dynamic relative valuation

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Dynamic Relative Valuation

Liuren Wu, Baruch College

Joint work with Peter Carr from Morgan Stanley

Derivatives 2013: 40 Years after the Black-Scholes-Merton ModelOctober 11, 2013

Liuren Wu (Baruch) Dynamic Relative Valuation 10/11/2013 1 / 21
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The standard approach of no-arbitrage pricing

Identify an instantaneous rate as the basis, fully specifythe dynamics andpricing of this basis, and representthe value of actual contracts as someexpectation of this basis over some periods.

Options: Choose the instantaneous variance rateas the basis, andfully specify its dynamics/pricing.

Interest rates: Choose the instantaneous interest rateas 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 to

specify 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 ofmany tradedsecuritiesmove over the next day.

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A new approach: Work with what the market knows better

Model what the market knows better the next moveofmany traded securities.

1 Find commonalities of traded securities via some transformation:

B(yt(C), t,Xt;C) = price

yt(C) transformation of security prices that are similar and movetogether across securities with different contract details (C)

Xt some dependence on other observables (such as the underlyingsecurity price for derivatives)

The market is reasonably good at finding this transformation

2 Specify how the transformations move together

dyt(C) =t(C) +t(C)dZt

One Borwnian motion drives all yt(C)

The levels of (t(C), t(C)) are known, but not their dynamics.

3 The new no-arbitrage pricing relation: What we assume about theirnextmovedictates how their values (yt(C)) compare right now.

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I. Options market: Model BMS implied volatility

Transform the option price via the Black-Merton-Scholes (BMS) formula

B(It(K,T), t, St;K,T) StN(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 the

same 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=vtdWt.

Leave the dynamics ofvt unspecified; instead, model the near-futuredynamics of the BMS implied volatility under the risk-neutral (Q) measure,

dIt(K,T) =tdt+tdZt, for all K>0 and T >t.

The drift (t) and volvol (t) processescan depend on K, T, and It.Correlation between implied volatility and return is tdt=E[dWtdZt].

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From NDA to the fundamental PDE

We 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 ordZ.

By Itos lemma, each option in this portfolio has risk-neutral drift given by:

Bt+tB+vt

2S2tBSS+tt

vtStBS+

2t2B .

No arbitrage and no rates imply that both option drifts must vanish, leadingto the fundamental PDE:

Bt=tB+vt

2S2tBSS+tt

vtStBS+

2t2B .

When t and tare independent of (K,T), the PDE defines a linearrelation between thetheta(Bt) of the option and its vega(B), dollargamma(S2tBSS), dollarvanna(StBS), andvolga(B).

We call the class of BMS implied volatility surfaces defined by the abovefundamental PDE as the Vega-Gamma-Vanna-Volga (VGVV) model.

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Our PDE is NOTa PDE in the traditional sense

Bt=tB+ vt2S2tBSS+ttvtStBS+

2t

2B .

Traditionally, 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 formula

that we use the transform the option price into implied volatility.The coefficients on traditional PDEs are deterministic; they are stochasticprocessesin 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 constrainton how the different stochasticquantities should relate to each other.

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From the PDE to an algebraic restriction

Bt=tB+vt

2S2tBSS+tt

vtStBS+

2t2B .

The value function B is well known, so are its various partial derivatives:

Bt = 2

2 S2BSS, B = S

2BSS,SBS = d2

S2BSS, B = d1d2S

2BSS,

where dollar gamma is the common denominator of all the partial

derivatives, a nice feature of the BMS formula as a transformation.The PDE constraint on Bis reduced to an algebraic restriction on theshape of the implied volatility surface It(K,T),

I2t2 tIt

vt

2 tt

vtd2+

2t2 d1d2 = 0.

If (t, t) 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(t, t, t, vt), which dictate the next moveof the implied volatilitysurface, but we do not need to specify their future dynamics.

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Proportional volatility dynamics, as an example

(t, t) 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) =et(Tt) (mtdt+ wtdZt) ,

with t,wt>0 and (t,mt,wt) independent ofK,T.

A proportional specification has more empirical support than a

square-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, ) It(K,T), with k= lnK/St and =T t.The implied variance surface (I2t(k, )) solves a quadratic equation:

0 = 14e2tw2t

2I4t(k, )+ 1 2etmt etwttvt

I2t(k, )

vt+ 2etwttvtk+ e2tw2tk2 .Liuren Wu (Baruch) Dynamic Relative Valuation 10/11/2013 8 / 21
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Unspanned dynamics

0 = 14e2tw2t

2I4t(k, )+ 1 2etmt

etwtt

vt I2t(k, ) vt+ 2etwttvtk+ e2tw2tk2 .

Given the current levels of the five stochastic processes (t, vt,mt, t,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 current volatility surface shape is only linked to the near-future

dynamics of the security return and implied volatility.

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!

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The implied volatility smile and the term structure

At a fixed maturity, the implied variance smile can be solved as

I2(k, ) =at+2

k+ vt

etwt

2

+ ct.

In the limit of= 0, I2t(k, 0) =vt+ 2tvtwk+ w

2k2.

The smile is driven by vol of vol (convexity), and the skew is driven by

the return-vol correlation.

Atd2 = 0, the at-the-money implied variance term structure is given by,

A2t() = vt

1 2etmt.

The 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 (,,,). Performing dailycalibration on Heston would result in the same degrees of freedom, but the

calculation is much more complicated and the processisinconsistent.Liuren Wu (Baruch) Dynamic Relative Valuation 10/11/2013 10 / 21
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Sequential multi-layer valuation

Suppose 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) =t(Vt(K,T) I(K,T)) dt+ wtI(K,T)dZt,where V(K,T) denotes the model valuation for the option contract,represented in the implied vol space.

tmeasures 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-improvingvaluations!

