Stochastic : Modeling and Asymptotic Approachesto Pricing & Portfolio Selection

Matthew Lorig ∗ Ronnie Sircar †

July 2014; revised February 2, 2015

Abstract Empirical evidence from equity markets clearly shows that the volatility of asset returns varies randomly in time. Typically, this randomness is referred to as stochastic volatility.In this article, we review how stochastic volatility can be modeled, and the use of asymptotic analysis to quantify (i) how the presence of stochastic volatility affects option prices, and (ii) how stochastic volatility affects investment strategies.

1Introduction

Understanding and measuring the inherent uncertainty in market volatility is crucial for portfolio optimization, risk management and derivatives trading. Theproblemismadedifficultsincevolatil- ity is not directly observed. Rather, volatility is a statistic of the observable returns of, for example, astock,andsoestimatesofitareatbestnoisy.Amongthemajor empirical challenges have been separating contributions of diffusive and jump components of log returns, typical time-scales of fluctuation and memory effects. Until recently, data was limited to low frequencies, typically daily. The availability of high-frequency data over the past twentyyearsbringswithitissuesofdeci- phering market micro-structure effects such as bid-ask bounce, which contaminate the potential usefulness of such large data sets, and we refer to the recent book A¨ıt-Sahalia and Jacod (2014) for an overview of the difficulties. The major problem that has been the driver of stochastic volatility models is the valuation and hedging of securities. This market grew in large part from the landmark paper by Black and Scholes (1973), which showed how to value simple options contracts when volatility is constant. Even at the time of their paper, Black and Scholes realized that the constant volatility assumption was a strong idealization. In an empirical paper Black and Scholes (1972), the authors tested their option price formulas and concluded: “we found that using past data to estimate the variance caused the model to overprice options on high-variance stocks and underprice options on low-variance stocks.” Indeed the overwhelming evidence from time-series data reveals that volatility exhibits unpredictable variation. In addition,aswewilldescribebelow,optionprices exhibit a significant departure from the Black-Scholes (constant volatility) theory, the “ skew”, which can be explained by allowing volatility to vary randomly in time.

∗Department of Applied Mathematics, University of Washington, Seattle, USA. Work partially supported by NSF grant DMS-0739195. †ORFE Department, Princeton University, Princeton, USA. Work partially supported by NSF grant DMS-1211906.

1 Asecondimportantproblemisportfoliooptimization:namely how to optimally invest capital between a risky stock and a riskless bank account. In a continuous time stochastic model with constant volatility, the pioneering work was by Robert Merton (Merton (1969) and Merton (1971), reprinted in the book Merton (1992)). Since then, understanding the effect of volatility uncertainty, stochastic growth rate, transaction costs, price impact, illiquidity and other frictions on the portfolio choice problem has generated considerable research. Here we will focus on the effect of stochastic volatility and present some new results in Section 3. The increased realism obtained by allowing volatility to be stochastic comes with increased computational difficulties. Moreover, there is no broad consensus concerning how to best model volatility. Here we will discuss some computationally efficient approaches, focusing particularly on asymptotic approximations.

1.1 Options and Implied Volatility The most liquidly traded derivatives contracts are call and put options, which give the option holder the right to buy (in the case of a call) or sell (in the case of a put) one unit of the underlying at a fixed K on a fixed date T .Herewearefocusingspecificallyon European-style options (i.e., no early ), which are typically traded on indices such as the S&P500. If St represents the price of a stock or index at time t,thenaEuropean-stylederivative has a payoffat time T which is a function h of ST .Inthecaseofcallsandputsthepayofffunctions are h(S)=(S K)+ and h(S)=(K S)+ respectively. − − 1.1.1 Black-Scholes model In the Black-Scholes model, the stock price S is a geometric Brownian motion described by the following stochastic differential equation (SDE)

dSt = µ dt + σ dWt, (1.1) St where W is a standard Brownian motion with respect to a historical (orrealworldorphysical) probability measure P.Heretheparametersaretheexpectedgrowthrateµ and the volatility σ,bothassumedconstant.TheremarkablefindingofBlackandScholes (1973) is that the no- arbitrage price of an option does not depend on µ,andso,topriceanoption,theonlyparameter that needs to be estimated from data is the volatility σ.Unlessotherwisestated,weshallassume throughout this article that interest rates are zero. It will be convenient to introduce the following notation:

τ := T t, x := log S ,k:= log K, − t where t is the current time, and K and T are the strike and expiration date respectively of a call or . Then, for fixed (t, T, x, k), the Black-Scholes pricing formula for a with time to expiration τ>0isgivenby

2 BS x k 1 σ τ u (σ):=e N(d (σ)) e N(d−(σ)),d±(σ):= x k , (1.2) + − σ√τ ! − ± 2 "

2 where N is the CDF of a standard normal random variable, and we have stressed the volatility argument σ in the notation. It turns out that the Black-Scholes price (1.2) can be expressed as the expected payoffof the option, but where the expectation is taken with respect to a different probability measure Q under which the stock price is a martingale (that is, it is a pure fluctuation process with no trend or growth rate). This means that there is a so-called risk-neutral world in which the stock price follows the dynamics dSt Q = σ dWt , St where W Q is a Brownian motion under Q,andthecalloptionprice(1.2)canbeexpressedasthe conditional expectation uBS(σ)=EQ[(S K)+ log S = x], T − | t where EQ denotes that the expectation is taken under the probability measure Q.

1.1.2 Implied Volatility The implied volatility of a given call option with price u (which is either observed in the market or computed from a model) is the unique positive solution I of

uBS(I)=u. (1.3)

It is the volatility parameter that has to be put into the Black-Scholes formula to match the observed price u. Note that the implied volatility I depends implicitly on the maturity date T and the log strike k as the option price u will depend on these quantities. The map (T,k) I(T,k)isknownasthe #→ implied volatility surface.IfmarketoptionpricesreflectedBlack-Scholesassumptions, I would be constant and equal to the stock’s historical volatility σ.However,inequitiesdata,thefunction I(T, )exhibitsdownwardslopingbehaviorink,whoseslopevarieswiththeoptionmaturitiesT , · as illustrated in Figure 1. This downward slope is known as the implied volatility skew. These features of the implied volatility surface can be reproduced by enhancing the Black- Scholes model (1.1) with stochastic volatility and/or jumps. One focus of this chapter will be to survey some approaches taken to capturing the implied volatility skew.

1.2 Volatility Modeling While the overwhelming evidence from time-series and optionpricedataindicatesthatthevolatility σ in (1.1) should be allowed to vary stochastically in time:

dSt = µ dt + σt dWt, St there is no consensus as to how exactly the (stochastic) volatility σt should be modeled.

