Risk-Neutral Pricing of European Call Options: A Specious Concept

Risk-neutral pricing of European call options is investigated from a mathematical point-of-view and is found to be a specious concept 1 . Risk-neutral pricing of European call options is an approximation in which all terms of order ( ) r α σ − are ignored, where r α− is the risk premium and σ is the volatility


Introduction
The concept of risk-neutral pricing of European call options is investigated from a mathematical approach.It is found that risk-neutral pricing used in the pricing of European call options is a specious concept [1] [2] [3] that is only approximately correct and that ignores terms of ( ) , where r α − is the risk premium and σ 2 is the variance of the one-day return of the asset that underlies the call option.The risk premium equals ( ) r α − with α the true drift rate of the underlying option and r the risk-free rate.

Background
First some notation and background information.Let the probability density function (pdf) for a random variable x be ( ) Specious "refers to something that appears at first encounter to be genuine or to be soundly argued or reasoned" [1].A specious argument is "an argument that seems correct only if you do not think about it carefully" [2].Something specious is "seemingly well-reasoned, plausible or true, but actually fallacious" [3].
Let t S be the value of an asset at time t.
and, for simplicity in notation, that When returns over non-lapping time periods are independent, the pdf for the n-day return is the n-fold convolution of the pdf of the 1-day return since the n-day return is the sum of n independent 1-day returns.Variances add under convolution, and hence the variance of n-day returns is n times the variance that describes the distribution of the 1-day returns.Since the normal distribution is stable under self-convolution, if the 1-day return distribution is a normal distribution with a mean μ and a variance σ 2 , then the pdf for an n-day return is a normal distribution with a mean nµ = and with a variance The returns of assets are better described by Student's t-distributions than by normal distributions.However, Student's t-distributions are not stable under self-convolution and have fat tails.The fat tails lead to integrals that diverge and these integrals are needed to price options.Both of these characteristics make pricing with Student's t-distributions difficult [4] [5] [6] [7] [8].For simplicity and to be specific, the pricing of European call options for returns with normal distributions is discussed in this note.In particular, attention is paid to the mathematical basis for risk-neutral pricing of European call options.The results apply to fat tailed distributions as well [9].
Following [10] [11], the price of a European call option at time T is

S
is the expectation of the maximum value of 0 or the difference ( ) of the value of the asset T S and the strike price T K at time T. At time 0 t = , when the option is purchased, the value of the option is ( ) as the sale of the option is a cash transaction and therefore the value of the option at time T, T C , is discounted by the time value of money, with r the risk-free rate.

Black-Scholes Option Pricing Formula
The Black-Scholes option pricing formula gives prices for European call options and is obtained under the constraints of: 1) no arbitrage; 2) the price of an asset is described by a geometric Brownian motion process with support [ ] , −∞ +∞ ; 3) risk-neutral pricing; and, 4) the future price is a martingale.Constraint 1) requires, from the arbitrage theorem [12] [13] [14], that Constraints 3) and 4) require { } ( ) where t S is the price of the asset at time t, 0 S is the price at time 0 t = , and r is the risk-free rate.Given that t S is a geometric Brownian motion process derived from a stochastic process , t n R with mean n µ and variance 2 n σ , i.e., the The value of a stock is expected to drift at the rate of ( ) r r α α = + − where r α − is called the risk premium and r is the risk-free rate.

Risk-Neutral Pricing
Assume that ( ) is the T-day return and that the T-day return is normally distributed with mean T α ′ and variance 2 T σ .Thus { } ( ) The price of a European call option with strike price T K is then [10] [11], assuming normal statistics for the returns with support [ ] , −∞ +∞ , The difference between the Black-Scholes option pricing formula and Equation (3) rests in the α in Equation (3).If one sets r α = in Equation (3), then one obtains the Black-Scholes formula.The Black-Scholes formula is obtained by using the concept of risk-neutral pricing, wherein one essentially sets the risk premium, r α − , equal to zero.Mathematically this approach can not be correct.Setting 0 r α − = violates the no arbitrage condition that ( ) . The distribution for t S is centred about ( ) 0 exp S t α (i.e., the mean of the pdf for the n-day return is ( ) Arbitrarily setting the mean of the distribution will change the value of the expectation and hence mis-price European call options.In general, for small T, ( ) and the error introduced by arbitrarily setting the mean of the distribution will be small.Application of Girsanov's theorem in risk-neutral pricing is discussed in Sec. 3.
A series expansion in α of 0 C about the risk-neutral value r α = shows that risk-neutral pricing underestimates the price of a call option when r α > .0 C α is the cost at time 0 t = of a European call option whereas 0 r C α = is the cost of a European call option using risk-neutral pricing (i.e., using the same assumptions that yield the Black-Scholes option pricing formula). ( ( ) ) ( ) and ( ) f a and ( ) F a the pdf and CDF for a normal distribution with a mean of zero and a variance of σ 2 .See Equation ( 8) for definitions of ( ) which follows from the definition . Thus the risk-neutral pricing underestimates the value of the call option and gives, on average, an advantage to the option buyer when r α > .The converse holds.The seller has, on average, an advantage under risk-neutral pricing when r α < .Neither party has, on average, an advantage when the true value of α is used to price an option.
The best estimate of a true value is typically the sample mean.as measured over one-year, and 0 50.00 S = for geometric Brownian motion with Gaussian increments.Note that risk-neutral pricing (i.e., the Black-Scholes formula) underestimates the price of a European call option when r α > and overestimates the price of a European call option when r α < .
The "success" of risk-neutral pricing owes to the fact that the magnitude of the random fluctuations are typically significantly greater than the magnitude of the risk premium, i.e., ( ) Note that ( ) F a is the cumulative density function (CDF), and that ( ) The standard deviation σ is a measure of the width of the distribution whereas the mean μ shifts the curve left or right.For a broad curve, small shifts left or right make a small difference.
In an Ito calculus formalism, risk-neutral pricing is explained as W is for a transformed process with non-zero mean such that { } ( ) Equation ( 11) is equivalent to the first line of Equation ( 9).In the Langevin formalism, the equation for the development in time of the average value of with solution ( ) ( ) In the event that α is a constant, ( ) ( )

