Probability-free models in option pricing: statistically indistinguishable dynamics and historical vs implied volatility
aa r X i v : . [ q -f i n . P R ] A p r Probability-free models in option pricing: statisticallyindistinguishable dynamics and historical vs implied volatility
Damiano Brigo ∗ Paper presented at the conference
Options: 45 Years after the publication of the Black-Scholes-Merton Model
Jerusalem, 4-5 December 2018.April 3, 2019
Abstract
We investigate whether it is possible to formulate option pricing and hedging modelswithout using probability. We present a model that is consistent with two notions of volatility:a historical volatility consistent with statistical analysis, and an implied volatility consistentwith options priced with the model. The latter will be also the quadratic variation of themodel, a pathwise property. This first result, originally presented in Brigo and Mercurio(1998, 2000) [8, 9], is then connected with the recent work of Armstrong et al (2018) [1],where using rough paths theory it is shown that implied volatility is associated with a purelypathwise lift of the stock dynamics involving no probability and no semimartingale theoryin particular, leading to option models without probability. Finally, an intermediate resultby Bender et al. (2008) [4] is recalled. Using semimartingale theory, Bender et al. showedthat one could obtain option prices based only on the semimartingale quadratic variation ofthe model, a pathwise property, and highlighted the difference between historical and impliedvolatility. All three works confirm the idea that while historical volatility is a statisticalquantity, implied volatility is a pathwise one. This leads to a 20 years mini-anniversary ofpathwise pricing through 1998, 2008 and 2018, which is rather fitting for a talk presented atthe conference for the 45 years of the Black, Scholes and Merton option pricing paradigm.
AMS Classification Codes : 62H20, 91B70
JEL Classification Codes : G12, G13
Keywords : Historical volatility, implied volatility, statistically indistinguishable models, optionpricing, rough paths theory, pathwise finance, pathwise option pricing.
In this work we focus initially on the following question. Take two models S and Y of stock pricedynamics under the objective (or statistical / historical / physical) probability measure P . Fix adiscrete time trading grid, starting from time 0 and up to a final time T >
0, with time step ∆.∆ can be as small as needed but it has to be fixed in advance, at time 0. We then consider pricingoptions on the stock, according to either model S or Y , via the continuous time theory of Black,Scholes and Merton (BSM) [5],[17] and subsequent extensions, especially by Harrison et al. [14],[15]. Our question is the following. Can we find situations where S and Y are statistically very ∗ Department of Mathematics, Imperial College, London amiano Brigo, Probability-free models in option pricing & historical vs implied volatilityamiano Brigo, Probability-free models in option pricing & historical vs implied volatility
0, with time step ∆.∆ can be as small as needed but it has to be fixed in advance, at time 0. We then consider pricingoptions on the stock, according to either model S or Y , via the continuous time theory of Black,Scholes and Merton (BSM) [5],[17] and subsequent extensions, especially by Harrison et al. [14],[15]. Our question is the following. Can we find situations where S and Y are statistically very ∗ Department of Mathematics, Imperial College, London amiano Brigo, Probability-free models in option pricing & historical vs implied volatilityamiano Brigo, Probability-free models in option pricing & historical vs implied volatility P ), having very close laws in the ∆ grid, but where theyimply very different option prices? Prices will be computed as expectations of discounted cashflows under the pricing (or risk-neutral/martingale) measure Q . If we do find such situations, canwe do this in a constructive way, rather than just proving they exist? To answer this question, we begin with two different models S and Y . We take S as the BlackScholes and Merton model, and we construct a second model Y whose marginal laws are the sameas S . Start from the Black-Scholes-Merton model for the stock price S given by dS t = µS t dt + ¯ σS t dW t (abbreviated BSM( µ, ¯ σ )) with initial condition S = s under the objective measure P . Here W is a Brownian motion under P , while s > µ and ¯ σ > Y , dY = u ( Y, . . . ) dt + σ t ( Y t ) dW t with local volatility σ and with the same margins as S , namely p S t = p Y t for all t ∈ [0 , T ]. Herefor a random variable X we denote by p X its probability density function. To find Y , invert theFokker Planck (FP) equation for Y and find the drift u such that the FP equation for the densityof Y has solution equal to p S t , namely the