Anticipating ECB monetary policy decisions through the RND extrapolation from Liffe 3M EURIBOR options.

PierPaolo Cassese

Assicurazioni Generali Finance Department November 2013

Contents

Abstract

1. Introduction

2. RNDs methodologies

1. Second call-put partial derivative with respect to the strike

price.

2. RNDs estimated with the lognormal distribution function.

3. Fitting the implied Black & Scholes volatility smile using the spline

4. Case studies

1. Lehman Brothers market reaction

5. Concluding remarks

References

The price of derivative contracts has always played a strategic role in extracting valid

information to be employed in one’s own investment strategies. Shimko (1993) was a

pioneer in this arena. Taking up where Shimko left off, this paper will demonstrate how to

extract Risk neutral density function using two different techniques from derivative

contracts. Shimko proposed a technique to build up the risk neutral density function

starting by employing the implied volatility smile of 3M Liffe EURIBOR derivative

contracts. The procedures adopted here are based on the implementation of an

interpolation model, a polynomial splines, applied to the implied volatility smile of future

Liffe EURIBOR 3 M from which RNDs will be created. The flexibility of the model, as will be

explained, is entirely attributed to the differentiability of the call and put prices found in

the Black-Scholes model, as well as the log-normality on which the Black-Scholes model

rests. Finally, the paper will include the impact the Lehman Brothers collapse had on the

Future EURIBOR 3 M.

 

Key words: implied volatility smile, risk neutral density function, second partial derivative

call-put vs strike price, ECB monetary policy expectations.

Abstract

3

• In this paper two different approaches to extrapolate the risk neutral density function

(hereafter referred to as RND) from 3M Liffe EURIBOR option prices traded on the

future prices of EURIBOR 3M will be presented.

• Financial operators are continuously in search of vital information from financial

instruments.

• A valid example is the forward interest rate market that represents the market

consensus with respect to the likely interest rate levels expected in future. Investors

follow derivative markets where they look at the implied volatility smile (hereafter

referred to as IVS) to quantify the sense that there is toward the future risk

perception.

• Two RND estimation techniques will be introduced in more detail further on. Firstly,

an explanation on how to extract RNDs from second partial derivatives of call-put

prices against their strike price(This methodology referred to as the “non-parametric

technique).

• The second methodology consists of using the lognormal density function from which

the RND can be extrapolated using four key measures: IV, future prices (converted to

the interest rate simply by subtracting the price of the future from one hundred), The

maturity of the options and, finally, the mean and standard deviation of lognormal

density function.

Introduction

4

RNDs methodologies

5

• RND is supposed to be the likelihood that the price of an asset follows a

possible dynamic. Quite commonly, RND is thought to be the martingale of

an asset price because the distribution, under which the asset price moves,

follows a martingale process. This theory states that the price of an asset

evolves as a diffusion process better known as geometric Brownian motion;

it is:

Where :

S = the price of the underlying asset

μ = mean of the distribution

σ =standard deviation of the distribution.

dW is a random variable ~N(0,1).

RND methodologies

6

Second call-put partial derivative with respect to the strike price.

7

• The paper written by Breeden and Litzenberger (1976) entitled “Prices of state contingent claims implicit in option prices”, accurately introduced the elementary formula to derive the RND from a combination of option strategies.

• Breeden and Litzenberger managed to demonstrate the thoroughness of this model by replicating the pay-off of a butterfly option strategy.

• The option spread strategy is managed in the following way:

Buy put option OTM with strike price X

Sell put option ATM with strike price Y

Buy call option OTM with strike price J

Sell call option ATM with strike price Y

Second call-put partial derivative with respect to the strike price.

8

• The price strategy could be compared to a variation of the strike price at T which can be expressed as:

• We should compute the price of an elementary claim as:

• Calculating the limit that tends to zero, both Breeden and Litzernberger showed that the result of this equation becomes:

• The above formula represents the second partial derivative of the call price with respect to its strike price.

• In consequence of thinking in the risk neutral world, the fundamental theorem of asset pricing evaluates the price of the call or put option as the expected payoff under a specific probability level, the risk neutral probability, discounted at the risk free interest rate Rf.

Second call-put partial derivative with respect to the strike price.

9

• The second derivative of the call price with respect to the exercise price is equal to the discounted risk neutral density function (RND).

• Considering the “Greeks” of the vanilla options, the second derivative of the call price with respect to the exercise price should be compared to:

Where d2 is:

For being compliant with the Black-Scholes model, the value of s should be thought constant.

Second call-put partial derivative with respect to the strike price.

10

• You have to choose only the option prices that are slightly ITM and far OTM because their liquidity is broadly wider than the ATM option.

• As long as the second partial derivative works correctly, it is necessary for the Black-Scholes call option pricing function to be twice as differentiable as the strike price.

• The monotonicity of both call and put prices are warranted, hence is easier to differentiate their prices against the strike price to get the RND.

Some steps to follow in order to compute the second

call-put partial derivative with respect to K

Second call-put partial derivative with respect to the strike price.

11

RNDs estimated with the lognormal distribution function.

12

• The lognormal distribution function is a reasonable technique selected by researchers among other possible methodologies adopted for this purpose.

• By creating a link with the Black-Scholes assumptions for which the price of a risky asset follows a lognormal dynamic, the RND will certainly be depicted using a lognormal distribution function.

• Lognormal distribution is also characterized by several parameters whose the most important are the mean and its standard deviation:

• Where ST is price of the underlying asset, in this case the level of interest rate is thought as:

RNDs estimation with the Lognormal distribution function.

