expectations of monetary policy in australia ... - Semantic Scholar

EXPECTATIONS OF MONETARY POLICY IN AUSTRALIA IMPLIED BY THE PROBABILITY DISTRIBUTION OF INTEREST RATE DERIVATIVES

Ramaprasad Bhar and Carl Chiarella School of Finance & Economics University of Technology, Sydney PO Box 123, Broadway, NSW 2007 AUSTRALIA

Fax: +61 2 9514 7711 E-mail: [email protected]

2

EXPECTATIONS OF MONETARY POLICY IN AUSTRALIA IMPLIED BY THE PROBABILITY DISTRIBUTION OF INTEREST RATE DERIVATIVES

Abstract:

This paper describes and compares different methods of extracting the implied probability distribution of the underlying interest rate futures from the prices of traded options on these futures as well as from past futures prices. These methods are applied to short-term contracts on bank accepted bills trading on the Sydney Futures Exchange. The information on the distribution of the underlying asset thus obtained is very important to the central bank authorities since this allows them to monitor market expectations regarding future price movements. Alternatively, market reaction to central bank monetary policy changes may be judged this way. It is also important to practitioners for use in pricing over the counter (OTC) or exotic products where the trading volume is not particularly high. In that situation, the information on the distribution recovered from highly traded products from the exchange may be used as representative for the OTC products as well. As an empirical application, the recovered information on distribution is analysed in the context of reductions in interest rates in Australia by the Reserve Bank between July 1996 and May 1997.

3

I.

Introduction

In many countries derivatives markets are mature and many emerging markets are developing their own derivatives exchanges as well. Trading volume in emerging market derivative instruments suggest increasing importance of these markets. Growth in derivative activities has also initiated much discussion about its proper management and monitoring within the central banks. It is also becoming increasingly important for the authorities to understand and extract any relevant information concerning market expectations contained in the prices of these derivatives. Such information may be useful for monetary policy formulation within the central banks.

Treasury bill futures have been examined for their ability to provide the market's expectation about future rates eg. Cole, Impson and Reichenstein (1991). These authors also examine whether such forecasts exhibit rational expectations. The volatility implied by option prices has also been studied extensively.

Xu and Taylor (1994) attempt to improve hedging

strategies by analysing the term structure of implied volatilities in foreign currency options. Heynen, Kemna and Vorst (1994) also examine the relationship between short-term and longterm implied volatilities in equity options.

Another area of research that is also becoming important in this context is that where the objective is to understand the probability distribution of the underlying asset on the maturity date of the option contract. This particular line of research is based upon the relationship, first described by Breeden and Litzenberger (1978), between the risk-neutral probability density function of the underlying asset of a European style option on the option maturity date and the various strike prices of the option. Precisely, the theory states that the risk-neutral density of the underlying asset is proportional to the second derivative of the call option price with respect to the strike price. It is this relationship that is analysed in this paper for the short-term interest rate futures contracts trading on the Sydney Futures Exchange.

The economic policy situation in Australia from about mid-1996 to mid-1997 provides an ideal environment to implement techniques for computing the distribution of interest rates

4

implied by the prices of actively traded derivatives and using this distribution to infer anticipation of monetary policy changes by the Reserve Bank of Australia, in particular cuts in interest rates. Interest rates had risen to historically high levels during 1980s, attaining a maximum of nearly 19% for short-term loans. As inflation was also high, the real rate was comparable to that in other OECD countries. The high nominal interest rates, high inflation era was accompanied by a long period of depressed economic activity. As the inflation rate declined during the 1990s and attained the level of about 4-5% by 1996 there was growing calls for a cut in official borrowing rates which would quickly feed through to an overall reduction in borrowing rates.