lognormal density of the original S .This was done in Brigo and Mercurio (1998, 2000) [8, 9] using previous results on diffusionswith laws on exponential families (Brigo (1997) [6] and Brigo (2000) [7]).We obtain the following model for Y . To avoid singularities of our model coefficient u near t = 0, we start with a regularization in a small interval [0 , ǫ ) that has the same dynamics as S ,and then we move to the different dynamics from time [ ǫ, T ]. We obtain d ¯ Y t = µ ¯ Y t dt + ¯ σ ¯ Y t dW t , ≤ t < ǫ, ¯ Y = s ,Y t = ¯ Y t for t ∈ [0 , ǫ ) ,dY t = u σt ( Y t , s , dt + σ t ( Y t ) dW t , Y ǫ = ¯ Y ǫ − , ǫ ≤ t ≤ T, (1) u σt ( x, y, α ) := 12 ∂ ( σ t ) ∂x ( x ) + 12 ( σ t ( x )) x (cid:20) µ ¯ σ − − σ ( t − α ) ln xy (cid:21) + x t − α ) " ln xy − µ ¯ σ − − σ ( t − α ) . We have introduced ¯ Y to avoid singularities in the drift coefficient of the SDE (1) near t = 0.The process Y , if the related SDE has a solution that is regular enough and admits densities,has the same marginal distribution as BSM( µ, ¯ σ ): p S t = p Y t for all t . We will show a fundamentalexample where everything works fine in Section 2.3 below. ∆ grid For our purposes of statistical indistinguishability, the above Y is not enough. A further funda-mental property of the BSM( µ, ¯ σ ) model is that its log-returns satisfyln S t + δ S t ∼ N (cid:18) ( µ −
12 ¯ σ ) δ, ¯ σ δ (cid:19) , δ > , t ∈ [0 , T − δ ] . amiano Brigo, Probability-free models in option pricing & historical vs implied volatilityamiano Brigo, Probability-free models in option pricing & historical vs implied volatility
12 ¯ σ ) δ, ¯ σ δ (cid:19) , δ > , t ∈ [0 , T − δ ] . amiano Brigo, Probability-free models in option pricing & historical vs implied volatilityamiano Brigo, Probability-free models in option pricing & historical vs implied volatility Y above do not share this property because identity of the marginallaws alone does not suffice to ensure it. We need equality of second order laws or of transitiondensities.To tackle this issue, we restrict the set of dates for which the log-return property must holdtrue. Modify the definition of Y so that, given T ∆ := { , ∆ , , . . . , N ∆ } , ∆ = T /N , ∆ > ǫ , wehave ln Y i ∆ Y j ∆ ∼ N (( µ −
12 ¯ σ )( i − j )∆ , ¯ σ ( i − j )∆) , i > j. (2)Limiting such key property to a finite set of times is not so dramatic. Indeed, only discrete timesamples are observed in practice, so that once the time instants are fixed, our process Y can notbe distinguished statistically from the Black-Scholes process S .The new definition of Y we propose now, to match log returns distributions in grids, is stillbased on our earlier Y . However, we use the earlier Y process locally in each time interval[( i − , i ∆). In such interval we define iteratively the drift u σ as in the earlier Y but we translateback the time–dependence of a time amount ( i − Y with the final value of Y in theprevious interval. We obtain, in each interval [ i ∆ , ( i + 1)∆): d ¯ Y t = µ ¯ Y t dt + ¯ σ ¯ Y t dW t , t ∈ [ i ∆ , i ∆ + ǫ ) , ¯ Y i ∆ = Y i ∆ − Y t = ¯ Y t for t ∈ [ i ∆ , i ∆ + ǫ ) ,dY t = u σt ( Y t , Y α ( t ) , α ( t )) dt + σ t ( Y t ) dW t , t ∈ [ i ∆ + ǫ, ( i + 1)∆) , Y i ∆+ ǫ = ¯ Y i ∆+ ǫ − (3)where ¯ Y = Y = s , u σt ( x, y, α ) was defined in the earlier Y and α ( t ) = i ∆ for t ∈ [ i ∆ , ( i + 1)∆).It is clear that the transition densities of S and Y satisfy p Y ( i +1)∆ | Y i ∆ ( x ; y ) = p S ( i +1)∆ | S i ∆ ( x ; y )by construction.Note also that the new process Y is not a Markov process in [0 , T ]. However, it is a Markovprocess in all time instants of T ∆ (∆ –Markovianity ).Finally, note that now the two models S (BSM( µ, ¯ σ )) and Y are statistically indistinguishable in T ∆ since there they share the same finite dimensional distributions. Any statistician who triedto estimate the two models from data could not find a way to distinguish them. Before turningto option prices implied by the two indistinguishable models, we need to show that we have notproduced an empty theory so far. In other terms, we need to give concrete examples of σ for whichour framework works rigorously. Such a fundamental case is addressed in the next section. σ t ( y ) = νy We take now σ ( Y ) = νY , so that also the volatility of Y is of BSM type, but with constant ν instead of ¯ σ . In this case the equation for u specializes to u νt ( y, y α , α ) = y (cid:20) ( ν − ¯ σ ) + µ ν ¯ σ + 1) (cid:21) + y t − α ) (1 − ν ¯ σ ) ln yy α , and in this fundamental case one can show that the SDE for Y has a unique strong solution [8],[9]. Moreover, the change of measure that replaces the drift u with rY is well defined and regular,so that it is possible to change probability measure from P to the equivalent pricing measure Q for the model Y .This is precisely what we are interested in, since changing measure leads to some quite in-teresting developments. Before turning to the change of measure, a final remark concerning theregularization in [0 , ǫ ) is in order. amiano Brigo, Probability-free models in option pricing & historical vs implied volatilityamiano Brigo, Probability-free models in option pricing & historical vs implied volatility
