pricing and hedging short sterling options using neural networks

ICMA Centre Discussion Papers in Finance, DP2011-18

Pricing and Hedging Short Sterling Options Using

Neural Networks

Fei Chen and Charles Sutcliffe

ICMA Centre - University of Reading

September 2011

ICMA Centre Discussion Papers in Finance DP2011-18

ICMA Centre · University of Reading

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Director: Professor John Board, Chair in Finance

The ICMA Centre is supported by the International Capital Market Association

Pricing and Hedging Short Sterling Options UsingNeural Networks

Fei Chen and Charles Sutcliffe* **

ABSTRACT

This paper compares the performance of artificial neural networks (ANNs) with thatof the modified Black model in both pricing and hedging Short Sterling options.Using high frequency data, standard and hybrid ANNs are trained to generate optionprices. The hybrid ANN is significantly superior to both the modified Black modeland the standard ANN in pricing call and put options. Hedge ratios for hedging ShortSterling options positions using Short Sterling futures are produced using thestandard and hybrid ANN pricing models, the modified Black model, and alsostandard and hybrid ANNs trained directly on the hedge ratios. The performance ofhedge ratios from ANNs directly trained on actual hedge ratios is significantlysuperior to those based on a pricing model, and to the modified Black model.

Key Words: Short Sterling, Neural Networks, Option Pricing, Option Hedging

14 September 2011

* The ICMA Centre, University of Reading, PO Box 242, Reading RG6 6BA, UK,[email protected]

** The ICMA Centre, University of Reading, PO Box 242, Reading RG6 6BA, UK,[email protected]

We wish to thank Owain ap Gwilym (Bangor) and Chris Brooks (Reading) for their comments onan earlier draft.

Pricing and Hedging Short Sterling Options Using Neural Networks

1. Introduction

Artificial neural networks (ANN) have proved useful in a large variety of financial applications

including business failure prediction, bond rating assessment, term structure forecasting, volatility

forecasting and financial instrument pricing and hedging. This study focuses on option pricing and

hedging. It compares the performance of a parametric option pricing model with that of an ANN in

pricing and hedging Short Sterling options traded on NYSE Liffe. The underlying asset for Short

Sterling options is futures on 3-month sterling interest rates.

Conventional parametric approaches to option pricing and hedging are based on theory. For some

types of option closed form pricing models have been derived, while the pricing of other options

relies on numerical procedures such as Monte Carlo simulation and the binomial model. These

pricing models are based on theoretical arguments using assumptions concerning the behaviour of

the underlying asset price and the riskless interest rate. It is well known that the conventional

parametric pricing models have systematic biases, due to the use of simplifying and unrealistic

assumptions (e.g. Black, 1975; Rubinstein, 1985; Backus et al., 2004).

ANNs are information processing tools commonly used for function approximation and

classification, and they offer an alternative way of developing option pricing and hedging models.

Their particular strength lies in their ability to approximate highly non-linear and multivariate

relationships without the restrictive assumptions implicit in parametric approaches. This property

of ANNs makes them attractive for problems such as pricing and hedging options. In addition, neural

networks are adaptive and respond to structural changes in the markets. The drawback of this

approach is that it is highly data driven, requiring large quantities of historical prices. Table 1

summarises the main differences and links between parametric models and ANNs.

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Table 1 Comparison of Parametric and ANN Models

Parametric Model ANN

Restrictive assumptions leading tosystematic bias

Recognizes patterns and relationshipswithout restrictive assumptions

Multivariate and non-linearEffectively approximates non-linearfunctions

Fails to adjust to changing marketbehaviour

Adaptive

Easy to apply and no historical datais required

Requires a large amount of data and issometimes unable to converge rapidly

Although many previous researchers have compared ANNs and parametric models in pricing options

and a few of them have also investigated hedging, this study is novel in a number of ways. First, no

earlier study has looked at the pricing of Short Sterling options. Second, most previous papers use

daily or weekly prices for the options and the underlying. However, when daily closing prices are

used, the closing times for trading in options and futures are non-synchronous. This study uses high

frequency data, making it possible to find virtually synchronous options and underlying futures

prices. Third, it is the first to look at hedging interest rate options. Fourth, it is only the second study

after Carverhill and Cheuk (2003) to train an ANN directly on the hedge ratios. Fifth, it is only the

second study after White (2000) to compare the performance of an ANN with the parametric model

of Black (1976), modified for futures-style margining. Finally, it is the first to examine hybrid

hedging models.

Section 2 summarises the previous literature, while section 3 sets out the rival pricing and hedging

models used in this study. In section 4 we discuss the dataset and the implied parameter estimates

we derive, and explain how the models are fitted to the data. Section 5 describes the estimation of

the ANN model, while section 6 has the results and section 7 concludes.

2 Literature Review

2.1 Pricing

Previous studies have used ANNs to price options on a variety of financial instruments. These 98

studies, listed in table 2, have generally found that ANNs produce prices closer to market prices than

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do parametric pricing models, such as Black-Scholes. As table 2 shows, 47 previous studies of the1

pricing of financial options by ANN have focused on equity indices (S&P 500, S&P 100, FTSE 100,

DAX, OMX, KOSPI 200, CAC 40, S. African All Share and S&P CNX Nifty), and 15 have

considered individual equity options. Seventeen studies have analysed the pricing of options on stock

index futures (S&P 500, SPI, TAIFEX, Ibex 35 and Nikkei 225), while two have looked at currency

options, and two have considered options on currency futures. With 12 studies using simulated data,

this leaves just three previous studies of the pricing of interest rate options. White (2000) studied

options on 3-month eurodollar futures which are traded on NYSE Liffe, and so use futures style

margining. Using high frequency data for January to July 1994 and the same inputs as Black-Scholes,

ANN option prices were found to be superior to those of the Modified Black (MB) model . Raberto2

et al (2000) investigated the pricing of options on German Treasury bond futures traded on NYSE

Liffe, but do not appear to have benchmarked their ANN option prices against a parametric model.