3 DTM = 25 DTM = 53

0.45 0.5

0.45 0.4

0.4

0.35 0.35

0.3 0.3

0.25

0.25 0.2

0.2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1 −0.5 0 0.5 1

DTM = 88 DTM = 116

0.5 0.5

0.45 0.45

0.4 0.4

0.35 0.35

0.3 0.3

0.25 0.25

0.2 0.2

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

DTM = 207 DTM = 298

0.5 0.5

0.45 0.45

0.4 0.4

0.35 0.35

0.3 0.3

0.25 0.25

0.2 0.2

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 −0.5 0 0.5

DTM = 389 DTM = 571

0.5 0.5

0.45 0.45

0.4 0.4

0.35 0.35

0.3 0.3

0.25 0.25

0.2 0.2

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4

Figure 1: Implied volatility from S&P 500 index options on May25,2010,plottedasafunctionof log- to maturity ratio: (k x)/(T t). DTM = “days to maturity.” − −

4 1.2.1 No-arbitrage Pricing and Risk-Neutral Measure In standard option-pricing theory, it is assumed that markets do not admit arbitrage. No arbitrage pricing implies that all traded asset prices (after discounting) are martingales under some proba- bility measure Q,typicallyreferredtoasarisk-neutral measure. Consequently, the price ut of an option at time t with payoff ϕ(ST )attimeT is given by

u = EQ[h(S ) F ], (1.4) t T | t where Ft is the history of the market up to time t.Typicallytherearenon-tradedsourcesofran- domness such as jumps or stochastic volatility. As a result, there exist infinitely many risk-neutral measures. The non-uniqueness of these measures is often referred to as market incompleteness, meaning not every derivative asset can be perfectly hedged. In practice, one assumes that the market has chosen a specific risk-neutral measure, which is consistent with observed option prices. In what follows, we will model asset dynamics under a unique risk-neutral pricing measure Q, which we assume has been chosen by the market. Under Q,wehave

S =eXt , dX = 1 σ2dt + σ dW Q, (1.5) t t − 2 t t t which describes the dynamics of Xt =logSt.Below,wereviewsomeofthemostcommonmodels of volatility and discuss some of their advantages and disadvantages.

1.2.2 Models

In local volatility (LV) models, the volatility σt of the underlying is modeled as a deterministic function σ( , )oftimet,andthetime-t value of the underlying X .Thatis, · · t 1 dX = σ2(t, X )dt + σ(t, X )d W Q, (local volatility) t −2 t t t Typically, one assumes that the function σ(t, )increasesasx decreases in order to capture the · leverage effect,whichreferstothetendencyforthevalueofanassettodecrease as its volatility increases. One advantage of local volatility models is that markets remain complete,meaningthatderiva- tives written on S can be hedged perfectly – just as in the Black-Scholes model. While market completeness is convenient from a theoretical point of view,itisnotnecessarilyarealisticprop- erty of financial markets. Indeed, if markets are complete, then one can ask: why do we need derivatives? Another advantage of local volatility models is that they canprovideaverytightfittooption prices quoted on the market. In fact, Dupire (1994) shows thatthereexistsalocalvolatilitymodel that can exactly match option prices quoted on the market and that there is an explicit formula for how to construct this model from observed call and put prices under the assumption that they can be interpolated across continuous strikes and maturities. However, a tight fit must be balanced with stability: local volatility models are notoriously badatprovidingstabilityandtypicallyneed to be re-calibrated hourly.

5 1.2.3 Stochastic Volatility Models In a stochastic volatility (SV) model, promoted in the late 1980s by Hull and White (1987), Scott (1987) and Wiggins (1987), the volatility σt of the underlying is modeled as a deterministic function σ( )ofsomeauxiliaryprocessY , which is usually modeled as a diffusion: · 1 dX = σ2(Y )dt + σ(Y )dW Q, t −2 t t t Q dYt = α(Yt)dt + β(Yt)dBt , (stochastic volatility) (1.6) d W Q,BQ = ρ dt, ⟨ ⟩t with ρ < 1. Here, BQ is a Brownian motion that is correlated with W Q.Onetypicallytakesthe | | correlation ρ to be negative in order capture the empirical observation that when volatility goes up, stock prices tend to go down, the leverage effect. In a single factor stochastic volatility setting such as described by (1.6), derivatives written on S cannot be perfectly hedged by continuously trading abondandtheunderlyingS alone. However, a derivative written on S can be perfectly replicated by continuously trading a bond, the underlying S and single option on S.Thus,assumingoptions can be traded continuously can complete the market.However,astransactioncostsonoptionsare much higher than on stocks, and as their liquidity is typically lower, this assumption typically is not made. Unlike the local volatility case, there is no explicit formula for constructing Y dynamics and avolatilityfunctionσ( )sothatmodel-inducedoptionpricesfitobservedmarketprices exactly. · It is common to assume that the volatility driving process Y is mean-reverting, or ergodic, meaning there exists a distribution Πsuch that the ergodic theorem holds:

1 t lim g(Ys)ds = g(y)Π(dy), t→∞ t #0 # for all bounded functions g. Equation (1.6) actually refers specifically to one-factor stochastic volatility models. One can always introduce another auxiliary process Z,andmodelthevolatilityσ as a function σ( , )of t · · both Y and Z.IfS, Y and Z are driven by three distinct Brownian motions, then continuously trading a bond, the underlying S and two options on S would be required to perfectly hedge further options on S. Multi-factor stochastic volatility models have the ability of fit option prices better than their one-factor counterparts. But, each additional factor of volatility brings with it additional computational challenges. Multi-factor and multiscale stochastic volatility models are discussed at length in Fouque et al. (2011).

1.2.4 Local-Stochastic Volatility Models As the name suggests, local-stochastic volatility (LSV) models combine features of both local volatil- ity and stochastic volatility models by modeling the volatility σ as a function σ( , , )oftimet, t · · · the underlying X and an auxiliary process Y (possibly multidimensional). For example 1 dX = σ2(t, X ,Y )dt + σ(t, X ,Y )dW Q, t −2 t t t t t Q dYt = f(t, Xt,Yt)dt + β(t, Xt,Yt)dBt , (local-stochastic volatility) (1.7) d W Q,BQ = ρ dt, ⟨ ⟩t

6 where ρ < 1. Note that the class of models described by (1.7) nests all LVmodelsandall(one- | | factor) SV models. However, while LSV models offer more modeling flexibility than LV or SV models separately, these models also present new computational challenges. Indeed, while there exists LV and SV models for which option prices can be computedinclosedform(orsemi-closed form, for instance up to Fourier inversion), explicit formulas for option prices are available in an LSV setting only when ρ =0.Weremarkthathavingaclosedformformulaorafastapproximation is crucial for the inverse problem of calibrating an LSV modelfromobservedoptionprices.

1.2.5 Models with Jumps Some authors argue that diffusion models of the form (1.5) are not adequate to capture the complex dynamics of stock price processes because diffusion models do not allow stock prices to jump. Discontinuities in the stock price process can be modeled by adding a jump term dJt to the process (1.5) as follows

1 2 z Q dXt = σ λ (e 1)F (dz) dt + σdW +dJt, !−2 t − # − " t and this type of model dates back to Merton (1976). Here, jumpsarriveasaPoissonprocesswith intensity λ and have distribution F ,andthedriftofX is compensated to ensure that S =eX is a martingale under Q.AswithSVmodels,jumpsrenderamarketincomplete.Addingjumps also to volatility can help fit the strong implied volatility smilethatiscommonlyobservedforshort maturity options, and we refer to Bakshi et al. (1997) for an analysis.

1.2.6 ARCH & GARCH Models Although our focus will be on the continuous-time models, it is worth mentioning that discrete-time models for stock returns are widely studied in econometrics literature. A large class of discrete- time models are the autoregressive conditional heteroskedasticity (ARCH) processes introduced by Engle (1982), later generalized under the name GARCH. The discrete-time models that are closest to the type of continuous-time stochastic volatility models (1.6) driven by diffusions are the EGARCH models developed in Nelson (1991, 1990). Those papersalsodiscussconvergenceofthe discrete-time EGARCH process to an exponential Ornstein-Uhlenbeck continuous-time stochastic volatility model.