12 ¯ σ )( i − j )∆ , ¯ σ ( i − j )∆) , i > j. (2)Limiting such key property to a finite set of times is not so dramatic. Indeed, only discrete timesamples are observed in practice, so that once the time instants are fixed, our process Y can notbe distinguished statistically from the Black-Scholes process S .The new definition of Y we propose now, to match log returns distributions in grids, is stillbased on our earlier Y . However, we use the earlier Y process locally in each time interval[( i − , i ∆). In such interval we define iteratively the drift u σ as in the earlier Y but we translateback the time–dependence of a time amount ( i − Y with the final value of Y in theprevious interval. We obtain, in each interval [ i ∆ , ( i + 1)∆): d ¯ Y t = µ ¯ Y t dt + ¯ σ ¯ Y t dW t , t ∈ [ i ∆ , i ∆ + ǫ ) , ¯ Y i ∆ = Y i ∆ − Y t = ¯ Y t for t ∈ [ i ∆ , i ∆ + ǫ ) ,dY t = u σt ( Y t , Y α ( t ) , α ( t )) dt + σ t ( Y t ) dW t , t ∈ [ i ∆ + ǫ, ( i + 1)∆) , Y i ∆+ ǫ = ¯ Y i ∆+ ǫ − (3)where ¯ Y = Y = s , u σt ( x, y, α ) was defined in the earlier Y and α ( t ) = i ∆ for t ∈ [ i ∆ , ( i + 1)∆).It is clear that the transition densities of S and Y satisfy p Y ( i +1)∆ | Y i ∆ ( x ; y ) = p S ( i +1)∆ | S i ∆ ( x ; y )by construction.Note also that the new process Y is not a Markov process in [0 , T ]. However, it is a Markovprocess in all time instants of T ∆ (∆ –Markovianity ).Finally, note that now the two models S (BSM( µ, ¯ σ )) and Y are statistically indistinguishable in T ∆ since there they share the same finite dimensional distributions. Any statistician who triedto estimate the two models from data could not find a way to distinguish them. Before turningto option prices implied by the two indistinguishable models, we need to show that we have notproduced an empty theory so far. In other terms, we need to give concrete examples of σ for whichour framework works rigorously. Such a fundamental case is addressed in the next section. σ t ( y ) = νy We take now σ ( Y ) = νY , so that also the volatility of Y is of BSM type, but with constant ν instead of ¯ σ . In this case the equation for u specializes to u νt ( y, y α , α ) = y (cid:20) ( ν − ¯ σ ) + µ ν ¯ σ + 1) (cid:21) + y t − α ) (1 − ν ¯ σ ) ln yy α , and in this fundamental case one can show that the SDE for Y has a unique strong solution [8],[9]. Moreover, the change of measure that replaces the drift u with rY is well defined and regular,so that it is possible to change probability measure from P to the equivalent pricing measure Q for the model Y .This is precisely what we are interested in, since changing measure leads to some quite in-teresting developments. Before turning to the change of measure, a final remark concerning theregularization in [0 , ǫ ) is in order. amiano Brigo, Probability-free models in option pricing & historical vs implied volatilityamiano Brigo, Probability-free models in option pricing & historical vs implied volatility As Y has the same margins as S , it has to be positive like S , so that Y t >
0. Then take Z ǫt := ln Y t : Z ǫt = Z ǫj ∆ + ( µ −
12 ¯ σ )( t − j ∆) (4)+ ¯ σ ( W t − W j ∆ ) for t ∈ [ j ∆ , j ∆ + ǫ ) , (cid:16) t − j ∆ ǫ (cid:17) β/ (cid:20) ¯ σ ( W j ∆+ ǫ − W j ∆ ) + ν R tj ∆+ ǫ (cid:16) u − j ∆ ǫ (cid:17) − β/ dW u (cid:21) for t ∈ [ j ∆ + ǫ, ( j + 1) ǫ ) . Here β = 1 − ν ¯ σ . In [8] we show that we can take ǫ → ZZ t = Z j ∆ + ( µ − ¯ σ t − j ∆) + ν Z tj ∆ (cid:20) t − j ∆ u − j ∆ (cid:21) β dW u , t ∈ [ j ∆ , ( j + 1)∆) . This process is well defined since the integral in the right-hand side exists finite almost surely eventhough its integrand diverges when u → j ∆ + . The above equation can be better compared to theBlack and Scholes process when written in differential form: dZ t = ( µ −
12 ¯ σ ) dt + β t − j ∆) β/ − (cid:18)Z tj ∆ ( u − j ∆) − β/ dW u (cid:19) ν dt + ν dW t t ∈ [ j ∆ , ( j + 1)∆) . The central term in the right hand side is needed to have returns equal to the Black and Scholesprocess even after changing the volatility coefficient from ¯ σ to ν . More precisely, the central termis the correction needed so that the exponential of Z will simultaneously have returns equal tothose of BSM( µ, ¯ σ ) and volatility coefficient equal to ν . Note that this term goes to zero for ¯ σ = ν .It is this term that makes our process non-Markov outside the trading time grid.Finally, we point out that for all ǫ > Q ǫ equivalent to P and such that Z ǫ is a Brownian motion under Q ǫ . As noted in [3], informally taking the limitfor ǫ ↓ t − γ for some 0 < γ <
1, and rougher trajectories than semimartingale ones.