Zhou, Yang and Han (2007b) analysed data on three convertible bonds traded on the Shanghai Stock

Exchange. Using Black-Scholes and the binomial model as benchmarks, their ANN prices were

superior.

2.2 Hedging

Hedge ratios can be derived analytically from the chosen parametric pricing model. Since ANNs are

differentiable functions of the input variables, option sensitivities (the Greeks) such as the hedge

ratio can also be derived analytical from the ANN pricing approximation. Alternatively, ANNs can

be trained directly on the desired hedge ratios. There have been 20 previous studies of hedging using

ANNs and they are listed in table 3, but none have analysed interest rate options. All 20 of these

studies derived the hedge ratios analytically from an ANN fitted to prices. In addition, Carverhill and

Cheuk (2003) also used a new ANN to model the hedge ratio directly. They found that the best delta

hedging performance was produced by the binomial model, followed by the new ANN, while hedge

ratios derived from the pricing ANN model were worst.

Most of these studies have been conducted by artificial intelligence researchers.1

The MB model is described below in section 3.1.1.2

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Table 2 Studies Using ANN to Price Financial Options

S&P 500 Andreou et al (2002, 2004, 2006,

2008, 2010)Dugas et al (2001)Garcia & Gençay (1998, 2000)Gençay & Gibson (2007)Gençay & Qi (2001)Gençay & Salih (2003)Ghaziri, Elfakhani & Assi (2000)Ghosn & Bengio (2002)Gradojevic et al (2009)Liu (1996)Qi & Maddala (1996)Saito & Jun (2000)Tzastoudis et al (2006)S&P 500 FuturesCarverhill & Cheuk (2003)Geigle & Aronson (1999)Hamid & Habib (2005)Hutchinson, Lo & Poggio (1994)S&P 100Blynski & Faseruk (2006)Kitamura & Ebisuda (1998)Malliaris et al (1993a, 1993b)Samur & Temur (2009)FTSE 100 Bennell & Sutcliffe (2004)Gregoriou et al (2007)Healy et al (2002, 2003, 2004,

2007)Mostafa & Dillon (2008)Niranjan (1996)Schittenkopf & Dorffner (2001)Ibex 35 FuturesMartel et al (2009)

SPI Futures Boek et al (1995)Lajbcygier (2002, 2003, 2004))Lajbcygier, Boek, Flitman &

Palaniswami (1996)Lajbcygier, Boek, Palaniswami &

Flitman (1995)Lajbcygier & Connor (1997a,

1997b)Lajbcygier & Flitman (1996)Lajbcygier, Flitman, Swan &

Hyndman (1997)DAXAnders, Korn & Schmitt (1998)Hanke (1999a, 1999b)Herrmann et al (1997a & 1997b)Krause (1996)Ormoneit (1999)OMXAmilon (2003)KOSPI 200Choi, Lee, Han & Lee (2004)CAC 40De Winne et al (2001)Nikkei 225 FuturesYao, Li & Tan (2000)TAIFEX FuturesLin & Yeh (2005)Tseng et al (2008)S. African All SharePires (2005)Pires & Marwala (2005)S&P CNX NiftyMitra (2006)Saxena (2008)

Individual EquitiesAmornwattana et al (2007)Dindar & Marwala (2004)Kakati (2005, 2008)Kelly (1994)Lachtermacher et al (1996)Liang, Zhang & Yang (2006) Meissner & Kawano (2001)Pande & Sahu (2006)Pires & Marwala (2004, 2007) Zapart (2002, 2003a, 2003b)Zhou, Yang & Han (2007a)Deustche Mark-US$Carelli, Silani & Stella (2000)Deustche Mark-US$ VolatilityKaraali et al (1997)Sterling-US$ FuturesTeddy, Lai & Quek (2006)Tung & Quek (2005)Eurodollar FuturesWhite (2000)German Treasury Bond FuturesRaberto et al (2000)Convertible BondsZhou, Yang & Han (2007b)Simulated DataBarucci et al (1996, 1997)Charalambous et al (2005)Galindo-Flores (2000)Hanke (1997)Le Roux & Du Toit (2001)Lu & Ohta (2003a, 2003b)Montagna et al (2003)Morelli et al (2004)Tsaih (1999)White (1998)

Table 3 Studies Using ANN to Hedge Financial Options

S&P500Andreou et al (2008, 2010) Garcia & Gençay (2000) Gençay & Qi (2001) S&P500 FuturesCarverhill & Cheuk (2003)Hutchinson, Lo & Poggio (1994) FTSE 100Mostafa & Dillon (2008)Schittenkopf & Dorffner (2001)

SPI FuturesLajbcygier & Connor (1997a) TAIFEX FuturesKo (2009)Ko et al (2005)DAXHerrmann & Narr (1997a, 1997b) Ormoneit (1999) OMXAmilon (2003)

Individual EquitiesKakati (2005)Kelly (1994) Simulated DataHanke (1997) Morelli et al (2004) Ibex 35 FuturesMartel et al (2009)

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3 The Rival Pricing and Hedging Models

The Black (1976) model is a well-established closed-form model for pricing and hedging short term

interest rate options, and options on short term interest rate futures. The performance of various

ANNs will be compared with that of the Modified Black (MB) model.

