A Subjective and Probabilistic Approach to Derivatives

0.19 MB·English

by  Ulrich Kirchner

A Subjective and Probabilistic Approach to Derivatives
U. Kirchner
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ICAP
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0 PO Box 1210, Houghton, 2041, South Africa
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[email protected]

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a January 11, 2010
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1
Abstract
tic approach with dynamic hedging. On the other

] hand one can simply calculate the expected pay-off
R
We propose a probabilistic framework for pricing for the log-normal probability distribution with an
P derivatives,whichacknowledgesthatinformationand expected return equal to the risk-free rate.
.
n beliefs are subjective. Market prices can be trans- The second approach is considerably simpler,
fi latedintoimpliedprobabilities. Inparticular,futures neater, anda directconsequenceof probabilisticrea-
- imply returns for these implied probability distribu- soning. However,itcannotexplainwhytheexpected
q
[ tions. We arguethat volatilityis notrisk,butuncer- return should be equal to the risk-free rate — and
tainty. Non-normal distributions combine the risk in presumably that is the reason why this derivation is
1
thelefttailwiththeopportunitiesintherighttail— hardly ever presented.
v
6 unifying the “risk premium” with the possible loss. We want to offer an alternative framework in-
1 Riskandrewardmustbepartofthesamepictureand spiredbythe secondapproach,which, inessence,ac-
6 expected returns must include possible losses due to cepts the subjectivity of information/beliefs with a
1 risks. Bayesian interpretation of probabilities [1] and rein-
1. We reinterpret the Black-Scholes pricing formulas terprets the meaning of asset prices, futures, and
0 as prices for maximum-entropy probability distribu- risk. It yields the Black-Scholes pricing equations
0 tions, illuminating their importance from a new an- (but without the usual assumptions like no trading
1
gle. costs) as maximum-entropy solutions for known first
:
v Using these ideas we show how derivatives can be two moments, but the concept extends beyond this
Xi priced under “uncertain uncertainty” and how this special case. We remove hedging arguments and re-
creates a skew for the implied volatilities. place them with probabilistic reasoning1.
r
a We argue that the current standard approach One could best describe our approach as a proba-
based on stochastic modelling and risk-neutral pric- bilistic one embracing subjectivity and Bayesian in-
ing fails to account for subjectivity in markets and ference, while the traditional approach is based on
mistreats uncertainty as risk. Furthermore, it is statistical/stochastic ideas and methods. These su-
founded on a questionable argument — that uncer- perficially appear as if they are objective in nature
tainty is eliminated at all cost. as they do not use prior distributions and do not ex-
plicitly involve subjective information. This, how-
1 Introduction ever,hidesthefactthatparticularpriors(usuallyas-
sumptionsaboutnormality)arealreadyassumedand
TherearetwowaystojustifytheBlack-Scholesequa- 1Given the solution hedging prescriptions can then be re-
tion. On the one hand there is the standardstochas- constructed.
1
hence limit the applicability of many methods. Also, risk. Consider for example the standard log-normal
one could argue that the neglect of available infor- case with
mationbesides historicdatais ratherashort-coming dS =µSdt+σSdX, (1)
than an advantage. A good overview of these issues
where X is a “normally distributed random vari-
is given in [1].
able”4. It is clear that dS becomes ‘increasingly de-
Thispaperisorganizedasfollows. Westartbyex-
terministic’ as σ goes to zero — for σ = 0 it just
posing short-comings of the traditional approach in
expresses the deterministic drift. Hence we want to
section 2. In section 3 we present our approach and
argue that dS is really a mixture of a certain and
discuss the interpretation of futures and risk. In sec-
uncertain component. As one cannot eliminate the
tion 4 we discuss the interpretation of Black-Scholes
dependence of the portfolio value change on the dif-
pricing in our context. In section 5 we present ap-
ferentcomponents ofdS separately,itis notpossible
plications of the probabilistic framework to pricing
to follow a hedging strategy which keeps the drift,
underanuncertainsecondmoment(uncertainuncer-
but eliminates the randomness.
tainty),whichnaturallyleadstoanimpliedvolatility
Now, by eliminating dS one also discards the non-
skew, derivative exposure and risk management.
random drift term and the resulting hedging is gen-
erally (except for a drift equal to the risk free rate)
2 The Traditional Approach suboptimal. This is most easily illustrated in ex-
treme (hypothetical) cases with huge (compared to
and its Problems
the volatility) positive or negative drift.
