Closed-End Formula for options linked to Target Volatility Strategies
CClosed-End Formula for options linked to Target VolatilityStrategies
Luca DI PERSIO a , Luca PREZIOSO b,c , and Kai WALLBAUM da Department of Computer Science, University of Verona, Strada le Grazie, 15, Verona, Italy b Department of Mathematics, University of Trento, via Sommarive, 14, Trento, Italy c LPSM, University of Paris Diderot, 5 Rue Thomas Mann, Paris, France d RiskLab, Allianz Global Investors, Seidlstrasse 24-24a, Munchen, GermanyFebruary 26, 2019
Abstract.
Recent years have seen an emerging class of structured financial products based onoptions linked to dynamic asset allocation strategies. One of the most chosen approachis the so-called target volatility mechanism. It shifts between risky and riskless assets tocontrol the volatility of the overall portfolio. Even if a series of articles have been alreadydevoted to the analysis of options linked to the target volatility mechanism, this paper is thefirst, to the best of our knowledge, that tries to develop closed-end formulas for VolTargetoptions. In particular, we develop closed-end formulas for option prices and some key hedgingparameters within a Black and Scholes setting, assuming the underlying follows a targetvolatility mechanism.
Key words.
Volatility target portfolio, generalized Black-Scholes model, European options, exactformulas, Greeks, Euler-Maruyama scheme, Milstein scheme.
AMS subject classification.
In the aftermath of the financial markets, risk management solutions became more and more impor-tant for institutional and retail investors. The low interest rate environment forced practitioners tothink of more efficient techniques how to use the available risk budgets in client portfolios. One ofthe most successful strategies, which were introduced in multi asset portfolios but also within struc-tured products offering at least partial capital protection is the so-called target volatility strategy,also known as
VolTarget strategy (VTS). This concept shifts dynamically between risk-free and riskyassets in order to generate a portfolio with a stable risk level independent of market volatilities.The approach assumes that market volatilities are a good indicator for asset allocation decisionsand the concept works well in rising markets with low volatility and in falling markets with highervolatilities. Practitioner often compare the concept with a constant portfolio protection insurance(CPPI) strategy, which also allocated dynamically between risky and riskless assets, but in thisconcept the investment process aims to achieve capital protection in general.In recent years, dynamic asset allocation process like VTS or CPPI Strategies have been used asunderlying of options and we saw a series of academic papers looking into option theory when the1 a r X i v : . [ q -f i n . P R ] F e b nderlying of the derivative follows a certain trading rule shifting between risky and riskless assets.Most of these papers took a numeric approach to determine option prices or hedging parameters.We refer, e.g. to Albeverio et al. [1, 2], Jawaid [11, 12], Zakamulin [20], who were especially lookingat VTS in different market models. Zagst et al. [8] focused on option on a CPPI and they alsodeveloped a closed-end formula of CPPI options in a Black-Scholes environment.This paper is the first attempt, to the best of ur knowledge, which considers closed-end formulasfor VTS-linked options. Our underlying environment can be compared to the one by Zagst et al.in [8], where the authors assume the risky asset to evolve as a Black-Scholes model. We extendsuch an analysis considering a generalized geometric Brownian motion framework, with randomdrift and diffusion, adapted to the real-world filtration of the probability space, then deriving aclosed-end formula for call and put options linked to VTS portfolios. We also consider the model’s Greeks , providing closed-end expressions for key hedging parameters of options linked to VTSs.We would llike to stress that our results constitute an important step for any practitioner, who ispricing and hedging options linked to VTS portfolios.The paper is organized as follows: in section 2 we analyzed VTS portfolios in the case that therisky-asset dynamics are described by a generalized geometric Brownian motion. We treated theevaluation problem for options that have, as underlying, VTS portfolios determined by standardVTS, preserving fixed volatility in time. In section 3 we considered a modification of the VTS, whichis placing an upper bound to the leverage effect caused by the continuous dynamic adjustmentsof the VTS. In this case we considered a risky asset evolving as a geometric Brownian motionwith time-dependent drift and volatility. For both strategies exact formulas for the price of calland put options are presented. In section 4 we analyzed the sensibility of the prices of optionswritten on VTS portfolios with respect to volatility and risky asset value. We gave emphasis onthe analysis of the Greeks (Vega, Delta and Gamma) with respect to changes in the underlyingvolatility, also providing several graphs to better highlight both the robustness and soundness of ourresults. In section 5 we present relevant simulations for the paths of a VTS portfolio, by exploitingboth Euler-Maruyama and Milstein discretization approach, when the risky asset is described by aHeston model.