Girsanov's Theorem
Girsanov's theorem [18] provides for a multiplicative transformation that alters the shape of the pdf.This allows the mean of a random variable to be set to an arbitrary value, without using an additive correction to the original problem or pdf ( [19], pp.36-39).In option pricing, the goal behind the multiplicative transformation is an economic one: to find a risk-neutral measure to ensure no risk-free arbitrages in pricing ( [19], p. 39).The multiplicative transformation must decrease probabilities for values greater than the desired mean and increase probabilities for values that are less than the desired mean.
Clearly Z is selected to give the desired mean to the equivalent measure.In the example given here, ( ) This result is Girsanov's theorem." Gardiner ([15], p. 247) shows simulations of a stochastic differential equation.
The simulations for the same noise but different drifts look qualitatively indistinguishable whereas the simulation with different noise looks different than the other simulations with same noise but different drifts: "it is quite credible that either could be a simulation of the other equation."Qualitatively indistinguishable appears to mean the rms noises about the local trend lines appear to be similar.This result is not surprising.The solution to Equation (9) or to Equation (11) in a Langevin approach is given by Equation (10) and can be recast as    .Figure 1 shows the first 201 points of Figure 2. The sharp rise in t S owes to the sequence of random draws.
A different seed for the pseudo random number generator gives drastically different results.
Figure 2  The simulations for the same noise but different drifts look qualitatively indistinguishable, as pointed out by Gardiner ([15], p. 247).
Gardiner ([15], p. 247) writes: "Girsanov's theorem is now the justification for use of the drift rate r instead of μ in the valuation of options using the risk-neutral procedure.
The noise term is identical for both cases, and in the case we can say that the two processes can be seen as arising from the choice of a different probability measure to the same set of sample paths.In some sense it can be shown that this is a rigorously justifiable procedure [10.11], although not everyone would accept that.However, the use of change of measure is now an accepted part of the procedure for valuing options and other derivatives when one goes beyond the simple geometric Brownian motion picture." Gardiner does not seem to be a true believer, and seems resigned to a deeply engrained status quo.
From [20]: "The relation (5) has been called by Cox-Ingersoll-Ross (1981) the "Local Expectation Hypothesis", a terminology which has led to some confusion.Note that the equilibrium process has not been changed, it is the same under both measures P and P  .Girsanov's Theorem allows us to replace the relation (4) through the equivalent simpler relation (5).In particular, no assumption has been made about the existence of risk-neutral investors.In a real economy neither a "representative" nor a "risk-neutral" investor will exist, since both assumptions would prevent the existence of a (stable) equilibrium.The great advantage of the representation (5) under the (martingale) measure P  is that we do not have to know anything about the individual expectations P and the investors' attitude towards risk.In summary: t x has not been changed.It is the same equilibrium price process as under P, but in simpler representation under P  .P  is called the "equivalent risk-neutral measure" or the "P-equivalent martingale measure".( [20], p. 72) It would appear that t x has been changed.The possible paths are the same, but more weight (probability) has been placed on lower yielding paths (assuming the risk premium is greater than zero).From [19]: "After the change of measure, we are still considering the same set of stock price paths, but we have shifted the probability on them.If r α > , as it normally is, then the change of measure puts more probability on the paths with lower return so that the overall mean rate of return is reduced from α to r." ([19], p. 217) "In finance, the change from the actual to the risk-neutral probability measure changes the distribution of asset prices without changing the asset prices themselves, ..." ( [19], p. 37) Altering the distribution changes the problem, unless the alteration is undone by an inverse transformation to return to the original frame of reference.
Essentially, the risk-neutral approach appears to be to multiply one term on one side of an equation by Z.This is not a valid mathematical approach.Consider In the development of the risk-neutral measure d t ′ W in Equation ( 9), it is considered that d t ′ W is a Weiner process (Brownian motion).Strictly speaking, Brownian motion is a zero mean process.d t ′ W is not a zero mean process.One might wish to apply a coordinate transformation such that d t ′ W is a zero mean Brownian motion in the transformed frame of reference.However, one must remember that one is working in a transformed coordinate system, and provide a reverse transformation at the end to obtain the answer in the original frame of reference.This is similar to the problem of relative motion.
Consider an airplane that can cruise at v km/hour with respect to the air mass in which the airplane is embedded (i.e., the local air) and assume that the local air is moving relative to the ground.If one is interested in the location of the plane after a given time, one can solve the problem by using a coordinate system that is embedded in the moving air.Relative to this coordinate system the plane has travelled a distance d vt = in time t.The choice of coordinate system makes the problem look simpler.However, to know the location of the plane relative to a reference coordinate system on the ground such as an airport, one must know the relationship between the reference coordinate system and the moving coordinate system.In a similar manner, to find the value of an asset or an option, one must know the risk premium.