It appears as if it is often not understood what it
The standard approach to derivatives pricing based
really means to have a drift in the distribution. This
on stochastic models and risk-neutral measures (see
is notjustsomethingobservedinhistoricaldata,but
appendix A for a short review) raises a number of
somethingweclaimtoknowaboutthepossibilitiesin
issues. Firstly, it does not allow for subjectivity as
thefuture. Ifweusealog-normaldistributionwitha
it assumes the existence of an objective “real distri-
1000% drift rate and 10% volatility then we also say
bution”, and hence does not take into account that
that it is practically impossible to observe a draw-
peoplehavedifferentinformationandbelieves2. This
down overthe next year (and even to see a return of
effectively leads to the idea that there is a “correct”
less than 900%) — and we claim to know this.
value for a derivative, independently of its current
Yetrisk-neutralpricingtellsustoignoretheknowl-
trading price — it would possibly be “mispriced”.
edge of the extreme drift and to hedge as if there is
We think that this is misleading. Every market
a 50/50 chance of a draw-dawn (below the risk-free
participant will assign a different value according to
rate). Having a short put-delta position will lead us
his state of knowledge and beliefs. If there are sell-
with near certainty to buy back the asset at higher
ers and buyers with different subjective valuations,
prices.
it might result in trades. These in turn might re-
Figure 1 illustrates this. The x-axis shows the un-
sultinobservedmarketprices. One shouldnote that
derlying asset value, with x = 1 being the current
the market price then does not necessarily represent
spotprice. Thesolidline intheupper graphisalog-
the subjective valuation of neither the buyer nor the
normal distribution with an expected return equal
seller3.
to the risk-free rate (here 10%). The lower graph
Secondly, even if we assume this awkward“under-
shows the correspondingBlack-Scholesput-delta po-
lying real distribution”, we find it difficult to fol-
sition for a 110% strike. For this case it makes sense
low the risk-neutral hedging argument, as the elim-
inated term does not only contain uncertainty and 4Itisquestionablewhethertheconceptofarandomvariable
can beconsistently defined [1]. Onecanarguethat “random-
2 A good pricing theory should be able to value an asset ness” is a result of insufficient information about the state of
evenforaninside-trader. thesystem. Arandomvariableresultsthenfromuncontrolled
3SeeappendixE. (andhenceunknown)initialconditions.
2
0.6 Again, risk-neutral pricing gives us counter-intuitive
risk-neutral log-normal
0.5 anotohreigr lloogg--nnoorrmmaall advisetoignoreourknowledgeoftheexpectedreturn.
0.4 Notethatthepointisnotwhetherweknowtheex-
0.3
pected return or not. The expected return is part of
0.2
whatwebelieveabouttheasset. Ifwearenotcertain
0.1
0 aboutitsvalueweshouldincorporatethisuncertainty
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x usingprobabilisticmethods,insteadofmodellingour
0 BS put delta for 110% strike knowledgeandthenthrowingpartofthemodelaway.
-0.2
-0.4
3 An Alternative Approach
-0.6
-0.8
-1 No “real” distribution InlinewiththeBayesian
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x interpretation of probabilities [1] we do not believe
thatthereisanobjective“real”distribution. Historic
data tells us what materialized, not what we should
Figure1: Threelog-normalprobabilitydistributions
believeaboutthefuture,andarguablynotevenwhat
(expressing our subjective state of knowledge) with
we should have believed (see appendix D). In fact,
different drift rates and the corresponding Black-
duetoselectioneffectshistoricdatacanbeextremely
Scholes delta for a put with strike at the risk-free
misleading —think ofhedge-fundmanagersshowing
rate. Risk-neutral hedging suggests we use the same
“consistent alpha” until they blow up. It appears
deltaindependentlyofthedrift,i.e.forallthreecases.