Let us start by considering a framework similar to the pioneering paper by Merton [16]. This meansthat through the paper we are going to consider a market in which two investment opportunitiesoccurs: a risk-free asset , also referred as money market or Government bond or simply bond, anda risky underlying asset , also called stock or share. Moreover we assume that the randomnessof the underlying asset is described by Black-Scholes-Merton stochastic differential equations andthat there exist continuously-trading perfect markets where the agents are not subjected to anytransaction costs to trade the risky asset for the riskless asset and vice versa.Let (Ω , F , {F ( t ) } t ≥ , P ) be a filtered, complete probability space, with right-continuous filtra-tion, supporting a Brownian motion W , and consider a market consisting of two investment op-portunities: a risky asset { S ( t ) } t ≥ , and a riskless asset { B ( t ) } t ≥ , evolving as a stochastic processsatisfying the generalized geometric Brownian motion and a deterministic function:d S ( t ) = S ( t ) (cid:0) µ ( t ) d t + σ ( t ) d W ( t ) (cid:1) , (1)d B ( t ) = r B ( t ) d t, t ≥ t , where t ≥ W is a Brownian motion F ( t )-adapted, r ∈ R + is a positive constant representing the risk-free rate, and µ and σ are stochastic processes adaptedto {F W } , the natural filtration generated by the Brownian motion, and represent the mean rate ofreturn and the percentage volatility of the risky asset respectively. Let s, b ∈ R + be the values attime t for the risky and riskless assets.Moreover, consider an investor holding a portfolio, starting with a positive position x investedin the riskless asset and a positive position y invested in the risky asset, and assume that he isable to transfer its capitals from an investment to another without paying any transaction costs.Therefore let us introduce the processes L representing the cumulative amount of riskless asset soldin order to buy risky asset, and M the process representing the cumulative amount of risky assetsold in order to buy riskless asset. Both L and M are assumed to be non-negative, non-decreasingand c`adl`ag.Finally, the portfolio value can be represented continuously in time by the couple ( X ( t ) , Y ( t )) t ≥ t ,starting at X ( t ) = x , Y ( t ) = y , with ( X, Y ) representing the amount of capital invested in theriskless asset and in the risky asset, respectively, and evolving according to the following stochasticdifferential equationsd X ( t ) = r X ( t ) d t + d M ( t ) − d L ( t ) , (2)d Y ( t ) = µ ( t ) Y ( t ) d t + σ ( t ) Y ( t ) d W ( t ) + d L ( t ) − d M ( t ) . (3)In what follows we are not introducing neither proportional or fixed transaction costs, leavingthese framework for a future study. Let us just outline that, if costs are taken into account, thencontinuous re-balancing would cause non-negligible expenses to the investor in the portfolio aimingto preserve a fixed volatility which implies to consider a volatility target interval, instead of apunctual volatility target.let us denote the total portfolio value of the investor at time t > t , by V ( t ), and let α ( t ) denotesthe percentage of portfolio invested at the same time in the risky-asset assumed to be an adaptedpredictable c`adl`ag processe, while 1 − α ( t ) will denote the portfolio weight invested in the riskless-asset, namely we define V ( t ) = X ( t ) + Y ( t ), while α ( t ) = Y ( t ) X ( t )+ Y ( t ) . Since the investments evolveaccording to (2) and (3), by substituting the risky asset dynamics (1), we derive the dynamics ofthe portfolio value process: (cid:40) d V ( t ) = V ( t ) (cid:8)(cid:0) α ( t ) ( µ ( t ) − r ) + r (cid:1) d t + α ( t ) σ ( t ) d W ( t ) (cid:9) , t > t ,V (0) = x + y =: v, (4)where α is controlled by the investor and adapted to the filtration F . We make two remarks: firstof all notice that the portfolio whose value is determined by (4) is self-financing, i.e. the dynamicsof (4) are equivalent to d V ( t ) = V ( t ) (cid:18) α ( t ) d S ( t ) S ( t ) + (1 − α ( t )) d B ( t ) B ( t ) (cid:19) , (5)moreover notice that V is a Markovian portfolio and that a priori we do not know the future valueof the wealth process since it has a random dynamic.Turning back to the Volatility Target (VT) investment strategy, let us recall that it is a dynamicasset allocation mechanism, where the amount invested in the risky asset is determined by a pre-defined volatility target level, denoted by (cid:98) σ representing the volatility of the underlying riskyasset, σ ( t ), see (1). By dynamically shifting between the two investment opportunities, whichevolve accordingly to equations (1), the investor aims at preserving a constant volatility level of3he resulting portfolio V (cid:98) σ ( t ), which can be used as underlying for derivatives, e.g., for Europeancall/put options. We resume this notion in the following definition: Definition 2.1 (VolTarget strategy portfolio)
Consider the stochastic process V (cid:98) σ evolving ac-cording to V (cid:98) σ ( t ) = X − (cid:98) α ( t ) + Y (cid:98) α ( t ) , t ≥ t , (6)where Y (cid:98) α = (cid:98) α V (cid:98) σ and X (cid:98) α = (1 − (cid:98) α ) V (cid:98) σ , and by (cid:98) α we meant the proportion of portfolio value α ( t )dynamically invested in the risky asset. We say that V (cid:98) σ is a VolTarget strategy portfolio if it isself-financing and the weight process is preserving a constant volatility equal to (cid:98) σ , where X and Y represent the amount of capital invested in the riskless and risky asset and evolve according to (2)and (3), respectively.We want to determine explicitly the equation for the control which preserves a fixed volatilityto the portfolio process (4). Proposition 2.2