3.1 Pricing Model

3.1.1 Modified Black Model. NYSE Liffe, which uses futures-style margining for all its option

contracts, trades American-style options on Short Sterling futures. Lieu (1990) modified the Black

(1976) model to price and hedge futures options with futures-style margining. As Lieu (1990) states,

under the futures-style margining system, both the long and the short positions post risk-based initial

margin on entering their option positions. During the life of the option it is marked to market daily,

and gains and losses are paid and collected on a daily basis, just like futures positions. As the entire

option premium is not paid at the time of purchase, option premiums are potentially higher because

the shorts require a higher price to compensate for the loss of interest on the full premium.

Chen and Scott (1993) have shown that the results derived by Lieu can be applied to interest rate

futures options, and that it is never optimal to exercise early an American-style option on an interest

rate future with futures-style margining. Therefore these American-style options can be priced as

though they are European-style using the MB model. The only difference between the Black (1976)

model and the MB model is that the interest rate drops out of the futures-style option formula. This

is because it is no longer necessary for longs to borrow to pay the option premium, or for shorts to

invest the premium. Because the discount factor is less than one, both call and put prices are higher

under futures-style margining. The MB model for pricing short-term interest rate options under

futures-style margining is as follows:

2 1C = [(100!X)×N(!d )]![(100!F)×N(!d )]

1 2P = [(100!F)×N(d )]![(100!X)×N(d )] (1)

1 f X 2 1where d = [ln(R /R )+0.5(S T)]/(S%T)] and d = d !S%T2

fwhere C is the call premium, P is the put premium, F is the futures price, X is the strike price, R is

Xthe interest rate implied by the futures price (i.e., 100!F), R is the interest rate implied by the strike

price (i.e., 100!X), S is the volatility of 3-month interest rates measured by the annual standard

deviation, T is the time to expiration in years, and N(d) is the cumulative probability distribution

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function for a standardized normal variable. Equation (1) requires the value of four parameters:

underlying futures price (F), strike price (X), maturity (T) and volatility (S). Volatility is the only

variable that cannot be directly observed in the market.

The partial derivatives (i.e. hedge ratios) of this MB model with respect to F are as follows:

C 1Delta = äC/äF = N(!d )(2)

P 1Delta = äP/äF = !N(d )

The goal of delta neutral hedging is to make the value of a portfolio of options, futures and bonds

immune to changes in the price of the underlying asset. However, a delta hedged portfolio involving

options is only immune to small changes in the underlying asset. To maintain delta neutrality the

portfolio position must be adjusted frequently.

3.1.2 Standard ANN Pricing Model. The standard ANN (SANN) uses actual option market prices

as the output. We chose to use the Multilayer Perceptron (MLP). MLP is also known as a back

propagation network which is by far the most popular type of neural network. Hornik et al. (1990)

showed that a one-hidden-layer MLP can approximate a large class of linear and non-linear functions

with arbitrary precision.

A back propagation network is trained using a two-step procedure. The activity from the input

pattern flows forward through the network, and the error signal flows backward to adjust the weights.

The objective function is to minimize the squared error between the observed option prices and the

ANN output, as shown in equation (3).

iMin 0.53 (d(i)!o(i)) (3)2

where d(i) is the desired output, and o(i) is the model output.

In order to construct an MLP, various decisions must be made including: the number of hidden

layers, the number of processing elements, and the transfer function. Choices must also be made for

the input and output variables and the length of the training and testing periods. Our standard ANN

(SANN) has three inputs: moneyness, volatility and time to maturity. The network is trained on

actual option prices from NYSE Liffe.

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3.1.3 Hybrid ANN Model. A hybrid ANN (HANN) is also used, where the target output is the

difference between the actual option market price and that estimated by the parametric model. The

idea of constructing a hybrid model is to use the MB model as a base, and allow the ANN to augment

its performance. In effect the HANN is modelling the bias in the parametric pricing model. This bias

is then added to the MB price to produce the final HANN model output. Previous studies have shown

that the hybrid model usually outperforms both the parametric and SANN models (see Lajbcygier

et al., 1997; Lajbcygier, 2003; Andreou et al., 2006). This superior performance is achievable

because the parametric model is highly accurate, thus providing a good building block for

statistically-based models, such as ANNs, to model their biases. In addition, accurate option pricing

can be achieved for illiquid options, such as deep-in-the-money or very long maturity options,

because, unlike ANNs, the accuracy of parametric models is unaffected by liquidity.

3.2 Hedging Model

Delta hedging performance is also examined in this study. The delta of the MB model is defined in

equation (2), and we use two ANN hedging models, based on either option prices (analytical hedging

models) or price changes (directly trained hedging models).

3.2.1 Analytical Hedging Model (SANN1 and HANN1). Delta measures the sensitivity of the option

price to underlying price movements. Once the network is trained on prices, delta can be analytically

derived from both the SANN and HANN models. For SANN, the partial derivatives of the output

(Price/Strike) with respect to one of the inputs (Moneyness) give the delta, called SANN1. The

HANN model output is the bias of the MB Model (MB Price/Strike)!(Market Price/Strike) called

HANN1. The HANN model delta is the MB delta minus the sensitivity of the ANN output to

moneyness.

3.2.2 Directly Trained Hedging Model (SANN2 and HANN2). In this case the hedge ratio is modelled

directly. SANN2 trains the network on the desired delta, and HANN2 trains the network on the

difference between the MB model delta and the desired delta. Changes in option prices and futures

prices for the same contract are used to compute the desired delta.

