Markovian projection for volatility calibration

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options are usually the most liquid options available in any market.

More often than not, they are the only options that are liquid enough to be used for model calibration. Thus, efficient methods for valuing European-style options are a critical requirement for any model. In this article, we develop the Markovian projection method, a very general and powerful approach for deriving accu- rate approximations of European-style option prices in a wide range of models.

Various ideas related to the Markovian projection method have already been used, and very successfully so, in a wide range of contexts. In Piterbarg (2004) and Piterbarg (2005b), closed-form approximations of European-style swaptions prices in a forward Libor model with stochastic volatility have been obtained. In Andreasen (2005), that has also been accomplished for a short- rate model with stochastic volatility. Piterbarg (2005a) and Piter- barg (2006b) have introduced a foreign exchange skew in the interest rate/forex hybrid model and demonstrated how closed- form approximations for forex options can be derived. The same techniques can easily be adapted to interest rate/equity hybrids, for example. In this article, we formalise the ideas from these papers as the Markovian projection method. Interest in the sub- ject appears widespread – while this article was in the final prepa- ration stages, two more working papers on the Markovian projec- tion method, Antonov & Misirpashaev (2006a) and Antonov &

Misirpashaev (2006b), were circulated.

The Markovian projection method is based on combining the pioneering ideas of Dupire (1997) on local and stochastic volatil- ity (and the recently rediscovered, in the quantitative finance con- text, results of Gyöngy, 1986) with the parameter-averaging tech-

niques that have been developed for stochastic volatility models relatively recently. Both Dupire and Gyöngy show that if two underlyings, each governed by its own stochastic differential equations (SDEs), are given, and the expected values of the SDE coefficients conditioned on the appropriate underlying (the so- called ‘Dupire’s local volatilities’) are equal, then European-style option prices on (or, equivalently, one-dimensional marginal dis- tributions of) the two underlyings are the same. This observation is the key tool to replace a model with complicated dynamics for the underlying with a simpler one. Once that is accomplished, the (local or stochastic volatility) model that is obtained is further simplified, by applying the parameter averaging techniques, to a form that admits closed-form solutions.

The Markovian projection method proceeds in four steps:

n 1. For the underlying of interest, its SDE, driven by a single Brownian motion, is written down by calculating its quadratic variance (and combining all dt terms that exist).

n 2. In this SDE, the diffusion and drift coefficients (if they exist) are replaced with their expected values conditional on the underlying. As shown below, this does not affect the values of European-style options.

n 3. The conditional expected values from Step 2 are calculated or, more commonly, approximated. Methods based on, or related to, Gaussian approximations are often used for this step.

n 4. Parameter averaging techniques (see Piterbarg, 2005b and 2005c) are used to relate the time-dependent coefficients of the SDE obtained in Step 3 to time-independent ones. This, typi- cally, allows for a quick and direct calculation of European-style option values.

One-dimensional distributions of Itô processes

Suppose a model is given. Let S(t) be the quantity of interest on which European-style options are to be valued. For example, it could be a forward price of an equity share, a forward forex rate, a Libor or a swap rate, and so on. The market observable may or may not be a ‘primitive’ variable in the model. For example, equity models are most often expressed in terms of the share price, and then the dynamics are imposed on S(·) directly. On the other hand, in interest rate modelling, the dynamics are often imposed on short rates or Libor rates. If S(·) is the swap rate, then it is not a ‘primitive’ variable in such a model; the dynamics of S(·) are implicitly defined by the model expressed in terms of other variables. In either case, an Itô process for S(·),

Markovian projection for volatility calibration

Vladimir Piterbarg looks at the ‘Markovian

projection method’, a way of obtaining closed-form approximations of European-style option prices on various underlyings that, in principle, is applicable to any (diffusive) model. The aim is to distil the essence of the method into a conceptually simple ‘plan of attack’

that anyone who wants to obtain European-style option approximations can follow

European-style

1

We do not include the dt term as most often there is a measure under which S(·) is a martingale, and it is

precisely the measure that we are interested in studying the dynamics of S(·) under (for example, the swap

measure for a swap rate, or the forward measure for a forward forex rate)

^risk.net 85

^{dS t} ( ) = Σ ( ) t,... dW t ( )

Here S(t, ...) is, in general, a stochastic process that may depend on S(·) and other sources of randomness.