A Thought Experiment
Consider pricing a very long lived option, one so long lived that the mean value t α is

Apply Girsanov's Theorem
Let us examine the justification for risk-neutral pricing, which is presented as Equation ( 9).As before, define a risk premium, move it to the noise term Now t ′ W is a non-zero mean noise term, except in the special case that 0 r α − =.Multiply the noise term by Z Z , c.f. Equation ( 16), to obtain

A Scaled Approach
Equation (21) suggests an approach to understand risk-neutral pricing of a European call option.Start with the expression for the value of the call option and manipulate to remove the drift owing to the risk premium in T S : where both T S and T K have been scaled by the same factor to remove drift in T S owing to the risk premium.The expectation in the last line might be anticipated at first look to be risk neutral.However,

∫
. In addition, the inverse scaling factor remains in the expression for 0 C as an overall multiplicative factor.The risk premium is thus required to price the option.If the risk premium is negligible, then the scaling factor is approximately unity and risk-neutral pricing of the European call option would be sufficiently accurate.The approach presented in this section is silent on the magnitude of the risk premium that is negligible-one would need to examine the expectation, as was done in Sec.

Conclusions
Risk-neutral pricing of European call options is mathematically an approximation.
Provided that ( ) r α σ − is small, then the drift rate is obscured over short time intervals by random fluctuations and one is justified in ignoring the risk premium r α − in the drift rate.It would seem honest to state this as the rationale behind risk-neutral pricing, rather than appealing to theorems and pretending that risk-neutral pricing is exact.
Risk-neutral pricing underestimates the price of a European call option when r α > and overestimates the price of a European call option when r α < .
It is interesting to note that risk-neutral pricing of European call options ignores r α − but takes great care to include 2 2 σ in the average drift rate.If the risk premium can be ignored, then likely 2 2 σ can also be ignored.

..
The probability that a measurement of the random variable x takes a value between x and d The cumulative density function (CDF) 1

Figure 1
presents simulations of t S for 200 time steps (200 days) for r α = and for 8r α = , with 1% r = per annum.The smooth lines are the expected values.All simulations used 0.01 σ = and the expected means for the different drift rates: a simulation with r α = (red curves) gives values that are less than the values obtained with 8r α = (blue curves).

Figure 2
Figure 2 presents simulations of t S for 1000 time steps (1000 days) for r α = and for 8r α = , with 1% r = per annum.The smooth lines are the

Figure 2 .
Figure 2. Simulations of t S for 1% r α = = for x are not the same unless 1 Z = .
2.2,to determine the magnitude.The rms fluctuations of the one day returns, σ, the relevant metric.It is the magnitude of the ratio ( ) Let the 1-day return be 1-day returns are Journal of Mathematical Finance normally distributed with drift rate μ and variance σ 2 , then the expectation of

Table 1
Journal of Mathematical Finance obscure the drift.Presumably fluctuations about the mean value are described well by the shape and scale parameters of the distribution and one should use the best available estimate of the location parameter of the distribution (i.e., the mean drift rate, α) to price an option.Consider a normal distribution with mean μ and variance σ 2 .If µ σ and thus the random fluctuations Table1.The costs of European options, 0 C , for various strike prices and values of α, r α = corresponds to prices obtained via risk-neutral pricing, i.e., by the Black-Scholes formula.T K The mean value of t S (10)(c.f.Equations (2),(10), and (13)).Care must be employed in obtaining and in interpreting results within the Ito formalism.One could attempt, in the Langevin picture, to hide the risk premium r α − in ( ) Langevin equations are first order differential equations with noise driving terms.The Langevin equations should be interpreted as integral equations ([16], p. 172; [17], Ch. 10.2; [15], p. 79).Average values found by the Langevin approach are identical to solutions found by Ito's calculus ([17], p. 189).The Langevin approach has the advantage that transformations obey the usual rules of calculus.Ito's lemma is not applied for each transformation in the Langevin approach ([17], pp.189, 282).
a scaled version of t S and is determined solely by the noise