to be a common mistake to interpret historic data
as representative of a “real distribution” for future
returns as suggested by statistics.
as the partial short position corresponds reasonably
An objectionoftenraisedagainstthe Bayesianap-
to the two possible outcomes.
proachisitsrelianceonsubjectivepriordistributions.
The two coloured lines in the upper graph are
We wanttopointoutthatevenusinghistoricdatais
two other log-normal distributions of the same (log-
asubjectivematter—wejudgesubjectivelywhether
return) standard deviation, but with different ex-
the data can simply be extrapolated to the future.
pected returns (drift). The blue graph represents a
This is fine if one is aware of above selection effects
case for which we are practically certain that the as-
andrepresentativityissuesandjudges themtobe in-
set price will have declined over the next year. For
significant. If,however,thistreatmentisduetoanin-
such a case it is clear that one would hedge a 110%
terpretationofthe standardstochasticapproachand
put by immediately shorting the full amount, as we
above issues are ignored, one makes dangerous im-
know that the asset will decline in value.
plicit assumptions about the connection of the past
Risk-neutral pricing would tell us though to price and the future. The markets have enough examples
and hedge the position as if the expected return was of these, like misjudged risks on mortgages, credit
actually the risk-free rate. If the blue graph repre- default swaps, and Ponzi schemes. In this context
sents our state of knowledge,then this does not look pricing derivatives off a “risk-neutral” distribution
like a good idea. appears like a safety precaution — too good things
Thecasefortheredlineinfigure1issimilar. Here (drift) should be balanced by bad things, which are
itisknownwithalmostcertaintythattheassetvalue notincludedinthe“realdistribution”ofthestandard
will rise above the strike level, and hence that we approach.
shouldnot havea shortposition atexpiry. It is clear Not everything is, and can possibly be in the his-
that in such a case we would value the put worthless toric data. This data is conditional on where we are
and we would also not hold a short delta position. nowanddrawingassumptionsaboutthefutureintro-
3
duces its own risks. This then has to be a subjective There is just an implied distribution, of which fu-
process involving all information available5 and not tures determine the expected return. This implied
just historic data. expected return is not necessarily the risk-free rate,
but a return compatible with hedging expenses (and
feasibility) (this is a well-known fact for commodi-
Subjective and Implied Probability Distribu-
ties withstoragecostslikeoil,butalsoforperishable
tions We argue that every market-participant has
agricultural products). By pricing options on such
their own subjective probability distribution based
assetsoffthe futures valueoneimplicitly usesthe fu-
on the information available to him. Because peo-
tures price as an expected return — now different to
ple have different information available (and differ-
the risk-free rate.
ent levels of rationality when forming beliefs about
thefuture)thedistributionsandhencevaluationsare
subjective. Interaction of Spot and Future Markets We
Thesedifferingvaluationsthenleadtotradingand first observe here that a future arbitrage trade does
the observed market prices determine implied prob- not only move the futures price, but also the spot
abilities — one can consider them the beliefs of the priceasassetswillhavetobeboughtorshorted. The
market as an “organism”6. Futures imply expected neteffectisamovetowardsequilibriumofbothmar-
returns for these implied distributions, but more de- kets. The less liquid market will adjust more than
tailed implied distribution features can be extracted the liquid one7.
from option prices (volatility smiles) [2]. Hence it is misleading to say that arbitrage does
Hence we argue that there are only the beliefs of implyfuturesprices. Arbitragetradesmovespotand
eachmarketparticipant(whichdonothavetobehu- futures marketstobe compatiblewithhedgingcosts.
man,ascomputertradingillustrates)andanimplied (See figure 2 for an analogy from physics.) In effect
probabilitydistribution(orforsparsedataratherim- this enforces an implied expected return compatible
plied probabilities), which represents the “beliefs of with the hedging cost. For assets with no storage
the market”. and transaction costs and available funding at the
risk-free rate, this is the risk-free rate. But if the
impliedexpectedreturnisthe risk-freerate,whereis
Futures Futuresarearguablythemostsimplecon-
the risk premium?
tracts and they tell us directly the implied expected
return.