To price a European-style option on S(·) with expiry T and strike K, we need to know the one-dimensional distribution of S(·) (at time T). On the other hand, the knowledge of prices of all European-style options for all strikes K for a given expiry T is equivalent to knowing the (one-dimensional) distribution of S(·) at time T. In view of these remarks, the importance of the follow- ing result is evident (Gyöngy, 1986, Theorem 4.6).

n Theorem 1. Let X(t) be given by:

^{dX t} ( ) ^{= α} ( ) t dt + β t ( ) dW t ( ) ⁽²⁾

where a(·), b(·) are adapted bounded stochastic processes such that (2) admits a unique solution. Define a(t, x), b(t, x) by:

a t, x ( ) = E α t ( ( ) X t ( ) ⁼ x )

b ² ( ) t, x = E β ( ² ( ) t X t ( ) = x )

Then the SDE:

dY t ( ) = a t,Y t ( ( ) ) dt + b t,Y t ( ( ) ) dW t ( )

Y 0 ( ) = X 0 ( ) ⁽³⁾

admits a weak solution Y(t) that has the same one-dimensional distributions as X(t).

Critically, since X(·) and Y(·) have the same one-dimensional distributions, the prices of European-style options on X(·) and Y(·) for all strikes K and expiries T will be the same. Thus, for the purposes of European-style option valuation and/or calibration to European options, a potentially very complicated process X(·) can be replaced with a much simpler Markov process Y(·), the Marko- vian projection

of X(·).

The process Y(·) follows what is known in financial mathemat- ics as a ‘local volatility’ process. The function b(t, x) is often called

‘Dupire’s local volatility’ for X(·), after Dupire, who directly linked E(b

(t)|X(t) = x) to prices of European-style options on X(·) when a(·) ≡ 0 (see Dupire, 1997), a link that can be seen to lead to the same result.

Efficient valuation and approximation methods for European- style options in local volatility models, that is, models of the type (3), are available (see, for example, Andersen & Brotherton- Ratcliffe, 2001).

For the variable S(·) considered above, the theorem immediately implies that we can replace (1) with the Markovian projection:

_{dS t} ( ) ^{= %Σ} ( t,S t ( ) ) dW t ( ) ⁽⁴⁾

to price European-style options, where S ^~ (t, x), the deterministic function of time and state, is obtained by:

^{%Σ 2} ( ) ^{t, x} = E Σ ( ² ( ) t,... S t ( ) = x ) ⁽⁵⁾

We emphasise that the Markovian projection is exact for Euro- pean-style options but, of course, does not preserve the dependence structure of S(·) at different times or, equivalently, the marginal dis- tributions of the process of orders higher than one. Thus, the prices of securities dependent on sampling of S(·) at multiple times, such as barriers, American-style options and so on, are different between the original model (1) and the projected model (4).

approaches later in the article. Next, however, we consider other ways that Theorem 1 can be used.

Applications to stochastic volatility models

Suppose a stochastic volatility model is given. Let S(·) be the vari- able of interest as before, governed in the model by equation (1).

The results of Theorem 1 allow us to replace the dynamics with local volatility dynamics to price European-style options. How- ever, substituting a local volatility model for a stochastic volatility model is not always ideal. The two types of model are very differ- ent, and that makes calculating the conditional expected value in (5) particularly difficult. This renders various possible approxima- tions rather inaccurate. As a general rule, when one model is approximated with another, it is always best to keep as many fea- tures as possible the same in the two models. For example, a sto- chastic volatility model should be approximated with a (poten- tially simpler) stochastic volatility model, a model with jumps by a (simpler) model with jumps, and so on.

Upon some reflection, it should be clear that Theorem 1 pro- vides us with a tool that is more general than just a way of approx- imating any model with a local volatility one. If two processes have the same Dupire’s local volatility, the European-style option prices on both are the same for all strikes and expiries. Consider a stochastic volatility model. Let X

(t) follow:

dX 1 ( ) t = b 1 ( t, X 1 ( ) t ) z 1 ( ) t dW t ( )

where z

(t) is some variance process. Suppose we would like to match the European-style option prices on X

(·) (for all expiries and strikes) in a model of the form:

dX 2 ( ) t ⁼ b 2 ( t, X 2 ( ) t ) z 2 ( ) t dW t ( )

where z

(t) is a different, and potentially simpler, variance proc- ess. Theorem 1 implies that b