We now consider option pricing based on the models S and Y . We will assume that interest rates r are deterministic and constant in time.Recall our two indistinguishable models under the measure P , and see what happens when wechange measure to Q : dS t = µS t dt + ¯ σS t dW P t ∆ − indistinguishable from dY t = u νt dt + ν Y t dW P t , but changing measure to Q dS t = rS t dt + ¯ σS t dW Q t very different from dY νt = rY νt dt + ν Y t dW Q t No arbitrage conditions are reflected in the fact that changing measure to Q enforces the drift r .If we now price a call option with the Q expectation of the discounted payoff we have E Q [ e − rT ( S T − K ) + ] = BlackScholesFormula(¯ σ ) , E Q [ e − rT ( Y T − K ) + ] = BlackScholesFormula( ν )where the remaining inputs of the Black and Scholes formula, namely s , r, T are the same forboth cases. Since the indistinguishability holds for every ν , we can take ν ↓ ν ↑ + ∞ . Thisway we find the following. amiano Brigo, Probability-free models in option pricing & historical vs implied volatilityamiano Brigo, Probability-free models in option pricing & historical vs implied volatility
We now consider option pricing based on the models S and Y . We will assume that interest rates r are deterministic and constant in time.Recall our two indistinguishable models under the measure P , and see what happens when wechange measure to Q : dS t = µS t dt + ¯ σS t dW P t ∆ − indistinguishable from dY t = u νt dt + ν Y t dW P t , but changing measure to Q dS t = rS t dt + ¯ σS t dW Q t very different from dY νt = rY νt dt + ν Y t dW Q t No arbitrage conditions are reflected in the fact that changing measure to Q enforces the drift r .If we now price a call option with the Q expectation of the discounted payoff we have E Q [ e − rT ( S T − K ) + ] = BlackScholesFormula(¯ σ ) , E Q [ e − rT ( Y T − K ) + ] = BlackScholesFormula( ν )where the remaining inputs of the Black and Scholes formula, namely s , r, T are the same forboth cases. Since the indistinguishability holds for every ν , we can take ν ↓ ν ↑ + ∞ . Thisway we find the following. amiano Brigo, Probability-free models in option pricing & historical vs implied volatilityamiano Brigo, Probability-free models in option pricing & historical vs implied volatility Statistically indistinguishable stock price models in the ∆ time grid imply option prices sodifferent to span the whole no arbitrage interval [( S − Ke − rT ) + , S ].Perhaps surprisingly, they span a range that is not related to ∆. Since models are equivalentin a ∆ grid, one might have expected that by tightening ∆ one might have had option prices in anarrower range. This is not the case.Our result shows that conjugating discrete and continuous time modeling (e.g. statistics andoption pricing) has to be done carefully and is subject to important assumptions. What if what we have seen is a way to account consistently for historical and implied volatility?Indeed, it is known that option prices trade independently of the underlying stock price, and wehave been able to construct a stock price process Y ν whose marginal distribution and transitiondensity depend on the volatility coefficient ¯ σ (historical volatility), whereas the correspondingoption price only depends on the volatility coefficient ν (implied volatility). As a consequence, wecan provide a consistent theoretical framework justifying the differences between historical andimplied volatility that are commonly observed in real markets. Since probability and statistics have proven to be deceptive when working in discrete time under P , we try now to remove probability and statistics from valuation. We will achieve this by usingideas from rough paths theory. Rough paths theory has been initiated and developed by manyresearchers over the years, here we recall briefly F¨ollmer [11], Lyons [16], Davie [10], Gubinelli [13]and Friz among others, see for example [12] and references therein. Applying ideas from roughpaths theory, and from [10, 13, 12] in particular, in Armstrong et al. (2018) [1] we manage tore-interpret the Black Scholes formula and option pricing in a purely pathwise sense.After writing [1] we found out that Bender et al. (2008) [4] had formulated option pricing,in the framework of semimartingale theory, relying only on the quadratic variation of price tra-jectories, which is a pathwise property . In our work [1] the analysis is brought one step furtherby abandoning the semimartingale setting. Using Davie’s rough differential equations and roughbrackets (see [12]) we abandon probability theory altogether. This entails the definition of Gu-binelli derivatives, which in the classical Black and Scholes framework are Gamma sensitivities ofthe options. Although ignored in the most classical formulas for portfolio dynamics, they play anactive role in the convergence of the integrals describing portfolio processes, when such convergenceis analysed pathwise as in [1].We now go back to the statistically indistinguishable