The official borrowing rate in Australia is set by the Reserve Bank of Australia. A decision to cut the official borrowing rate would be made by the Reserve Bank Board1 at one of its monthly meetings. Economic and political commentators analyse in the minutest detail nuances (apparent or otherwise) in pronouncements of Reserve bank officials, Treasury officials and Government ministers for evidence that at such-and-such a meeting of the Reserve Bank Board the decision to cut the official borrowing rate will be taken. The success of such commentators in predicting interest rate cuts could at best be described as mixed.

By early 1996 the conjunction of inflation rate, Government borrowing requirement and wages growth were such that a cut in official borrowing rate seemed an appropriate policy. Indeed there was a high expectation on the part of financial commentators, business leaders and other community groups that the Reserve Bank Board would take such a policy decision. The issue was the timing of such decisions and the amount of the rate cut. Rates in fact were cut in July 1996, November 1996, December 1996 and May 1997 by 50 basis points each time. In this paper we seek to quantify the expectations of financial market traders about this series of rate cuts via a study of the probability distribution of interest rates implied in the prices of actively traded derivative securities. This is achieved by application of the methodology for inferring the risk-neutral distribution of the underlying asset developed by Neuhaus (1996) and Stutzer (1996). 1 The Reserve Bank Board consists of Reserve Bank officials, Australian Treasury officials, businesspersons, union officials, academics and some other community representatives. It is presided over the Governor of the Reserve Bank.

5

In section II the two models normally used to describe prices of options on interest rate futures are outlined and the difficulties that are encountered in applying the Breeden and Litzenberger (1978) theory are pointed out. Section III then discusses the possible solution to these problems as suggested by other researchers. In section IV another practical approach is suggested and the data and results of application of these techniques are discussed in section V. Section VI concludes the paper.

II.

Interest Rate Futures Option Pricing Models

This section outlines two commonly used models for pricing European options on interest rate futures, namely the Black (1976) model and the more recent Heath-Jarrow-Morton (1992) model. Detailed descriptions of both these models can be found in Jarrow and Turnbull (1996, page 373). Let the price of a European call option with maturity T and strike price K written on a futures contract with delivery time T F ≥ T be denoted by c. Under the assumptions of Black-Scholes model it is shown that c can be expressed (Black (1976)) as,

c B = B(0,T)[F(0,T F)N(d 1) - KN(d 2)]

(1)

where F(0, T F) is the futures price at time 0, B(0,T) is the price of a zero coupon bond maturing at time T with a payoff of $1, N(.) represents the cumulative normal distribution, and, d 1 = [ln[F(0,T F) / K] + σ 2 T / 2] / (σ T), d 2 = d 1 - σ T

(2)

with σ denoting futures price volatility which is assumed constant.

The European call option as above can also be priced using the Heath-Jarrow-Morton (1992) model details of which can be found in Musiela, Turnbull and Wakeman (1993). The HeathJarrow-Morton model is identified by a suitable specification of the volatility of forward interest rates as opposed to Black's model, which requires the volatility of the underlying futures price. To be consistent with equation (1) above, assuming a constant volatility in this

6

case also, the European call option price is given by,

c HJM = B(0,T)[F(0,T F ,T B) exp(θ)N(d) - KN(d - σ1)]

(3)

where F(0, T F ,T B) is the futures price at time 0 with delivery time T F and written on a discount bill that matures at time T B ≥ T F and,

σ1 d ≡ ln[F(0,T F ,T B) / K] + θ + σ12 / 2, θ ≡ (T B - T F) T 2 σ 2 / 2,

(4)

σ12 ≡ (T B - T F ) 2 T σ 2 where σ is the volatility of the forward rate.

The risk-neutral probability density function can now be obtained with the help of second derivative with respect to the strike price, K, from either equation (1) or (3). One practical problem is that options trading takes place only on a finite set of discrete strike prices thus making it difficult to infer these derivatives on a continuous basis.