For (cid:98) α ( t ) = (cid:98) σ/σ ( t ) we have that the process whose dynamics are given by (6) isa VTS portfolio.Proof. By (4), for α ( t ) = (cid:98) α ( t ), we haved V (cid:98) σ ( t ) = V (cid:98) σ ( t ) (cid:18)(cid:18) (cid:98) σσ ( t ) ( µ ( t ) − r ) + r (cid:19) d t + (cid:98) σ d W ( t ) (cid:19) , (7)i.e. by Definition 2.1 we reach our thesis. (cid:3) Notice that we are not considering the equation involving the underlying (1) and the amountsof capital that have to be invested in the risky and riskless asset respectively. In Proposition 2.2 wesaw that in order to obtain a VTS portfolio the investor has to keep this ratio inversely proportionalto the actual value of the volatility rate of the risky asset, i.e. equal to (cid:98) α ( t ) = (cid:98) σ/σ ( t ), which isstochastic. Let X = Φ( V T ) be a contingent claim with maturity T and with underlying portfolio V , where Φis a contract function . In what follows we provide an arbitrage-free price Π( t ; X ) for such a claim,sometimes also denoted as Π( t ; Φ) or Π( t ).Let us heuristically assume for the moment that there exists a function F ∈ C , ([0 , T ] × R + )such that Π( t ) = F ( t, S ( t )) , then by the Black-Scholes equation we would have absence of arbitrage if F is solution to thefollowing PDE (cid:40) ∂ ( t ) F ( t, s ) + r s ∂ ( s ) F ( t, s ) + s (cid:98) σ ∂ ss F ( t, s ) − r F ( t, s ) = 0 F ( T, s ) = Φ( s ) , (8)for t ∈ [0 , T ] and s ∈ R + . In the next subsections we are going to remove the above mentionedheuristic assumption to derive pricing formulas for contingent claims written on the VTS portfolio.Let us note that the associated PDE (8), can be solved `a la Feynman-Kaˇc, i.e. F ( t, s ) = e − r ( T − t ) E Q t,s [ N ( S T )] , for t ∈ [0 , T ] , s > , Q con-ditioned by S ( t ) = s , see, e.g., [5, Ch. 14] for what concerns the existence of a unique risk-neutralmeasure. Let Q be the unique equivalent martingale measure , namely the unique measure under which S ( t ) /B ( t ) is a local martingale, and let W Q be a Brownian motion under Q . Then, by Girsanovtheorem, we have that the risky asset process S satisfies the following SDEd S ( t ) = S ( t ) (cid:16) r d t + σ ( t ) d W Q ( t ) (cid:17) . (9)Since the underlying risky asset is governed by a geometric Brownian motion with dynamics givenby equation (9), we can apply the Itˆo-D¨oblin formula to log( S ( t )) obtaining S ( t ) = S (0) exp (cid:26)(cid:90) t ( r − σ ( s ) /
2) d s + (cid:90) t σ ( s ) d W Q ( s ) (cid:27) . Since the volatility in equation (9) is stochastic, we cannot say much about the distribution oflog( S ( t ) /S (0)), which would have been Gaussian, in the special case of deterministic volatility.Let us consider a European call, resp. put, option with payoffΦ call ( V T ) = ( V T − K ) + , (10)Φ put ( V T ) = ( K − V T ) + , (11) T ≥ t being its maturity time, while K represents its strike price. Through the next propositionsand corollaries we will determine the price at the starting time t ≥ Proposition 2.3
Assuming that the risky asset dynamics follow a generalized geometric Brownianmotion with random F W -adapted drift and volatility, see equation (1) , the price at time t of a calloption with payoff (10) , denoted as Φ call , linked to the VTS portfolio V (cid:98) σ ( t ) , see equation (7) , isgiven by the following explicit formula Π( t , Φ call ( V (cid:98) σ ( T ))) = v N ( d ( t )) − K e − r ( T − t ) N ( d ( t )) , (12) where we recall that the proportion (cid:98) α ( t ) of portfolio value invested in the risky-asset is as theone defined in Proposition 2.2, N is the cumulative distribution function for the standard normaldistribution, v = V (cid:98) σ ( t ) is the starting value of the portfolio and we defined the following parameters d ( t ) = − z (cid:98) σ ( t ) + (cid:98) σ ( T − t ) √ T − t ,d ( t ) = − z (cid:98) σ ( t ) √ T − t ,z (cid:98) σ ( t ) = 1 (cid:98) σ log (cid:18) Kv (cid:19) + (cid:18) (cid:98) σ − r (cid:98) σ (cid:19) ( T − t ) . roof. Notice that while the underlying risky asset has non constant volatility, see eq. (7), thedynamics for the VTS portfolio are simpler. In fact, we can easily obtain the following explicitsolution V (cid:98) σ ( t ) = v exp (cid:18)(cid:18) r − (cid:98) σ (cid:19) ( t − t ) + (cid:98) σ W Q ( t − t ) (cid:19) , for t ≥ t . Therefore, we have that V (cid:98) σ ( T ) > K iff W Q ( T − t ) > (cid:98) σ log (cid:18) Kv (cid:19) + (cid:18) (cid:98) σ − r (cid:98) σ (cid:19) ( T − t ) =: z (cid:98) σ ( t ) . Denoting by f N (0 ,t ) ( x ) the probability density function of the Gaussian random variable havig mean0 and variance t , i.e. : f N (0 ,t ) ( x ) = 1 √ π t e − x t , and by N ( x ) the cumulative distribution function of a standard Gaussian random variable, thenwe have that the price of the call option on the portfolio value at time t equalsΠ( t , Φ call ( V (cid:98) σ ( T ))) = E (cid:104) e − r ( T − t ) ( V (cid:98) σ ( T ) − K ) + (cid:12)(cid:12)(cid:12) F t (cid:105) = e − r ( T − t ) (cid:90) + ∞ z (cid:98) σ ( t ) (cid:26) v exp (cid:18)(cid:18) r − (cid:98) σ (cid:19) ( T − t ) + (cid:98) σ x (cid:19) − K (cid:27) f N (0 ,T − t ) ( x ) d x = e − r ( T − t ) v e (cid:16) r − (cid:98) σ (cid:17) ( T − t )+ (cid:98) σ / T − t ) (cid:18) − N (cid:18) z (cid:98) σ ( t ) − (cid:98) σ ( T − t ) √ T − t (cid:19)(cid:19) − K e − r ( T − t ) (cid:18) − N (cid:18) z (cid:98) σ ( t ) √ T − t (cid:19)(cid:19) = v N (cid:18) − z (cid:98) σ ( t ) + (cid:98) σ ( T − t ) √ T − t (cid:19) − K e − r ( T − t ) N (cid:18) − z (cid:98) σ ( t ) √ T − t (cid:19) . (cid:3) Corollary 2.4