2AsymptoticRegimesandApproximations

Given a model and its parameters, computing expectations of the form (1.4) one time is straightfor- ward using Monte Carlo methods or a numerical solution of the associated pricing partial integro- differential equation (PIDE). However, when the computation is part of an iterative procedure to calibrate the model to the observed implied volatility surface, it becomes important to have a fast method of computing option prices or model-induced implied volatilities. As a result, a number of different efficient approximation methods have developed. Broadly speaking, there are two methods of setting up asymptotic expansions for option pricing and implied volatility. In contract asymptotics,oneconsidersextremeregimesspecifictotheoption contract, in other words, large or small time-to-maturity (T t), or large or small strikes K.Inthe −

7 model asymptotics approach, one views the complicated incomplete market modelasaperturbation around a more tractable model, often the Black-Scholes model(1.1).

2.1 Contract Asymptotics The foundational papers in this approach appeared in 2004. The main result from Lee (2004), commonly referred to as the moment formula relates the implied volatility slope at extreme strikes to the largest finite moment of the stock price. We refer to the recent book Gulisashvili (2012) for an overview of this and related asymptotics. The approach pioneered by Berestycki et al. (2004) uses largedeviationscalculationsforthe short-time regime. The regime where the time-to-maturity islargeisstudiedbyTehranchi(2009). However, for the rest of this article we concentrate on model asymptotics as they are adaptable to other option contracts and, moreover, are amenable to nonlinear portfolio optimization problems as we discuss in Section 3.

2.2 Model Asymptotics

We will present the analysis in terms of the log-stock price Xt =logSt,andconsideraEuropean x option with payoff ϕ(XT )attimeT ,whereϕ(x)=h(e ). There may be other factors such as stochastic volatility driving the stock price dynamics, andwedenotethesebythe(possiblymulti- dimensional) process Y .If(X, Y )is(jointly)aMarkovprocess,thenthetimet price u(t, x, y)of the European option is an expectation of the form

u(t, x, y)=EQ[ϕ(X ) X = x, Y = y]. T | t t

Here, we are using the Markov property of (X, Y )toreplacethefiltrationFt in (1.4) with the time-t values of (X, Y ). Under mild conditions on the processes (X, Y )andthepayofffunctionϕ,thefunctionu(t, x, y) is sufficiently smooth to be the solution of the Kolmogorov backward equation (KBE)

(∂t + A(t))u =0,u(T,x)=ϕ(x), (2.1) where the operator A(t)isthegenerator of (X, Y )(whichmayhavet-dependence from the co- efficients of (X, Y )). The operator A(t) is, in general, a second order partial integro-differential operator. Unfortunately, equation (2.1) rarely has a closed-form solution – especially when we include realistic features such as jumps and stochastic volatility. As such, one typically seeks an approximate solution to (2.1). We will discuss an approach tothisusingperturbation theory (also referred to as asymptotic analysis). Perturbation theory is a classical tool developed to solve problems arising in physics and engi- neering. We describe its use here to find an approximate solution to (typically) a PIDE starting from the exact solution of a related PIDE. More specifically, suppose the integro-differential operator A(t)in(2.1)canbewrittenintheform

∞ n A(t)= ε An(t), (2.2) n$=0

8 where each An(t)inthesequence(An(t)) is an integro-differential operators, and ε>0isa(typically small) parameter. Formally, one seeks an approximate solution to (2.1) by expanding the function u as a power series in the parameter ε

n u = ε un. (2.3) n$=0 Inserting the expansion (2.2) for A(t)andtheexpansion(2.3)foru into the PIDE (2.1), and collecting like powers of ε,onefinds

O(1) : (∂t + A0(t))u0 =0,u0(T,x,y)=ϕ(x), O(ε): (∂ + A (t))u = A (t)u ,u(T,x,y)=0, t 0 1 − 1 0 n ...... ∞ n O(ε ): (∂ + A (t))u = A (t)u − ,u(T,x,y)=0. t 0 n − k n k n $k=1

The approximating sequence of functions (un)arethenfoundbysolvingtheabovenestedsequence of PIDEs. This method is most useful when the fundamental solution Γ0 (also referred to as the Green’s function)correspondingtotheoperatorA0(t), is available in closed form. It is the solution of

(∂ + A (t))Γ (t, x, y; T,ξ,ω)=0, Γ (T,x,y; T,ξ,ω)=δ(x ξ)δ(y ω), t 0 0 0 − − where δ( )istheDiracdeltafunction(orpointmassatzero). · Upon finding Γ0,theapproximatingsequenceoffunctions(un)canbewrittendowndirectly:

u0(t, x, y)=P0(t, T )ϕ(x):= dξdω Γ0(t, x, y; T,ξ,ω)ϕ(ξ), (2.4) # T n un(t, x, y)= dt1P0(t, t1) Akun−k(t1,x,y). (2.5) # t $k=1

Here the operator P0(t, T )isreferredtoasthesemigroup generated by A0(t). A ∞ A Finding an appropriate decomposition of the generator (t)= n=0 n(t)isabitofanart. In general the most appropriate decomposition will depend strongly% on the underlying process X (from which A(t)isderived).Asastartingpoint,itwillhelptoidentifyoperators A0 for which the fundamental solution Γ0 can be written in closed form or semi-closed form, and we shalldiscuss some examples in Section 2.4.

2.3 Implied Volatility Asymptotics Models are typically calibrated to implied volatilities rather than to prices directly. As such, it is useful to have closed-form approximations for model-induced implied volatilities. In this section, we will show how to translate an expansion for option prices intoanexpansionforimpliedvolatilities. Throughout this section we fix a model for X =logS,atimet,amaturitydateT>t,theinitial values X = x and a call option payoff ϕ(X )=(eXT ek)+.Ourgoalistofindtheimplied t T −

9 volatility for this particular call option.Toeasenotation,wewillsuppressmuchofthedependence on (t, T, x, k). However, the reader should keep in mind that the implied volatility of the option under consideration does depend on (t, T, x, k), even if this is not explicitly indicated. Assume that the option price u has an expansion of the form

n BS u = u0 + ε un, where u0 = u (σ0), for some σ0 > 0. (2.6) n$=1 We wish to find the implied volatility I corresponding to u,whichistheuniquepositivesolution of (1.3). To find the unknown implied volatility I,weexpanditinpowersofε as follows:

∞ ε ε n I = I0 + E , where E = ε In. n$=1

BS Expanding u (I)aboutthepointI0 we find

BS BS ε u (I)=u (I0 + E ) ∞ (Eε)n = uBS(I )+ ∂ uBS(I ) 0 n! σ 0 n$=1 BS BS 2 1 2 2 BS = u (I0)+εI1∂σu (I0)+ε I2∂σ + 2! I1 ∂σ u (I0) + ε3 I ∂ + 1 2I I ∂2 + 1&I3∂3 uBS(I )+' . (2.7) 3 σ 2! 1 2 σ 3! 1 σ 0 ··· & ' Inserting the expansion (2.6) for u and the expansion (2.7) for uBS(I)intoequation(1.3),and collecting like powers of ε we obtain

BS O(1) : u0 = u (I0), BS O(ε): u1 = I1∂σu (I0), O 2 1 2 2 BS (ε ): u2 = I2∂σ + 2! I1 ∂σ u (I0) O 3 & 1 ' 2 1 3 3 BS (ε ): u3 = I3∂σ + 2! 2I1I2∂σ + 3! I1 ∂σ u (I0). & ' BS Using u0 = u (σ0), we can solve for the sequence (In)recursively.Wehave

O(1) : I0 = σ0, (2.8) O 1 (ε): I1 = BS u1, ∂σu (I0) 1 O(ε2): I = u 1 I2∂2 uBS(I ) 2 ∂ uBS(I ) 2 − 2! 1 σ 0 σ 0 ( & ' ) 1 O(ε3): I = u 1 2I I ∂2 + 1 I3∂3 uBS(I ) . (2.9) 3 ∂ uBS(I ) 3 − 2! 1 2 σ 3! 1 σ 0 σ 0 ( & ' ) Note that the implied volatility expansion involves both model independent and model dependent terms:

n BS ∂σ u (I0) un model independent : BS model dependent : BS . (2.10) ∂σu (I0) ∂σu (I0)

10 The model independent terms are always explicit. For example, using (1.2), a direct computation reveals

2 BS 2 3 BS 4 2 2 2 2 ∂σu (σ) (k x) tσ ∂σu (σ) t (k x) 3(k x) (k x) t σ BS = − 3 , BS = + −2 6 −4 − 2 + . ∂σu (σ) tσ − 4 ∂σu (σ) −4 t σ − tσ − 2σ 16

Higher order terms are also explicit. Whether or not the modeldependenttermsin(2.10)canbe computed explicitly (i.e., meaning without numerical integration or special functions) depends on the specific form the sequence (un)takes.