Furthermore, the

assumptions underlying the models in equations (1) and (3) would suggest that options with the same time to maturity but with different strike prices should imply the same volatility of the underlying asset. But there is considerable evidence (see for example Ritchken (1996), page 221) against this in most asset classes and this dependence of the implied volatility on the strike price is known as the volatility smile effect (Dupire (1994)). Therefore, it is necessary to account for these effects to be able to extract reliable information on the riskneutral probability distribution of the underlying asset on the maturity date of the option. The next section discusses the approaches taken by other researchers in extracting information on the risk-neutral probability distribution and an alternative approach to account for the volatility smile effect is suggested in section IV.

III.

Review of Related Literature

The price of a European futures call option in the risk-neutral world can be expressed as,

7

+∝

c = ∫ p(F T , T; F t , t)Max (0, FT - K) dF T

(5)

-∝

where p(F T ,T; F t , t) is the transitional probability density of F at time T conditional on Ft , t. It follows that (with subscript K denoting partial derivative with respect to K), +∝

c K = - ∫ p(F T , T; F t , t)dF T , and

(6)

K

c KK = p(K, T; F t , t).

(7)

Thus, the probability of the futures price lying within a certain range can be found by evaluating an integral of the form in equation (6). The problem of a finite number of traded strike prices may be avoided by employing an approximation suggested by Breeden and Litzenberger (1978). Neuhaus (1996) shows that if the difference between adjacent strike prices is always ∆K, then the probability density at a specific strike price Ki can be approximated by a second order finite difference of the form,

c - 2 c i + c i +1 . p K i ≈ i -1 (∆ K ) 2

(8)

Therefore, the probability that the futures price on the maturity of the option lies within a range of K i ± ∆K is given by

c - 2 c i + c i +1 Prob[(K i - ∆K) ≤ FT ≤ (K i + ∆K)] ≈ i -1 . (∆K ) 2

(9)

In this approach only three call option prices are necessary where the strike prices are ∆K apart. To generate a complete histogram, however, a large number of strikes are needed,

8

otherwise the sum of all these probabilities may not add up to 1. Neuhaus (1996) suggests that instead of inferring the probability density function the implied cumulative probability distribution may be inferred as follows:

+∝

Prob[FT ≥ K] = ∫ p(F T , T; F t , t)dF T = - c K

(10)

K

which implies that the first derivative of the option price with respect to the strike price supplies information about the probability when the underlying futures price is greater than or equal to the strike price, K. This first derivative can be approximated as,

c -c Prob[FT ≥ K i] ≈ i -1 i +1 2(∆K )

(11)

using option prices from adjacent strikes (∆K apart) and is based upon the fact that call option prices monotonically decline with increasing strike prices. Thus, Prob[K i ≤ FT ≤ K i +1] = Prob[FT ≥ K i] - Prob[FT ≥ K i +1] [ c - c ] - [ c i - c i + 2] . ≈ i -1 i +1 2(∆K )

(12)

Neuhaus (1996) also shows that under mild assumptions, namely when the range of strike prices is sufficiently large and the options at the boundaries are traded according to their intrinsic values, the probabilities calculated as suggested add up to 1 as required. He then applies the method to bond futures options in Germany and monitors the changes in expectations and the uncertainty in the market by examining the interquartile range as the dispersion parameter.

Malz (1996), on the other hand, investigates the risk-neutral probability distribution of future exchange rates using over the counter (OTC) data on currency options with the Black-Scholes model adapted for currencies. He uses price data on certain option combinations to model the volatility smile and is thus able to draw inferences on the probability distribution with a relatively small amount of data. To be precise, he needs only three different strikes to be able

9

to estimate the complete distribution.