(t, x) should be chosen such that:

E b ( ₂ ² ( t, X ₂ ( ) t ) z ₂ ( ) t X ₂ ( ) t = x )

= E b ( ₁ ² ( t, X ₁ ( ) t ) z ₁ ( ) t X ₁ ( ) t = x )

or, rearranging the terms:

b t x b t x z t X t x

z t X t

2 2

1 2 1 1

2 2

, ,

( ) = ( ) ( ( ) ( ) = )

( ) ( )

E

E ( == ^x ) ⁽⁶⁾

To apply this formula, the conditional expected values need to be calculated. Note that it is beneficial to choose z

as similar to z

as possible. Then, various approximations used to calculate both conditional expected values will tend to cancel each other out (since the ratio of similar terms appears in (6)), thus increasing the accuracy of calculating b

(t, x).

It is generally the case that a realistic stochastic volatility model with a local volatility component should have a local volatility function that does not have a lot of curvature (for example, a lin- ear or a power function of the underlying). Moreover, such a choice makes many calculations and approximations much easier, because then the local volatility functions can be described by just two numbers, a level and a slope (at the forward value of the underlying, for example). In practice, this means that when a sto-

2

Often we will use the same letter to denote both the original process and its Markovian projection

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chastic volatility model is constructed to fit implied volatility smiles (be that market volatilities or implied volatilities from a different model), it is better to choose the stochastic volatility that explains as much of the curvature of the implied volatility smile as possible. Then the local volatility function will be ‘almost lin- ear’, and the benefits mentioned above will apply. For the con- crete example of equation (6), one should typically try to find z

(·) that has the same term variance as z

(·), as it is the volatility of stochastic variance that defines the curvature of the smile.

Gaussian approximations for conditional expected value calculations

Recall the set-up discussed above, namely the general equation of the type (1) for an underlying of interest. Calculations of condi- tional expected values, as in (5) or (6) are often the most difficult part of the Markovian projection method. Tractable formulas are hard to find for any type of process (an example of how difficult it could be even for relatively simple processes can be seen in Atlan, 2006), with one notable exception. If in (5) the process (S(t), S

(t, ...) is jointly Gaussian, then the calculations are straightforward.

While this is the case in only the trivial situation of S

(t, ...) being deterministic, the availability of closed-form solutions in the Gaussian case gives rise to the approximation in which the actual dynamics of S(t) and S

(t)(= S

(t, ...)) are replaced by the Gaussian ones. While the details vary from case to case, the general approach proceeds as follows.

Let the dynamics of (S(t), S

(t)) be written in the following form:

dS t ( ) ^{= Σ} ( ) t dW t ( )

dΣ ² ( ) t ^{= η} ( ) t dt + ε t ( ) dB t ( ) ⁽⁷⁾

where S(t), h(t) and e(t) are adapted stochastic processes (that may depend on S(·), S

(·), etc) and W(t), B(·) are both Brownian motions. Then the conditional expected value in (5) can be approximated by:

%Σ ² ( ) t, x ^@ E Σ ( ² ( ) t S t ( ) ⁼ x )

= Σ ² ( ) t + r t ( ) ( x − S ₀ )

r t ( ) = ∫ ₀ ^t ε ( ) s Σ ( ) s ρ ( ) s ds Σ ² ( ) s ds

0

∫ t

(8)

where e _ (t), S _

(t), r _ (t) and S _

(t) are, respectively, deterministic approximations to e(t), S

(t), r(t) =

〈dW(t), dB(t)〉/dt, and S(t). In particular, one can take:

ε ( ) t = Eε t ( )

Σ ² ( ) t = EΣ ² ( ) t

ρ ( ) t = E dW t ( ( ) ,dB t ( ) / dt )

Σ ( ) t = Σ ² ( ) t

We note that if more is known about the form of the dynamics (7), other deterministic approximations to e(t) (and other param- eters) can be more useful.

It is important to point out that the approximation (8) is linear in x. Hence, the Gaussian approximation works best when the local volatility functions that are used are either linear or close to linear. This provides another angle to the comment at the end of

the previous section, namely that the most practical technique for calculating conditional expected values, the Gaussian approxima- tion, is not going to work well when the curvature in the volatility smile is explained by the local volatility function, rather than by the stochastic volatility component.