2.4 Tractable models

We say that a model X is tractable if its generator A0(t)admitsaclosed-form(orsemi-closedform) fundamental solution Γ0.AlargeclassoftractablemodelsaretheexponentialL´evymodels. In this class, a traded asset S =eX is described by the following L´evy-ItˆoSDE

Q dXt = µ dt + σd Wt + z dNt(dz). (2.11) #R * Here, W Q is a standard Brownian motion and N is an independent compensated Poisson random measure * dN (dz)=dN (dz) ν(dz)dt. t t − The last term in (2.11) can be understood* as follows: for any Borel set A the process N(A)isa Poisson process with intensity ν(A). Thus, the probability that X experiences a jump of size z A ∈ in the time interval [t, t +dt)isν(A)dt.InorderforS to be a martingale, the drift µ must be given by 1 µ = σ2 ν(dz)(ez 1 z). −2 − #R − −

The generator A0 of X is given by

1 2 2 A0 = µ∂x + σ ∂ + ν(dz)(θz 1 z∂x), (2.12) 2 xx # − − where the operator θz is a shift operator: θzf(x)=f(x + z). When X has no jump component (i.e., when ν 0), the generator A has a fundamental ≡ 0 solution Γ0 that can be written in closed-form as a Gaussian kernel: 1 (x y + µ(T t))2 Γ0(t, x; T,y)= exp − − . 2πσ2(T t) !− 2σ2(T t) " − − + More generally, when jumps are present, the generator A0 has a fundamental solution Γ0 which is available in semi-closed form as a Fourier transform:

1 iξ(x−y)+(T −t)Φ0(ξ) Γ0(t, x; T,y)= dξ e , (2.13) 2π #

11 where

1 2 2 iξz Φ0(ξ)=iµξ σ ξ + ν(dz)(e 1 iξz). − 2 # − −

The function Φ0 is referred to as the characteristic exponent of X,sinceitsatisfies i i EQ[e ξXT X = x]=e ξx+(T −t)Φ0(ξ). | t

Using (2.4) and (2.13) we can express the action of the semigroup operator P0(t, T )onageneral function ψ as follows:

1 iξ(x−y)+(T −t)Φ0(ξ) P0(t, T )ψ(x)= dy Γ0(t, x; T,y)ψ(y)= dy dξ e ψ(y) # 2π # # 1 = dξ eiξx+(T −t)Φ0(ξ)ψ(ξ), 2π # , where ψ is the Fourier transform of ψ:

, i ψ(ξ):= dy e− ξyψ(y). # , Because the semigroup operator P0(t, T )correspondingtoA0 is well-understood in the expo- A A ∞ nA A nential L´evy setting, if one can write the generator of a process X as = n=0 ε n with 0 given by (2.12), then one can use (2.4)-(2.5) to find approximate solutions to the% full pricing PIDE (∂t + A)u =0.

2.5 Model Coefficient Polynomial Expansions Model coefficient expansions are developed in a series of papers Lorig et al. (2015c, 2014a, 2015b, 2014b, 2015a). The authors’ method (which we shall henceforth refer to as “LPP”) can be used to find closed-form asymptotic approximations for option prices and implied volatilities in a general d-dimensional Markov setting. Here, for simplicity, we focusontwosimplecases:(i)general 2-dimensional local-stochastic volatility (LSV) models, and (ii) general scalar L´evy-type models.

2.5.1 LSV models We consider a general class of models, in which an asset S =eX is modeled as the exponential of a Markov diffusion process X that satisfies the SDEs 1 dX = σ2(X ,Y )dt + σ(X ,Y )dW Q, t −2 t t t t t Q dYt = f(Xt,Yt)dt + β(Xt,Yt)dBt , d W Q,BQ = ρ dt, ⟨ ⟩t where W Q and BQ are correlated Brownian motions. This is as in the model discussed in Section 1.2.4 except we remove the explicit t-dependence in the coefficients for simplicity. Note that the drift of X is 1 σ2,whichensuresthatS =eX is a Q-martingale and so the stock price is arbitrage − 2 free.

12 The generator A of X is given by

A = a(x, y)(∂2 ∂ )+f(x, y)∂ + b(x, y)∂2 + c(x, y)∂2 , (2.14) xx − x y yy xy where we have defined 1 1 a(x, y):= σ2(x, y),b(x, y):= β2(x, y),c(x, y):=ρσ(x, y)β(x, y). 2 2 For general coefficients (σ,β, f )thereisnoclosedform(orevensemi-closedform)expression of Γ A A ∞ A the fundamental solution corresponding to .Thus,weseekadecompositionof = n=0 n for which the order zero operator A0 admits a closed form fundamental solution Γ0. % The LPP approach is to expand the coefficients of A in polynomial basis functions where the zeroth order terms in the expansion are constant. Specifically,

∞ χ(x, y)= χ (x, y),χa, b, c, f , n ∈{ } n$=0 where χ is a constant and χ (x, y)dependspolynomiallyonx and y for every n 1. For example, 0 n ≥ Taylor series:

n k n−k 1 k n−k χ (x, y)= − (x x¯) (y y¯) , − = ∂ ∂ χ(¯x, y¯), (2.15) n Xk,n k − − Xk,n k k!(n k)! x y $k=0 − or Hermite polynomials:

n χ (x, y)= − H − (x, y), − = H − ,χ . n Xk,n k k,n k Xk,n k ⟨ k,n k ⟩ $k=0

In the Taylor series example, (¯x, y¯)isafixedpointinR2.IntheHermitepolynomialexample,the brackets , indicate an L2 inner product with a Gaussian weighting. The Hermite polynomials ⟨· ·⟩ (H )formacompletebasisinthisspaceand(properlyweighted)are orthonormal H ,H = n,m ⟨ n,m i,j⟩ δn,iδm,j . Upon expanding the coefficients of A in polynomial basis functions, one can formally write the operator A as

∞ n A = An, with An = Ak,n−k, (2.16) n$=0 $k=0 where

2 2 2 A − = a − (x, y)(∂ ∂ )+f − (x, y)∂ + b − (x, y)∂ + c − (x, y)∂ . k,n k k,n k xx − x k,n k y k,n k yy k,n k xy Note that the operator A in (2.16) is of the form (2.2) if one sets ε =1. Moreover, the order zero operator

A = A = a (∂2 ∂ )+f ∂ + b ∂2 + c ∂2 . 0 0,0 0,0 xx − x 0,0 y 0,0 yy 0,0 xy

13 has a fundamental solution Γ0 which is a Gaussian density:

1 1 T −1 ξ Γ0(t, x, y; T,ξ,ω)= exp (η m) C (η m) , η = , 2π C !−2 − − " !ω" | | + with covariance matrix C and mean vector m given by

2a c x (T t)a C =(T t) 0,0 0,0 , m = − − 0,0 . − ! c 2b " !y +(T t)f " 0,0 0,0 − 0,0

Since Γ0 is available in closed form, one can use (2.4) and (2.5) to find u0 and the sequence of functions (un)n≥1 respectively. After a bit of algebra, one can show that

un(t, x, y)=Lnu0(t, x, y), (2.17) where the operator Ln is of the form

L = η(n) (t, x, y)∂k∂m(∂2 ∂ ). n k,m y x x − x $k,m

(n) The precise form of the coefficients (ηk,m)willdependonthechoiceofpolynomialbasisfunction. If the coefficients of A are smooth and bounded, then the small-time-to-maturity accuracy of the price approximation is

n (n+3)/2 sup u(t, x, y) u¯n(t, x, y) = O((T t) ), u¯n(t, x, y):= uk(t, x, y). x,y | − | − $k=0 The proof can be found in Lorig et al. (2015a). The LPP price expansion also leads to closed-form expressions for implied volatility. Indeed, for BS European call options, one can easily show that u0 = u ( 2a0,0). Therefore, the series expansion for u is of the form (2.6), and hence (I0,I1,I2,I3)canbecomputedusing(2.8)–(2.9).Moreover,+ BS the model dependent terms (un/∂σu )appearingin(2.8)–(2.9)canbecomputedexplicitlywith no numerical integration. This is due to the fact that un can be written in the form (2.17). For details, we refer the reader to Lorig et al. (2015b).

Example: Heston Consider the Heston (1993) model, under which the risk-neutral dynamics of X are given by 1 dX = eYt dt +eYt/2dW Q, t −2 t dY = (κθ 1 δ2)e−Yt κ dt + δ e−Yt/2dBQ, t − 2 − t d W Q,BQ = ρ& dt. ' ⟨ ⟩t The generator of (X, Y )isgivenby 1 1 A = ey ∂2 ∂ + (κθ 1 δ2)e−y κ ∂ + δ2e−y∂2 + ρδ∂ ∂ . 2 x − x − 2 − y 2 y x y & ' & '

14 Thus, using (2.14), we identify 1 1 a(x, y)= ey,b(x, y)= δ2e−y,c(x, y)=ρδ, f(x, y)= (κθ 1 δ2)e−y κ . 2 2 − 2 − & ' We fix a time to maturity τ and log-strike k.AssumingaTaylorseriesexpansion(2.15)ofthe coefficients of A with (¯x, y¯)=(x, y), the time t levels of (X, Y ), one computes

y/2 I0 =e , 1 1 I = e−y/2τ δ2 +2( ey + θ) κ +eyδρ + e−y/2δρ(k x), 1 8 − − 4 − −3y/2 & y/2 ' e 2 e I = − τ 2 δ2 2θκ + τ 2 5κ2 5δκρ + δ2 1+2ρ2 2 128 − 96 − − & ' & & '' e−y/2 + τ 4τθκ2 τδ3ρ +2τδθκρ+2δ2 8+τκ+ ρ2 192 − − 1 & & 1 '' + e−3y/2τδρ 5δ2 +2(ey 5θ) κ eyδρ (k x)+ e−3y/2δ2 2 5ρ2 (k x)2. 96 − − − 48 − − & ' & ' The expression for I3 is also explicit, but omitted for brevity. In Figure 2 we plot the approximate implied volatility (I0 + I1 + I2 + I3)aswellastheexactimpliedvolatilityI,whichcanbecomputed using the pricing formula given in Heston (1993) and then inverting the Black-Scholes formula numerically.

2.5.2 L´evy-type models In this Section, we explore how the LPP method can be applied tocomputeapproximateoption prices in a one-dimensional L´evy-type setting. Specifically, we consider an asset S =eX ,whereX is a scalar L´evy-type Markov process. Under some integrability conditions on the size and intensity of jumps, every scalar Markov process on R can be expressed as the solution of a L´evy-ItˆoSDE of the form:

Q dXt = µ(Xt)dt + 2a(Xt)dWt + zdNt(Xt−, dz), + # * where W Q is a standard Brownian motion and N is a state-dependent compensated Poisson random measure * dN (x, dz)=dN (x, dz) ν(x, dz)dt. t t − Note that jumps are now described* by a L´evy kernel ν(x, dz), which is a L´evy measure for every x R.InorderforS to be a martingale, the drift µ must be given by ∈ µ(x)= a(x) ν(x, dz)(ez 1 z). − − #R − − The generator A of X is

2 A = µ(x)∂x + a(x)∂ + ν(x, dz)(θz 1 z∂x). xx # − −

15 T t =0.25 T t =0.50 − −0.215

0.215 0.210

0.210 0.205

0.205 0.200 0.200 0.195 0.195 0.190

!0.2 !0.1 0.1 0.2

!0.2 !0.1 0.1 0.2 T t =0.75 T t =1.00 − − 0.210 0.210

0.205 0.205

0.200 0.200

0.195 0.195

0.190 0.190

!0.2 !0.1 0.1 0.2 !0.2 !0.1 0.1 0.2

Figure 2: Exact (solid) and approximate (dashed) implied volatilities in the . The horizontal axis is log-moneyness (k x). Parameters: κ =1.15, θ =0.04, δ =0.2, ρ = 0.40 − − x =0.0, y =logθ.

16 For general drift µ(x)variancea(x)andL´evykernelν(x, dz)thereisnoclosedform(oreven semi-closed form) expression for the fundamental solution ΓcorrespondingtoA.Thusweseeka A A A decomposition of generator = n n for which 0 has a fundamental solution Γ0 available in semi-closed form. % Once again, the LPP approach of expanding the coefficients and Levy kernel of A in polynomial basis functions will lead to the desired form for the expansion of A.Specifically,let

∞ ∞ ∞ µ(x)= µn(x),a(x)= an(x),ν(x, dz)= νn(x, dz), $n=0 n$=0 n$=0 where µ0(x)=µ0, a0(x)=a0 and ν0(x, dz)=ν0(dz), and higher order terms µn(x), an(x)and νn(x, dz)dependpolynomiallyonx.Forexample 1 Taylor series : a (x)= ∂na(¯x) (x x¯)n, n n! x · − Hermite polynomial : a (x)= H ,a H (x), n ⟨ n ⟩· n and similarly for µ and ν.UponexpandingthecoefficientsofA in polynomial basis functions, one A A ∞ A A can formally write the operator as = n=0 n,where 0 is given by % 2 z∂x A0 = µ0∂x + a0∂ + ν0(dz)(e 1 z∂x), (2.18) xx # − − and each A for n 0isoftheform n ≥ 2 An = µn(x)∂x + an(x)∂ + νn(x, dz)(θz 1 z∂x). xx # − −

Comparing (2.18) with (2.12), we see that A0 is the generator of a L´evy process. Thus, using (2.13), the fundamental solution Γ0 corresponding to A0 can be written as a Fourier integral

1 iξ(x−y)+(T −t)Φ0(ξ) Γ0(t, x; T,y)= dξ e , 2π # where

2 iξz Φ0(ξ)=iµ0ξ a0ξ + ν0(dz)(e 1 iξz). − # − −

Since Γ0 is available in closed form, one can use (2.4) and (2.5) to find u0 and un respectively. Explicit computations are carried out in Lorig et al. (2015c). In this case, it is convenient to express un(t, x)asan(inverse)Fouriertransformofun(t, x). Defining , −iξx Fourier transform : un(t, ξ)=F[un(t, )](ξ):= dx e un(t, x), · # , −1 1 iξx Inverse transform : un(t, x)=F [un(t, )](x):= dx e un(t, ξ), · 2π # , ,

17 we have n T (t1−t)Φ0(ξ) iξz un(t, ξ)= dt1 e νk(i∂ξ, dz) e 1 iξz un−k(t1,ξ) # # d − − $k=1 t R0 ( ) , n T , (t1−t)Φ0(ξ) 2 + dt1 e µk(i∂ξ)(iξ)un−k(t1,ξ)+ak(i∂ξ)(iξ) un−k(t1,ξ) , # $k=1 t & ' , , where u0(t, ξ)isgivenby

iξx+(T −t)Φ0(ξ) , u0(t, ξ)=e ϕ(ξ).