4 Data

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Short Sterling futures traded on NYSE Liffe are listed on a quarterly expiry cycle of March, June,

September and December. In addition two serial months are traded on two of the next three months,

so that at any time 23 delivery months are available, with the nearest three delivery months being

consecutive. Options on Short Sterling futures are listed on the nearest ten futures, with identical

expiration dates and times. Every day, ten contracts with different maturities are traded - eight on a

quarterly cycle, and two serial options. The strike price intervals are 0.125 for all serial delivery

months and the nearest four quarterly delivery months, and 0.25 for the remaining quarterly delivery

months. This study uses tick data on Short Sterling call and put options and Short Sterling futures

traded on NYSE Liffe for 4th January 2005 to 29th December 2006. Table 4 summarises the options

and futures data.

Table 4 Observations in the Options and Futures Database3

Call Options

Ask Bid TradeSpreadTrade

BlockTrade

Volatility All

2005 18055 16374 3615 6005 2161 - 46210

2006 504652 450177 3735 7070 2970 - 968604

Put Options

2005 16762 16766 2452 3340 828 - 40148

2006 517241 446903 2239 3306 1488 - 971177

Futures

2005 12697759 12579266 772710 566368 346 560 26617009

2006 22701523 22738706 1012472 907558 728 587 47361577

There are five types of options data: Ask, Bid, Trade, Spread Trade and Block Trade, where Ask and

Bid are quote prices, and all other data types are trades. For pricing and hedging purposes only

options trade data is considered, giving 39,209 observations, of which 25,556 are call prices and

13,563 are put prices. When matching each options trade with an underlying futures price all the

available futures data, including quote prices, is considered . If there is no futures quote or trade price4

The massive increase in the number of recorded quotes in 2006 is because it includes quotes of the same price3

at the same second, whereas in 2005 two or more quotes at the same price occurring simultaneously were

recorded as only one quote.

Although subject to bid-ask bounce, trades represent economic transactions, while quotes can be4

unrepresentative of market prices. The unreliability of quotes is particularly important for option quotes a long

way from the money. This moneyness-related problem is not relevant for futures quotes, making them more

reliable than options quotes. There is a need to use synchronous futures and options prices, but the modest

number of options trades makes it impossible to generate a large sample of synchronous trade prices. Since there

is a very large number of futures quotes with tight bid-ask spreads, options trade prices were synchronised with

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available at the same second as an options trade, the time-weighted average of the futures price

immediately before and after the option trade is used. For 98% of the options trades, the time

difference between the options and underlying futures prices is less than one minute.

For Short Sterling options with futures-style margining, the European-style put-call parity gives the

following lower boundary conditions on options prices:

C > F ! X (4)

P > X ! F (5)

Of the 39,209 options trades, 448 trades fail to satisfy the boundary condition in either equation (4)

or (5), and are deleted. The futures data from 1 February 2005 to 4 February 2005 is missing fromst th

the raw database supplied by NYSE Liffe and so this period is dropped from the analysis. In addition,

options with a time to maturity of less than seven calendar days are deleted. The cleansed database

has 38,531 call and put trades.

The MB model requires four inputs (volatility, time to maturity, the futures price and the strike

price). These four inputs are also used for the pricing ANNs. In common with most previous studies,

the variables are transformed using the homogeneity hint (Garcia and Gençay, 1998). Provided

returns on the underlying asset are distributed independently of its price, the MB model is

homogeneous of degree one in the underlying and the strike, and so both sides of equation (1) can

be divided through by the strike price. Therefore, the inputs are volatility, time to maturity and

f X Xmoneyness (R /R ), while the output is C/R . Time to maturity (T) is measured as the number of

calendar days until expiration, divided by the number of calendar days in that year.

To price an option, the most important input is the expected volatility (S). Some studies of S&P 500

options use historical volatility estimates, or the VIX. The present study uses the volatility implied

by the price of recent options trades. Computing implied volatilities at time t involves solving the

MB model in equation (1) iteratively for volatility, given the observed values of the other inputs at

time t. For a specified expiration date and strike price, the implied volatilities for calls and puts are

computed using option trades over the past 15 days (i.e. t!15 days) up to five minutes before the

futures quotes and trades. This enables very closely synchronized options and futures prices to be obtained.

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trade being priced (i.e. t!5 mins). This five minute lag is to permit the option pricing model to be

implemented in real time. The volatility used in the MB and ANN models is the equally weighted

average of these implied volatilities. Some options, such as serial trades, are highly illiquid. If less

than ten implied volatility estimates are available, the option trade is excluded. This leads to the

exclusion of 1,042 trades, and the database for pricing and hedging options now contains 37,489

option observations.

For options on Short Sterling futures, the current minimum price movement is 0.005% (0.5 basis

points). Since the contract size of the future is £500,000, this is equivalent to (£500,000×0.005%÷4),

i.e., £6.25 . The raw data was recorded to four decimal places until May 2006, and to three decimal5

places thereafter but without a decimal point. Therefore in the raw data base, the minimum price

movement is five units before May 2006 and 50 units afterwards. To convert the price data into

percentages it was divided by 1,000 before May 2006, and then by 10,000, so that the inputs

(underlying and strike) are measured as (100!Yield).

The data is further filtered for the pricing and hedging models. We set the maximum time to maturity

at 360 days, maximum option prices at 0.5 (99.97% quantile), the maximum moneyness at 1.2 for

calls and the minimum moneyness at 0.85 for puts. Therefore, the sample space is restricted by

excluding deep-in-the-money options (normally with extreme prices) and very long date options.