Sometimes it is beneficial to apply approximations that are not Gaussian, but linked to them. For example, a lognormal approxi- mation can be used in place of the Gaussian one. Then, the approximation is based on the known expression for E(X|Y) when (X, Y) are jointly lognormal. Other twists involving ‘mapping functions’ Y → f(Y) are also possible.

Examples

Here, we present a number of examples to demonstrate typical applications of the method we developed.

n Example 1: options on an index in local volatility models.

Consider a collection of N assets S(t) = (S

(t), ... , S

(t))

each driven by its own local volatility model:

dS n ( ) t = ϕ _n ( S n ( ) t ) dW n ( ) t n = 1,..., N

with a vector of Brownian motions (W

(t), ... , W

(t)) with an N × N correlation matrix:

dW i ,dW j = ρ _ij dt, i, j = 1,..., N

We restrict the set of local volatility functions ϕ

(x) to the set of functions described by its value and the first-order derivative at the forward. In effect, we require ϕ

(x) to be linear, or close to linear (such as constant elasticity of variance). As always, this is not really a restriction as using more complicated local volatility functions is not a good idea anyway.

Define an index S(t) by:

S t ( ) ⁼ w n S n ( ) t ⁼ w ^T

n=1

∑ N S t ( )

An index is a weighted average of the assets. We would like to derive an approximation formula to (European-style) options on the index S(t). This problem was considered first in Avellaneda et al (2002).

Applying Itô’s lemma to S(t) we find that:

dS t ( ) = w n ϕ _n ( S n ( ) t ) dW n ( ) t

n=1

∑ N

If we define:

σ ² ( ) t = w _n w _m ϕ _n ( S _n ( ) t ) ^ϕ m ( S _m ( ) t ) ^ρ nm n,m=1

∑ N

dW t ( ) ^{= 1} _σ

( ) t ^{w n ϕ n} ( ^{S n} ( ) ^t ) dW n ( ) t

n=1

∑ N

then:

dS t ( ) = σ ( ) t dW t ( )

and, by calculating its quadratic variation, we see that dW is a standard Brownian motion. The process s(t) is, of course, a com- plicated function of S(t). Using our earlier results and, in particu- lar, Theorem 1, for European-style option valuation, we can replace this SDE with a simple one:

dS t ( ) ^{= ϕ} ( t,S t ( ) ) dW t ( )

where:

^risk.net 87

ϕ ² ( ) t, x = w _n w _m ρ _nm E ϕ ( _n ( S _n ( ) t ) ^ϕ m ( S _m ( ) t ) S t ( ) = x )

n, m=1

∑ N

By linearising s

(t) in S(t) and applying Gaussian approximations to calculating conditional expected values of the type E(S

(t) − S

(0)|S(t) = x), we find that for European-style option pricing, the dynamics of the index S(t) can be approximated with the follow- ing SDE:

dS t ( ) = ϕ ( ) S t ( ) dW t ( )

where ϕ(x) is such that:

ϕ ( S 0 ( ) ) ⁼ p, ′ ϕ ( S 0 ( ) ) ⁼ q with:

p ² = w _n w _m p _n p _m ρ _nm

n, m=1

∑ N

q = 1

p w _n p _n ρ _n ² q _n

n=1

∑ N

ρ _n = 1

p w m p m ρ _nm

m=1

∑ N

where:

p ⁿ @ ϕ _n ( S _n ( ) 0 ) , q _n @ ′ ϕ _n ( S _n ( ) 0 ) , n = 1,..., N Details are available in Piterbarg (2006a).

We note that the approximations we develop work best when all q

s are ‘not very different’. This would be the case when, for example, a swap rate is represented as a weighted average of con- tiguous Libor rates, and we can expect their skews to be similar (see Example 3 below). In other cases, when q

s can be signifi- cantly different, as for example in a forex rate basket, non-linear approximations of Avellaneda et al (2002) could be more accurate (but are more complicated).

n Numerical results. We demonstrate the approximation with a numerical example. Five assets (n = 5) in the index have the parameters shown in table A. For an option on the index with five years to maturity (T = 5), the index parameters in table B are obtained by the Markovian projection method. Table C compares prices of options on the index calculated with Monte Carlo simulation and the Markovian projection approximation.

