Value At Risk (VAR) With Respect To Single Risk Factor

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Value At Risk (VAR) With

Respect To Single Risk Factor

University of Fort Hare East London, South Africa

În articol se prezint un subiect privind dezvoltarea unei valori de risc, model privind un singur factor de risc. Rezult c ecuaţia diferenţial stohastic şi variantele sale pot ﬁ considerate cazuri speciale ale valorii de risc dezvoltate. Folosind valoarea de risc, principala ecuaţie (rezultat) s-a obţinut cu referire la nivelul de încredere “c” al unei poziţii care const din “N” a aceloraşi instrumente care depind numai de un singur factor de risc important “S”. Rezultatele analizei indic c se pot scrie dou formule pentru riscul implicat atât în poziţia lung cât şi scurt în unul şi acelaşi factor de risc.

Abstract: A Working Paper

The paper reports on developing a value at risk (VaR) model with re-spect to a single risk factor. In the process, it shows how stochastic differential equation (SDE) and its variants can be considered as special cases of the VaR framework developed. Using VaR, the main result (equation) was obtained with respect to the conﬁ dence level c of a position consisting of N of the same instruments depending solely on a single underlying risk factor S. The results of the analysis indicate that one can write down the two formulas for risk in-volved in both a long and short position in one and the same risk factor.

Key words: Risk, VaR, Single Risk factor, Market risk, Forecast ho-rizon, Conﬁ dence level, Standard normal distribution, stochastic differential equation, volatility.

1.1 Introduction

This paper reports on developing a value at risk model in respect of risk issues associated with ﬁ nancial institutions (FIs). The research acted as a feasibility analysis on how to model a risk factor using the stochastic dif-ferential equation approach in VaR estimation.

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VaR and its applicability and relevance of particular framework, such as using the VaR for a single risk factor. The section proceeds with the understanding of the concept of value at risk with respect to single risk factor (SRF) followed by its mathematical treatment. The last section summarises on the lessons learnt.

2.1 What is risk and value at risk (VaR)?

There are numerous defi nitions of risk, which are informed princi-pally by the context in which they are applied. Financial and many other insti-tutions need to adopt a defi nition that best contextualises risk in their specifi c environment (Chappell and Dowd, 1999). What really is risk? According to literature (Agarwal and Naik, 2004; McNeil and Frey, 2000; Sahalia and Lo, 2000; Chappell and Dowd 1999; Basel, 1996; Morgan, 1996; Holton, 1997), risk can be broadly defi ned as the degree of uncertainty about future net returns. Arguably, a common classifi cation thus refl ects the fundamen-tal sources of this uncertainty; noting that fi nancial institutions are subject to many sources of risk. Accordingly, the above cited literature disguises four main types of risk. Credit risk relates to the potential loss due to the inability of counterparty to meet its obligations. It has three basic components: credit exposure, probability of default and loss in the event of default. Operational risk takes into account the errors that can be made in instructing payments of settling transactions, and includes fraud and regulation risks. Liquidity risk is caused by an unexpected large and stressful negative cash fl ow over a short period. If, a fi rm has illiquid assets and suddenly needs some liquidity, it may be compelled to sell some of its assets at a discount. Market risk estimates the uncertainty of future earnings, due to the changes in market conditions.

However, the most prominent of these risks noted by a number of recent scholars (Poon, Rockinger and Tawn, 2004; Penza and Bansal, 2000; Basel, 1996) in fi nancial trading is market risk. This arguably is due to the fact that it refl ects the potential economic loss caused by the decrease in the mar-ket value of a portfolio. The question then is; how is this potential economic loss estimated? Value at Risk (VaR) has become the standard measure that fi nancial analysts use to quantify this known risk associated with potential economic loss. For preliminary understanding, what really constitutes VaR? Yamai and Yoshiba, (2005) explain that it may be broadly noted as a number such that there is a probability say p of exhibiting a worse return over the next say k days, where p and k must be predetermined by the risk manager. Quite intuitive? This will be elaborated further.

Nothing from the fact that VaR is quite an intuitive concept inferring from the above, its measurement is a very challenging statistical problem. Al-though the existing models1 for calculating VaR employ different

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be summarised in three points: (1) Mark-to-market the portfolio (2) Estimate the distribution of portfolio returns (3) Compute the VaR of the portfolio. It is imperative to note that the main differences among the VaR methods are related to point 2: that is the way they address the problem of how to estimate the possible changes in the value of the portfolio. In this research, the concern is about similar difference. Thus, exploring how to estimate VaR with respect to single risk factor.