In the traditional approach it is argued that this 3.1 Risk and Risk Premium
is true, but it is the expected return of the “risk-
Part of the confusions seems to originate from the
neutral” distribution (which obviously should be the
mislabelling of uncertainty as risk. It is clear that
risk-freerate)andthe“real”distributioncouldhavea
there are situations where we are not certain about
differentexpectedreturn,andinfactgenerallyshould
the precise outcome, yet we are certain that we will
have to allow for a risk premium.
nothaveanadverseoutcome. Suchasituationisrep-
We disagree and argue that there is nothing like a
resented by the green line in figure 1 — here we are
“real” distribution that exists objectively.
almost certain not to lose money (with a long posi-
5 Interesting feedbacks ariseas people use historicdata as tion),butthereisasignificantuncertainty(standard
information. If it is known how people use historic data this deviation) about the precise outcome.
influencestheinterpretationofthehistoricdataitself. Wejust Should uncertaintythenattractarisk-premiumas
wanttomentionheremomentum effects.
6 In fact, the economy as a system seems to satisfy most is for example assumedin Modern PortfolioTheory?
criteria for a living organism. Even Darwinian evolution can
be observed in the way business structures, political systems, 7 Note that the spot market isnot always the moreliquid
andcommercialproductschange overtime. one. Oneexampleareilliquidgrainmarkets.
4
1.2 1.2-3-3 -2.5-2.5 -2-2 -1.5-1.5 -1-1 ppaarrtt-0.5-0.5iiaall ddiissttrrii 0 0bbuuttiioonn 0.5 0.5 1 1 1.2 1.2
1 1 1 1
0.8 0.8 0.8 0.8
pp 0.6 0.6 0.6 0.6
closed 0.4 0.4 0.4 0.4
0.2 0.2 0.2 0.2
0 0-3-3 -2-2 -1-1 0 0 1 1 0 0
lloogg rreettuurrnn
Figure 3: A distribution, appearingclose to normal,
as one might estimate from historic returns. There
appears to be a meanabove the risk-free rate (10%),
which would traditionally be interpreted as a risk-
closed
premium for the uncertainty (variance or volatility).
Figure 2: An example from physics illustrating how
a common equilibrium is established due to interac- We would argue in two ways: Firstly, uncertainty
tion. Under the influence of gravity two columns of isnotwhatwarrantsapremiumasitisaninsufficient
liquids interact. The flow of liquids is the analogue descriptionofrisk. Andsecondly,therisk-premiumis
to the flow of assets through time (e.g.being removed amisleadingconceptasitsuggeststhatthepremium
from the spot market and stored for later disposal) ispartofthedistribution,butnottheriskwereceive
betweenthefutureandspotmarkets. A“flowagainst it for.
time” corresponds to short positions flowing forward We believe risk is best defined as the probability
in time (like the travelling of ‘holes’ in semiconduc- of an adverse outcome as suitably defined for a cer-
tors). If, for example, the left side corresponds to tain situation8. Hence the risk should be part of the
the spot market and the boundary between the liq- distribution describing our beliefs about all possible
uids represents the asset price, then the arrow on outcomes. If we are aware of possible negative out-
toprepresentsassetsbeing carriedto the future, and comeswewilldemandcompensatingpossiblepositive
the arrow at the bottom represents borrowed cash outcomes.
positions to fund the arbitrage. The analogy could Figure 3 shows a partial probability distribution
be taken further with friction and viscosity allowing which seems to have a significant “risk-premium” of
slightly different levels on each side. about 15% above the risk-free rate of 10% (all log
returns). Traditionally that would be seen as a com-
pensation for the uncertainty (standard deviation)
present.
