricing) The price of the European option is defined as the expectation under the risk neutral probability q of the discounted pay-off, that is: N   X N i ϕ(P (T, S, i)) D(0) = q (1 − q)N −i i B(T, i) i=0

Proof. We have just to apply at time 0 the relation given in proposition 3.5 and using the fact that the model is a binomial model then by the backward induction we get the result:  N    X N i ϕ(P (T, S, i)) 1 q (1 − q)N −i ϕ(P (T, S)) F0 = D(0) = EQ B(T ) i B(T, i) i=0

Definition 3.7. (Put-Call Bond Option) In the Ho and Lee model, the value of bond option is characterized by the following properties: 1. Call bond option: C(0) =

PN




2. Put bond option: P (0) =

PN




N i=0 i

N i=0 i

q i (1

q i (1

−

q)N −i

−

q)N −i



max(P (T, S, i) − K, 0) B(T, i)





max(K − P (T, S, i), 0) B(T, i)



Proposition 3.8 (Put-Call Parity). Let be P (t, S) the price of S-bonds if C(t) and P (t) designs the prices at time t of the call and put options respectively based on P (t, S) with exercise date T , then there exists a link between the prices of European call and put options, namely: C(t) + KP (t, S) = P (t) + P (t, S) Proof. Consider C(t) and P (t) European call and put option based on P (t, S) with the same expiry time T and strike K such that S > T . Now let us construct two portfolios h1 and h2 , that is: • h1 is constituted by one call option from S-bond and K-units of the T -bonds: Vt (h1 ) = C(t) + KP (t, T ) • h2 is constituted by one put option from S-bond and one unit of the S-bonds: Vt (h2 ) = P (t) + P (t, S) Then at the time T , the respective values of the two portfolios become: • VT (h1 ) = C(T ) + KP (T, T ) = max{P (T, S) − K, 0} + K = max{P (T, S), K} • VT (h2 ) = P (T ) + KP (T, S) = max{K − P (T, S), 0} + P (T, S) = max{P (T, S), K} And we get VT (h1 ) = VT (h2 ), but according to the law of one price, the values of portfolios h1 and h2 might be the same at any earlier time. It follows that, for all t: C(t) + KP (t, S) = P (t) + P (t, S)

Section 3.2. Numerical Approach of the Ho and Lee Model

3.1.2

Page 27

Interest Rate Option

In reality, interest rates are not trade-able which mean we cannot buy or sell an interest rate but despite of existence the relationship between the bond and its interest rate, so theoretically it is possible to use this equivalence to construct an derivative based on the interest rate. Proposition 3.9. If f , a positive function of P (t, T ), defines an interest rate (short rate, forward rate, spot rate, . . .), then the value of the European option based on the interest rate f (P (T, S)) is given by the following formula:   B(t) D(t) = EQ ϕ(f (P (T, S))) Ft B(T ) In other words, the value of the pay-off at time t of the contract of interest rate f (P (T, S)) which is written at time T and pays ϕ(f (P (T, S))) at the expiry time S is given by D(t). log(P (T − δt, T )) with the contract which will pay (r(T ) − K)+ = δt max{r(T ) − K, 0} at the time T . Then the pay-off at time t of this contract is:   B(t) + D(t) = EQ (r(T ) − K) Ft B(T )

Example 3.10. Consider r(T ) = −

Usually we call this contract as a caplet maturing at T with ceiling K where the interest r(T ) is paid at time T for a loan of one at time T -δt.

3.2 3.2.1

Numerical Approach of the Ho and Lee Model Algorithm of the Model

In this part, we are going to present the algorithm for the programming of the model. Algorithm 1 Algorithm of the model 1: Introduce all initial condition and the market, i.e: input all values of the parameters and define the family of initial bond price. 2: Applying theorem 2.15 to define the general formula of zero-coupon coupon bond price and after set the movements step up and step down. 3: Increment all interest rates model as: forward rate, short rate and spot rate with the movements way. 4: Define the derivative security function related to the bond or the interest rate. Then compute the price of the contract at the initial time(date where we do the contract). 5: Build and plot the graph of all functions defined in above.

