Chapter 9: ONE STEP

BINOMIAL TREES

9.1 A ONE-STEP INTEREST RATE

BINOMIAL TREE

• 9.1.1 Continuous Compounding

• 9.1.2 The Binomial Tree for a Two-Period Zero

Coupon Bond

9.1 A ONE-STEP INTEREST RATE

BINOMIAL TREE• An interest rate tree starts with the specification of the dynamics of the

short-term interest rate in conjunction with the data we have from the yield curve:

Table 9.1 Interpolated Treasury STRIPS on Jan 8, 2002

• These dynamics reflect our predictions of future interest rates• We take the short-term interest rate as exogenous, in the sense that it is

driven by monetary policy choices, and market participants cannot affect it• In principle, this characteristic is really appropriate for the overnight

(Federal funds) rate, but for now we keep matters simple, and we take the 6-month rate as the exogenous rate

9.1 A ONE-STEP INTEREST RATE

BINOMIAL TREE

9.1 A ONE-STEP INTEREST RATE

BINOMIAL TREE

• r0 is the 6-month continuously compounded rate at i = 0, r1,u is the interest rate in period i = 1 after an up (u) move in interest rates, and r1,d is the interest rate in period i = 1 after a down (d) move in interest rates

• In this scenario we consider there is a 50-50 chance of rates going either way

9.1.1 Continuous Compounding

• The model could use any rate, at any compounding frequency• We chose the continuously compounded (c.c.) rate because:▫ There is always a one-to-one relation between the c.c. rate and

any other rate:

er ×Δ = 1 + rn × Δ▫ When applying the model we might consider very high frequency

trees; using c.c. allows us to only vary the time interval between nodes without changing the rate itself

▫ Many interest rate securities and derivatives have cash flows that depend on different compounding frequencies; using c.c. allows us to make this change easily

▫ This makes clearer the link between these models and the continuous time models discussed later on

9.1.2 The Binomial Tree for a

Two-Period Zero Coupon Bond

• Let Pi,j(k) be the bond price in period i, in node j, and

with maturity in period k

• From the previous tree we can obtain the process for

a zero coupon bond maturing at k = 2

Table 9.3

The one-year zero

coupon bond

binomial tree

9.2 NO ARBITRAGE ON A

BINOMIAL TREE

• 9.2.1 The Replicating Portfolio Via No Arbitrage

• 9.2.2 Where is the Probability p?

9.2 NO ARBITRAGE ON A

BINOMIAL TREE

• We now exploit the binomial tree for the one-year zero coupon bond in Table 9.3 to obtain the fair, no-arbitrage price of additional securities whose final payoff depends on the interest rate

• Consider the following option:

(Payoff at i =1) = 100 × max(rK – r1,0)

where rK is the strike price (assume rK = 2%)

• So for the tree we have that:

(Payoff at i =1 if r1,u) = $0.00

(Payoff at i =1 if r1,d) = $1.05

9.2 NO ARBITRAGE ON A

BINOMIAL TREE (cont.)• Consider the following portfolio:Buy 0.8700 of bonds with k = 2 → –0.8700 × $97.8925 = -$85.1703Short 0.8554 of bonds with k = 1 → +0.8554 × $99.1338 = $84.8007

$0.3697

• What is the value of this portfolio at i = 1?Value of Portfolio if r1,u → 0.8700 × P1,u(2) – 0.8554 × $100 = $0.00Value of Portfolio if r1,d → 0.8700 × P1,d(2) – 0.8554 × $100 = $1.05

• Note that the value of the portfolio is identical to the value of the option▫ In other words the portfolio replicates the payoff of the option

• If they both have the same payoff, then it follows that, assuming there are no arbitrage opportunities, they should have the same price (which means that the option’s price $0.3697)

9.2 NO ARBITRAGE ON A

BINOMIAL TREE (cont.)

9.2 NO ARBITRAGE ON A

BINOMIAL TREE (cont.)

• A replicating portfolio of a security with payoffs V1,u

and V1,d in the two nodes u and d at time i = 1 is a

portfolio of bonds that exactly replicates the values of

the security at time i = 1.

• That is, if Пi,j denotes the value of the portfolio at

time i in node j, we have П1,u = V1,u and П1,d = V1,d.