The main objective of this paper therefore was to develop a model using the VaR approach to gauge the impact of modelling the complex, inde-pendent set of variables with various aspects of managing an identiﬁ ed risk factor. For the purpose of the feasibility of the model, and its understanding, functionality and dynamics of the VaR model, assumptions are made on one risk factor. The question then is-why the need for VaR?

2.1.1 The concept of Value at Risk: Relevance and Applicability The are several methods of measuring risk as noted above, one of such methods is volatility; a term that characterises the extent to which the re-turn of an amount of say R2m will ﬂ uctuate from now to a given time period: thus indicating a statistical measure of dispersion.

However, the statistical challenge of volatility is the direction of an investment (amount of say R2m), thus a stock is volatile, because, it suddenly jumps up high- signifying high losses: Noting that investors are not distressed about gains but rather losses (Jorion, 2001; McNeil and Frey 2000; Chappell and Dowd, 1999; Cvitanic, and Karatzas, 1999; Venkataraman, 1997). Thus in most instances, it is a relevant fact that risk is about losing money as already indicated above and VaR characterizes this fact. By this analogy, VaR (assum-ing it is a threshold value) answers the question of a) what is my worst-case scenario? or b) how much could I lose in a really bad month?

As an illustration before the mathematical treatment; see these “if and what” questions as a summary of VaR;

a) If an amount says Rxxx has a one-day 5% VaR of R1 million, there is a 5% probability that the portfolio will fall in value by more than R1 million over a one day period, assuming all is constant withing the set period3.

b) What is the most I can - with a 95% or 99% level of conﬁ dence - expect to lose in Rands over the next month?

c) What is the maximum percentage I can - with 95% or 99% conﬁ -dence - expect to lose over the next year?

What do the above mean to us?

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important step in VaR calculation is to defi ne the three parameters thus: Step 1: VaR forecast horizon- a) the time over which the VaR is calculated b) confi dence level- the probability that the realised changed in portfolio will be less than the VaR prediction (c) the base currency in this case Rands. Noting that for a given portfolio, once the cash fl ows have been defi ed and marked-to-market they need to be mapped to risk metrics vertices.

Step 2: Having mapped all the positions, a decision must be made as to how to compute VaR. If the user is willing to assume that the portfolio return is approximately conditionally normally distributed, then download4 the appropriate data ﬁ les (instrument level VaRs and correlation) and compute VaR using the standard risk metrics approach.

Step 3: If the users portfolio is subject to nonlinear risk to the extent that the assumption of conditional normality is no longer valid, the user can choose between some methodologies5 such as- delta-gamma and structured Monte Carlo. The former is an approximate of the latter. Inferring from the above, one can see how the “VaR question” has three elements of relevance and applicability; a relatively high level of conﬁ dence (typically either 95% or 99%), a time period (a day, a month or a year) and an estimate of invest-ment loss (expressed either in Rand or percentage terms). Thus, common pa-rameters for VaR are 5% and 1% probabilities and one day, one month or one year time horizons, although other combinations are in use. Now, how is VaR estimated given such information with respect to single risk facto? The fol-lowing discussion and modelling answer the objective of the paper, which is underpinned by the above, given information.

2.1.2 Discussing and Modelling VaR with respect to Single Risk factor

In most instances, ﬁ nancial institutions use VaR to evaluate multiple risks, but in this paper, the essence is to use it (VaR) to evaluate a single risk factor. It is imperative to note that more than one VaR model is currently being used and most practitioners have selected an approach based on their speciﬁ c needs, the types of positions they hold, their willingness to trade off accuracy for speed (or vice versa) and many other considerations. Both recent and past studies (Duarte, Longstaff and Yu, 2005; Yamai and Yoshiba, 2005; McNeil and Frey 2000;

Runggaldier and Zaccaria, 2000; Uryase, 2000; Shiryaev, 1999) re-veal that the different models (cf. section 2.1.1) used today differ on basically two spectra, thus;

a) How the changes in the value of ﬁ nances are estimated as a result of market conditions and or regulations

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What leads to the variety of choices of models currently employed is the fact that the choices made on the two spectra mentioned can be mixed and matched in different ways. The following section (2.1.2.1) will address the presentations of the general and detailed account of calculating VaR in respect of the above spectra. This section provides a step-by-step analysis as well the method that is necessary to compute VaR with respect to single risk factor. Thus the following hypothetical description shows how to compute VaR un-der the assumption that these returns are normally distributed.