models S and Y above, implying arbitrar-ily different option prices. We connect them to our discussion of pathwise properties. Since theresults from [4] and [1] suggest that option pricing ultimately depends only on pathwise features ofprice trajectories, it is with the lenses of pathwise analysis that we can distinguish the P dynamicsof S and of Y , by looking at their quadratic variation. While probability and statistics could notdistinguish between S and Y , the difference is instead captured by this aspect of price trajectories.Paraphrasing: • Probability and Statistics do not allow one to distinguish between S and Y in T ∆ . • Prices of options written on S and Y are different, so that option prices allow one to distin-guish between S and Y . • One then conjectures that option prices cannot be properties of S and Y based on probabilityor statistics. With prices being traditionally associated with expectations of discounted cashflows, this seems initially counterintuitive. amiano Brigo, Probability-free models in option pricing & historical vs implied volatilityamiano Brigo, Probability-free models in option pricing & historical vs implied volatility
We now consider option pricing based on the models S and Y . We will assume that interest rates r are deterministic and constant in time.Recall our two indistinguishable models under the measure P , and see what happens when wechange measure to Q : dS t = µS t dt + ¯ σS t dW P t ∆ − indistinguishable from dY t = u νt dt + ν Y t dW P t , but changing measure to Q dS t = rS t dt + ¯ σS t dW Q t very different from dY νt = rY νt dt + ν Y t dW Q t No arbitrage conditions are reflected in the fact that changing measure to Q enforces the drift r .If we now price a call option with the Q expectation of the discounted payoff we have E Q [ e − rT ( S T − K ) + ] = BlackScholesFormula(¯ σ ) , E Q [ e − rT ( Y T − K ) + ] = BlackScholesFormula( ν )where the remaining inputs of the Black and Scholes formula, namely s , r, T are the same forboth cases. Since the indistinguishability holds for every ν , we can take ν ↓ ν ↑ + ∞ . Thisway we find the following. amiano Brigo, Probability-free models in option pricing & historical vs implied volatilityamiano Brigo, Probability-free models in option pricing & historical vs implied volatility Statistically indistinguishable stock price models in the ∆ time grid imply option prices sodifferent to span the whole no arbitrage interval [( S − Ke − rT ) + , S ].Perhaps surprisingly, they span a range that is not related to ∆. Since models are equivalentin a ∆ grid, one might have expected that by tightening ∆ one might have had option prices in anarrower range. This is not the case.Our result shows that conjugating discrete and continuous time modeling (e.g. statistics andoption pricing) has to be done carefully and is subject to important assumptions. What if what we have seen is a way to account consistently for historical and implied volatility?Indeed, it is known that option prices trade independently of the underlying stock price, and wehave been able to construct a stock price process Y ν whose marginal distribution and transitiondensity depend on the volatility coefficient ¯ σ (historical volatility), whereas the correspondingoption price only depends on the volatility coefficient ν (implied volatility). As a consequence, wecan provide a consistent theoretical framework justifying the differences between historical andimplied volatility that are commonly observed in real markets. Since probability and statistics have proven to be deceptive when working in discrete time under P , we try now to remove probability and statistics from valuation. We will achieve this by usingideas from rough paths theory. Rough paths theory has been initiated and developed by manyresearchers over the years, here we recall briefly F¨ollmer [11], Lyons [16], Davie [10], Gubinelli [13]and Friz among others, see for example [12] and references therein. Applying ideas from roughpaths theory, and from [10, 13, 12] in particular, in Armstrong et al. (2018) [1] we manage tore-interpret the Black Scholes formula and option pricing in a purely pathwise sense.After writing [1] we found out that Bender et al. (2008) [4] had formulated option pricing,in the framework of semimartingale theory, relying only on the quadratic variation of price tra-jectories, which is a pathwise property . In our work [1] the analysis is brought one step furtherby abandoning the semimartingale setting. Using Davie’s rough differential equations and roughbrackets (see [12]) we abandon probability theory altogether. This entails the definition of Gu-binelli derivatives, which in the classical Black and Scholes framework are Gamma sensitivities ofthe options. Although ignored in the most classical formulas for portfolio dynamics, they play anactive role in the convergence of the integrals describing portfolio processes, when such convergenceis analysed pathwise as in [1].We now go back to the