Assuming that the risky asset dynamics follow a generalized geometric Brownianmotion with random F W -adapted drift and volatility, see equation (1) , the price at time t of a putoption with payoff (11) , denoted as Φ put , linked to the VTS portfolio V (cid:98) σ ( t ) , see equation (7) , isgiven by the following explicit formula Π( t , Φ put ( V (cid:98) σ ( T ))) = K e − r ( T − t ) N ( − d ( t )) − v N ( − d ( t )) , (13) where the parameters d , d and z (cid:101) σ are defined as in Proposition 2.3.Proof. By the put-call parity formula, see, e.g., [19, 4.5.6], we have that the difference betweenthe price of a call option and the price of put option with same strike price, time to expiration andunderlying, equals the difference between the actual price of the underlying, represented by the VTportfolio in our setting, and the discounted strike price, namelyΠ( t , Φ call ( V (cid:98) σ ( T ))) − Π( t , Φ put ( V (cid:98) σ ( T ))) = v − K e − r ( T − t ) , therefore, we have (13), since N ( − x ) = 1 − N ( x ) for each x ∈ R . (cid:3) VolTarget Strategy with maximum allowed Leverage Factor
In what follows we are going to study a more interesting strategy from the practitioner point ofview. In particular, we introduce a parameter L ≥ L : (cid:101) α ( t ) := min { L ; (cid:98) σ/σ ( t ) } . (14)From now on, we will distinguish the notations for standard VTSs by the one for VTSs withmaximum allowed leverage factor (MLVTS, in short), by marking the volatility and weight symbolswith an hat and a tilde, respectively. In particular, while (cid:98) σ and (cid:98) α refere to standard VTS portfolios, (cid:101) σ and (cid:101) α are referred to MLVTS portfolios.This limitation is imposed in order to prohibit VTSs that finance by loans a large portion ofthe risky investment. The typical setup occurring within real world scenarios is L = 2, see [2], forfurther details.The next proposition gives an analytical expression for the value of a European call optionwritten on the MLVTS portfolio, with limited leverage, and time dependent volatility. In particular,we are considering a particular case of equation (1):d S ( t ) = S ( t ) (cid:0) µ ( t ) d t + σ ( t ) d W ( t ) (cid:1) , (15)where µ, σ : R + → R + are deterministic functions of time, allowing, instead, the percentage driftterm to be stochastic and F W -adapted. Proposition 3.1
Assuming that the risky asset dynamics follows a geometric Brownian motionwith time-dependent drift and volatility, see equation (15) , the price at time t of a call option withpayoff (10) , denoted as Φ call , linked to the MLVTS portfolio V (cid:101) σ ( t ) , is given by the following explicitformula Π ( t , Φ call ( V (cid:101) σ ( T ))) = v N (cid:16) (cid:101) d ( t ) (cid:17) − K e − r ( T − t ) N (cid:16) (cid:101) d ( t ) (cid:17) (16) where the proportion of portfolio value invested in the risky-asset is (cid:101) α ( t ) := min { L ; (cid:98) σ/σ ( t ) } , N isthe cumulative distribution function for the standard normal distribution, v = V (cid:98) σ ( t ) is the startingvalue of the portfolio and we defined the following parameters (cid:101) d ( t ) = − (cid:101) z (cid:98) σ ( t ) + ς ( t ) (cid:112) ς ( t ) , (cid:101) d ( t ) = − (cid:101) z (cid:98) σ ( t ) (cid:112) ς ( t ) ,z (cid:101) σ ( t ) = log (cid:18) Kv (cid:19) − r ( T − t ) + ς ( t )2 ,ς ( t ) = (cid:90) Tt (cid:101) σ ( s ) d s, (cid:101) σ ( t ) = min { L σ ( t ) , (cid:98) σ } . Proof.
For this strategy we have that the portfolio value has not a constant volatility and it hasthe following expression V (cid:101) σ ( t , t ) = v exp (cid:18) r ( t − t ) − ς ( t ) / (cid:90) tt min( L σ ( s ) , (cid:98) σ ) d W Q ( s ) (cid:19) , (cid:102) W ( t − t ) := (cid:82) tt min( L σ ( s ) , (cid:98) σ ) d W Q ( s ) ∼ N (0 , ς ( t )), which means that its probabilitydensity function is f N (0 ,ς ( t )) ( x ) = 1 (cid:112) π ς ( t ) exp (cid:18) − x ς ( t ) (cid:19) . Therefore we have that V (cid:101) σ ( T ) > K iff (cid:102) W ( T − t ) > log (cid:18) Kv (cid:19) − r ( T − t ) + ς ( t )2 =: z (cid:101) σ ( t ) , and have we have that the considered option value equals toΠ ( t , Φ call ( V (cid:101) σ ( T ))) = E (cid:104) e − r ( T − t ) ( V (cid:101) σ ( T ) − K ) + (cid:12)(cid:12)(cid:12) F t (cid:105) = e − r ( T − t ) (cid:90) + ∞ z (cid:101) σ (cid:110) v exp (cid:16) r ( T − t ) − ς ( t ) / x (cid:17) − K (cid:111) f N (0 ,ς ( t )) ( x ) d x = e − r ( T − t ) v e r ( T − t ) − ς ( t ) / ς ( t ) / (cid:32) − N (cid:32) z (cid:101) σ ( t ) − ς ( t ) (cid:112) ς ( t ) (cid:33)(cid:33) − K e − r ( T − t ) (cid:32) − N (cid:32) z (cid:101) σ ( t ) (cid:112) ς ( t ) (cid:33)(cid:33) = v N (cid:32) − z (cid:101) σ ( t ) + ς ( t ) (cid:112) ς ( t ) (cid:33) − K e − r ( T − t ) N (cid:32) − z (cid:101) σ ( t ) (cid:112) ς ( t ) (cid:33) . (cid:3) Remark 3.2
Notice that the price of this call option depends on the future volatility, but, sinceit is deterministic, it is not an issue, and indeed we have obtained exact formulas.