2.1.2.1 VaR with respect to single risk factor

Before the section starts with the concepts, notice that the notations6 togeth-er with othtogeth-er assumptions are as follows: htogeth-ere as always the notation N(x, y) denotes the normal distribution with mean x and variance y, i.e. N (0, 1) denotes the standard normal distribution. The sign ‘~’ in these circumstances is to be read as ‘is distributed as’. σ meaning volatility, while dInS(t) denotes an inﬁ tesimal random change. With a Wiener process W as a stochastic process which changes randomly by an amount dW over a time interval dt. These changes dW are normally distributed with a mean of zero and with a variance equal to the length of the time interval passed during the change, i.e. with a variance equal to dt. Or written more compactly as:

dW ~ X√dt with X ~ N (0, 1) 3.12

Note that in the above case, 3.12 means in words: ‘dW is distributed as √dt times a random number X. This random number X is in turn distributed according to the standard normal distribution’. With above in mind and other notations being introduced along the way, it will sufﬁ ce to begin.

In order to illustrate the practical application of the concepts just intro-duced, let’s now consider a single risk factor S whose (inﬁ nitesimal) changes are governed by a geometric Brownian motion (GBM) as given in Equations 21.7. In other words, the risk factor satisﬁ es the stochastic differential equation

dInS (t) = μdt + σdW where dW ~ N (0, dt) 21.7

The evolution of S in a ﬁ nite time interval of length t is the solution to the stochastic differential equation 21.7. This solution has already been given in equation 3.24 (see appendix 1.3). It is a stochastic process which describes the distribution of the market parameter S (t + t) a ﬁ nite time step

t later:

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The volatility in this instance σ of the risk factor appearing here is theoretically the same volatility as appears in the differential equation, Equa-tion 21.17. In practice, it is usually calculated as the standard deviaEqua-tion of the relative price changes over a ﬁ nite time span t (for example, through the analysis of histoirical time series),

S (t + t) - S(t)

____________ ≈ In {S(t)}

S(t)

or obtained from the volatility values made available by commercial data providers or internet sites as explained earlier on in section 2.1.1.

As opposed to Equation 21.7 which only holds for inﬁ nitesimal time steps dt, the ﬁ nite time span t in Equation 21.8 can be taken to be arbitrary long. Obviously, since the variance of the normally distributed stochastic component W of the risk factor equals t, the risk can be kept small by keep-ing the length of this time interval small. t should, however, be chosen large enough so that the position concerned can be liquidated within this time span. For this reason, t is sometimes referred to as the liquidation period. In Capital Adequacy Directive CAD II, liquidation periods of t = 10 days are required for internal models; some commercial data providers have, for instance, data for t = 1 day and t = 25 days available.

Now lets consider the value at risk with respect to a speciﬁ ed conﬁ -dence c of a portfolio consisting of a single position in N of one risk factor S. The value of this portfolio at time t is V = N S (t). The change in the value V (t) caused by the change in the risk factor is

V(t) = N S(t) Where S(t) = S(t + t) - S(t)

This case is by no means as special as it may seem. The change in S induces a change in V ampliﬁ ed by the constant factor N. The factor N is, so to speak, the sensitivity of V with respect to S and V is a linear function of S. The interpretation of N as the ‘number of instruments’ in the portfolio is not essential in my deliberations. The same results hold for any portfolio whose change in value is a linear function of the change in the risk factor (or at least approximately so), as is the case for the delta-normal approximation in the variance-covariance method introduced below. The value change V can be explicitly derived from Equation 21.8 as:

V = N S(t + t) – NS(t)

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This allows us to express the probability required in equation 21.3 (see formulary in appendix 2) i.e., the probability that V ≤ - VaR (c), as

cpf _V ( V ≤ - VaR) = cpf _V { N S(t) (eμ t + σ W - 1) ≤ - VaR}

The strategy is now to write the unknown cumulative probability function cpf V in terms of a probability whose distribution is known. The only stochastic variable involved is the Brownian motion whose distribution is given by

W ~ N(0, t) → W ~ X √ t where X ~ N(0, 1)

Lets begin by rewriting the event that V ≤ - VaR with the purpose of isolating the stochastic component W.

cpf _V ( V ≤ - VaR)