3.2.2

Simulation

Model 1: Caplet and Cap Here, we want to calculate the price of the contract for a loan of 1000000 euros during 8 years with interest rate paid annually and a strike value K = 4.5% (resp K = 3.5%). For a given Tmax , the

Section 3.2. Numerical Approach of the Ho and Lee Model

Page 28

sum of each caplet between the initial time until the Tmax determines the Cap which is the loan of one unity of money payable at time T max with interest rate payable at each interval δt. In your model, we take δt = 1 and given by the market the initial price P (0, T ) follows the deterministic relation, for all 0 6 T 6 Tmax : P (0, T ) = e−r0 T , where r0 is the initial short rate and its value is 2.5%. First let us see the evolution of the zero-coupon price P(t,T) and the caplet with the exercise time T = 8: Now by computational(using the program define in the appendix), we find the price of the cap

Figure 3.1: Binomial lattices for P (t, 8) and Caplet(t, 8) for the Ho and Lee model for a loan of 1 euro such that: Cap1 (8) =

8 X

Caplet(0, k) = 0.000767302

k=0

Therefore the price of the cap for a loan of 1000000 euros during 8 years with K = 4.5% is equal to: 767.3 euros. Respectively, we find also for K = 3.5% the price of the cap for a loan of 1000000 euros during 8 years is given by: 5271.2 euros. Model 2: Calibration of the model Consider that the T -bond price evolves on the period 0 to 30 years and the initial price P (0, T ) is governed by the following relation, for all T :
−T P (0, T ) = 1.1 − 0.05.e−0.18T   0.01 √ with two model parameters: q = 0.6 and λ = exp . Then the resulting binomial tree is q(1−q)

plotted in the graph below: We observe that for some nodes, the bond prices exceed the value 1 and the interest rates are negative. This gives rise to arbitrage opportunities. However, using the method of calibration in remark 2.17, we see that there exists λmax which defines an interval in such away that to avoid the arbitrage opportunities,

Section 3.2. Numerical Approach of the Ho and Lee Model

Page 29

Figure 3.2: Binomial lattices for P (t, 30) and r(t) for the Ho and Lee model the volatility parameter must be belong it. Since δt = 1 and Tmax = 30, by definition we have: 1 λmax





 = 

 1−q   P (0, Tmax ) −q P (0, Tmax − 1)

Tmax − 1

Then with precision: λmax = 1.009029561571911415, in addition, arbitrage free is determined by the interval of acceptance ]1, λmax ]. Now let us see the model after calibration and compare it with the result obtained previously. We consider λ = 1.009, this give us the following graph:

Figure 3.3: Binomial lattices for the news P (t, 30) and r(t) for the Ho and Lee model We note that the for all 0 6 t 6 30, the price of zero-coupon bond is always less than 1 and about the interest rate, the condition of non negative interest rate is also satisfied. Finally we can now affirm that the model satisfied the no arbitrage property.

Conclusion In summary, we conclude that the Ho and Lee model is a model of term structure of interest rates, which describes the evolution of the interest rate following the time and fit the yields curves. It is a exogenous model governed by the volatility factor and the risk neutral probability. In contrast with the model with constant interest rate models, as the Black Scholes or the CRR model, Ho and Lee use a different approach in taking into account all possible risks of interest rate due to economic perturbations. It gives a more realistic aspect of the market. We have seen that the model exclude all arbitrage possibilities. Moreover the uniqueness of the risk neutral measure ensures the completeness of the market under the Ho and Lee model. However, the model presents some disadvantages: first negative rates can appear in the model for some cases but also all zero-coupon rates have the same volatility therefore the model can only simulate the interest with constant volatility which makes less flexible since it does not reflect the complexity of interest rate modelling. On the other hand, the model is very useful in practice compared to other interest rate model as the Health, Jarrow and Morton model (1993) or Vascicek (1992) such as the computation side is feasible and offers high efficiency in the analysis of risk rate and interest rate option pricing. Finally to ensure this efficiency, the model needs also to be calibrate or adjusted in order to avoid negative interest rates otherwise the no arbitrage opportunity does not hold.

30

Appendix A. The Code of the Program for the Ho and Lee Model from __future__ import division from scipy import * from scipy import stats from scipy import linalg from scipy import random import Gnuplot import sys import math ################ Input the data of the market ##### print(’\n’) example=input("Choice the example: ") print(’\n’) # Define the family of the initial price : P0=[] # Example 1 if example==1: Tmax=4 N=Tmax T1=2 T2=4 K=0.8 q=0.5 l=1/0.95 P0=[0.9048,0.8187,0.7408,0.6703] # Example 2 if example==2: Tmax=30 N=Tmax T1=30 T2=T1+1 K=0.8 q=0.6 l=1.009 #l=exp(0.01/sqrt(q*(1-q))) for w in xrange(1,T2+1): P0.append((1+0.1-(0.05*exp(-0.18*w)))**(-w)) # Example caplet and cap if example==3: Tmax=8 N=Tmax 31

Page 32 q=0.5 l=1.01 r=0.025 K=0.045 T1=Tmax T2=Tmax+1 for w in xrange(1,T2+1): P0.append((1+r)**(-w)) ################ Zero-Coupon Bond ################# def P(t,T,s): # Create the price function if 0