13

Some steps to follow to compute the Lognormal

RND

• Compute mean and standard deviation of the lognormal distribution function for the pre-selected underlying asset.

• Since RND is extracted from a constant horizon time period, it is necessary to fix the next expiry date both for the call or put option contract before carrying out the calculus of the RND. Indeed, the shape of the RND will be modified by this parameter depending on the mean and standard deviation.

• Converting the price of the future on Liffe EURIBOR 3 M into the level of interest rate.

RNDs estimation with the Lognormal distribution function.

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Fitting the implied Black-Scholes volatility smile using the spline.

15

• The Black-Scholes model is primarily concerned with the concept of implied volatility. This measure is the unknown variable with which the entire model is calibrated.

• Both traders and market participants use IV such as a reliable perception of the future market behavior.

• The input of the interpolation technique is surely the volatility surface typically built by using the pair "IV vs strike prices".

• An important variant should be considered to fit IV:

By switching FROM IV vs strike prices TO IV vs d call/put options.

• This is due to the fact that strike prices could change more frequently than the d call-put options, whereas the d is strictly embedded in the domain zero-one.

• Reducing the domain in which the IV might be chosen, it will allow the shape of the RND to be more flexible and smoother.

Fitting the implied Black-Scholes volatility smile using the spline.

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Procedure for fitting the IV

• Downloading the IV for call/put options prices strictly for OTM and ITM options.

• By switching FROM IV vs strike prices TO IV vs d call/put options.

• Using a polynomial function with fourth order degrees with respect to the selected d.

• Once that the right IV has been chosen, IV should be converted back into the space IV-K

Fitting the implied Black-Scholes volatility smile using the spline.

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Case study: Lehman Brothers market reaction

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• To show the robustness of the RND approach, here it was decided to bring up an example that it replicates pretty good the theory behind the RND.

• Following the chronological events happened promptly after the Lehman Brothers collapse, the monetary authorities had immediately decided to cut off the key interest rates.

• As occurred in US with the FED’s FOMC, also the ECB Governing Council had reduced the massive scare diffused into the markets by stimulating the bond markets.

• The expectation of cutting the key ECB interest rates was already anticipated in the main statistics of RNDs of LIFFE Euribor 3M.

Case study: Lehman Brothers market reaction.

19

• Looking at the chart, from the end of September, it is rather visible which trend both the Liffe 3M EURIBOR and the Forward rate had undertaken.

• Both lines sharply went up and down respectively in conjunction with the Lehman Brothers collapse.

• Both values had anticipated, with high likelihood, the effective cut of the ECB interest rate.

• On the 8th October 2008, ECB brought the main interest rate at 3.75%, at the same level achieved by the EURIBOR 3M forward rate.

Case study: Lehman Brothers market reaction.

20

Case study: Lehman Brothers market reaction.

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• The slides basically show a double and straightforward methodology starting by an alternative interpolation technique that carries to the true value from the options volatility surface.

• The two methods bring at the same results, even though by comparing the shapes of RNDs, it has been emphasized that the lognormal function assures more smoothness for RNDs than that extracted with the second partial derivative.

• By checking with scrupulousness the RND that comes out from second partial derivative, it is easily visible a peak at its top. The vertex is due to the perfect match with the RND that belongs to the call prices with the second half of RND that is linked to the put prices. The lack of the curvature at the top of the RND depends on the choice did up front;

• For preparing the fair RND, those options that are far out of the money and nearly in the money have to be selected for this purpose.

• The main task to be delivered to the people who are searching the RND in order to forecast the evolution of interest rates or whatever they are interested in, is not merely to rely on the outcome, but to put beside it both the consistency of the data quality and the market consensus.

Concluding remarks.

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[1] Mark J.P Anson ”CAIA LEVEL1 An introduction to Core Topics in Alternative Investments”.SECOND EDITION 2013.

[2] Bhupinder Bahra.“Implied risk-neutral probability density function from option prices:theory and application”. Bank

of England Working paper.1997.

[3] Morten Bergendahl Nitteberg. “Implied Risk-Neutral Densities:An application to the WTI Crude Oil market”November

2011. Pg.11

[4] Allan Bodskov Andersen and Tom Wagener. “Extracting risk neutral probability densities by fitting implied volatility

smiles: some methodological points and an application to the 3M EURIBOR futures option prices” European Central Bank

Working paper n° 198. December 2002.

[5] Campa, Chang and Reider (1997).”Implied exchange rate distributions:Evidence from OTC option markets”.NBER

Working Paper no.6179.

[6] Clews, Panigirtzouglou and Proudman.(2000), “Recent developments in extracting information from option markets”.

Bank of England Quarterly Bulletin. February, pp 50-60.

[7] Hull ”Option futures and other derivatives”. Chapter 13th VII Edition.

[8] Hull ”Option futures and other derivatives”. Chapter 16th VII Edition.

[9] Matlab Polynomial Fitting toolbox.

[10] Rupert de Vincent-Humphreys and Joseph Noss. “Estimating probability distributions of future asset prices:

empirical transformations from option-implied risk-neutral to real-world density functions”. Working Paper No. 455 June

2012.

Bloomberg

Thompson Reuters Datastream

European Central Bank Data Warehouse

References

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Anticipating ECB monetary policy decisions through the RND extrapolation from Liffe 3M EURIBOR options.

PierPaolo Cassese

Assicurazioni Generali Finance Department November 2013