Corollary 3.3
Assuming that the risky asset dynamic follows a geometric Brownian motion withtime-dependent drift and volatility, see equation (15) , the price at time t of a put option withpayoff (11) , denoted as Φ put , linked to the MLVTS portfolio V (cid:101) σ ( t ) , is given by the following explicitformula Π ( t , Φ put ( V (cid:101) σ ( T ))) = K e − r ( T − t ) N (cid:16) − (cid:101) d ( t ) (cid:17) − v N (cid:16) − (cid:101) d ( t ) (cid:17) (17) where (cid:101) d and (cid:101) d are defined as in Proposition 3.1, and the proportion of portfolio value invested inthe risky asset is (cid:101) α ( t ) := min { L ; (cid:98) σ/σ ( t ) } .Proof. Direct consequence of the put-call parity formula. (cid:3)
In this section we will move on the quantitative study of the prices of options on a VTS portfolioin continuous time. In particular, we will explicitly derive the associated
Greeks’ values, the latterbeing those quantities representing derivatives price sensitivness to their characterizing underlyingparameters’ changes in time. 8n what follows we are going to consider a risky asset evolving as in the Black-Scholes model.Therefore, the price formulas (12), (13), (16) and (17) for call and put options, with VTS andMLVTS underlying portfolios, reduce toΠ ( t , Φ call ( V (cid:98) σ ( T ))) = v N ( d ) − K e − r ( T − t ) N ( d ) , Π ( t , Φ call ( V (cid:101) σ ( T ))) = (cid:40) v N ( (cid:101) d ) − K e − r ( T − t ) N ( (cid:101) d ) , for σ < (cid:98) σ/L,v N ( d ) − K e − r ( T − t ) N ( d ) , for σ > (cid:98) σ/L, Π ( t , Φ put ( V (cid:98) σ ( T ))) = K e − r ( T − t ) N ( − d ) − v N ( − d ) , Π ( t , Φ put ( V (cid:101) σ ( T ))) = (cid:40) K e − r ( T − t ) N ( − (cid:101) d ) − v N ( − (cid:101) d ) , for σ < (cid:98) σ/L,K e − r ( T − t ) N ( − d ) − v N ( − d ) , for σ > (cid:98) σ/L, where d = 1 (cid:98) σ √ T − t (cid:0) log( v/K ) + ( r + (cid:98) σ /
2) ( T − t ) (cid:1) ,d = 1 (cid:98) σ √ T − t (cid:0) log( v/K ) + ( r − (cid:98) σ /
2) ( T − t ) (cid:1) , (cid:101) d = 1 σ L √ T − t (cid:0) log( v/K ) + ( r + L σ /
2) ( T − t ) (cid:1) , (cid:101) d = 1 σ L √ T − t (cid:0) log( v/K ) + ( r − L σ /
2) ( T − t ) (cid:1) . Since VTS portfolios are meant to preserve a fixed volatility level, the most representative Greekvalue is the
Vega one, since it represents the sensitivity of the option price to the risky asset’svolatility.
Proposition 4.1
The Vega of a call and put option, with payoff (10) and (11) , on VTS andMLVTS portfolios with weight strategies (cid:98) α = (cid:98) σ/σ and (cid:101) α := min { L ; (cid:98) σ/σ } respectively, are respec-tively given by ν { Φ call ,V (cid:98) σ } = ∂ σ Π ( t , Φ call ( V (cid:98) σ ( T ))) = 0 ,ν { Φ call ,V (cid:101) σ } = ∂ σ Π ( t , Φ call ( V (cid:101) σ ( T ))) = v √ π exp (cid:18) − (cid:101) d (cid:19) L √ T − t , for σ < (cid:98) σL , , for σ > (cid:98) σL , (18) with ν { Φ put ,V (cid:98) σ } = ν { Φ call ,V (cid:98) σ } and ν { Φ put ,V (cid:101) σ } = ν { Φ call ,V (cid:101) σ } , where (cid:101) d = log( v/K ) + (cid:16) r + L σ (cid:17) ( T − t ) L σ √ T − t . Proof.
Let us consider the MLVTS for the case in which σ < (cid:98) σL , then we recall that the price ofthe call option simplifies toΠ ( t , Φ call ( V (cid:98) σ ( T ))) = v N ( (cid:101) d ) − K exp( − r ( T − t )) N ( (cid:101) d ) , (cid:101) d = − log( K/v ) + ( r + L σ /
2) ( T − t ) L σ √ T − t , (cid:101) d = − log( K/v ) + ( r − L σ /
2) ( T − t ) L σ √ T − t . Then, computing the partial derivative with respect to σ , we have ∂ σ Π ( t , Φ call ( V (cid:101) σ ( T ))) = 1 √ π (cid:32) v e − (cid:101) d / (cid:32) L (cid:112) T − t − (cid:101) d σ (cid:33) + K e − (cid:101) d / − r ( T − t ) (cid:32) L (cid:112) T − t + (cid:101) d σ (cid:33)(cid:33) = 1 √ π v e − (cid:101) d / (cid:32) L (cid:112) T − t − (cid:101) d − (cid:101) d σ (cid:33) = v √ π exp (cid:32) − ( (cid:101) d ) (cid:33) L (cid:112) T − t , where the second row steams from the identity exp( (cid:101) d − (cid:101) d ) = vK exp( r ( T − t )), and the last by (cid:101) d − (cid:101) d = L σ √ T − t .Similar computations also work for the Vega of put options. (cid:3) Remark 4.2 let us underline that if σ < (cid:98) σ/L , then the MLVTS call and the standard leverage callshare the same price, which can be expressed in terms of the standard call option price, denotedas Π S := Π ( t , Φ call ( S ( T ))); hnce stating the dependence with respect the risky asset’s volatilityΠ V (cid:101) σ ( σ ) = Π S ( L σ ) , so that, computing the partial derivative with respect to the volatility, we have ν { Φ call ,V (cid:101) σ } = ∂ σ Π V (cid:101) σ ( σ ) = L ∂ σ Π S ( L σ ) , namely the same expression as in equation (18).In figure 1 we provide a comparison between the graphs of a Vega for a call option written ona portfolio adopting a MLVTS, and the Vega for a standard call option. The left graph representsVegas for at-the-money options. Here, one can notice that, while for volatilities higher than σ > (cid:98) σL ,the MLVTS Vega is null, for small volatilities the MLVTS Vega is even higher than the Vega forstandard call options. The right graph in figure 1 represents the comparison taking into accountthe sensitivity of Vega with respect to the portfolio value. Notice that, while for a standard calloption the highest Vega is reached for the underlying share’s value equal to v ∗ = s ∗ = K e − ( T − t ) ( r − σ ) , for the MLVTS call option, it is reached in v ∗ = K e − ( T − t ) ( r − L σ ) . Finally, in figure 2, we summarized the dependence of Vega with respect to both the volatility andthe underlying portfolio value. 10igure 1: The plots represent the behavior of Vega values of an option written on the MLVTSportfolio, with maximum allowed leverage given by the weight strategy (cid:101) α = min( L, (cid:98) σ/σ ), high-lighting its dependence on the values of the volatility σ (left) and on the values of v (right). Theparameters are set up as r = 5% , v = 12 , K = 10 , t = 0 , T = 1 , (cid:98) σ = 20% , L = 2. For the volatilitydependence (left), the MLVTS Vega line (in blue) is also compared with the dotted line of anhypothetical portfolio holding L times its wealth in the risky asset (in cyan) and the Vega for astandard call option whose underlying is simply the risky asset. Instead, for the portfolio initialvalue dependence (right), we considered σ = 0 .