Figure 4 then shows the full distribution, which is
actually a maximum-entropydistribution for a mean
8 Or through a probability distribution for adverse out-
comes.
5
1.2 1.2-3-3 -2.5-2.5 -2-2 -1.5-1.5 -1-1 ff-0.5-0.5uullll ddiissttrrii 0 0bbuuttiioonn 0.5 0.5 1 1 1.2 1.2 sdcisatlreisbuwtiitohnsthoevesrhlaorge-spriinceisssauned) hoenneceshlooug-ldretluoronks.at
Secondly, the log-normal probability distribution
1 1 1 1
is a normal distribution for log-prices, and hence it
0.8 0.8 0.8 0.8 is the maximum-entropy distribution for known vari-
ance and mean. This means the distribution makes
pp 0.6 0.6 0.6 0.6 the “least assumptions” besides what is known (the
first two moments). So whether the actual log-
0.4 0.4 0.4 0.4 returns “are normal” or not is not the question —
if we have only given the first two moments as a de-
0.2 0.2 0.2 0.2 scription of our beliefs then the normal distribution
will be the right choice.
0 0-3-3 -2-2 -1-1 0 0 1 1 0 0
lloogg rreettuurrnn Thirdly, if we have no unique extra information
(and historic data is available to everyone) then we
might want to go with the “markets belief” of an
Figure 4: The full distribution contains the risk
expected return as priced in by the futures.
in the left tail. This balances with the peak above
the risk-freerateto give anexpectedreturncloseror Given above, the Black-Scholes pricing formulas
equal to the risk-free rate. can easily be derived as the present value of the ex-
pected pay-off under a log-normal probability distri-
bution with a mean value of the risk-free rate (see
equal to the risk-free rate and given values for the
appendix B). This derivation is in fact considerably
second, third, and fourth moment. Here we see that
simpler than the stochastic differential equation ap-
the “risk-premium” is really compensated by a fat
proach10 — for example there is no need to invoke
left-hand tail. The expected value contains both —
Itoˆ’s lemma11.
the risk-reward, if risks do not materialize, and the
risk-penalty, if risks do materialize. Of course, that is not to say that we couldn’t do
Hencewearguethat“realdistributions”(inpartic- better. Firstly, we might not be certain of the stan-
ular log-normalwith a drift higher than the risk-free dard deviation, in which case our result incorporates
rate) are the wrong way to incorporate risk-reward, a wrong sense of certainty in a parameter — see sec-
as they incorporate a risk premium but do not ac- tion5.1below,where weshowthatanuncertainsec-
count sufficiently for the downside-risk. ondmomentleadstoaskew. Secondly,wemighthave
more information about the probability distribution,
e.g., higher moments. Not using such information
4 Connection to Classical Pric-
will of course be suboptimal.
ing Ideas
Sogiventheseconcerns,whyisBlack-Scholesdoinga
reasonable9 job? To answer this question let us first 10It is interesting though that the Greens-function to the
Black-Scholes partial differential equation is nothing else but
note a number of facts.
the log-normal distribution with the risk-free rate as mean.
Firstly, one can argue that due to scale-invariance Sothewholeeffectofthedetour over thestochastic pdeisto
(as the actual value of a share is meaning less — it eliminatethedrift.
11TheequivalentofItoˆ’slemmainaprobabilistictreatment
9 Andreasonabledoesnotmeangreat. Theobservedskew istrivial: Iftheprobabilitydistributionforln(x)isaGaussian
showsthattheBlack-Scholespricingequationsareinsufficient. withmeanνandstandarddeviationσthentheexpectedvalue
Seesection5.1,whichaddressestheseissues. See[3]forinter- of x is eν+σ2/2. If the expected value of x is known to be er
estingcommentsonthehistoryandrelevanceofBlack-Scholes. thenwehavetosetν=r−σ2/2.