2.6 Small ‘vol of vol’ expansion, , Lewis (2000) considers a stochastic volatility model of the form 1 dX = Y dt + Y dW Q, t −2 t t t + Q dYt = α(Yt)dt + εβ(Yt)dBt , d W Q,BQ = ρ dt. ⟨ ⟩t The parameter ε is referred to as the volatility of volatility or vol of vol for short. The generator A of (X, Y )isgivenby 2 A = A0 + εA1 + ε A2, where y 1 A = ∂2 ∂ + α(y)∂ , A = ρ√yβ(y)∂ ∂ , A = β2(y)∂2. 0 2 x − x y 1 x y 2 2 y & ' Thus, A is of the form (2.2). Moreover, the solution of (∂ + A )Γ =0, Γ (T,x,y; T,ξ)=δ(x ξ), t 0 0 0 − is given by a Gaussian 1 (ξ x σ2(t, T )/2)2 Γ0(t, x, y; T,ξ,ζ)= exp − − δ(ζ η(t)), 2πσ2(t, T ) !− 2σ2(t, T ) " −

2 T + where σ (t, T )= t η(s)ds,andη(s)isthesolutionofthefollowingODE: - dη(s) = α(η(s)),η(t)=y. ds

Since Γ0 is available in closed form up to finding η,onecanuse(2.4)and(2.5)tofindu0 and the sequence of functions (un)respectively.Inparticular,forEuropeancallsonefinds σ2(t, T ) u = uBS(¯σ(t, T )), σ¯(t, T )= 0 . T t − As the order zero price is equal to the Black-Scholes price, computed with volatilityσ ¯(t, T ), one can apply the implied volatility asymptotics of Section 2.3 to find approximate implied volatilities. Explicit expressions for approximate implied vols are giveninLewis(2000)forthecaseα(y)= κ(θ y)andβ(y)=yγ . −

18 2.7 Separation of time-scales approach In Fouque et al. (2000, 2011), the authors develop analytic price approximations for interest rate, credit and equity derivatives. The authors’ approach (henceforth referred to as FPSS) exploits the separation of time-scales that is observed in volatilitytime-seriesdata.Morespecifically, the authors consider a class of multiscale diffusion models in which the volatility of an under- lying is driven by two factors, Y and Z,operatingonfastandslowtime-scalesrespectively. Lorig and Lozano-Carbass´e(2015) extend the FPSS method to models with jumps. Here, for simplicity we consider an exponential L´evy-type model withasingleslowly-varyingfactor,which drives both the volatility and jump-intensity. Consider a model for a stock S =eX ,whereX is modeled under the risk-neutral pricing measure as the solution of the following L´evy-ItˆoSDE

Q dXt = µ(Zt)dt + σ(Zt)dWt + s dNt(Zt−, ds), #R 2 * Q dZt = ε c(Zt) ε Γ(Zt) g(Zt) dt + εg(Zt)dBt , (risk-neutral measure Q) ( − ) d W Q,BQ = ρ dt. ⟨ ⟩t Here, W Q and BQ are correlated Brownian motions and N is a compensated state-dependent Poisson random measure with which we associate a L´evy kernel ζ(z)ν(ds). The drift µ(z)isfixed * by the L´evy kernel and volatility so that S =eX is a martingale: 1 µ(z)= σ2(z) ζ(z) ν(ds)(es 1 s) . −2 − #R − − Note that the volatility σ and L´evy kernel ζν are driven by Z,whichisslowly-varying in the following sense. Under the physical measure P,thedynamicsofZ are given by

2 dZt = ε c(Zt)dt + εg(Zt)dBt, (physical measure P) * where B = BQ t Γ(Z )ds is a standard P-Brownian motion (Γis known as the market price of t t − 0 s - risk associated* with B). The infinitesimal generator of Z under the physical measure * 1 A = ε2 g2(z)∂2 + c(z)∂ Z !2 z z " is scaled by ε2,whichisassumedtobeasmallparameter: ε2 << 1. Thus, Z fluctuates over an intrinsic time-scale 1/ε2,whichislarge. Due to the separation of time-scales the (X, Z)processhasageneratorthatnaturallyfactors into three terms of different powers of ε,justasin(2.2):

2 A = A0 + εA1 + ε A2, where A0, A1 and A2 are given by

A 1 2 2 0 = µ(z)∂x + σ (z)∂x + ζ(z) ν(ds)(θs 1 s∂x), (2.19) 2 #R − −

19 A = Γ(z)g(z)∂ + ρg(z)σ(z)∂ ∂ , 1 − z x z 1 A = g2(z)∂2 + c(z)∂ . 2 2 z z

Here, the shift operator θs acts on the x variable. Comparing equations (2.12) and (2.19), we observe that, for fixed z,theoperatorA0 above is the generator of a L´evy process. As the operator A0 in (2.19) acts only on the variable x,thevariablez serves only as a parameter. Thus we have

1 iξ(x−y)+(T −t)Φ0 (ξ,z) Γ0(t, x, z; T,y)= dξ e , 2π # where

1 2 2 iξs Φ0(ξ,z):=iµ(z)ξ σ (z)ξ + ζ(z) ν(ds)(e 1 iξs). − 2 # − − Because A is of the form (2.2) (set A =0forn 3), and since the fundamental solution Γ n ≥ 0 corresponding to A0 is known, we can use (2.4) and (2.5) to write u0 and u1 explicitly. For an option with payoff ϕ(XT )wehave

1 iξ(x−y)+(T −t)Φ0(ξ,z) 1 iξ(x−y)+(T −t)Φ0(ξ,z) u0(t, x, z)= dxϕ(y) dξ e = dξ e ϕ(ξ), # 2π # 2π # where ϕ is the Fourier transform of ϕ. , Asimilarcomputationyieldsthefollowingexpressionforu1: , (T t)/2 iξx+tΦ0(ξ,z) u1(t, x, z)= − dξ e ϕ(ξ)Mξ(z), 2π #R where ,

3 2 2 s iξs Mξ(z)=V1(z)( iξ + ξ )+U1(z) ξ ν(ds)(e 1 s)+iξ ν(ds)(e 1 iξs) − ! #R − − #R − − "

2 s iξs + V0(z)( ξ iξ)+U0(z) iξ ν(ds)(e 1 s)+ ν(ds)(e 1 iξs) , − − !− #R − − #R − − " and 1 1 V (z)= g(z)ρσ(z)∂ σ2(z),V(z)= g(z)Γ(z)∂ σ2(z), 1 2 z 0 −2 z U (z)=g(z)ρσ(z)∂ ζ(z),U(z)= g(z)Γ(z)∂ ζ(z). 1 z 0 − z If the coefficients of A and the payofffunction are smooth and bounded, then one can establish the following accuracy for the first order price approximation

u(t, x, z) (u (t, x, z)+εu (t, x, z)) = O(ε2), | − 0 1 | which holds pointwise. Note that, to compute the approximatepriceofanoptionu0 + εu1, one does not need full knowledge of (g, σ, Γ,ζ,ρ,z,ε). Rather, at order ε,theinformationcon- tained in these five functions and two variables is entirely captured by four group parameters (εU1(z),εU0(z),εV1(z),εV0(z)). The values of these four parameters can be obtained by calibrating to observed call and put prices, as described in detail in Lorig and Lozano-Carbass´e(2015).

