Dynamic Relative Valuation - Faculty.baruch.cuny.edu

Dynamic Relative ValuationLiuren Wu, Baruch CollegeJoint work with Peter Carr from Morgan StanleyOctober 15, 2013Liuren Wu (Baruch)Dynamic Relative Valuation10/15/20131 / 20

The standard approach of no-arbitrage pricingIdentify an “instantaneous” rate as the basis, fully specify the dynamics andpricing of this basis, and represent the value of actual contracts as someexpectation of this basis over some periods.Options: Choose the instantaneous variance rate as the basis, andfully specify its dynamics/pricing.Interest rates: Choose the instantaneous interest rate as the basis,and fully specify its dynamics/pricing.The challenge: One needs to think far far into the future.Under this approach, to price a 60-year option or bond, one needs tospecify how the instantaneous rate moves and how the market pricesits risk over the next 60 years.The market has a much better idea about how the prices of many tradedsecurities move over the next day.Liuren Wu (Baruch)Dynamic Relative Valuation10/15/20132 / 20

A new approach: Work with what the market knows betterModel what the market knows better — the next move of many traded securities.1Find commonalities of traded securities via some stablization transformation:B(yt (C ), t, Xt ; C ) priceyt (C ) — transformations that are stable across both time (t) andcontract specification (C ) — We henceforth label it as spec rate.There shall exist a model under which yt (C ) is a fixed constant overboth time t and across contracts C .Xt — Dependence on other observables (such as the underlyingsecurity price for derivatives)2Specify how the spec rates move together at time t (but nothing further)dyt (C ) µt (C )dt ωt (C )dZtOne Brownian motion drives all yt (C ).The levels of (µt (C ), ωt (C )) are known, but not their future dynamics.3The new no-arbitrage pricing relation: What we assume about their nextmove (dyt (C ) at t) dictates how their values (yt (C ) at t) compare rightnow, but nothing more.Liuren Wu (Baruch)Dynamic Relative Valuation10/15/20133 / 20

I. Options market: Model BMS implied volatilityTransform the option price via the Black-Merton-Scholes (BMS) formulaB(It (K , T ), t, St ; K , T ) St N(d1 ) KN(d2 )Zero rates for notional clarity.BMS implied vol It (K , T ) is the commonality of the options market.The implied volatilities of different option contracts (underlying thesame security) share similar magnitudes and move together.A positive quote excludes arbitrage against cash and the underlying.It is the industry standard for quoting/managing options. Diffusion stock price dynamics: dSt /St vt dWt .Leave the dynamics of vt unspecified; instead, model the near-futuredynamics of the BMS implied volatility under the risk-neutral (Q) measure,dIt (K , T ) µt dt ωt dZt , for all K 0 and T t.The drift (µt ) and volvol (ωt ) processes can depend on K , T , and It .Correlation between implied volatility and return is ρt dt E[dWt dZt ].Liuren Wu (Baruch)Dynamic Relative Valuation10/15/20134 / 20

From NDA to the fundamental PDEWe require that no dynamic arbitrage (NDA) be allowed between any option at(K , T ) and a basis option at (K0 , T0 ) and the stock.The three assets can be combined to neutralize exposure to dW or dZ .By Ito’s lemma, each option in this portfolio has risk-neutral drift given by:Bt µt Bσ vt 2ω2St BSS ρt ωt vt St BSσ t Bσσ .22No arbitrage and no rates imply that both option drifts must vanish, leadingto the fundamental “PDE:” Bt µt Bσ vt 2ω2St BSS ρt ωt vt St BSσ t Bσσ .22When µt and ωt are independent of (K , T ), the “PDE” defines a linearrelation between the theta (Bt ) of the option and its vega (Bσ ), dollargamma (St2 BSS ), dollar vanna (St BSσ ), and volga (Bσσ ).We call the class of BMS implied volatility surfaces defined by the abovefundamental PDE as the Vega-Gamma-Vanna-Volga (VGVV) model.Liuren Wu (Baruch)Dynamic Relative Valuation10/15/20135 / 20

Our PDE is NOT a PDE in the traditional sense Bt µt Bσ vt 2ω2St BSS ρt ωt vt St BSσ t Bσσ .22Traditionally, PDE is specified to solve the value function. In our case, thevalue function B(It , t, St ) is definitional and it is simply the BMS formulathat we use the transform the option price into implied volatility.The coefficients on traditional PDEs are deterministic; they are stochasticprocesses in our “PDE.”Our “PDE” is not derived to solve the value function, but rather it is usedto show that the various stochastic quantities have to satisfy this particularrelation to exclude NDA. Our “PDE” defines an NDA constraint on how the different stochasticquantities should relate to each other.Liuren Wu (Baruch)Dynamic Relative Valuation10/15/20136 / 20

From the “PDE” to an algebraic restriction Bt µt Bσ vt 2ω2St BSS ρt ωt vt St BSσ t Bσσ .22The value function B is well known, so are its various partial derivatives:BtSBσS2 σ2 S 2 BSS ,Bσ d2 τ S 2 BSS , Bσσ στ S 2 BSS , d1 d2 τ S 2 BSS ,where dollar gamma is the common denominator of all the partialderivatives, a nice feature of the BMS formula as a transformation.The “PDE” constraint on B is reduced to an algebraic restriction on theshape of the implied volatility surface It (K , T ), vtIt2ω2 µt It τ ρt ωt vt τ d2 t d1 d2 τ 0.222If (µt , ωt ) do not depend on It (K , T ), we can solve the whole impliedvolatility surface as the solution to a quadratic equation.We just need to know the current values of the processes(µt , ωt , ρt , vt ), which dictate the next move of the implied volatilitysurface, but we do not need to specify their future dynamics.Liuren Wu (Baruch)Dynamic Relative Valuation10/15/20137 / 20