6
5 Applications
5.1 Unknown Variance
0.03
Letusassumeweaccepttherisk-freerateofreturnas
our expected future asset return, but we are not cer- 0.025
tainaboutthesecondmoment(variance)foracertain
date. How can we value an European option under 0.02
such circumstances?
Inaprobabilisticframeworkthiscanbedealtwith prob 0.015
easily by marginalizing over the second moment pa-
rameter (Black-Scholes ‘volatility’). Starting with 0.01
the definition of the derivatives value as the present
value of the expected cash-flows12 we find 0.005
∞
V = e−rt dxf(x)p(xI) 00 0.5 1 1.5 2
| volatility
Z0
∞ ∞
= e−rt dσ p(σ I) dxf(x)p(xσI)
| |
Z0 Z0
∞ Figure5: Log-normalprobabilitydistributionforthe
= dσ p(σ I)Vbs(r,σ), (2) standard deviation (Black-Scholes volatility).
|
Z0
where f(x) is the pay-off function, Vbs(r,σ) is the
Black-Scholespriceforgivenrisk-freerateandsecond
moment σ2 (‘variance’), and p(σ I) is expressing our
|
beliefs about the possible values of the (square-root
of the) second moment.
1.4
Figures 5, 6, and 7 give a graphical example of
above relationship. Figure 5 shows a probability dis- 1.2
tribution describing our beliefs about what the stan-
dard deviation (Black-Scholes ‘volatility’) could be. 1
As the standard deviation is a positive quantity we
0.8
considerthelog-standarddeviationmotivatedbythe put price
scale-invarianceargument[1]. Thesecondordermax- 0.6
imumentropydistributionforthestandarddeviation
isthen,justasforthepriceintheBlack-Scholescase, 0.4
a log-normal distribution. The resulting put option
0.2
valuations and the implied volatility skew are pre-
sented in figures 6 and 7. 00 0.5 1 1.5 2 2.5 3
One could now consider introducing additional rel strike
“Greeks” for the parameters of the distribution of
the standard deviation. If, as in the example, a
log-normal distribution was used then the resulting Figure 6: Example put prices for certain standard
Greeks follow directly from differentiating (2). deviation (green line) and for a log-normalprobabil-
itydistributionforthestandarddeviation(blueline).
12 Weassumeherethattherearenodividendsandallother
parametersareknown,i.e.expectedreturn,timetoexpiry,and
risk-freerate.
7
where V is the value based on our subjective proba-
i
0.65 bility distribution.
We suggest to maximize the objective function
0.6 ξ(n ,...,n ) = Π Πm subject to constraints on
1 N
−
selected risk and possibly exposure measures (e.g.no
0.55 short positions). Two examples of possible risk con-
straints are
volatility 0.5
the probability of a loss (including initial costs)
•
0.45 mustbe smallerthany (this measureis onlyde-
pendent on the relative number of contracts)
0.4
if there is a negative final portfolio value, its ex-
•
0.350 0.5 1 1.5 2 2.5 3 pected absolute value is less than z (expected
rel strike value conditional on that there is a loss).
In two dimensions this optimization can easily be
Figure 7: The resulting implied probability skew done graphically.
given the probability distribution of figure 5 for the
second moment.
6 Conclusion
5.2 Exposure Management We criticised the stochastic risk-neutral pricing ap-
proach for its lack of subjectivity and its mistreat-
Let us assume we have a set of derivatives (on the
ment of uncertainty as risk. In particular, risk-
same underlying) we can invest in. Prices of these
neutral pricing is founded on the questionable argu-
aredeterminedbythemarket,andhencearecompat-
ment that any drift is worth sacrificing to eliminate
ible with the implied probability distribution p (x).
m whatever uncertainty there is.