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Applications

1 Implied 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.

2 Extracting 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-specificoption expected volatilitysurface ...

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Extensions

1 Non-zero financing rates: Treat S and Bas the forward values of theunderlying and the option, respectively.

2 Options on single names withpotential defaults, upon which stock pricedrops to zero (Merton (1976))

The BMS implied volatility transformation is no longer as tractable:

0 =Bt+tB+vt

2

S2tBSS+ttvtStBS+

2t

2

B +KN(d2).

The last term induced by default does not cancel.

We can choose the transformation through the Merton (1976) model:Let Mt(Merton implied volatility) as the input in the Merton modelto match the market price, conditional on being known:

B(St, t,Mt) =StN(d1)eKN(d2), d1/2 = ln St/K+ 12M2t

Mt

.

Bt= 2

2 S2BSS eKN(d2) has an extra term that cancels out

the default-induced term

3 It is a matter of choosing the right transformation...Liuren Wu (Baruch) Dynamic Relative Valuation 10/11/2013 13 / 21
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Contract versus instantaneous specification

Traditional modeling strives to identify an instantaneous rate as the basis forall traded securities.

The consistency is naturally guaranteed relative to this basis if all securitiesare priced from it.

The challenge is to fully specify the instantaneous rate dynamics that canreasonably price short-term as well as long-term contracts, with reasonable

tractability and stability in parameter identification.

Our approach directly models contract-specific quantities (such as theimplied volatility of an option contract) and link the contract values togetherwithout going through the basis dynamics.

We can focus on what we know better: the near-term moves of marketcontracts instead of the dynamics of some unobservable instantaneousrate over the next few decades.

The derived no-arbitrage relation shows up in extremely simple terms.

No parameters need to be estimated, only states need to be extracted.

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II. Bonds market: Model YTM

For bond pricing, standard approach starts with the full dynamics and

pricing of the instantaneous interest rate.We directly model the yield to maturity(yt) of each bond as thecommonality transformation of the bond price,

B(t, yt) j

Cjeytj

where Cjdenotes the cash flow (coupon or principal) at time t+j.

Assume the following risk-neutral dynamics for the yields of bond m,

dymt =mtdt+

mt dWt

where we use the superscript m to denote the potentially bond-specificnature of the yield and its dynamics.

Again, (t, t) is just a generic representation, one can be more specific interms of which process is global and which is contract-specific...

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Dynamic no-arbitrage constraints on the yieldcurve

Assume dynamic no-arbitrage between bonds and the money marketaccount, we obtain the following PDE,

rtB=Bt+ Byt+1

2Byy

2t,

With the solution to B(t, y) known, the PDE can be reduced to a simplealgebraic relation on yield:

ymt rt=mtm 1

2(mt )

22m,

where m and 2m denote cash-flow weighted average maturity and maturity

squared, respectively,

m =j

Cjeytj

Bm j,

2m=

j

Cjeytj

Bm 2j.

The relation is simple and intuitive: The yield curve goes up with itsrisk-neutral drift, but comes down due to convexity.


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Applications

1 Decomposing the yield curve:

Assume a mean-reverting square-root dynamics on the YTM:

dymt =t(t ymt )dt+tymt dWt

The no-arbitrage yield curve

ymt = rt+ttm

1 +tm+ 12

2t2m

The yield curve starts at rt at m = 0 and moves toward its risk-neutraltarget tas average maturity increases, subject to a convexity effect

that drives the yield curve downward in the very long run.The target tis a combination of expectation and risk premium.

No fixed parameters to be estimated, only the current states of somestochastic processes to be extracted.


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Applications

1 Decomposing the yield curve:

2 Extracting risk premium from long bonds:Assuming long-dated bonds move like a random walk under P, therisk-neutral dynamics become

dymt =tmt dt+

mt dWt,

with tdenoting the market price of bond risk.

The no-arbitrage relation is:

rt=ymt tmt m+

1

2(mt )

22m,

Based on the observed yield-to-maturity of each bond and a volatility

estimator for each bond, we can infer the market price of bond risk as

mt = ymt rt+ 12 (mt )22m

mt m

To the extent that the market price estimator mt differs acrossdifferent bonds, the difference represents relativevalueopportunities.


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III. Defaultable bonds

For corporate bonds, directly modeling yield to maturity should work thesame as for default-free bonds.

rtB=Bt+ Byt+1

2Byy

2t + (R 1)B.

The PDE can be reduced to an algebraic equation that defines thedefaultable yield term structure:

ymt (r+ (1 R)) =mt m 1

2(mt )

22m.

One can also directly model the credit spread if one is willing to assumedeterministic interest rates...


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Conclusion: Choose the right representation

At the end of the day, we are choosing some transformation of the prices of tradedsecurities to obtain a better understanding of where the valuation comes from.

Academics often try to go bottom up in search for the building blocksthat can be used to value anything and everything.

Arrow-Debrew securities, state prices, instantaneous rates...

Practitioners tend to be more defensive-minded, often trying to reduce the

dimensionality of the problem via localization and facilitatemanagement/monitoring via stabilization

Most transformations, such as implied volatility, YTM, CDS arecontract-specific very localized.

They also stabilize the movements, standardize the value forcross-contract comparison, and preclude some arbitrage possibilities.

We start with these contract-specific transformations, and derivecross-contract linkages by assuming how these transformed quantitiesmove together in the next instant, but nothing further.

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