08, i.e. σ < (cid:98) σ/L . The MLVTS Vega line (in blue)is the same as the Vega of an hypothetical portfolio holding L times its wealth in the risky asset.This line is compared with the Vega of a portfolio investing all its capital in the risky asset. Weremark that we obtained a greater Vega value for the MLVTS option than the one for the standardoption. This is due to the fact that we considered a relatively small volatility, namely less than (cid:98) σ/L ). Instead, if we would have considered a volatility greater than (cid:98) σ/L , the Vega value for theoption written on the MLVTS would have been identically zero.Figure 2: The two surfaces are the Vega for a portfolio adopting a MLVTS (left figure) and theVega for a portfolio investing L times its wealth v in the risky asset (right figure). One can noticethat the MLVTS hedges well the portfolio against volatility variations when the volatility is high(higher than (cid:101) σ/L ). The parameters are set as r = 5% , t = 0 , T = 1 , (cid:98) σ = 20% , L = 2 , K = 10.11 .2 Delta Before dealing with the Delta of options written on VTS and MLVTS portfolios, it is worth to startanalyzing the sensitivity of the VTS portfolio with respect to small changes in the risky asset price.To perform this, we write the VTS portfolio dynamics asd V (cid:98) σ ( t ) = ϕ (cid:98) σ ( t ) d S ( t ) + ψ (cid:98) σ ( t ) d B ( t ) , (19)where we defined ϕ S and ϕ B as the instantaneous number of shares and bonds held in the portfolio.By the self-financing equation (5), we have that ϕ (cid:98) σ ( t ) = V ( t ) (cid:98) α ( t ) S ( t ) ,ψ (cid:98) σ ( t ) = V ( t ) (1 − (cid:98) α ( t )) B ( t ) , which means that the Delta of the VT portfolio is∆ V (cid:98) σ = V ( t ) (cid:98) σS ( t ) σ . (20) Remark 4.3
Let us underline that the price of an option written on the VTS portfolio may beequivalently determined solely by the dynamics of the VTS portfolio and the actual time, or by therisky asset dynamics, the bond dynamics and the actual time. Namely, we may denote the price ofa generic option on a VTS portfolio as Π( t, V ) or Π( t, S, B ). Therefore, within the first setting, byequation (19) and Itˆo-D¨oeblin formula, see [19, Ch. 4], we havedΠ( t, V ) = ∂ t Π( t, V ) d t + ∂ V Π( t, V ) d V t + 12 ∂ V V Π( t, V ) d[ V, V ] t = ∂ t Π( t, V ) d t + ∂ V Π( t, V ) (cid:0) ϕ ( t ) d S t + Ψ( t ) d B t (cid:1) + 12 ∂ V V Π( t, V ) ϕ ( t ) d[ S, S ] t , (21)where [ V, V ] denotes the quadratic variation of the stochastic process V , see, e.g., [19, 3.4.2].Instead, considering the option price as a function of time, risky asset price and bond price, byItˆo-D¨oeblin formula, we havedΠ( t, S, B ) = ∂ t Π( t, S, B ) d t + ∂ S Π( t, S, B ) d S t + ∂ B Π( t, S, B ) d B t + 12 ∂ SS Π( t, S, B ) d[ S, S ] t , (22)therefore, combining equations (21) and (22), we derive a simpler expression for Delta and Gammaof an option on VTS portfolios ∂ S Π( t, S, B ) = ∂ V Π( t, V ) ϕ ( t ) ,∂ SS Π( t, S, B ) = ∂ V V Π( t, V ) ϕ ( t ) . Proposition 4.4
The Delta of a European call option with payoff (10) on VTS and MLVTSportfolios with weight strategies (cid:98) α = (cid:98) σ/σ and (cid:101) α := min { L ; (cid:98) σ/σ } are respectively given by ∆ { Φ call ,V (cid:98) σ } = ∂ S Π ( t , Φ call ( V (cid:98) σ ( T ))) = v (cid:98) σs σ N ( d ) , (23)12 { Φ call ,V (cid:101) σ } = ∂ S Π ( t , Φ call ( V (cid:101) σ ( T ))) = (cid:40) L vs N ( (cid:98) d ) , for σ < (cid:98) σL , v (cid:98) σs σ N ( d ) , for σ > (cid:98) σL , (24) where d = log( v/K ) + (cid:16) r + (cid:98) σ (cid:17) ( T − t ) (cid:98) σ √ T − t , (cid:101) d = log( v/K ) + (cid:16) r + L σ (cid:17) ( T − t ) L σ √ T − t . While the Delta of a European put option with payoff (11) is ∆ { Φ put ,V (cid:98) σ } = ∂ S Π ( t , Φ put ( V (cid:98) σ ( T ))) = v (cid:98) σs σ ( N ( d ) − , (25)∆ { Φ put ,V (cid:101) σ } = ∂ S Π ( t , Φ put ( V (cid:101) σ ( T ))) = (cid:40) L vs ( N ( (cid:98) d ) − , for σ < (cid:98) σL , v (cid:98) σs σ ( N ( d ) − , for σ > (cid:98) σL . (26) Proof.