statistically indistinguishable models S and Y above, implying arbitrar-ily different option prices. We connect them to our discussion of pathwise properties. Since theresults from [4] and [1] suggest that option pricing ultimately depends only on pathwise features ofprice trajectories, it is with the lenses of pathwise analysis that we can distinguish the P dynamicsof S and of Y , by looking at their quadratic variation. While probability and statistics could notdistinguish between S and Y , the difference is instead captured by this aspect of price trajectories.Paraphrasing: • Probability and Statistics do not allow one to distinguish between S and Y in T ∆ . • Prices of options written on S and Y are different, so that option prices allow one to distin-guish between S and Y . • One then conjectures that option prices cannot be properties of S and Y based on probabilityor statistics. With prices being traditionally associated with expectations of discounted cashflows, this seems initially counterintuitive. amiano Brigo, Probability-free models in option pricing & historical vs implied volatilityamiano Brigo, Probability-free models in option pricing & historical vs implied volatility • [1] show that, in general, option prices can be derived based only on purely pathwise prop-erties, without using probability theory anywhere. Bender et al. [4] had obtained a similarresult earlier but within semimartingale theory. • In particular, in models S and Y option prices are completely specified by probability-freepathwise properties of S and Y , confirming the above conjecture.We now summarize how in [1] we were able to give a probability-free, and more specificallysemimartingale-free derivation of option pricing based on purely pathwise properties. We denote in general X s,t = X t − X s . Recall the BSM( µ, ¯ σ ) model dS t = S t [ µdt + ¯ σdW P t ] , dB t = rB t dt, ≤ t ≤ T where we also included the risk-free bank account numeraire B . In the classic theory of stochasticdifferential equations (SDEs), the above equation is a short form for an integral equation S t − S = Z t µS u du + Z t ¯ σS u dW u . The last integral is an Ito stochastic integral. If one tries to re-formulate the above equationwithout probability, one will not be able to use stochastic integrals any more, and as a result onewill need to define the integral R t ¯ σS u dW u pathwise. To do this, one will need to add informationon the price trajectory in the form of a lift. One needs to provide the input S s,t = Z ts S s,u dS u . This is really an input and is not defined a priori based only on properties of S : if the signal S has finite p -variation for 2 < p <
3, as in case of paths in the Black Scholes model, it is toorough to define the above intergral as a Stiltjes or Young integral. One needs therefore to add S as an input. To understand why this is important, we now explain how introducing S helps indefining integrals of the type R F ( S r ) dS r . Consider R F ( S r ) dS r and try to write it as a Youngintegral. Take Taylor expansion F ( S r ) ≈ F ( S u ) + DF ( S u ) S u,r . The Young integral can be seenas approximating F ( S r ), in each [ u, t ] ∈ π with the zero-th order term F ( S u ), where π is thepartition for the discrete sums approximating the integral, and | π | is the mesh size. Hence Z T F ( S r ) dS r = lim | π |→ X [ u,t ] ∈ π Z tu F ( S u ) dS r = lim | π |→ X [ u,t ] ∈ π F ( S u ) S u,t . The limit is on all partitions whose mesh size tends to zero. If we cannot use a Young integralbecause S is too rough, we can try a first order expansion for F ( S ) rather than a zero-th orderone. Z T F ( S r ) d S r = lim | π |→ X [ u,t ] ∈ π Z tu ( F ( S u ) + DF ( S u ) S u,r ) dS r == lim | π |→ X [ u,t ] ∈ π ( F ( S u ) S u,t + DF ( S u ) S u,t ) . (5)This intuition can be made rigorous with the relevant norms and metrics, see for example [12].What attracted initially the interest of the author, prompting the initial work that later leadto the collaboration culminating in [1], is the fact that in a delta-hedging context, one can writethe replication condition for a call option in a Black-Scholes type model as Z T ∆( r, S r ) dS r + Z T η ( r, S r ) dB r = ( S T − K ) + − V amiano Brigo, Probability-free models in option pricing & historical vs implied volatilityamiano Brigo, Probability-free models in option pricing & historical vs implied volatility