20 2.8 Comparison of the Expansion schemes The approximation methods presented in Sections 2.5, 2.6 and 2.7 exploit different small parameters in A and therefore work best (i.e., provide the most accurate approximations) in different regimes. Rigorous accuracy results must be obtained on a case-by-casebasis.However,itisgenerallytrue n A¯ n A that the accuracy ofu ¯n := k=0 uk depends on how well n := k=0 approximates . For example, the Taylor% series expansion described in Section% 2.5 works best if the coefficients of A are slowly varying. In this case, the coefficients can be well-approximated by a Taylor series of low order. Thus, a highly accurate approximation of A can be obtained with only a few terms. The approximation considered in Section 2.6 works best when the diffusion coefficient of the volatility-driving process Y is small in comparison to the drift coefficient of Y and the drift and diffusion coefficients of X. Finally, the approximation considered in Section 2.7 when the drift coefficient of the volatility- driving process Z is small in comparison to the diffusion coefficient of Z,whichinturnissmallin comparison to the drift and diffusion coefficients of X.Aspreviouslymentioned,thismeansthat the intrinsic time-scale of Z must be slow in comparison to the intrinsic time-scale of X.

3MertonProblemwithStochasticVolatility:ModelCoefficient Polynomial Expansions

AlandmarkpairofpapersonoptimalinvestmentstrategybyRobert Merton analyzed the problem of how an investor should optimally allocate his wealth between a riskless bond and some risky assets, in order to maximize his expected utility of wealth. This problem, and its variation are now referred to as the Merton problem. In the original papers, each of the risky assets follows geometric Brownian motion with constant volatility. This modeling assumption is convenient from thestandpointofanalytictractability,but is not realistic in practice, as it does not allow for stochastic volatility. Because of this, there has been much interest in analyzing how an investor’s optimalinvestmentstrategychangesinthe presence of stochastic volatility. Here we focus on asymptotic methods for analyzing the stochastic control problem associated with portfolio optimization. Analysis with multiscale stochastic volatility models described in Section 2.7 was presented in the case of simple power utilities in (Fouque et al., 2000, Section 10.1), and expansions for a hedging problem in Jonsson and Sircar (2002b,a) in the dual optimization problem, both for fast mean- reverting stochastic volatility. In Fouque et al. (2012), expansions are constructed directly in the primal problem under both fast and slow volatility fluctuations. Indifference pricing approximations with exponential utility and fast volatility were studied in Sircar and Zariphopoulou(2005).Inthis section, we present some new approximations for the Merton problem with stochastic volatility.

3.1 Models & Dynamic Programming Equation The polynomial expansion techniques outlined in Section 2.5canbeextendedtofindexplicit asymptotic solutions to the Merton problem in a general local-stochastic volatility (LSV) setting, i.e., σt = σ(t, St,Yt). To fix ideas, however, we consider the simpler stochastic volatility (SV) setting, in which a risky asset S,underthephysicalprobabilitymeasureP,isthesolutionofthe

21 following SDE dS = µ(Y )S dt + σ(Y )S dBS, dY = c(Y )dt + β(Y )dBY , d BS,BY = ρ dt. t t t t t t t t t t ⟨ ⟩t Here, BS and BY are standard Brownian motions with correlation ρ.LetW denote the wealth process of an investor who holds π units worth of currency in S at time t,andhas(W π )units t t − t of currency in a bond. For simplicity, we assume the risk-freerateofinterestiszero.Assuch,the wealth process W satisfies πt S dWt = dSt = πtµ(Yt)dt + πtσ(Yt)dBt . St Observe that S does not appear in the dynamics of the wealth process W . An investor chooses πt to maximize his expected utility of wealth at a time T in the future, where utility is measure by a smooth, increasing and strictlyconcavefunctionU : R R,andthe + → objective to maximize is E U(WT ). Increasing describes preference for more wealth than less, while concavity captures risk aversion, more concave being more risk averse. The analysis is illustrated with power utility functions in Section 3.3. We define the investor’s value function u by

u(t, y, w):=supE[U(WT ) Yt = y,Wt = w], π∈Π | where Πthe set of admissible strategies: T 2 2 Π:= π adapted : E πt σ (Yt)dt< and Wt 0a.s. , / #0 ∞ ≥ 0 where adapted means adapted to the filtration generated by (BS,BY ). Assuming that u C1,2([0,T], R, R ), the value function solves the Hamilton-Jacobi-Bellman ∈ + partial differential equation (HJB-PDE) problem Y π (∂t + A )u +maxA u =0,u(T,y,w)=U(w), (3.1) π∈R Y π where A + A is the generator of (Y,W)assumingaMarkovinvestmentstrategyπt = π(t, Yt,Wt). Specifically,& the operators' AY and Aπ are given by 1 AY = c(y)∂ + β2(y)∂2, y 2 y 1 Aπ = π(t, y, w)µ(y)∂ + π2(t, y, w)σ2(y)∂2 + π(t, y, w)ρσ(y)β(y)∂ ∂ . w 2 w y w The optimal strategy π∗ is given by ∗ Aπ µ(∂wu)+ρβσ(∂y∂wu) π =argmax u = 2 2 , (3.2) π∈R − σ (∂wu) where, for simplicity (and from now on), we have omitted the arguments (t, y, w). Inserting the optimal strategy π∗ into the HJB-PDE (3.1) yields Y ∂t + A u + N(u)=0, (3.3) where N(u)isanonlinearterm: & ' 2 2 N 1 2 (∂wu) (∂wu)(∂y∂wu) 1 2 2 (∂y∂wu) (u)= 2 λ 2 ρβλ 2 2 ρ β 2 . − ∂wu − ∂wu − ∂wu Here, we have introduced the Sharpe ratio λ(y):=µ(y)/σ(y).