The two option combinations that are frequently traded on the OTC currency market are known as strangle and risk-reversal. Both involve two out of the money options with the same delta, usually 25-delta options (i.e. the strike price is chosen such that the delta is 25%). The risk reversal price (RR) is quoted as the volatility spread between a 25-delta put and a call. The strangle price (STR) is quoted as the average deviation of the 25-delta put and call volatilities from the at the money (ATM) volatility. Malz finds that most of the information about the volatility smile can be captured by a functional form, (δ being the Black-Scholes’s

σ(ATM, RR, STR, δ) = α ATM + RR β (δ - 0.5) + STR ( γ (δ - 0.5) ) 2

(13)

option delta) where α, β, γ are the parameters and this implies that the volatility smile has three components. Malz also shows a simple scheme to estimate the three parameters from observed prices of these instruments.

Malz’ (1996) procedure starts with a particular value for the strike and calculating its delta using the Black-Scholes model and then computing the implied volatility using equation (13). But since delta itself is a function of implied volatility, these two equations have to be solved together. Malz also notes that the only assumption in his procedure is the form of the volatility smile, which is largely consistent with the market prices of options and is unlikely to lead to large error in the estimation of the probability density function.

In a recent paper, Stutzer (1996) outlines a canonical valuation method that uses historical time series to predict the risk neutral probability distribution of the asset at any future date. This method is different from the two procedures discussed so far in that there is no need for any option prices although it is possible to apply the methodology with option prices as well. Stutzer's method is summarised here in the context of the distribution of interest rate futures at a specified later date. In this respect the principle of risk-neutral valuation is important which suggests that if the price process P(t) of an asset is such it allows no arbitrage then there exists a probability π* such that the discounted price is a martingale under π* (see

10

Huang and Litzenberger (1988), page 231). π* is referred to as an equivalent martingale probability measure as opposed to the actual or historical probability π . The current price P(0) may be expressed in terms of price at a future date T, P(T), as (see Stutzer (1996), equation(2))

     P(T)   P(T) * dπ  = Eπ  T  P(0) = E π*  T dπ      ∏ (1 + r(t))   ∏ (1 + r(t))   t =1   t =1 

(14)

where E is the mathematical expectation operation taken with respect to the probability measure given by the subscript, r(t) represents the one period risk-less rate of interest and the quantity (d π */dπ) is the Radon-Nikodym derivative for the change of measure.

Stutzer's (1996) canonical valuation approach exploits the relationship given by the equation (14). It requires an historical time series of asset prices (i.e. the interest rate futures prices in this paper) denoted by P(t), t = -1, -2, ... -H for a suitable horizon length H. The first step is to construct (H-T) historical T period returns as, R(-h) = P(-h) / P(-h - T), h = 1,2,..., H - T. From these returns the (H - T) possible values of the asset prices T period in future may be obtained as,

P(h) = P(0)R(-h), h = 1,2,..., H - T .

(15)

Thus, previous realised returns are used to construct possible T period prices each with an estimated actual probability π! (h) = 1/ (H - T) . It is claimed (Stutzer (1996), page 1638) that

under the general assumption of an ergodic Markov chain for the returns π! is a consistent estimator of the actual unknown distribution. The next step is to transform the estimated empirical probabilities π! of prices T periods ahead to an estimated equivalent martingale probability measure by imposing the constraint given by equation (14). Assuming a constant risk free rate, r say, this constraint may be written as:

11

R(-h) π* (h) πˆ (h). T ˆ (h) h =1 (1 + r ) π

H -T

1= ∑

(16)

The aim now is to choose strictly positive probabilities π* such that equation (16) is satisfied. This linear inverse problem can be solved by solving the following convex minimisation problem (see Stutzer (1996), equation (5)):

π! * = arg min{π * (h) > 0, ∑ h π * (h) = 1} I( π * , π! ) =

H -T

∑ π * (h)log

h=1

π *(h) . π! (h)

(17)

The objective function I( π * , π! ) in equation (17) is the distance of the positive probabilities

π* to the empirical probabilities π! and is known as the Kullback-Leibler Information Criterion (see Stutzer (1996), page 1639).