By Itˆo calculus’ chain rule (see Remark 4.3)∆ { Φ call ,V (cid:98) σ } = ∂ V Π ( t , Φ call ( V (cid:98) σ ( T ))) ∆ V (cid:98) σ . (27)Moreover, it is straightforward to obtain ∂ V Π ( t , Φ call ( V (cid:98) σ ( T ))) = N ( d ) + v N (cid:48) ( d ) ∂ v d − K e − r, ( T − t ) N (cid:48) ( d ) ∂ v d = N ( d ) + 1 √ π √ T − t (cid:98) σ v (cid:16) v e − d − K e − r ( T − t ) e − d (cid:17) = N ( d ) + e − d √ π √ T − t (cid:98) σ v (cid:18) − Kv e − r ( T − t ) e − ( d − d ) (cid:19) = N ( d ) , (28)where the last equality holds since e − ( d − d ) = v/K e r ( T − t ) . Therefore, substituting (20) and(28) in equation (27), we obtain (23).The same arguments works for ∆ V (cid:101) σ , just taking care that in this case we have∆ V (cid:101) σ = L vs , for σ < (cid:98) σL , since for σ < (cid:98) σL , (cid:101) α = L .For the put option case, we have that ∂ V Π ( t , Φ call ( V (cid:98) σ ( T ))) = N ( d ) −
1. Therefore, analogouslyas before, we obtain equation (25). (cid:3)
In figure 3 we compared the Deltas for calls and puts written on a VTS portfolio with callsand puts written on the risky asset. One can notice that Deltas for VTS-linked options present anasymptotic behavior for both low and high volatilities. This is because, for extreme low volatilites,the VTS portfolio finances a great amount of shares through short selling the riskless asset, while,for extreme high volatilities, the VTS portfolio invests only a small proportion of its value in therisky asset. See Remark 4.3 for the mathematical explanation of this effect.For MLVTS portfolios the analysis of the Deltas with respect to the volatility of the risky assetis not significantly different than the Deltas for standard options, since it suffices to consider thechange of variable σ L = L σ in the standard Delta value and to boost the latter by the maximumleverage parameter L . 13igure 3: The right graphs represent the behavior of the Delta of standard call (top figure) andput (bottom figure) options with respect to different volatility values. The graphs on the leftrepresent the Delta for call and put options written on VTS portfolios. We considered strike prices K in order to obtain an in-the-money option, an at-the-money option, an at-the-money-forward option and an out-of-the-money option. Notice that the Delta for the VTS-linked options exhibitstwo asymptotes: the vertical one corresponding to null volatility, and the horizontal one whichcorrespond to a volatility value that goes to infinity. The parameters are fixed as s = v = 10, (cid:98) σ = 0 . µ = 8%, r = 5%, T = 1, t = 0 and the volatilities start at σ = 0 . .3 Gamma The computation of the Gamma for options on VTS and MLVTS portfolios can be derived as insection 4.2, see, in particular, Remark 4.3.
Proposition 4.5
The Gamma of an option with payoff (10) on the VTS and MLVTS portfolioswith weight strategies (cid:98) α = (cid:98) σ/σ and (cid:101) α := min { L ; (cid:98) σ/σ } are respectively given by Γ { Φ call ,V (cid:98) σ } = ∂ SS Π ( t , Φ call ( V (cid:98) σ ( T ))) = v (cid:98) σs σ √ T − t f N (0 , ( d ) , (29)Γ { Φ call ,V (cid:101) σ } = ∂ SS Π ( t , Φ call ( V (cid:101) σ ( T ))) = (cid:40) L vs σ √ T − t f N (0 , ( (cid:98) d ) , for σ < (cid:98) σL , v (cid:98) σs σ √ T − t f N (0 , ( d ) , for σ > (cid:98) σL , (30) with Γ { Φ put ,V (cid:98) σ } = Γ { Φ call ,V (cid:98) σ } and Γ { Φ put ,V (cid:101) σ } = Γ { Φ call ,V (cid:101) σ } , where by f N (0 , we denote the probabilitydensity function of a standard normal random variable and d = log( v/K ) + (cid:16) r + (cid:98) σ (cid:17) ( T − t ) (cid:98) σ √ T − t , (cid:101) d = log( v/K ) + (cid:16) r + L σ (cid:17) ( T − t ) L σ √ T − t . Proof.
By Remark 4.3, we have ∂ SS Π( t, S, B ) = ∂ V V Π( t, V ) ϕ ( t ) . Computing ∂ V V Π( t, V ): ∂ V V Π( t, V ) = ∂ V [ ∂ V Π ( t , Φ call ( V (cid:98) σ ( T )))]= ∂ V N ( d )= N (cid:48) ( d ) ∂ V d = f N (0 , ( d ) 1 (cid:98) σ v √ T − t , and since ϕ ( t ) = v (cid:98) σs σ , we obtain (29).The Gamma for put VTS-linked options is the same as the Gamma for call VTS-linked options,since the second partial derivatives w.r.t. the portfolio value of the price of the two options are thesame. (cid:3) In figure 4, we compared the Gamma for standard European options with the Gamma forEuropean options written on VTS portfolios. Notice that, while the Gamma for standard Europeanoptions exhibits two asymptotes only when the underlying risky asset is at-the-money-forward (ATMF), i.e. S = K e − r ( T − t ) which impliesΓ S = 1 s σ √ T − t e − σ T − t , , for the Gamma of VTS-linked options we have always two asymptotes, since for low volatilities alsotheir Gamma is amplified, in fact even more than the Delta.15igure 4: The graph on the right represents the behavior of the Gamma of standard call/putoptions with respect to different volatility values, while the one on the left represents the Gammafor call/put options written on VTS portfolios. As in figure 3, we considered strike prices K in orderto obtain an in-the-money option, an at-the-money option, an at-the-money-forward option and an out-of-the-money option. Notice that, once again, also this Greek for the VT options exhibitsasymptotes, while this is the case for standard options only when the underlying asset is ATMF.The parameters are fixed as v = s = 10, (cid:98) σ = 0 . µ = 8%, r = 5%, T = 1, t = 0 and the volatilitiesstart at σ = 0 . To better explain how VTS and MLVTS portfolios work, let us assume that the dynamics of therisky asset evolve according to the Heston model, see, e.g., [10] and [3], for further details andcontrol-theory related problems. The, we haved S t = µ S t d t + √ ν t S t d W (1) t , (31)d ν t = κ ( θ − ν t ) d t + ξ √ ν t d W (2) t , (32)where W (1) and W (2) are ρ -correlated Brownian motions, ν evolves as a Cox-Ingersoll-Ross (CIR)process representing the instantaneous variance of the risky asset, θ is the long-variance, κ is therate at which ν reverts to θ , ξ is the volatility of the volatility, and we assume that the Fellercondition holds: 2 κ θ > ξ , in order to guarantee the process ν to be strictly positive.Let us consider underlying risky asset’s parameters calibrated to values observed in the realdata, as in the papers by Morellec et al. [17], and CIR parameters as in the seminal paper bySamuelson [18]. Consider the parameters values as shown in Table 1, namely an adaptation of theones in [17, 18], to show representative scenarios explaining the effect of the VTS and the MLVTS.We partition the time-interval [0 , T ] into N equal subintervals of width T /N t < t < · · · < t N = T κ θ ξ ρ ν µ S r B V T S, ν ) t . By a modified Euler-Maruyama scheme ,see below, we approximate the path of the corresponding VTS and MLVTS portfolio V ∆ t (cid:98) σ ( t n +1 ) = V ∆ t (cid:98) σ ( t n ) (cid:26) α ( t n ) S ( t n ) ∆ S n + 1 − α ( t n ) B ( t n ) ∆ B n (cid:27) , for n ∈ { , . . . , N − } , (33)where ∆ S n := S ( t n +1 ) − S ( t n ), ∆ B n := B ( t n +1 ) − B ( t n ) and V ∆ t (0) = v . Proposition 5.1
Let
T > be fixed. The numerical scheme (33) is strongly convergent to thesolution to (4) ., i.e. lim ∆ t → E [ | V T − V ∆ tT | ] = 0 . Proof.