These account for 5.8% of the call volume and 5.6% of the put volume.

5 Estimating the ANN Models

The data is separated into six categories for pricing: all calls, all puts, out-of-the-money (OTM) calls

and puts, and in-the-money (ITM) calls and puts. Table 5 summaries the training and testing sample

sizes.

Division by four is because the tick size of 0.005% is a 3 month rate, which needs to be increased to an annual5

rate for the computation of the monetary value corresponding to one tick.

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Table 5 Sample Size Summary

No. of Obs. Size of Training Sample No. of Testing Periods

All Call 23226 6000 43

All Put 12104 6000 15

OTM Call 19314 6000 33

OTM Put 8681 6000 6

ITM Call 3747 2000 4

ITM Put 3307 2000 3

For most categories, the size of the training sample is 6,000 observations. As we only have 3,747

ITM calls and 3,307 ITM puts, the training sample is 2,000 for the ITM options. The out-of-sample

testing samples are the 400 subsequent observations after the end of each training sample. Using a

rolling windows approach, the training and testing samples are then rolled forward by 400

observations, and the ANN re-estimated. For each training sample, weights and bias information is

used to derive the hedge ratios. The ANN design uses a three layer MLP with 10 neurons in the

hidden layer. A tan-sigmoid transfer function is applied for the hidden layer and a linear function is

applied for the output layer of the pricing models (SANN1 and HANN1). A tan-sigmoid transfer

function is applied for both the hidden and output layers of the hedging models (SANN2 and

HANN2).Given a sufficient number of neurons in the hidden layer, this network can approximate

any function with a finite number of discontinuities arbitrarily well. There are a number of back

propagation algorithms in the Matlab neural network toolbox, and we use the Levenberg-Marquardt

algorithm as it is much faster than the other methods. Cross validation has been widely used in

training neural networks to prevent over-fitting, and since it is part of the Matlab “train” function,

it is applied automatically in our analysis.

6 Results

6.1 Pricing Model Performance

Using the homogeneity hint improves the MLP performance, relative to using the underlying and

strike as separate inputs; i.e. C/K = f(S, T, M) and P/K = f(S, T, M) with the inputs of moneyness (M),

volatility (S) and time to maturity (T). Compared with other input combinations, this specification

gives the best results for all categories of option. Volume was tried as an additional input, but pricing

accuracy becomes slightly worse.

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Although ANNs are trained to minimise the sum of the squared errors, there is no agreement on the

appropriate measure of pricing accuracy. In this study, four alternative summary measures of

performance are used: (a) mean squared error (MSE); (b) mean absolute error (MAE); © the

correlation between actual and computed prices; and (d) mean error (ME)

MSE = (3(y!x) /n (6)2

MAE = (3|y!x|)/n (7)

ME = (3(y!x))/n (8)

where x is the actual value of the dependent variable, y is the estimated value of the dependent

variable, and n is the number of observations.

Table 6 shows the performance of MB, SANN and HANN in terms of MSE, MAE, correlation and

ME for the overall out-of-sample period. The figures in bold in table 6 indicate the estimation

technique with the best score. Each ANN is trained to minimise the total squared error, and so the

MSE is used as the prime performance measure. Using the MSE criterion HANN is best in every

case. For the MAE criterion HANN is best in five cases. For correlation, MB is best in all cases.

Using ME, SANN is the best in three cases, while HANN is the best for the other three. Thus,

overall, HANN appears to be the preferable technique.

To test whether these results are statistically significant, a paired t-test is the obvious choice.

However, this test requires that the differences between the errors from the two models are normally

distributed. As the data does not satisfy this assumption, the bootstrap method is applied as a non-

parametric alternative to the paired t-test. The bootstrap is a procedure that involves choosing

random samples with replacement from a data set. In our case the MSE numbers in table 6 indicate

that HANN is the best model, but how certain is this conclusion? We define the difference in the

MSEs as the test statistic. The sign column in tables 7 and 8 are the signs of the MSE of model A

minus the MSE of model B, and P indicates a positive difference, while N indicates a negative

difference. To determine whether the sign is significant, squared errors for the two models are re-

sampled 1,000 times and a distribution for the test statistic produced. Tables 7 and 8 have the

bootstrap test results for calls and puts. For each category, we compare the three models pairwise.

For example, we first compare the MB model with the SANN model in the All Call Option category.

The sign is positive, which means that the mean error of MB (Model A) is higher than SANN (Model

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B). To test whether the difference is significant, we compute the 95% confidence interval. If both

the lower bound and the higher bound are positive, we conclude that the test statistic is significantly

positive at the 95% confidence level and Model B is preferred. If the lower bound is less than zero

and the higher bound is greater than zero, we conclude that the difference in MSE is not significantly

different from zero. If both bounds are negative, we conclude that the difference is negative and

Model A is preferred.