Since π! (h) = 1/ (H - T) , the constrained

minimisation problem in equation (17) is the same as the constrained maximisation of Shannon entropy - ∑ h π * (h)log π * (h) (see Stutzer (1996) for additional reference to the maximum entropy principle). Using Lagrange multipliers the solution to equation (17) results

 R(-h)  exp γ * T ) (1 + r    , h = 1,2,..., H - T π! * (h) =  * R(-h)  ∑ h exp γ   (1 + r ) T 

(18)

in, where the Lagrange multiplier γ * has to be found for the equation (18) to be useful. Ben-Tal (1985) shows that the Lagrange multiplier γ * is the solution to the following unconstrained optimisation problem:

γ * = arg

  R(-h)   min  . ∑ h exp γ  T γ   (1 + r )  

(19)

12

Thus γ * obtained from equation (19) when substituted in equation (18) the risk-neutral probability distribution implied by the prices given by equation (15), T period in future, is obtained. An empirical application of this approach is discussed in section V.

Before proceeding to the next section it is further stressed that the procedures discussed above deal with only the risk-neutral probability distribution which is different from the historical or "real world" probability distribution due to the existence of a risk premium. However, Rubinstein (1994) shows, using an example, that risk-neutral probabilities are a close approximation of the probabilities assumed by the market participants even in the presence of a risk premium. Neuhaus (1996) also demonstrates a useful application of the risk-neutral distribution from the perspective of the central bank for the purposes of monetary policy analysis and development.

Stutzer (1996) also extends his approach to risk-neutral

distribution of asset prices to the pricing of options written on the asset.

Other approaches based on density approximation methods have also been reported in the literature. For example Abken, Madan and Ramamurtie (1996) adopt a procedure where contingent claims are priced as elements of a separable Hilbert space. They specialise the Hilbert space basis to the family of Hermite polynomials and derive a four-parameter representation of the risk-neutral density function. Zou and Derman (1996), on the other hand, utilise an Edgeworth expansion to approximate the density function and apply this to valuation of path dependent options on indices. Buchen and Kelly (1996) adopt the principle of maximum entropy to estimate the distribution of an underlying asset from a set of option prices.

IV.

Volatility Smile in the Australian Short-Term Interest Rate Futures Options

The existence of a smile effect in Australian equity option prices has been examined by Oliver (1995) who used polynomial regressions to estimate the relationship between implied volatility and intrinsic value (ratio of asset price to strike price). Oliver reports that a quadratic polynomial offers the best explanation of the volatility smile. Shimco (1990) also employs a smooth quadratic function of strike price to the implied volatilities of option

13

prices. It still requires arbitrary rules to handle cases outside the traded range of strikes.

The smile effect in Australian interest rate futures options has not been reported so far. Black's (1976) futures option pricing model is often used in this market and this model also assumes constant volatility of the underlying asset similar to that in the Black-Scholes equity option pricing model. This paper investigates the dependence of implied volatility on strike price using the Black (1976) model. The HJM (1992) model, on the other hand, has not been pursued for this purpose since the volatility component in this model relates to the forward interest rate process and not to the underlying futures (see equation (3) and (4)). A simple function of the intrinsic value is fitted to the observed smile effect in the implied volatility.

Figure 1 shows the dependence of the implied volatility (obtained by using equation (1)) for the call options prices on 25 July 1996. These options are written on the 90-day bank bill futures trading on the Sydney Futures Exchange and deliverable on the second Friday in September 1996. The fitted equation is also described in the figure along with the estimated coefficients and the corresponding standard errors. This relationship may be exploited to extract information on probability distribution as discussed in the next section.

Figure 1 Dependence of Implied Volatility on Strike Price Implied Volatility (Black'76)

0.012 0.010 0.008 0.006 0.004 0.002 0.000 0.998

1.000

1.002

1.004

1.006

1.008

1.010

1.012

Rat io o f Fut ures t o St rik e P rices

This relationship can be adequately explained by σ imp = a ( F / K) + b ln( F / K) where F is the futures price and K is the strike price. The estimated coefficients are a=0.000823 (0.00047), b=1.003775 (0.08528) . The numbers in parentheses are standard errors.