= cpf _V { eμ t + σ W - 1 ≤ - VaR/N S(t) }

= cpf _V { eμ t + σ W ≤ 1 - VaR/N S(t) }

= cpf _V { μ t + σ W ≤ In [1 – VaR/N S(t)] }

= cpf _V [ W ≤ In [1 - VaR/N S (t)] - μ t ] ___________________

σ

The probability that W is less than or equal to a certain value is de-pendent on the distribution of W alone and not on that of V. Thus we can therefore simply replace cpf _V with cpf _W: cpf _V ( V ≤ - VaR)

= cpf _V { W ≤ (1/σ) In [1 - VaR/N S(t) ]- (μ/σ) t }

= cpf _V { √ tX ≤ (1/σ) In [1 - VaR/N S(t) ]- (μ/σ) t } = cpf _V (X ≤ a)

having deﬁ ned the parameter ‘a’ as follows

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The probability that X is smaller than a particular variable is depen-dent on the distribution of X alone and not that of W allowing cpf _W to be simply replaced by cpf_x. The probability distribution of X is just the standard normal distribution N (0, 1). Thus the needed cumulative probability function can be written as

cpf _V ( V ≤ - VaR) ∞

= cpf_X (X ≤ a)

= =

The value at risk with respect to the conﬁ dence c is the value sat-isfying Equation 21.3 (see appendix 2). Inserting the above result into this requirement now yields

c = 1 - cpf_X ( V ≤ - VaR) = 1 –

The parameter ‘a’ introduced in Equation 21.10 is thus the (1-c) per-centile of the standard normal distribution

21.11

This value can be determined for any arbitrary conﬁ dence level. Ex-ample of the boundary of a one-sided conﬁ dence interval of the standard nor-mal distribution is

c= 95% = 0.95 → a = ≈ -1.65

c= 99% = 0.99 → a = ≈ -2.325 21.12

Both of this conﬁ dence is frequently used in practice. CAD II7 re-quires c = 99% for internal models, where as some commercial data providers make available data at c = 95% conﬁ dence level.

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a = {In [1 - VaR/N S(t)]- μ t }/σ√ t) aσ√( t) + μ t = In [1 - VaR/N S(t) ]

exp{aσ√( t) + μ t }= 1 - VaR/N S(t)

VaR/N S(t) = 1 – exp{aσ√( t) + μ t}

VaR = N S(t) (1 – exp{aσ√( t) + μ t})

Now using the main result Equation 21.11 we ﬁ nally obtain the value at risk with respect to the conﬁ dence level c of a position consisting of N of the same instruments depending solely on a single underlying risk factor S:

VaR = N S(t) (1 – exp{ μ t +

σ√( t) }) 21.13

Thus, under the assumption that S behaves as the random walk in Equation 21.7, the probability is c that any loss due to this investment over the time span t is smaller than this value at risk.

Now consider the risk of short position consisting of N of the risk fac-tor. The change in the portfolio’s value is given by an equation analogous to Equation 21.9, viz

V = N S(t) (exp{ μ t + μ W}- 1)

and the cumulative distribution faction needed in Equation 21.3 is now cpf_X ( V ≤ - VaR)

= cpf _V { - N S(t) (eμ t + σ W - 1) ≤ - VaR}

= cpf _V {N S(t) (eμ t + σ W - 1) ≥ VaR}

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cpf _V ( V > - VaR) . For the short position, this is given by

cpf _V ( V > - VaR)

= cpf _V {-N S(t) (eμ t + σ W - 1) > - VaR} =cpf _V {N S(t) (eμ t + σ W - 1) < VaR}

=cpf _W { W < (1/σ) In [1 + VaR/N S(t) ]- (μ/σ) t} = cpf _x {X< (1/σ√ t) In [1 + VaR/N S(t) ]- (μ/σ) √ t}

In the second last step, cpf _v was once again replaced with cpf _w since the probability that W is smaller than a particular ﬁ xed value is depen-dent on the distribution of W alone, and not on that of V. The same reason-ing applies in replacreason-ing cpf _w with cpf _x in the last step.