The Euler-Maruyama scheme associated to equation (4) is V ∆ t (cid:98) σ ( t n +1 ) = V ∆ t (cid:98) σ ( t n ) (cid:40) (cid:34) (cid:98) σ (cid:112) ν ( t n ) ( µ − r ) − r (cid:35) ∆ t + (cid:98) σ ∆ W (1) n (cid:41) , (34)which i strongly convergent to (4). Moreover, we have thatlim ∆ t → E (cid:34) ∆ S n S ( t n ) − µ ∆ t (cid:112) ν ( t n ) − ∆ W (1) n (cid:35) = 0 . (35)Substituting (35) in (34), we obtain (33). (cid:3) Even if the Euler-Maruyama scheme is strongly convergent, it is just of order 0.5, therefore,aiming at achivieng a higher convergence order, we also considered the following modification ofthe
Milstein scheme : V ∆ t (cid:98) σ ( t n +1 ) = V ∆ t (cid:98) σ ( t n ) (cid:40) α ( t n ) ∆ S n S ( t n ) + (1 − α ( t n )) ∆ B n B ( t n ) − α ( t n ) (1 − α ( t n ))2 (cid:34)(cid:18) ∆ S n S ( t n ) (cid:19) − ν ( t n ) ∆ t (cid:35)(cid:41) , which in general converges strongly to the solution with order 1. Let us underline that the sensitivity analysis provided along the previous section is well captured byfigure 6. For example, one can notice that the tendency of the path is more or less met, dependentlyon the volatility instantaneous value, see Section 4.2 for the sensitivity analysis of the VTS portfoliowith respect to small changes in the underlying risky asset, i.e. the Delta analysis. Moreover, it isclearly visible that the white noise is affecting the VTS portfolio value linearly, i.e. the Vega for theVTS standard portfolio is null, see Section 4.1. 17igure 5: The top graphs represent the asset price and the volatility values, simulated as a realiza-tion of a Heston model. Here the volatility (top-right figure) is more frequently greater than thetarget volatility (cid:98) σ = 0 .
2, and in these cases the risky proportion (cid:98) α is less than one (bottom-rightfigure). In the bottom-left figure is represented the corresponding realization of the VTS portfolio.For the discretization scheme (33) we considered a time step ∆ t = 10 − .18igure 6: The left graphs from the top to the bottom represent the risky asset dynamics, the VTportfolio and the MLVTS portfolio, for (cid:98) σ = 0 . L = 2. The top-right figure represents thevolatility of the risky asset (in black), the volatility of the VTS portfolio (in blue) and the effect ofthe leverage limitation in the MLVTS (in red). In the bottom-left figure are highlighted in red thepath section of the MLVTS portfolio in which the leverage effect intervenes. For the discretizationscheme (33) we considered a time step ∆ t = 10 − .19 Extension to the transaction case and concluding remarks
The present paper presents a first attempt to consider options linked to VTSs from an analyticalperspective. We develop closed-end formulas for call and put options linked to VolTarget concepts,as well as for the associated sensitiveness,
Greeks , parameters.The results agree with what we would expect from a practitioner view. One can see, how aVolTarget approach can simplify option pricing for structured products and why also key hedgingparameters look much easier than for standard options with changing volatility pattern. Furtheranalysis should be done relaxing some of the assumptions we made to derive our results, e.g.dropping the non transaction costs hypotesis.As an example, we already started to look into the aspect of transaction costs and how these canbe embedded into our framework. We hereby point out two possible ways that can be undertaken.The first one consists in a modification of the chosen VTS, i.e. the VTS portfolio will no longerpursue a constant volatility, instead it will aim to have a volatility belonging to a desired interval.Moreover, one could consider to deal with a structural modification, namely considering a restrictionof the admissible time interval in which we will have portfolio weight adjustments, to consider adiscrete subset. Namely, the VTS will pursue the target volatility only in a discrete set of timepoints, instead of considering continuous adjustments. Such modifications are required, when theasset dynamics are assumed to not have a constant volatility, in order to avoid the cumulatedtransaction costs to be theoretically infinity, which could happen even when the transaction costsare relatively small, see, e.g., [6, 7, 9, 14, 15].We will also investigate how a dynamic asset allocation strategy can be developed within realworld scenarios, when a rather constant volatility level can be considered.
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