14

V.

Data and Results

In this paper the implied risk-neutral probability distribution in interest rate futures prices is investigated using daily prices on the 90-day bank bill futures and call options on these futures traded on the Sydney Futures Exchange. These are very liquid and important shortterm interest rate futures contracts for the Australian market.

As mentioned in the

introduction during the period July 1996 to May 1997, the Reserve Bank of Australia lowered the discount rate four times on 31 July, 6 November, 11 December and 23 May by 50 basis points each time. This provides an opportunity to monitor the distributions of futures prices around those days following the methods outlined earlier and to make some inferences about market expectations of these policy decisions.

[ Insert Figure 2 about here ]

Figures 2 depicts the results obtained by application of Stutzer's (1996) approach using only past prices of the 90-day bank bill futures maturing in September 1996, December 1996 and June 1997 corresponding to the dates around which the Reserve Bank of Australia lowered its discount rates. Each panel of this figure attempts to capture the probability of an interest rate move (ie. positive or negative basis point changes) from 55 to 5 trading days prior to the date on which the rate cut is announced. The common feature in all four panels is that the probability of a cut generally increased from 55 days prior to 5 days prior. It is also of interest to note that the market had a much better expectation of the last two rate cuts (11 December 1996 and 23 May 1997). Although the interest rate futures market exhibits the market's view on rate movement the methodology adopted here is able to quantify such view in terms of probability.

During the period being analysed most financial market

commentators expected an interest rate cut. However, our analysis shows that as far as the rate cuts on 31 July 1996 and 6 November 1996 are concerned there was substantial probability of this not happening according to the information contained in the futures prices.


15

Figure 3 reveals interesting information regarding the market's expectation about interest rate movements both prior to and subsequent to the dates on which the Reserve Bank of Australia announced rate cuts. Here we use the approach suggested by Neuhaus (1996) which monitors the width of the gap between the 25% and 75% thresholds as implied by futures options prices. The gradual decline in this width before the 31 July 1996 cut indicates that the uncertainty in rate movement is lessening as well. Thus the sudden jump just prior to 31 July suggests that this action by the Reserve Bank was not anticipated in the market and that uncertainty about rate changes moved to a lower level. Similarly, the action of Reserve Bank on 6 November was largely anticipated by the market as revealed by the upper right panel of Figure 3.

The continual decline in the band width immediately subsequent to this cut

indicates declining uncertainty about further rate cuts on the part of the market. This information is consistent with that in the lower left panel of Figure 2 which indicates that by this time the market held a consistently high expectation of the 11 December cut. The lower left panel of Figure 3 indicates a large increase in uncertainty about future rate immediately subsequent to the 11 December cut. This no doubt reflects differing opinions at the time. One view was that the Reserve Bank of Australia announced three rate cuts in five months and this was sufficient given the economic fundamentals. Another view was that another rate cut would be undertaken to give a boost to the economic activity. By February 1997, pronouncements by the Governor of the RBA indicated that further rate cuts were not likely2. However, the lower panels in Figures 2 and 3 indicate that the market was not so surprised3. The lower panel of Figure 2 indicates a high, although slightly declining, expectation of the cut and the lower panel of Figure 3 indicates declining uncertainty leading up to the cut. The steep decline in uncertainty past the cut probably indicates that the market felt that the cycle of rate cuts had finally come to an end.


2 see Australian Financial Review, February 20, 1997. 3 These figures contrast with quotes from some market traders at the time claiming the market put a slight probability on a rate cut – see Australian Financial Review, 6 May 1997.