Our valuations and associated risks though are de-
In this paper an alternative approach was pre-
termined by our subjective probability distribution
sented which acknowledges that information and ex-
p(xI).
| pectationsaresubjective. Marketparticipantsshould
Consider a portfolio consisting of n contracts of
i use Bayesianprobabilisticreasoningto rationallyex-
instrument i. The current portfolio market value is
press their beliefs in terms of probabilities.
Observed marketprices can be translatedinto im-
N
Πm = n Vm, (3) plied probability distributions, for which futures im-
i i
ply expected returns.
i=1
X
Givenno significantextrainformationit is reason-
where Vim is the current market value of instrument able for market participants to adopt this expecta-
i. The value of the portfolio to us is13 tion. We showed that if a market participant is ad-
ditionally sure about the standard deviation then he
N
could rationally (due to a maximum entropy argu-
Π= n V , (4)
i i
ment)useBlack-Scholespricingtosubjectivelyvalue
i=1
X his position. If he is not sure about the standard
13 Here we make the simplifying assumption that Vi does deviation he would value his position according to a
not depend onthe nj. This istrueformostinvestments, but skew.
not in general. Consider the value of bottled water to you in
We commented on possible applications to deriva-
thedesert,whereyouaredehydrated. Thefirstbottlewillbe
considerablymoreworthtoyouthenthe1,000’sbottle. tives exposure and risk management. Furthermore,
8
we considered the case of uncertain standard devia- to benecessary,becauseotherwisetherewouldbe no
tion. Foralog-normalprobabilitydistributionforthe risk-premium. Hence the dogma that futures tell us
standard deviation we illustrated how a “volatility- nothing about the “real” distribution.
smile” was implied.
While basedonsimple probabilisticprinciples,our
B Deriving Black-Scholes from
approach is quite a deviation from what are current
standard methods in finance (and particularly in fi- the log-normal Distribution
nancialmathematics)andwebelievethatitoffersan
interestingdifferentperspectiveonfinancialmarkets. Let ν and σˆ2 be the firstmoment and secondcentral
moment of the log-price probability distribution for
sometimetfromnow,i.e.the distributionexpressing
A Origins of risk-neutral Pric-
our beliefs about future market levels. Having given
ing nootherinformationourbestguessisthemaximum-
entropy distribution for the log-price given the first
All well known textbooks present the standard ap- two moments as constraints. This is a Gaussian dis-
proachbased on stochastic differential equations, for tribution for the log-price, or the log-normal distri-
example [4, 5]. bution for the price, which is given by
Risk-neutralpricinghasitsoriginsinthestochastic
1 [ln(x) ν]2
approach to derivatives. Hence it is based on the p(x)= exp − . (6)
idea thatthere is a “real”distributiondescribingthe xσˆ√2π ( − 2σˆ2 )
underlying return process (the “random variable”).
Substituting z = ln(x)−ν we find
In the Black-Scholes framework the “random pro- σˆ
cess”oftheunderlyingassetpriceisdescribedbythe
K 1 z(K)
infinitesimal log-normalrandom price movements p(x)dx = e−z2/2 dz
Z0 √2π Z−∞
dS =µSdt+σSdX, (5) = N([ln(K) ν]/σˆ), (7)
−
where S is the asset price, µ is the drift, σ is the where N(K) is the cumulative normal distribution
volatility, and dX is a normally distributed random function
variable. This is the only uncertain/random contri- N(x)d=ef 1 x dz e−z22. (8)
bution to the price evolution. Hence it is argued 2π Z−∞
that a portfolio is “risk-free” if this term is elimi- Similarlyonefinds withz asaboveandtheinverse
natedthroughasuitable(dynamic)hedgingstrategy, x = exp(σˆz +ν) (note the usual completing of the
in which case the portfolio should earn cash returns. square in the exponent)
Assuming this is done one arrives at the risk-neutral
K z(K) exp(σˆz+ν)
pricing formulas, which do not depend on the drift xp(x)dx = e−z2/2dz
rate µ. Z0 Z−∞ √2π
Thisconceptisextendedtootherdistributions,for ν+σˆ2 1 z(K) −(z−σˆ)2
whichthe risk-neutralmeasureis linkedto the “real- = e 2 e 2 dz
√2π −∞
world” measure through a measure transformation, Z
which sets the expected return to the risk-free rate. = eν+2σˆ2N ln(K)−ν σˆ . (9)
σˆ −
Thisthenleadstothenotionthattherearetwodis- (cid:18) (cid:19)
tributions — the “real” distribution underlying the As K this gives the first moment m of x (ex-
1
processandthe “risk-neutral”distributionaccording pected→va∞lue) as
towhichderivativesshouldbepriced. Itisimportant
to understandthatthe “real”distributionis thought m =eν+σˆ2/2. (10)
1
9
Let us assume now that the first moment m is and ∆ the number of units of the underlying asset.