3, as in case of paths in the Black Scholes model, it is toorough to define the above intergral as a Stiltjes or Young integral. One needs therefore to add S as an input. To understand why this is important, we now explain how introducing S helps indefining integrals of the type R F ( S r ) dS r . Consider R F ( S r ) dS r and try to write it as a Youngintegral. Take Taylor expansion F ( S r ) ≈ F ( S u ) + DF ( S u ) S u,r . The Young integral can be seenas approximating F ( S r ), in each [ u, t ] ∈ π with the zero-th order term F ( S u ), where π is thepartition for the discrete sums approximating the integral, and | π | is the mesh size. Hence Z T F ( S r ) dS r = lim | π |→ X [ u,t ] ∈ π Z tu F ( S u ) dS r = lim | π |→ X [ u,t ] ∈ π F ( S u ) S u,t . The limit is on all partitions whose mesh size tends to zero. If we cannot use a Young integralbecause S is too rough, we can try a first order expansion for F ( S ) rather than a zero-th orderone. Z T F ( S r ) d S r = lim | π |→ X [ u,t ] ∈ π Z tu ( F ( S u ) + DF ( S u ) S u,r ) dS r == lim | π |→ X [ u,t ] ∈ π ( F ( S u ) S u,t + DF ( S u ) S u,t ) . (5)This intuition can be made rigorous with the relevant norms and metrics, see for example [12].What attracted initially the interest of the author, prompting the initial work that later leadto the collaboration culminating in [1], is the fact that in a delta-hedging context, one can writethe replication condition for a call option in a Black-Scholes type model as Z T ∆( r, S r ) dS r + Z T η ( r, S r ) dB r = ( S T − K ) + − V amiano Brigo, Probability-free models in option pricing & historical vs implied volatilityamiano Brigo, Probability-free models in option pricing & historical vs implied volatility V is the option price at time 0 and ∆ and η are the amounts of stock and cash in the self-financing replicating strategy. In particular, ∆ is the sensitivity of the option price with respectto its underlying S . The above integral R T ∆( r, S r ) dS r is an Ito integral. If one tries to re-writethe replication condition using a purely pathwise integral, using rough integration as in [12], it ispossible to do so by recalling that, according to (5), Z T ∆( r, S r ) d S r = lim | π |→ X [ u,t ] ∈ π (∆( u, S u ) S u,t + D S ∆( u, S u ) S u,t ) . In an option pricing setting, the term D S ∆( u, S u ) =: Γ( u, S u ) turns out to be the second derivativeof the option price with respect to its underlying term, the so called gamma of the option. Hencelim | π |→ X [ u,t ] ∈ π (∆( u, S u ) S u,t + Γ( u, S u ) S u,t ) + Z T η ( r, S r ) dB r = ( S T − K ) + − V . This immediately prompted the author to notice that gamma played an explicit role in the repli-cation condition when the condition is expressed using purely pathwise integrals rather than Itointegrals. Traders have been always using gamma as a correction term, but the above limit showsa fundamental role for gamma already at the level of the replication condition. On the contrary,the Ito integral R T ∆( r, S r ) dS r can be written as a limit where gamma does not appear. Theauthor further proposed to interpet the lift S as related to a non-standard covariance swap. Thisis discussed further in [1] and will be expanded in future work.Let us now go back to BSM and [1]. Before proceeding further, we need to explain that in [1]we do not use really full rough path theory. We presented above a sketch of how one makes senseof the BSM SDE for S in a purely pathwise way, but in effect we never use the full SDE. Indeed,we noted in the discussion above that only the quadratic variation of the SDE solution mattersin determining the option price. Similarly, in the no-semimartingales pathwise case, we will notreally need the pathwise analogous of the BSM SDE, but just a no-semimartingale analogous ofthe quadratic variation. More precisely, take S t as a path of finite p variation with 2 < p < q variation for all q >
2, so S is potentially rougher than BSM).Consider the lifted S t := ( S t , S t ), where S is our input for R S dS . From a technical point of viewin [1] we work with reduced rough paths, obtained from the pair ( S, S ) by considering only thesymmetric part of S (this distinction is essential in the multi-dimensional case, although here weare discussing the one-dimensional setting). This is equivalently described by the rough bracketdefined in [ S ] u,t = S u,t S u,t − S u,t . We refer again to [12] for the details, and point out the early work of F¨ollmer [11] in this regard.If [ S ] u,t is regular enough to define a measure of [ u, t ] with density a ( S t ) with a ( x ) also regular,then the classic partial differential equation for the option price is defined entirely in terms ofthe purely pathwise bracket [ S ], involving no probability theory and no semimartingale theory inparticular. It follows that the option price itself will not depend on the probabilistic setting butonly on path properties.The purely pathwise property [ S ] u,t takes the place of implied volatility in determining theoption price as a path property rather than a statistical property. The latter would be associatedwith historical volatility as a standard deviation (statistics). The first result we reported in this paper is the result of Brigo and Mercurio (1998) [8], where theprocess Y had simultaneously historical volatility ¯ σ as a statistical property and implied volatility ν as a pathwise property (quadratic variation). This is consistent with the later result of Bender,Sottinen and Valkeila (2008) [4] who note: amiano Brigo, Probability-free models in option pricing & historical vs implied volatilityamiano Brigo, Probability-free models in option pricing & historical vs implied volatility