16

Finally, Figure 4 depicts the risk-neutral probability distribution obtained by following the approach similar to that in Malz (1996) and uses a curve fitting technique to approximate the observed smile effect in Australian short term interest rate futures options. It also shows the distribution when volatility is assumed constant (ie. log normal distribution). The effect of the volatility smile is to steepen the curve ie. higher probability of futures prices when it is deep in the money.

VI.

Conclusions

This paper describes and compares three different approaches to recover the risk-neutral probability distribution of the short-term interest rate futures trading on the Sydney Futures Exchange. It is shown that this information on distribution is useful in analysing market's expectations regarding the round of interest rate cuts announced by the Reserve Bank of Australia during 1996-97.

This technique has important implications for the monetary

authority since it might help them to decide on an appropriate timing for action on the official interest rate. Although the interest rate futures market is indicative of the direction of the rate movement, the techniques discussed here allows quantification of such indications. Also, the methodologies discussed here may be compared with those based on density approximation eg. Zou and Derman (1996), Buchen and Kelly (1996).

As possible extensions of the research presented here, the methodology due to Stutzer (1996) may be adapted for pricing options on interest rate futures and its predictive ability compared with some suitable benchmark. Besides the approach based on modelling the volatility smile effect may be improved to monitor the changes in market expectations on interest rate movement and the presentation of this information.

17

References

Abken, P., Madan, D.B. and Ramamurtie, S. (1996), "Estimation of Risk-Neutral and Statistical Densities By Hermite Polynomial Approximation: With an Application to Eurodollar Futures Options", Federal Reserve Bank of Atlanta, Working paper 96-5. Ben-Tal, A. (1985), "The Entropy Penalty Approach to Stochastic Programming", Mathematics of Operations Research, 10, 263-279. Black, F. (1976), "The Pricing of Commodity Contracts", Journal of Financial Economics, 3, 167-179. Breeden, D.T. and Litzenberger, R.H. (1978), "Price of State Contingent Claims Implicit in Option Prices", Journal of Business, 51, 621-651. Buchen, P. W. and Kelly, M. (1996), “The Maximum Entropy Distribution of an Asset Inferred from Option Prices”, Journal of Financial and Quantitative Analysis, 31 (1), 143-159. Cole, C.S., Impson, M. and Reichenstein, W. (1991), "Do Treasury Bill Futures Rates Satisfy Rational Expectation Properties?", The Journal of Futures Markets, 11(5), 591-601. Dupire, B. (1994), "Pricing and Hedging with Smiles", Risk, 7, 18-20. Heath, D., Jarrow, R. and Morton, A. (1992), "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation", Econometrica, 60(1), 77-105. Heynen, R., Kemna, A. and Vorst, T. (1994), "Analysis of the Term Structure of Implied Volatilities", Journal of Financial and Quantitative Analysis, 29(1), 31-56. Huang, Chi-fu and Litzenberger, R.H. (1988), Foundation for Financial Economics, North Holland, The Netherlands. Jarrow, R. and Turnbull, S. (1996), Derivative Securities, South Western College Publishing, An International Thomson Publishing Company. Malz, A.M. (1996), "Recovering the Probability Distribution of Future Exchange Rates from Option Prices", Presented at the Imperial College/Chemical Bank conference, London, 27-29 March. Musiela, M., Turnbull, S.M. and Wakeman, L.M. (1993), "Interest Rate Risk Management", Review of Futures markets, 12(1), 221-261.