1
known to be equal to the present value x times the The portfolio value is then given by
0
risk-free growth factor, i.e.
Π=V +∆S. (15)
m =x ert. (11)
1 0
Differentiating with respect to the underlying asset
Substituting from above and solving for ν (the first value S and demanding that the portfolio value is
moment of the log-price) gives invariant gives
ν =ln(x )+rt σˆ2/2. (12) ∂Π ∂V
0 − 0= = +∆, (16)
∂S ∂S
For this value of ν let us define
and hence
∂V
ln K rt+σˆ2/2 ∆= . (17)
d d=ef[ln(K) ν]/σˆ = x0 − (13) −∂S
− 2 − (cid:16) (cid:17) σˆ
Forexample,fortheunknownvariancecaseofsection
and d d=efd +σˆ. 5.1thisgives(assumingthatp(σ I)isindependentof
1 2 |
the spot level)
The value ofan Europeanput with strike K is the
present value of the expected cash-flow. Hence ∞
∆= dσ p(σ I)∆bs(r,σ), (18)
K − |
V = e−rt dx(K x)p(x) Z0
p
−
Z0 where p(σ I) is the probability distribution for the
= e−rt KN( d ) eν+σˆ2/2N( d ) standardd|eviationand∆bs(r,σ)istheBlack-Scholes
2 1
− − −
delta for given standard deviation and risk-free rate.
= e−rtK(cid:16)N( d ) x N( d ), (cid:17)(14)
2 0 1
− − −
where we used (7) and (9) to evaluate the integral. D Real Uncertainties
Given the second moment as an annualized variance
σˆ2 = σ2t this agrees with the Black-Scholes pricing One might believe that generally probabilities are
formula (for no dividends). about inferring a correct, but unknown value. For
The calculation for European call and binary example,peoplemightaskthemselveswherethemar-
put/call are very similar. ket will be on a certain date, and then see whether
they guessed the “correct” value.
The problemherearisesasitisimplicitly assumed
C Reconstructing Hedging
that the “correct value” exists objectively without
Prescriptions being dependent on the actualbetting game and our
associated behaviour.
In this paper we emphasized that risk is not uncer- Letsconsideraspecificexample. Apparentlythere
tainty. However, if one wants to eliminate uncer- wasalotterysysteminIrelandwherethe“luckynum-
taintyonecanderivea“classichedgingprescription” bers” were assigned by a machine at the till. It so
fromagivenpricingequation. This isverysimilarto happenedthatthelaterwinnerwasallowedbysome-
theusualprocedure,exceptthatthepricinghasbeen oneelsetogoaheadandgethisnumbersfirst. Hence
found independently. the winner claimed later that if that other person
Let us assume we want to create a portfolio of a would have not let him go first, he would have not
derivative and its underlying asset such as to have received the winning numbers.
it momentarily invariant under asset price changes. This certainly would be true if the numbers would
Let V(S) be our subjective value of the derivative have been drawn already at that point in time (and
10