Analogous to Equation 21.10, we introduce the following deﬁ ni-tion8:

ã = (1/σ√ t) In [1 + VaR/N S(t) ]- (μ/σ) √ t} 21.14

It then follows that the deﬁ nition given in 21.2 (see appendix 2) for the value at risk becomes

c = cpf _V ( V >- VaR) = cpf_x (X < ã) =

The parameter ã is thus the c percentile of the standard normal

distri-bution

With a symmetry of the standard normal distribution N (-x) = 1 – N(x) Implies that the percentile obeys the relation9

=

It follows that we can now use the same percentiles as given for in-stance in Equation 21.12 for two commonly used conﬁ dence levels. Inserting

ã =

and solving Equation 21.14 for the value at risk of a short

position yields

VaR = -N S(t) (1 – exp{ ãσ√( t) + μ t })

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we can now write down the two formulas for risk involved in both a long and short position in one and the same risk factor:

VaRlong (c) = N S(t) (1 – exp{ μ t + σ√ t })

VaRshort (c) = -N S(t) (1 – exp{ μ t - σ√ t }) 21.15

3.1 Summary –lessons learnt

These two VaRs in Equation 21.15 are not equal in magnitude! Al-though the probability distribution used in the derivation, namely the standard normal distribution (0, 1) has density function symmetric about 0, the risk of a long position is different to that of a short position!

This is ﬁ rstly due to the drift and secondly to the fact that the normal distribution refers to the distribution of the logarithmic (i.e. relative) price changes of the underlying, as opposed to the VaR’s in Equation 21.15 which contains the lognormally distributed (absolute) price changes themselves.

Notes

1. Parametric (RsikMetrics and GARCH), Nonparametric (Historical Simulation and the Hybrid model) and Semiparametric (Extreme Value Theory, CAVaiR and quasi-maximum likelihood GARCH) models.

2. The number and types of methodologies to VaR estimation is growing exponen-tially and it is impossible to take all of them into account. In particular, the above mentioned methodologies would not be discussed here since this paper’s approach solely concentrates on VaR with respect to single risk factor. I refer the interested reader to the excellent web site www.gloriamundi.org for a compressive listing of VaR contributions.

3. The reason for assuming all is constant witching the set period and to restricting loss to things measured in daily, monthly or yearly accounts is to make the loss observable. In some extreme fi nancial events it can be impossible to determine losses, either because market prices are unavailable or because the loss-bearing institution breaks up. Some longer-term consequenc-es of disasters, such as lawsuits, loss of market confi dence and employee morale and impairment of brand names can take a long time to play out, and may be hard to allocate among specifi c prior decisions. VaR marks the boundary between normal days and extreme events. Institutions can lose far more than the VaR amount; the only thing we can say is they won’t do so very often.

4. That is if the information is readily not available; some ﬁ nancial service providers may offer data for general and public use such as Reuters etc.

5. These are yet of the ways of estimating VaR.

6. This section heavily depends on basic understanding of Stochastic differential equations; (see for instance Deutsch, 2004; McNeil and Frey 2000 ; Øksendal, 2003)

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4 is ﬁ xed by the FSA after auditing of internal bank procedures.

8. The difference to an in Equation. 21.10 lies in the sign of the VaR

9. This cab be seen quite easily: let c = N(x). by deﬁ nition, x is the percentile as-sociated with c. or equivalently, the inverse of the cumulative distribution function gives the percentile Q c = x = N -1_{(N(x)) = N}-1_{(c) . applying N -1 to the symmetry equation N(-x) =} 1-N(x) gives N -1 (N(-x)) = N -1 (1- N(x)) → -x = Q_{1 – N(x)} . Substituting c for N(x) and Qc for x immediately yields -Qc = Q _{1 - c}

Bibliography

- Ait Sahalia, Y. and Lo, A. Nonparametric Risk Management and Implied Risk Aver-sion, Journal of Econometrics 94, 2000, 9–51.

- Chappell, D. and Dowd k. Conﬁ dence Intervals for VaR, Financial Engineering News March, 12, 1999.

- Holton, G. A. Subjective Value at Risk, Financial Engineering News 1, No1, 1997, 1, 8–9, 11.

- Deutsch, H. P. Derivatives and internal models. 3rd ed - Palgrave Houndmills, Bas-ingstoke, Hampshire, 2004

- Duarte, J. Longstaff F. A. and Yu F. Risk and Return in Fixed-Income Arbitrage: Nickels in Front of a Steamroller? Journal of Banking and Finance 20, No3, 2005, 769 - 811. Jorion, P. Value at Risk: The new benchmark for managing risk, 2nd ed:

McGraw-Hill.USA. 2001

- McNeil, A. J. and Frey, R. Estimation of Tail-Related Risk Measures for Heterosce-dastic Financial Time Series: An Extreme Value Approach. Journal of Empirical Finance, 7, 2000,pp 271–300.

- Morgan, J. P. RiskMetrics – Technical Documen,(4th ed), New York, 1996. - Øksendal, B. K. Stochastic Differential Equations: an Introduction with Applica-tions 6th (ed) Strauss GmbH, Morlenbach, 2000.