The value of the option at i = 0 equals the value of

the portfolio П0 = V0

9.2.1 The Replicating Portfolio

Via No Arbitrage• Consider a portfolio (П) with N1 units of the bond with maturity i = 1 and

N2 units of the bond with maturity i = 2, at time i = 0 it’s value is:

П0 = N1 × P0(1) + N2 × P0(2)

• At time i = 1, the portfolio value will be:

П1,u = N1 × 100 + N2 × P1,u(2)П1,d = N1 × 100 + N2 × P1,d(2)

• Assume that we are trying to replicate a security that takes values V1,u and V1,d, so we have that:

П1,u = N1 × 100 + N2 × P1,u(2) = V1,u

П1,d = N1 × 100 + N2 × P1,d(2) = V1,d

a system of 2 equations with 2 unknowns (N1 and N2)

9.2.1 The Replicating Portfolio

Via No Arbitrage (cont.)• Subtracting one from the other, we get:

N2 × (P1,u(2) – P1,d(2)) = (V1,u – V1,d)• Solving for N2:

• Given N2, the solution for N1 is:N1 = [V1,u – N2 × P1,u(2)] / 100

• We then get:V0 = П0 = N1 × P0(1) + N2 × P0(2)

• This leads to a first recipe for pricing derivatives:1. Compute N1 and N2 from the previous equations2. Compute the price of the derivative security V0 as shown above

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Via No Arbitrage (cont.)• An example:▫ Consider the option payoff presented above, which implies

V1,u = $0 and V1,d = $1.05 ▫ We can compute the replicating portfolio by using the

binomial tree of the 2-period bond in Table 9.3▫ Applying the formulas we obtain:

which is the bond portfolio described in Table 9.4 ▫ The negative sign on N1 indicates a short position in the

bond with maturity i = 1

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Via No Arbitrage (cont.)

9.2.1 The Replicating Portfolio

Via No Arbitrage (cont.)• Another example: Consider a swap that pays at time i = 1 the amount 100 / 2 × (r1

− c), with c = 2%; then, the value of the swap from the fixed rate payer perspective is:

V1,u = 100 × (3.39% – 2%) / 2 = $0.695V1,d = 100 × (0.95% – 2%) / 2 = -$0.525

▫ We can obtain the replicating portfolio by choosing N1 and N2

▫ In this case, the replicating portfolio calls for a long position of 1.001 in the short-term bond, and a short position of 1.011 in the long(er)-term bond

Π1,u = N1 × 100 + N2 × P1,u(2) = 1.001 × 100 − 1.011 × 98.3193 = $0.695Π1,d = N1 × 100 + N2 × P1,d(2) = 1.001 × 100 − 1.011 × 99.5261 = −$0.525

▫ Because this portfolio indeed replicates the payoff from the swap, the value at i = 0 of the swap is

Π0 = N1 × P0(1) + N0 × P0(2) = 1.001 × 99.1338 − 1.011 × 97.8925 = $0.259

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Via No Arbitrage (cont.)• And yet another example:

▫ Consider an option with the following payoff at time i = 1 (strike price K = $99.00):

(Payoff at i = 1) = max(P1(2) – K,0)▫ In this case, the payoff at i = 1 from the option is

V1,u = max(98.3193− 99.00, 0) = 0V1,d = max(99.5261− 99.00, 0) = $0.5261

▫ Once again, we can obtain the replicating portfolio by choosing N1 andN2:

▫ Once again, we can check that the replicating portfolio in fact replicates:

Π1,u = N1 × 100 + N2 × P1,u(2) = −0.429 × 100 + 0.436 × 98.3193 = $0Π1,d = N1 × 100 + N2 × P1,d(2) = −0.429 × 100 + 0.436 × 99.5261 = $0.5261

▫ Because this portfolio indeed replicates the payoff from the option, the value of the option at i = 0 is

Π0 = N1 × P0(1) + N2 × P0(2) = −0.429 × 99.1338+ 0.436 × 97.8925 = $0.185

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9.2.2 Where is the Probability p?

• The prices of the derivative securities were obtained without any reference

to the probability p of an up movement in interest rates

▫ For example, how is it possible that the price of an option that pays when

interest rates go down is independent of the probability that the rate will in fact

go down?