Table 6 Pricing Performance of ANNs and MB

MSE MAE Correlation ME

AllCall

Options

MB 0.0093% 0.6452% 98.6864% 0.2363%

SANN 0.0087% 0.6095% 98.4930% 0.1144%

HANN 0.0082% 0.5796% 98.5620% 0.1029%

AllPut

Options

MB 0.0148% 0.9368% 99.2792% 0.6621%

SANN 0.0124% 0.8116% 99.1469% !0.0439%

HANN 0.0112% 0.7796% 99.2310% 0.0150%

OTMCall

Options

MB 0.0083% 0.5844% 97.3122% 0.1589%

SANN 0.0084% 0.5608% 96.8118% 0.0657%

HANN 0.0078% 0.5236% 97.0618% 0.0911%

OTMPut

Options

MB 0.0100% 0.7070% 97.7356% 0.4021%

SANN 0.0103% 0.6793% 97.3966% !0.3110%

HANN 0.0091% 0.6209% 97.7037% !0.2883%

ITMCall

Options

MB 0.0187% 1.0941% 98.9414% 0.9262%

SANN 0.0167% 1.0493% 98.3328% !0.0552%

HANN 0.0164% 1.0570% 98.3271% !0.1077%

ITMPut

Options

MB 0.0128% 0.8969% 99.5818% 0.5243%

SANN 0.0111% 0.8022% 99.4794% !0.0453%

HANN 0.0105% 0.7845% 99.5075% !0.0646%

Exactly the same orderings are found for both call and put options. For all call options and all put

options categories, the HANN model is clearly superior to SANN, which is superior to MB. For

OTM call and put options, HANN is superior to both MB and SANN, while for ITM call and put

options MB is inferior to both HANN and SANN. So for OTM calls and puts HANN is clearly

superior, while for ITM calls and puts it is not inferior to any other pricing model. Therefore, for

pricing Short Sterling call and put options, the hybrid ANN is superior to the standard ANN, and

both are superior to the MB model.

13

Table 7 Significance Tests for Calls

Model A Model B Sign2.5% LowerBound

2.5%UpperBound

Significance Rank

All Call Options

MB SANN P 4.42E!06 8.71E!06 Plus SANN > MB

MB HANN P 9.05E!06 1.29E!05 Plus HANN > MB

SANN HANN P 2.85E!06 6.13E!06 Plus HANN > SANN

OTM Call Options

MB SANN N !3.22E!06 2.61E!06 Insignificant Insignificant



ITM Call Options



SANN HANN P !4.40E!06 9.20E!06 Insignificant Insignificant

Note: The test statistic is the MSE of Model A minus the MSE of Model B. The Sign column is the sign of the MSE of

model A minus the MSE of model B, and P indicates a positive difference, while N indicates a negative difference. The

significance column is the significance of the test statistics. Plus indicates that the test statistic is significantly positive,

while Insignificant indicates that the test statistic is not significantly different from zero.

Table 8 Significance Tests for Puts


2.5%UpperBound

Significance Rank

All Put Options




OTM Put Options

MB SANN N !1.13E!05 4.55E!06 Insignificant Insignificant



ITM Put Options



SANN HANN P !5.60E!07 1.22E!05 Insignificant Insignificant

See notes for table 7.

14

6.2 Hedging Model Performance

Option pricing theory is based on replicating an option with a portfolio of other assets. In a stock-

style margining system, options can be replicated using positions in the underlying security and a

money market account (or a bond position). In a futures-style margining system, a bond position is

not required, and options can be hedged using just futures contracts. Rival hedge ratios (deltas) are

obtained from the MB model and an ANN. Portfolios of options and futures are formed which hedge

against changes in the underlying futures price, with the hedged portfolio being rebalanced in

discrete time.

ANN hedge ratios can be obtained in two different ways. The partial derivatives with respect to the

underlying futures price can be computed analytically from an ANN pricing model - i.e. analytical

hedging models. In this case the ANN is treated as a function, and the chain rule of differentiation

applied. This can be done because the transfer functions for both the hidden and output layers are

differentiable. SANN1 and HANN1 deltas are derived from the pricing models SANN and HANN

respectively. In the case of HANN1, the output from the pricing model is the bias of the MB model.

Therefore, the HANN1 delta is the partial derivative of the bias with respect to the underlying futures

price, plus the MB delta. Alternatively, a new ANN can be trained directly to target the desired hedge

ratios, i.e. directly trained hedging models. We fit a standard (SANN2) and a hybrid (HANN2) ANN.

HANN2 aims to adjust the MB hedge ratios using a non-parametric enhancement. The idea of

HANN2 is the same as the other hybrid models where the MB deltas are an additional input to the

ANN, and the MB hedge ratio biases are the output.

6.2.1 Analytical Hedging Model (SANN1 and HANN1). Deltas can be derived analytically from the

two ANNs used for pricing. Figures 1 to 4 show the scatter plot for SANN1 deltas and HANN1

deltas compared with MB deltas. Figures 1 and 2 are deltas estimated for calls, and figures 3 and 4

are deltas estimated for puts. The x-axis is the MB delta and the y-axis is the ANN delta. The

diagonal line indicates y=x. Deltas derived from the pricing models are more or less in line with the

MB deltas for both call and put options, especially when options are OTM. When options are ITM,

the ANN deltas tend to be bigger, and can be far away from the corresponding MB delta. As the non-

parametric model has no constraint on the range of delta, it can occasionally be greater than one, or

less than minus one.

15

The distributions of the estimated deltas are also shown in figures 1 to 4. We can see that the ANN

delta distributions have much longer tails, but are still similar to the MB delta distributions,

especially for puts. For both calls and puts, the MB deltas have a slightly lower correlation with the

SANN1 deltas (calls 0.988; puts 0.989) than with the HANN1 deltas (calls 0.990; puts 0.993).

Hedging performance will be discussed in section 6.2.2, where the analytical hedging models will

be compared with the directly trained hedging models.