2, so S is potentially rougher than BSM).Consider the lifted S t := ( S t , S t ), where S is our input for R S dS . From a technical point of viewin [1] we work with reduced rough paths, obtained from the pair ( S, S ) by considering only thesymmetric part of S (this distinction is essential in the multi-dimensional case, although here weare discussing the one-dimensional setting). This is equivalently described by the rough bracketdefined in [ S ] u,t = S u,t S u,t − S u,t . We refer again to [12] for the details, and point out the early work of F¨ollmer [11] in this regard.If [ S ] u,t is regular enough to define a measure of [ u, t ] with density a ( S t ) with a ( x ) also regular,then the classic partial differential equation for the option price is defined entirely in terms ofthe purely pathwise bracket [ S ], involving no probability theory and no semimartingale theory inparticular. It follows that the option price itself will not depend on the probabilistic setting butonly on path properties.The purely pathwise property [ S ] u,t takes the place of implied volatility in determining theoption price as a path property rather than a statistical property. The latter would be associatedwith historical volatility as a standard deviation (statistics). The first result we reported in this paper is the result of Brigo and Mercurio (1998) [8], where theprocess Y had simultaneously historical volatility ¯ σ as a statistical property and implied volatility ν as a pathwise property (quadratic variation). This is consistent with the later result of Bender,Sottinen and Valkeila (2008) [4] who note: amiano Brigo, Probability-free models in option pricing & historical vs implied volatilityamiano Brigo, Probability-free models in option pricing & historical vs implied volatility “[...] the covariance structure of the stock returns is not relevant for option pricing, but thequadratic variation is. So, one should not be surprised if the historical and implied volatilitiesdo not agree: the former is an estimate of the variance and the latter is an estimate of the[semimartingale] quadratic variation”. This has been further generalized in Armstrong et al. (2018) [1] where historical volatility is astatistics of the variance too, while implied volatility is associated with a pathwise lift involvingno semimartingale theory and no probability.It is perhaps fitting that this 20 years anniversary of pathwise pricing, running through 1998,2008 and 2018, is occurring at the conference on the 45th anniversary of the Black, Scholes andMerton option pricing theory.
Acknowledgments
The author is grateful to Claudio Bellani for checking the draft of the paper and for many helpfulsuggestions and to Mikko Pakkanen for referring him to the work of Bender, Sottinen and Valkeila.The author is further grateful to the organizers and participants of the conference “Options:45 Years after the publication of the Black-Scholes-Merton Model”, held in Jerusalem on 4–5December 2018, for their comments and suggestions.
References [1] Armstrong, J., Bellani, C., Brigo, D., and Cass, T. (2018). Gamma-controlled pathwise hedg-ing in generalised Black-Scholes models. https://arxiv.org/abs/1808.09378v1
Updated in April 2019 with the title “Option pricing models without probability”, https://arxiv.org/abs/1808.09378v2 [2] Bayer, C., Friz, P. and Gatheral, J. (2016). Pricing under rough volatility. QuantitativeFinance, 16(6):118, 2016.[3] Bellani, C. (2018). Connection between the result in Brigo Mercurio (2000) and rough volatil-ity. Internal note, Imperial College London.[4] Bender, C., Sottinen, T., and Valkeila, E. (2008). Pricing by hedging and no-arbitrage beyondsemimartingales. Finance and Stochastics, Vol. 12(4), pp 441–468[5] Black, F. and Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. TheJournal of Political Economy, Vol. 81, No. 3, pp. 637–654[6] Brigo, D. (1997). On nonlinear SDEs whose densities evolve in a finite–dimensional family.In: Stochastic Differential and Difference Equations, Progress in Systems and Control Theory23: 11–19, Birkh¨auser, Boston.[7] Brigo, D (2000). On SDEs with marginal laws evolving in finite-dimensional exponentialfamilies, Statistics and Probability Letters, 49: 127 – 134[8] Brigo, D. and Mercurio, F. (1998). Discrete time vs continuous time stock price dynamicsand implications for option pricing. arXiv.org and SSRN.com[9] Brigo, D. and Mercurio, F. (2000). Option pricing impact of alternative continuous timedynamics for discretely observed stock prices. Finance & Stochastics, 4, pp. 147-159[10] Davie, A. M. (2007). Differential equations driven by rough paths: an approach via discreteapproximation. Appl. Math. Res. Express AMRX 2, 40. amiano Brigo, Probability-free models in option pricing & historical vs implied volatilityamiano Brigo, Probability-free models in option pricing & historical vs implied volatility