- Penza, P. and Bansal, V. K. Measuring Market Risk with Value at Risk, Wiley, New York, 2000

- Poon, S. H., Rockinger, M., and Tawn, J. Extreme Value Dependence in Financial Markets, Diagnostics, Models, and Financial Implications. 17, No2, 2004,pp 581 - 610.

- Runggaldier, W. J. and Zaccaria, A. A Stochastic Control Approach to Risk Manage-ment under Restricted Information, Mathematical Finance 10, No2, 2000,pp 277–288.

- Shiryaev, A. N. Essentials of Stochastic Finance: Facts, Models, Theory. World Scientiﬁ c Publishing Co. Pte. Ltd., Singapore, 1999

- Venkataraman, S. Value at Risk for a Mixture of Normal Distributions: The Use of Quasi-Bayesian Estimation Techniques. Economic Perspectives,1997,pp 2–13.

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APPENDIX

Appendix 1.0: Ito Processes and Stochastic Analysis

In the beginning of this paper, risk factors were introduced and a motivation and derivative of an intuitive model (the random walk) was provided to de-scribe the risk factor. These appendices and the entire paper is devoted to the more theoretical fundamentals underlying these concepts introduced, namely to stochastic analysis. Stochastic analysis is a branch of mathematics dealing with the investigation of stochastic processes. Noting that this subject is quite theoretical.

Appendix 1.2: General diffusion Process

All stochastic processes in this paper which were used to model risk factor satisfy – as long as there is only one single random variable involved- an equation of the following form:

dS(t) = a(S, t)dt + b(S, t) dW with dW~ X√dt, X ~ N(0,1) 3.15

Here, w denotes – as always- a Brownian motion and X a standard normally distributed random variable. Processes satisfying an equation of this type are called diffusion processes of Ito process. The parameters a(S, t) and b(S, t) are called the drift rate and volatility of the Ito processes. They may depend on the time t, on the stochastic process S or on both. The interpretation of the variable S depends on the particular application under consideration. For instance in some cases, the logarithm of a risk factor may be modelled as the stochastic variable. In such cases the modes may be of the form 3.15 with stochastic variable being given by In(S), where b(S, t) = σ and a (S, t) = μ.

Appendix 1.3: The process for the Risk Factor over a ﬁ nite Time Interval

Equations 3.15 describe the infi nitesimal changes in S and thus de-termine the differential of S. I am therefore dealing with (partial) differential equation. Because it contains the stochastic component dW, it is referred to as a stochastic partial differential equation (SPDE). With the aid of Ito’s lemma and the general diffusion process, Equation 3.15, an equation for fi nite changes in S (over a fi nite, positive time span t) can be derived by solving the SPDE

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For this purpose, the paper uses the process 3.15 with a(y, t) = 0 and b(y, t) = 1, i.e. simply dy(t) = dW(t). Now lets construct a function S of the stochastic variable y by

S(y, t) = S₀ exp(μt + σy)

Where y (t) = W (t) is the value of the Wiener process at time t, and S0 is an arbitrary factor. Ito’s lemma gives us the process for S induced by the process dy:

dS = ( ) dt + b(y, t) dW

where

Thus dS = (μ + S dt + σSdW

This corresponds exactly to the process in Equation 3.15. This means that the process S thus constructed satisﬁ es the stochastic differential equation 3. 19, ie, is a solution of this SPDE. Simply making the substitution t→ t + t I then obtain

S(t + t) = S₀ exp(μt + μ t + σy(t + t)) = S₀ exp(μ t + σW (t + t)) = S₀ exp(σW(t) + μ t + σ W)

Where in the second step we absorbed exp (μt) in the (still arbitrary) S₀ and in the third step we adopted the notation W for a change in a Brown-ian motion after the passing of the ﬁ nite time interval t:

W = W(t + t) – W(t) → W ~ N(0, 1) 3.23

The ﬁ rst term in the exponent refers to (already known) values at time t. It can also be absorbed into the (still arbitrary) factor S₀ , i.e.

S( t + t) = S₀ exp(μ t + σ W)

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S( t + t) = S (t) exp (μ t + σ W) where W ~ N(0, t) 3.24

Analogously, Equation 3.20 gives the corresponding change in S over a ﬁ nite, positive time span:

S( t + t) = S (t) exp( (

+

where W ~ N(0, 1)

3.25

Appendix 2: Where Conﬁ dence, Percentile and Risk:

C = P ( V > - VaR ( c ) ) 21.2