18

Neuhaus, H. (1996), "The Information Content of Derivatives for Monetary Policy: implied Volatilities or Implied Probabilities?", Presented at the Imperial College/Chemical Bank conference, London, 27-29 March. Oliver, B. (1995), "An Examination of Volatility Smiles in the Australian Options Market", Working paper, School of Commerce, La Trobe University, Victoria, Australia. Ritchken, P. (1996), Derivative Markets, Theory, Strategy, and Applications, Harper-Collins College Publishers, New York. Rubinstein, M. (1994), "Implied Binomial Trees", Journal of Finance, 49, 771-818. Shimco, D. (1990), "Bounds of Probability", Risk, 6, 33-37. Stutzer, M. (1996), "A Simple Nonparametric Approach to Derivative Security Valuation", Journal of Finance, Vol. LI (5), 1633-1652. Xu, X, and Taylor, S.J. (1994), "The Term Structure of Volatility Implied by Foreign Exchange Options", Journal of Financial and Quantitative Analysis, 29(1), 57-74. Zou, J.Z. and Derman, E. (1996), "Monte Carlo Valuation of Path Dependent Option on Indexes with a Volatility Smile", Paper presented at the First Annual Computational Finance Conference, International Association of Financial Engineers, Stanford University, August.

Figure 2 Probability of a Fall in Interest Rate on Target Dates as Measured From Various Prior Trading Days (Horizontal Axis) Target Date 31 July 1996 Target Date 6 November 1996 0.60

0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00

0.50 0.40 0.30 0.20 0.10

15

5

15

5

Target Date 11 December 1996

25

0.00

5

25

10

35

15

35

20

45

25

45

30

55

35

55

40

65

45

75

50

85

55

Target Date 23 May 1997

0.70

0.80

0.60

0.70 0.60

0.50

0.50 0.40

0.40

0.30

65

75

5

15

25

35

45

0.00 55

0.00 65

0.10 75

0.10 85

0.20

85

0.30

0.20

20

0.100

0.058

0.000

0.068

0.600

0 .0 6 4

0.500

0 .0 6 2

0.062

0.400

0.060 0.300 0.058 0.056

0.200

0.054 0.100 0.052

Futures Yield

0.064

Width in Basis Points

0.066

0 .4 5 0 .4 3 0 .4 1 0 .3 9 0 .3 7 0 .3 5 0 .3 3 0 .3 1 0 .2 9 0 .2 7 0 .2 5

25 % T h resh o ld

0 .0 6 0

W idt h

0 .0 5 8 0 .0 5 6 0 .0 5 4 0 .0 5 2

7 5 % T h resh o ld

0 .0 5 0 30-May

28-May

26-May

22-May

20-May

16-May

14-May

12-May

8-May

6-May

2-May

30-Apr

20-Dec

18-Dec

16-Dec

12-Dec

10-Dec

6-Dec

4-Dec

2-Dec

28-Nov

26-Nov

22-Nov

20-Nov

0.000

28-Apr

0 .0 4 8

0.050 18-Nov

Futures Yield

12-Nov

Target Date 23 may 1997


Target Date 11 December 1996

14-Nov

0.060

8-Nov

0.200

6-Nov

0.062

4-Nov

0.300

7-Oct

9-Aug

14-Aug

6-Aug

1-Aug

29-Jul

24-Jul

19-Jul

16-Jul

8-Jul

11-Jul

3-Jul

28-Jun

0.00 25-Jun

0.065 20-Jun

0.05 17-Jun

0.067

0.064

31-Oct

0.10

0.400

29-Oct

0.15 0.069

0.066

25-Oct

0.20

0.071

0.500

23-Oct

0.073

0.25

0.068

21-Oct

0.30

0.600

17-Oct

0.075

0.070

15-Oct

0.35

Futures Yield

0.077


Futures Yield

0.40

11-Oct

0.45 0.079

9-Oct

Target Date 31 July 1996


Figure 3 Confidence Interval of Interest rate Movement Around Target Dates Target Date 6 November 1996

21

Figure 4 Probability Distribution Function For A Range Of Futures-Strike Price Ratio

Cumulative Distribution

0.30

0.20

0.10

0.00 0.90

1.00

1.10

1.20

1.30

1.40

1.50

Futures-Strike Price Ratio With 'Smile' Effect

Constant Volatility (0.22)

Call option prices of 25 July 1996 on September 1996 90-day bank bill futures used. Smile effect approximated as explained in Figure 1.