• The price of the derivative security is computed from a portfolio of other

bonds

• The prices of these bonds do depend on the probability that market

participants assign to an increase in future interest rate

• Everything else equal, if market participants have a lower expectation of

the 6-month rate next period, then the long-term bond would have a higher

price, which in turn would increase the price of the option

• Yet, for given bond prices, the price of the option can be computed only by

the replication of its payoff and thus the exact knowledge of the probability

p is not necessary

9.3 DERIVATIVE PRICING AS

PRESENT DISCOUNTED VALUES OF

FUTURE CASH FLOWS

• 9.3.1 Risk Premia in Interest Rate Securities

• 9.3.2 The Market Price of Interest Rate Risk

• 9.3.3 An Interest Rate Security Price Formula

• 9.3.4 What If We Do Not Know p?

9.3.1 Risk Premia in Interest

Rate Securities• Computing the present value of P1(2) we get:

PV of E[P1(2)] = e-r0 × E[P1(2)]= 0.9913 × (p × 98.3193 + (1 – p) × 99.5261)= 98.0658

• This is higher than the observed price (from the yield curve):P0(2) = 97.8925 < e-r0 × E[P1(2)] = 98.0658

• The price is lower because of a risk premium embedded in the price of longer term bonds▫ What risk? Clearly, there is no default risk in U.S. Treasuries

• An investment in Treasury securities is risky because an investor may suffer capital losses if the bond is sold before maturity

• The Dollar Risk Premium from investing in the long term bond with maturity i = 2 is defined as: e-r0 × E[P1(2)] – P0(2)

9.3.2 The Market Price of

Interest Rate Risk• Recall that given N2, we can define:

N1 × 100 = [V1,u – N2 × P1,u(2)] = [V1,d – N2 × P1,d(2)]

• Equivalently:

N1 = {E[V1]– N2 × E[P1(2)]} / 100

where: E[V1] = pV1,u + (1 – p)V1,d and

E[P1(2)] = pP1,u(2) + (1 – p)P1,d(2)

• Substituting this equation in V0 = N1 × P0(1) + N2 × P0(2), we get:

• Substitute N2 = (V1,u – V1,d) / (P1,u(2) – P1,d(2)), and after a few operations you get:

• This is a key relation between securities in no-arbitrage pricing

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9.3.2 The Market Price of

Interest Rate Risk (cont.)

• Note the following:▫ The left hand side (LHS) only involves zero coupon prices and the right

hand side (RHS) only involves derivative security prices▫ The expressions on the LHS for the two period bond is identical to the

expression on the RHS for the derivative security▫ The numerators of both expressions are simply the (dollar) risk premium

investors require from holding bonds (LHS) or the derivative security (RHS)

▫ The denominators represent the (dollar) risk of an investment in bonds (LHS) or in the derivative security (RHS), as it is given by the fluctuations of the security across the two possible states

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9.3.2 The Market Price of

Interest Rate Risk (cont.)

• So we have that, all interest rate securities on a binomial

tress have the same ratio between risk premium and risk:

• where λ0 is common across all interest rate securities

and is called the market price of (interest rate) risk

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9.3.3 An Interest Rate Security

Price Formula• If we know λ0 at time i = 0, we can compute the price of

any security as:V0 = e-r0 × Δ × E[V1] – λ0 × (V1,u – V1,d)

• How can we compute λ0? Using information on the bond with maturity i = 2

• This leads to a second recipe for pricing derivatives:1. Compute the market price of risk λ0 from the known bond

data2. Compute the price of the interest rate security from the

pricing formula shown above

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9.3.4 What If We Do Not Know p?