22 3.2 Asymptotic Approximation For general β,c,λ there is no closed form solution of (3.3). Hence, we seek an asymptotic { } approximation for u.Tothisend,usingequation(2.15)fromSection2.5.1asaguide, we expand the coefficients in (3.3) in a Taylor series about an arbitrary pointy ¯.Specifically,foranyfunction χ : R R we may formally write → ∞ 1 χ(y)= εnχ (y),χ(y):= ∂nχ(¯y) (y y¯)n,ε=1, (3.4) n n n! y · − n$=0 where we have once again introduced ε for purposes of accounting. We also expand the function u as a power series in ε

∞ n u = ε un,ε=1. (3.5) n$=0 Now for each group of coefficients appearing in (3.3), we insertanexpansionoftheform(3.4),and we define

A := c ∂ +(1 β2) ∂2,n0 N. (3.6) n n y 2 n y ∈{ }∪ We also insert into (3.3) our expansion (3.5) for u. Next, collecting terms of like powers of ε,weobtainatlowestorder

2 2 A 1 2 (∂wu0) (∂wu0)(∂y∂wu0) 1 2 2 (∂y∂wu0) (∂t + 0)u0 ( 2 λ )0 2 (ρβλ)0 2 ( 2 ρ β )0 2 =0, − ∂wu0 − ∂wu0 − ∂wu0 with u0(T,y,w)=U(w). We can look for a solution u0 = u(t, w)thatisindependentofy,and then we have 2 1 2 (∂wu0) ∂tu0 ( 2 λ )0 2 =0,u0(T,w)=U(w). (3.7) − ∂wu0 We observe that (3.7) is the same nonlinear PDE problem that arises when one considers an underlying that has a constant drift µ0 = µ(¯y), diffusion coefficient σ0 = σ(¯y)andSharperatio λ0 = λ(¯y)=µ(¯y)/σ(¯y). It is convenient to define the risk-tolerance function

∂wu0 R0 := − 2 , ∂wu0 and the operators = Rk∂k ,k=1, 2,. . Dk 0 w ··· We now proceed to the order O(ε)terms.Usingu0 = u0(t, w)and(3.7)weobtain

(∂ + A )u +(1 λ2) u + λ2 u +(ρβλ) ∂ u = ( 1 λ2) u , t 0 1 2 0 D2 1 0D1 1 0D1 y 1 − 2 1D1 0 u1(T,y,w)=0, which is a linear PDE problem for u1.

23 We can re-write equations (3.8)-(3.9) more compactly as

(∂t + A0 + B0(t))u1 + H1 =0,u1(T,y,w)=0, (3.10) where the linear operator B(t)andthesourcetermH1 are given by

B (t)= 1 λ2 + λ2 +(ρβλ) ∂ ,H=(1 λ2) R ∂ u . 0 2 0D2 0D1 0D1 y 1 2 1 0 w 0

Observe that (3.10) is a linear PDE for u1. The following change of variables (see Fouque et al. (2012)) will be useful for solving the PDE problem (3.10). Define

u (t, y, w)=q (t, y, z(t, w)),z(t, w)= log ∂ u (t, w)+ 1 λ2(T t). (3.11) 1 1 − w 0 2 0 −

Inserting (3.11) into (3.10), we find that q1 satisfies

0=(∂t + A0 + C0)q1 + Q1,q1(T,y,z)=0, (3.12) where the operator C0 is given by

C 1 2 2 0 = 2 λ0∂z +(ρβλ)0∂y∂z, (3.13) and the function Q1 satisfies H1(t, y, w)=Q1(t, y, z(t, w)). Now, from (3.6) and (3.13), we observe that the operator (A0 +C0)istheinfinitesimalgenerator 2 ′ of a diffusion in R whose drift vector and covariance matrix are constant. The semigroup P0(t, t ) generated by (A0 + C0)isgivenby

P0(t, T )G(y,z):= dη dζ Γ0(t, y, z; T,η,ζ)G(η,ζ), #R2 where Γ0,thefundamental solution corresponding to (∂t + A0 + C0), is a Gaussian kernel:

1 1 T −1 Γ0(t, y, z; T,η,ζ)= exp m C m , (2π)3 C !−2 " | | + with covariance matrix C and vector m given by

2 (β )0 (ρβλ)0 η y (T t)c0 C =(T t) 2 , m = − − − . − !(ρβλ)0 (λ )0 " ! ζ z " − By Duhamel’s principle, the unique classical solution to (3.12) is given by

T q1(t)= ds P0(t, s)Q1(s), #t In the case of a general utility function, (3.7) is easily solved numerically, for instance by solving the fast diffusion (or Black’s) equation for the risk tolerancefunctionR0 (see Fouque et al. (2012)). Then u1 can also be computed numerically using the formulas above. Inthecaseofpowerutility there are explicit formulas, as given in Section 3.3.

24 Having obtained an approximation for the value function u u +u we now seek an expansion ≈ 0 1 for the optimal control π∗ π∗ + π∗.Insertingtheexpansion(3.4)ofthecoefficientsandthe ≈ 0 1 expansion (3.5) for u into (3.2), and collecting terms of like powers of ε we obtain

O µ0(∂wu0) (1) : π0 = 2 2 , (3.14) −(σ )0(∂wu0) 2 2 O (σ )1 (∂wu1) (∂wu0) (ε): π1 = π0 2 π0 2 µ1 2 2 − (σ )0 − (∂wu0) − (σ )0(∂wu0) (∂wu1) (∂y∂wu1) µ0 2 2 (ρβσ)0 2 2 . (3.15) − (σ )0(∂wu0) − (σ )0(∂wu0) Higher order terms for the both the value function u and the optimal control π∗ can be obtained in the same manner as u1 and π1. Analysis of the asymptotic formulas for different utility functions and stochastic volatility models is presented in more detailinLorigandSircar(2014).

3.3 Power Utility Finally, we consider a utility function U from the Constant Relative Risk Aversion (CRRA), or power family: w1−γ CRRA utility : U(w):= ,w>0,γ>0,γ =1, 1 γ ̸ − where γ is called the risk aversion coefficient. Here all the quantities above can be computed explicitly. The explicit solution u0 to (3.7) is 1 1 γ u (t, w)=U(w)exp λ2 − (T t) . 0 !2 0 ! γ " − " w The risk-tolerance function is R0 = γ ,andthetransformationin(3.11)isthen 2γ 1 z(t, w)=γw +(T t) − 1 λ2. − ! γ " 2 0

An explicit computation reveals that u1 is given by

u1(t, y, w)=q1(t, y, z(t, w)) 1 γ ′ = − u (t, w) 1 λ2(¯y) (T t)(y y¯)+ 1 (T t)2 c + 1−γ ρβ λ . γ 0 2 − − 2 − 0 γ 0 0 & ' ( ( )) For the sp ecific casey ¯ = y,theaboveexpressionsimplifiesto

1 γ ′ u (t, y, w)= − u (t, w) 1 λ2(y) 1 (T t)2 c + 1−γ ρβ λ . 1 γ 0 2 2 − 0 γ 0 0 & ' ( ( )) Using these explicit representations of u0 and u1 the expressions (3.14) and (3.15) for the optimal stock holding approximations become

∗ µ0 π0 = 2 , γσ0

25 ′ µ (¯y) µ ′ (1 γ)(T t) ′ π∗(t, y)=(y y¯) 0 σ2(¯y) + − − ρβ 1 1 λ2(¯y) . 1 − ! γσ2 − γσ4 " γσ 0 γ 2 0 0 & ' 0 ( & ' ) ∗ For the sp ecific casey ¯ = y,theformulaforπ1 simplifies to

(1 γ)(T t) ′ π∗(t, y)= − − ρβ 1 1 λ2(y) . 1 γσ 0 γ 2 0 & ' 4Conclusion

Asymptotic methods can be used to analyze and simplify pricing and portfolio optimization prob- lems, and we have presented some examples and methodologies.Akeyinsightistoperturbproblems with stochastic volatility around their constant volatility counterparts to obtain the principle effect of volatility uncertainty. These approaches reduce the dimension of the effective problems that have to be solved, and often lead to explicit formulas that can be analyzed for intuition. In the context of portfolio problems accounting for stochastic volatility, recent progress has been made in cases where there are transaction costs (Bichuch and Sircar (2014); Kallsen and Muhle-Karbe (2013)), stochastic risk aversion that varies with market conditions (Dong and Sircar(2014)),andundermorecomplex local-stochastic volatility models (Lorig and Sircar (2014)).

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