Relativistic Black-Scholes model arXiv:1307.5122v1 [q-fin ...
arxiv.org/pdf/1307.5122
arXiv
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by M Trzetrzelewski - 2013 - Related articles
Jul 19, 2013 - the Euclidean version of the Dirac equation. Therefore the ... current market data are used against the Black-Scholes formula one finds that σ must ..... tion of the Black-Scholes equation, using the standard hedging ar- gument.
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Finance's objects of interest – markets, money, assets and securities – are also ... the sublime Dirac equation, difficult to interpret with pictures, expresses (i) the ...
The Intelligent Young Person's Guide to Pricing and Hedging
www.emanuelderman.com/media/The_Young_Persons_Guide.pdf
BREAKING THE CYCLE | Edge.org
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Among many unrealistic assumptions made in the Black-Scholes model [1], one is particularly problematic - constant volatility σ. When the current market data are used against the Black-Scholes formula one finds that σ must in fact depend on the strike K, and time to expiry T, in order to make the pricing formula work. Therefore the market data imply that σ is not constant but a function σI(K,T) - called im- plied volatility.
completely different before the market crash in late 80’. In the equity market before 1987, the implied volatility was indeed fairly constant - why it is not constant nowadays [2] ? One could explain this problem by blaming everything on yet an- other unrealistic assumption of the Black-Scholes model - that the underlier St undergoes the geometric Brownian motion dSt/St = µdt + σdWt, µ ∈R,σ > 0 (1) (where Wt is a Wiener process). It follows form (1) that log-returns (i.e. returns of lnSt) have Gaussian distribution. However it is very well known [3] that the actual log-returns are not distributed like that - instead they exhibit fat tails (Figure 1a). Therefore a rather nat-
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-0.04 -0.02 0.00 0.02 0.040 10 20 30 40 50 60 DAILYLOGRETURNSUNTIL2013 NORMALIZEDFREQUENCY aLREALHDJILVS.GAUSSIANDISTRIBUTIONS
-0.04 -0.02 0.00 0.02 0.040 10 20 30 40 50 60 DAILYLOGRETURNSUNTIL1987 NORMALIZEDFREQUENCY bLREALHDJILVS.GAUSSIANDISTRIBUTIONS
Figure 1: Distribution of daily log returns for Dow Jones (dotted) and the corresponding Gaussian distribution (continuous). a) since 27 May 1896 to 10 May 2013, b) since 27 May 1896 to 2 Jan 1987. The mean and the height of the Gaussian distribution are adjusted accordingly.
ural way to generalize (1) is to replace Wt with the process whose PDF exhibits fat tails corresponding to the ones observed in the mar- kets. However a careful inspection shows that this cannot be the main reason of the volatility smile observed today. The point is that even before 1987 the log-return distribution revealed fat tails (see Figure 1b; note that Mandelbrot’s paper [3] was published in 1963) but at the same time the Black-Scholes model was working well. This is clearly an issue. If fat tails are the reason of all these discrepancies then why the constant volatility assumption was correct before 1987?
Because of practical reasons the models that consider generaliza- tions of Wt are not very popular and the development in this subject
2
went in a completely different direction. Instead of changing Wt, fi- nancial practitioners prefer to leave Wt unchanged and assume that σ is a function σ = σ(S,t) - called local volatility [4]. Then the smile is explained by assuming that σ increases for large |lnSt| - if this is the case then the tails of the Gaussian distribution will become fatter
This brings us to further generalization by assuming that σ itself is a stochastic process [6]
dσt = α(σt,t)dt + β(σt,t)dWt (2)
(here α and β are some deterministic functions).This generalization is counter intuitive: the amplitude σ, that multiplies the random factor dWt, is stochastic now, but shouldn’t dWt contain all the randomness? Moreover, stochastic volatility models also fail in certain situations e.g. in the limit T → 0 where T is the time to maturity [7]. This could be a motivation to generalize further and introduce jumps i.e. discontinuous moves of the underling St [8].1
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