Proportional volatility dynamics, as an example(µt , ωt ) are just generic representations of the drift and diffusion processesof the implied volatility, we can be more specific if we think we know more.As an example,dIt (K , T )/It (K , T ) e ηt (T t) (mt dt wt dZt ) ,with ηt , wt 0 and (ηt , mt , wt ) independent of K , T .A proportional specification has more empirical support than asquare-root variance specification.The exponential dampening makes long-term implied volatility lessvolatile and more persistent.(mt , wt ) just specify our views on the trend and uncertainty over thenext instant.We can re-cast the implied volatility surface in terms of log relative strikeand time to maturity, It (k, τ ) It (K , T ), with k ln K /St and τ T t.The implied variance surface (It2 (k, τ )) solves a quadratic equation: 0 14 e 2ηt τ wt2 τ 2 It4 (k, τ ) 1 2e ηt τ mt τ e ηt τ wt ρt vt τ It2 (k, τ ) vt 2e ηt τ wt ρt vt k e 2ηt τ wt2 k 2 .Liuren Wu (Baruch)Dynamic Relative Valuation10/15/20138 / 20

Unspanned dynamics0 1 2ηt τ 2 2 4wt τ It (k, τ ) 1 2e ηt τ mt τ 4e vt 2e ηt τ wt ρt vt k e 2ηt τ wt2 k 2 . e ηt τ wt ρt vt τ It2 (k, τ )Given the current levels of the five stochastic processes (ρt , vt , mt , ηt , wt ),the current shape of the implied volatility surface must satisfy the abovequadratic equation to exclude dynamic arbitrage.The current shape of the surface does NOT depend on the exact dynamicsof five stochastic processes.The dynamics of the five processes are not spanned by the currentshape of the implied volatility surface.The dynamics of the five processes will affect the future dynamics of thesurface, but not its current shape.The current shape of the implied volatility surface is determined by 5 statevariables, but with no parameters!Liuren Wu (Baruch)Dynamic Relative Valuation10/15/20139 / 20

The implied volatility smile and the term structureAt a fixed maturity, the implied variance smile can be solved ass 2 ρ vt22k ηt τ ct .I (k, τ ) at τewt In the limit of τ 0, It2 (k, 0) vt 2ρt vt wk w 2 k 2 .The smile is driven by vol of vol (convexity), and the skew is driven bythe return-vol correlation.At d2 0, the at-the-money implied variance term structure is given by,vtA2t (τ ) .1 2e ηt τ mt τThe slope of the term structure is dictated by the drift of the dynamics.The Heston (1993) model generates the implied vol surface as a function of1 state variable (vt ) and 4 parameters (κ, θ, ω, ρ). Performing dailycalibration on Heston would result in the same degrees of freedom, but thecalculation is much more complicated and the process is inconsistent.Liuren Wu (Baruch)Dynamic Relative Valuation10/15/201310 / 20

Sequential, mutually-consistent, self-improving valuationSuppose you have generated valuations on these option contracts based oneither the “instantaneous rate” approach or some other method (includingthe earlier example of our VGVV models).One way to gauge the virtue of a valuation model is to assess how fastmarket prices converge to the model value when the two deviate.One can specify a co-integrating relation between the market price and themodel value,dIt (K , T ) κt (Vt (K , T ) I (K , T )) dt wt I (K , T )dZt ,where V (K , T ) denotes the model valuation for the option contract,represented in the implied vol space.κt measures how fast the market converges to the model value.A similar quadratic relation holds: The valuation Vt (K , T ) is regarded as anumber that the market price converges to, regardless of its source.Our approach can be integrated with the traditional approach byproviding a second layer of valuation on top of the standard valuation.It can also be used to perform sequential self-improving valuations!Liuren Wu (Baruch)Dynamic Relative Valuation10/15/201311 / 20

Applications1Implied volatility surface interpolation and extrapolation:Much faster, much easier, much more numerically stable than anyexisting stochastic volatility models.Performs better than lower-dimensional stochastic volatility models,without the traditional parameter identification issues associated withhigh-dimensional models.Reaps the benefits of both worlds: First estimate a low-dimensionalstochastic vol model, with which one can perform long-run simulations,and add the layers of VGVV structure on top of it.Sequential multi-layer, mutually consistent, self-improving modelinglocalizes modeling efforts, and satisfies the conflicting needs of differentmarket participants.2Extracting variance risks and risk premiums:The unspanned nature allows us to extract the levels of the 5 statesfrom the implied volatility surface without fully specifying the dynamics.One can also extract variance risk premiums by estimating ananalogous contract-specific option expected volatility surface .Liuren Wu (Baruch)Dynamic Relative Valuation10/15/201312 / 20

Extensions12Non-zero financing rates: Treat S and B as the forward values of theunderlying and the option, respectively.Options on single names with potential defaults, upon which stock pricedrops to zero (Merton (1976))The BMS implied volatility transformation is no longer as tractable: ω2vt 2St BSS ρt ωt vt St BSσ t Bσσ λKN(d2 ).22The last term induced by default does not cancel.0 Bt µt Bσ We can choose the transformation through the Merton (1976) model:Let Mt (Merton implied volatility) as the σ input in the Merton modelto match the market price, conditional on λ being known:B(St , t, Mt ) St N(d1 ) e λτ KN(d2 ),d1/2 ln St /K λτ 21 Mt2 τ .Mt τ23Bt σ2 S 2 BSS λe λτ KN(d2 ) has an extra term that cancels outthe default-induced termIt is a matter of choosing the right transformation.Liuren Wu (Baruch)Dynamic Relative Valuation10/15/201313 / 20