Figure 1 SANN1 Deltas Versus MB Deltas (Calls)Figure 2 HANN1 Deltas Versus MB Deltas (Calls)Figure 3 SANN1 Deltas Versus MB Deltas (Puts)Figure 4 HANN1 Deltas Versus MB Deltas (Puts)

6.2.2 Directly Trained Hedging Model (SANN2 and HANN2). As the aim of delta hedging is to

replicate changes in option prices (ÄP) using delta multiplied by the change in the underlying futures

price (ÄF), a delta of (ÄP/ÄF) serves as a perfect hedge ratio. Two new ANNs are trained to directly

estimate deltas - a standard ANN model (SANN2) and a hybrid ANN model (HANN2). The input

variables for SANN2 are the same as for SANN1: moneyness, time to maturity and volatility, while

the output is the desired delta, defined as the actual (ÄP/ÄF). For HANN2 the MB delta is included

as an input (as it is for HANN1), while the output is the MB delta’s bias, relative to the actual delta.

The aim of training each ANN is to learn how the hedge ratio varies with moneyness, time to

maturity and volatility. These sensitivities are defined for very small changes in prices, and so high

frequency data is used to train the ANNs. However, most institutions do not implement intraday

hedging because of the considerable transaction costs, and so hedging performance is tested using

daily data. The ANNs are trained using the data previously used for the two pricing ANNs

(henceforward, the high frequency database), while testing uses a daily database constructed from

this high frequency database. This data is used to train and test the four ANNs (SANN1, HANN1,

SANN2 and HANN2).

In the pricing section we considered 23,226 call and 12,104 put observations. The training data

comprises the odd observations (11,613 calls and 6,052 puts) from the high frequency database, and

the resulting models are tested on daily observations constructed from the even observations (11,613

calls and 6,052 puts) of the high frequency database. This odd-even method of splitting the data for

training and testing is often used in ANN applications (see Wikel et al., 1996, Mirsepassi, 2004, and

16

Bowers and Shedrow, 2000), and is superior to simply using the first half for training, and the second

half for testing. While the size of the training data set is still half the data, the daily out-of-sample

dataset is much bigger than half the available contract days. This is because, after removing the odd

data for training, as long as contracts generally have more than one observation per day, this permits

the computation of a daily value of (ÄP/ÄF) for most contracts.

To get the desired delta (ÄP/ÄF), the high frequency data is sorted by strike and maturity, so that

observations of the same contract are “stacked” to produce a single sequence of observations. The

odd sequence of observations is then differenced to produce actual values for (ÄP/ÄF). As these

ratios are much noisier than option prices, some restrictions are applied to this high frequency data

on which the ANNs are trained to assist in finding a function that links the inputs with the output.

Since delta is a slope measure for a continuous option pricing function, the time difference between

observations is restricted to a maximum of two days. As there can be two observations at the same

time, a minimum time change of 0.01 days is also applied. Sometimes the option price makes a big

movement while the futures price hardly moves, and vice versa, and so the actual delta can be as high

as 10 (or !10), or as low as 0. To find a relatively accurate sensitivity formula, only actual values

of (ÄP/ÄF) in the same range as the theoretical delta (0 to 1 for calls, and !1 to 0 for puts) are

considered as the output when training each ANN. Only price changes with the same strike and

maturity are considered. The application of these restrictions reduces the number of values of

(ÄP/ÄF) to 2,153 calls and 1,211 puts. The four ANNs were trained on this high frequency data set.

Since the performance of daily hedges is being examined, testing the ANNs is conducted using a

daily data set. The even high frequency data (11,613 calls and 6,052 puts) have not been used in

constructing the high frequency data set used for training the models, and so are out-of-sample. For

contracts with more than one observation in a day, the even observation at, or closest to, 15:00 is

chosen. Some options are not traded every day and the time difference between two observations for

a contract can be as long as a month, so the hedging rebalance period between the daily prices is

restricted to be no more than two days. Applying this restriction produces 2,816 daily values of

(ÄP/ÄF) for calls and 1,675 for puts. The four ANNs are then tested on these daily data sets.

17

Table 9 Hedging Data Set Summary

Out-of-SampleWhole Sample

Effective Training Effective Test

Calls 2,153 2,816 23,226

Puts 1,211 1,675 12,104

The six correlations between the deltas of each of the analytical hedging models (MB, SANN1 and

HANN1) and the two directly trained hedging models (SANN2 and HANN2) are between 0.75 and

0.90, while the correlation between SANN2 and HANN2 is 0.81 for calls and 0.85 for puts. This

indicates that, while the deltas for the three analytical hedging models were very similar with a

correlation of about 0.99, those for the directly trained hedging models are more dissimilar. There

is also a difference in the mean deltas and their variances, as shown in table 10. The directly trained

hedging models have larger absolute deltas of just under a half, with a much lower variance.

Table 10: Means and Variances of the Deltas

Call Deltas Put Deltas

Mean Variance Mean Variance

MB 0.292 0.042 !0.403 0.046

SANN1 0.271 0.045 !0.396 0.056

HANN1 0.277 0.031 !0.344 0.046

SANN2 0.434 0.008 !0.459 0.007

HANN2 0.436 0.007 !0.445 0.015

Hedging performance is measured using MSE, MAE, correlation and ME as in the pricing section.

Error is defined as ÄP!(delta×ÄF). Table 11 has the hedging performance for MB, SANN1,

HANN1, SANN2, and HANN2 out-of-sample. Apart from the MAE measure, the directly trained

models (SANN2 and HANN2) are clearly superior for both calls and puts. Table 12 contains the

pair-wise rankings of the five models that are statistically significant at the 5% level. These

statistically significant rankings enable the creation of the two quasi-orderings (one per data set)

which appear in figure 5.