• What if we make a mistake and miscalculate p?• In fact, as it turns out, even if we make a mistake in

computing p in the original tree, the pricing of the interest rate securities is not affected

• The key is to realize that λ0 also depends on p, so if we miscalculate p, we will also miscalculate the risk adjustment▫ As the following table shows as p varies so does λ0

• It turns out that one error exactly counterbalances the other

• In other words, p must not be true but it should be consistent within the model

9.3.4 What If We Do Not Know p?

9.4 RISK NEUTRAL PRICING

• 9.4.1 Risk Neutral Probability

• 9.4.2 The Price of Interest Rate Securities

• 9.4.3 Risk Neutral Pricing and Dynamic

Replication

• 9.4.4 Risk Neutral Expectation of Future Interest

Rates

9.4.1 Risk Neutral Probability

• Since when p changes λ0 varies, in theory we could find a value of p (which we can call p*) that makes λ0 = 0▫ Since λ0 is common to all securities so should p*

• When this occurs, from known bond prices, we get:P0(2) = e-r0 × Δ E*[P1]

= e-r0 × Δ (p* × P1,u(2) + (1 – p*) × P1,d(2))

• Which leads to:

• The risk neutral probability p* is the particular value of the probability p such that every interest rate security is given by the present value of expected payoff, where the present value is computed using the risk free rate

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Securities

• This leads to a third recipe for pricing derivatives:

1. Compute the risk neutral probability p*

2. Compute the price of the interest rate security from

the pricing formula: V0 = e-r0 × Δ × E*[V1]

9.4.3 Risk Neutral Pricing and

Dynamic Replication• The simplicity of the risk neutral methodology is its main

virtue• It is important to realize that there is no underlying

assumption that market participants are risk neutral▫ They are not, as in fact under the true probability, market

participants would require a risk premium to hold long-term bonds

• Underlying its logic is the existence of the replicating portfolio

• What is key to realize, however, is that the dynamic replication strategy exists among any two interest rate securities ▫ Example: next page using very liquid swap contracts

9.4.3 Risk Neutral Pricing and

Dynamic Replication

9.4.3 Risk Neutral Pricing and

Dynamic Replication

• An example:▫ We are going to use a portfolio with N2 swaps, and N1 of 1-

period bonds

▫ The methodology is the same as in Section 9.2.1 with the only difference that instead of the value of bonds, we must use the value of the swap along the way (let Vsw

ij be the value of the swap in node ij)

▫ Previously we obtained Vsw0 = $0.259 so we get Table 9.8

▫ We obtain that the portfolio of a swap and short-term bond is given by:

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Dynamic Replication

▫ The value of this portfolio is

Π0 = N1 × P0(1) + N2 × Vsw0

= 0.006 × $99.1338− 0.861 × $0.259 = $0.3697

▫ The value of the portfolio is identical to the value of the option that was obtained by using the portfolio of bonds, as in Table 9.4, it also replicates:

Π1,u = N1 × P1,u(1) + N2 × Vsw1,u

= 0.006× 100 − 0.861 × $0.695 = $0

Π1,d = N1 × P1,d(1) + N2 × Vsw1,d

= 0.006 × 100 − 0.861× (−$0.525) = $1.05

9.4.4 Risk Neutral Expectation

of Future Interest Rates• The expected future interest rate under the risk neutral

probability is given by

• E*[r1] = p* × r1,u + (1 – p*) × r1,d

= 0.6448 × 3.39% + 0.3552 × 0.95% = 2.5234%

• This number is higher than the true expected interest rate computed, which was equal to E[r1] = 2.17%▫ Risk neutral pricing includes the risk premium in the probability

of an up move (from p to p*) and thus increases the predicted future interest rate

▫ However, this does not mean that the market participants expect the interest rate in six months to be 2.5234% instead of 2.17%

• How close is the risk neutral expectation of interest rates to the forward rate?

9.4.4 Risk Neutral Expectation

of Future Interest Rates (cont.)• The forward rate is given by:

f(0,1,2) = -0.5 × ln(P0(1) / P0(2)) = 2.52%

• This has two implications:▫ Forward rates are not equal to the market expectation of future interest rates

Thus if today we observe high forward rates we should think about two possibilities: Market participants expect higher future interest rates

They are strongly averse to risk, and thus the price of long term bonds is low today

▫ The forward rate is not even equal to the risk neutral future interest rate, although they are quite close Recall that risk neutral pricing is based on the notion of dynamic replication,

which involves trading in securities

Interest rates are related to securities prices through a convex relation Thus the divergence between rates, is because there is a convexity adjustment

missing to make both interest rates the same

Please check the link at: Forward rate versus risk neutral expected future interest rate