In every case the two best models are the direct trained hedging models (SANN2 and HANN2),

which are significantly superior to both the analytical hedging models (SANN1 and HANN1) and

the MB model. For call options SANN2 and HANN2 are equally attractive, while for puts SANN2

18

dominates HANN2. Since HANN2 has more information than SANN2, its lack of dominance for

hedging is surprising.

Table 11 Out-of-Sample Hedging Performance

Hedging MSE MAE Correlation ME

Calls

MB 0.087% 0.987% 63.360% 0.442%

SANN1 0.090% 0.992% 61.972% 0.450%

HANN1 0.087% 1.001% 62.801% 0.431%

SANN2 0.071% 1.168% 72.916% 0.198%

HANN2 0.069% 1.124% 72.285% 0.269%

Puts

MB 0.228% 1.552% 70.297% 1.005%

SANN1 0.237% 1.571% 70.209% 1.037%

HANN1 0.203% 1.472% 68.503% 0.946%

SANN2 0.143% 1.519% 72.619% 0.759%

HANN2 0.160% 1.505% 67.623% 0.725%

Table 12: Significance Tests for Out-of-Sample Testing

Out-of-sample Call (Hedging Model 2)

Model A Model B Sign2.5%

Lower Bound

2.5% Upper Bound

Significance Rank

MB SANN1 !1 -6.13E-05 4.62E-06 Insignificant Insignificant

MB HANN1 1 -1.34E-05 1.77E-05 Insignificant Insignificant

SANN1 HANN1 1 -9.33E-06 7.27E-05 Insignificant Insignificant

MB SANN2 1 3.94E-06 3.37E-04 Plus SANN2 > MB

MB HANN2 1 5.66E-05 3.10E-04 Plus HANN2 > MB

SANN2 HANN2 1 -4.84E-05 7.59E-05 Insignificant Insignificant

SANN1 SANN2 1 2.44E-05 3.71E-04 Plus SANN2 > SANN1

HANN1 HANN2 1 5.39E-05 3.03E-04 Plus HANN2 > HANN1

HANN2 SANN1 !1 -3.43E-04 -7.44E-05 Minus HANN2 > SANN1

HANN1 SANN2 1 5.06E-05 3.50E-04 Plus SANN2 > HANN1

19

Out-of-sample Put (Hedging Model 2)


2.5% UpperBound

Significance Rank

MB SANN1 !1 -2.35E-04 8.14E-05 Insignificant Insignificant


SANN1 HANN1 1 1.82E-04 5.17E-04 Plus HANN1 > SANN1

MB SANN2 1 4.51E-04 1.28E-03 Plus SANN2 > MB


SANN2 HANN2 !1 -3.37E-04 -1.57E-05 Minus SANN2 > HANN2

SANN1 SANN2 1 5.36E-04 1.42E-03 Plus SANN2 > SANN1

HANN1 HANN2 1 1.43E-04 7.62E-04 Plus HANN2 > HANN1

HANN2 SANN1 !1 -1.25E-03 -3.70E-04 Minus HANN2 > SANN1

HANN1 SANN2 1 2.73E-04 9.67E-04 Plus SANN2 > HANN1

Note: The Sign column are the signs of the MSE of model A minus the MSE of model B, while 1 indicates a positive

difference and -1 indicates a negative difference. The Significance columns gives the significance of the test statistics.

Plus indicates that the test statistic is significantly positive. Insignificant indicates that the test statistic is not significantly

different from zero. Minus indicates that the test statistic is significantly negative.

Figure 5 Quasi-Order Diagrams for Hedging Models

7 Conclusions

The aim of this research is to investigate the use of artificial neural networks (ANN) as a tool for

pricing and delta hedging interest rate options, and Short Sterling options were selected for testing.

The MB model is widely used to price such options, and is used to benchmark the performance of

the ANNs. In our analysis we have used the average historical implied volatility as an estimate for

expected volatility. Neural networks are fitted using the Levenberg-Marquardt training algorithm.

It is shown that ANNs are a suitable tool for both pricing and hedging Short Sterling options,

capturing the nonlinear behaviour of option prices and hedge ratio deltas with a high degree of

accuracy. The results for pricing call and put options show that the hybrid ANN model is

significantly superior to the MB model. The standard ANN is indistinguishable from the MB model

for OTM options, but outperforms the MB model when both ITM and OTM options are tested.

We considered the performance of both analytical hedging ANN models and directly trained hedging

ANN models in hedging Short Sterling options. The analytical hedging models are based on the

ANN pricing model, and their hedging performance is practically indistinguishable from that of the

20

MB model. In contrast to Carverhill and Cheuk (2003) and S&P500 futures options, for Short

Sterling options we find that the directly trained hedging ANN models provide significantly better

hedging results than the MB and analytical models. Surprisingly, although the hybrid ANN is

superior for pricing, HANN2 is not dominant for hedging.

Neural networks are therefore a promising alternative to closed-form models in both pricing and

hedging options, and may play an important role for exotic options for which no closed-form model

exists. However, as fitting an ANN is data intensive, there may be problems in pricing and hedging

newly traded or illiquid options.

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Figure 1 SANN1 Deltas Versus MB Deltas (Calls)

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Figure 2 HANN1 Deltas Versus MB Deltas (Calls)

Figure 3 SANN1 Deltas Versus MB Deltas (Puts)

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Figure 4 HANN1 Deltas Versus MB Deltas (Puts)

Figure 5 Quasi-Order Diagrams for Hedging Models

Puts Quasi Ordering Out of SampleCalls Quasi Ordering Out of Sample

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pricing and hedging short sterling options using neural networks

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