Results are expressed in relative volatilities. Agreement is excel- lent throughout, with maximum error of about 0.01% in rela- tive volatility.

n Example 2: spread options with stochastic volatility. Con- sider the problem of valuing spread options, that is, (European) options on S(t) =

S

(t) − S

(t), where each of the assets S

, S

fol- lows its own stochastic volatility process (as pointed out in Pit- erbarg, 2006c, spread options are very sensitive to stochastic variance decorrelation, an often overlooked fact). In particular, we define:

dS i ( ) t = ϕ _i ( S i ( ) t ) z i ( ) t dW i ( ) t

dz i ( ) t ^{= θ} ( 1 − z i ( ) t ) dt + η i z i ( ) t dW 2+i ( ) t , z i ( ) 0 ⁼ 1

i = 1,2

(9) with the correlations given by:

dW i ( ) t ,dW j ( ) t ^{= ρ} ij dt, i, j = 1,...,4

The spread S(t) is the underlying of interest; the first step is to write an SDE for it. We have:

^{dS t} ( ) ^{= σ} ( ) t dW t ( ) ⁽¹⁰⁾

where:

σ

( ) t ^{= ϕ}

( S

( ) t ) ^z

( ) t ⁻ 2ρ

ϕ

( S

( ) t ) ^ϕ

( S

( ) t ) ^z

( ) t z

( ) t + ϕ

( S

( ) t ) ^z

( ) t

dW t ( ) ⁼

1

σ ( ) t ( ^ϕ

( ^S

( ) ^t ) ^z

( ) t dW

( ) t ^{− ϕ}

( S

( ) t ) ^z

( ) t dW

( ) t )

(11)

Using the same notations as in the previous example, we set:

p i = ϕ _i ( S i ( ) 0 ) , q i = ′ ϕ _i ( S i ( ) 0 ) , i = 1,2

As explained previously, directly replacing s

(t) (in (10)) with its expected value conditional on S(·) is not a good idea when sto- chastic volatility is involved.

We shall try to find a stochastic volatility process z(·) such that the curvature of the smile of the spread S(·) is explained by it, and the local volatility function is only used to induce the volatility skew.

To identify a suitable candidate for z(·), let us consider what the expression for s

(t) would be if the local volatility functions for the components, ϕ

(x), i = 1, 2, were constant functions. We see that in this case, the expression for s

(t) in (11) would not involve the processes S

(·), i = 1, 2 and thus would be a good candidate for the stochastic variance process. Hence, we define (the division by p

=

s

(0) is to preserve the scaling z(0) = 1):

Weights, w

20% 20% 20% 20% 20%

Spots, S

(0) 80% 90% 100% 110% 120%

Relative volatilities, p

/S

(0) 8% 9% 10% 11% 12%

blends, q

S

(0)/p

30% 40% 50% 60% 70%

b. Parameters of the index

expiry in years, T 5

index spot, S(0) 100%

index relative volatility, p/S(0) 7.95%

index blend, qS(0)/p 58.73%

c. Values of options on the index calculated by Monte carlo versus Markovian projection, in implied relative volatilities for different strikes

Strike Monte carlo Markovian projection difference

50% 9.32% 9.32% –0.01%

75% 8.48% 8.47% 0.01%

100% 7.96% 7.96% 0.00%

125% 7.62% 7.61% 0.01%

150% 7.37% 7.36% 0.01%

3

It is better to use (6) rather than (5); the approximation (8) should then be applied to both the

numerator and denominator separately

cutting edge. calibRation

^{z t} ( ) = 1

p ² ( p ₁ ² z ₁ ( ) t − 2 p ₁ p ₂ ρ ₁₂ z ₁ ( ) t z ₂ ( ) t + p ₂ ² z ₂ ( ) t )

(12) where:

^{p = σ 0} ( ) ⁼ ( p 1 2 − 2 p 1 p 2 ρ ₁₂ + p 2 2 ) ^{1/2 (13)}

Using methods similar to the case of the basket (see details in Piterbarg, 2006a), we find that for European-style option pricing the dynamics of the spread S(t) can be approximated with the fol- lowing SDE:

dS t ( ) = ϕ ( ) S t ( ) z t ( ) dW t ( )

where z(·), given by (12), the function ϕ(·), satisfies:

ϕ ( S 0 ( ) ) ⁼ p, ′ ϕ ( S 0 ( ) ) ⁼ q with:

q @ 1

p ( p 1 ρ ₁ ² q 1 − p 2 ρ ₂ ² q 2 )

and r

, r

are defined by:

^{ρ 1 = 1 p} ( ^{p 1 − p 2 ρ 12} ) , ρ 2 = 1

p ( p 1 ρ ₁₂ − p 2 )

(14)

To proceed, we still need to derive a simple form for the sto- chastic variance process z(·), defined by (12). Ideally, we should try to write the SDE for it in the same form as the SDEs for z

s in (9). This cannot be done exactly, but we can show (see Piterbarg, 2006a) that approximately:

dz t ( ) ^{= θ} 1 + γ θ − z t ( )

  

 dt + η z t ( ) dB t ( )

where:

η ² = 1

p ² ( ( p 1 η ₁ ρ ₁ ) ^{2 −} 2 p ( 1 η ₁ ρ ₁ ) ( p 2 η ₂ ρ ₂ ) ^ρ 34 + ( p 2 η ₂ ρ ₂ ) ² )

dB t ( ) = 1

ηp ( p ₁ η ₁ ρ ₁ dW ₃ ( ) t − p ₂ η ₂ ρ ₂ dW ₄ ( ) t )

and g is given by:

γ @ p ¹ p 2 ρ ₁₂

4 p ² ( η ₁ ² − 2η 1 η ₂ ρ ₃₄ + η ₂ ² )

(15) The (approximation of the) correlation between dW and dB _

is given by:

ρ = dW ,dB / dt

= 1 p ² η ( p 1 2 η ₁ ρ ₁ ρ ₁₃ − p 1 p 2 ( η ₁ ρ ₁ ρ ₂₃ + η ₂ ρ ₂ ρ ₁₄ ) ⁺ p 2 2 η ₂ ρ ₂ ρ ₂₄ )

n Numerical results. We test the performance of the approxi- mation on an example with asset parameters from table D, and correlations as in table E. Using the Markovian projection method, the parameters of the spread in table F are obtained.

Table G compares prices of options on the spread calculated with Monte Carlo simulation and Markovian projection approximation. Results are expressed in absolute volatilities.

Agreement is very good, with maximum error of about 0.07%

in volatility.

n Example 3: swap rate volatility in a forward Libor model.

Let us demonstrate the power of the method by deriving the swap rate volatility formula in a skew-extended forward Libor model. A similar formula (in a stochastic volatility case) was derived in Piterbarg (2005b) without explicitly using the Mark- ovian projection method.

Let 0 = T

< T

< ··· < T

be a tenor structure, and L _

(t) = (L

(t), ... , L

(t))

be a vector of spanning forward Libor rates, L

(t) = τ

(P(t, T

)/P(t, T

) − 1), τ

= T

− T

. The skew-extended for- ward Libor model is defined by the following dynamics:

dL n ( ) t = ψ _n ( t, L n ( ) t ) dW n T

( ) t , n = 1,..., N − 1 where W

(t) is a Brownian motion for rate L

under the T

forward measure for each n = 1, ... , N − 1. We denote r

= 〈dW

, dW

〉 /dt.

A swap rate S(t) = S

(t) is defined by:

S t ( ) = P t,T ( ) ^{n −} P t,T ( n+m )

A t ( )

A t ( ) = τ _k

k=n n+m−1

∑ P t,T ( k+1 )

d. Parameters of the underlyings for the spread

asset index n 1 2

Spots, S

(0) 100% 90%

Relative volatilities, p

/S

(0) 10% 9%

blends, q

S

(0)/p

100% 0%

Mean reversion of volatility, q 10% 10%

Volatilities of variance, h

100% 100%

e. correlations r

between spot and volatility factors

S

S

z

z

S

100% 70% –25% –25%

S

70% 100% –25% –25%

z

–25% –25% 100% 90%

z

–25% –25% 90% 100%

F. Parameters of the spread obtained by the Markovian projection method

expiry in years T 5

Spread spot S(0) 10.0%

Spread relative volatility p/S(0) 72.3%

Spread blend pS(0)/q 6.9%

Spread mean reversion of variance q 10.0%

Spread volatility of variance h 98.6%

Spread spot/volatility correlation r –6.7%

Spread mean reversion level 1 + g/q 154.3%

g. Values of options on the spread calculated by Monte carlo versus Markovian projections, in implied volatilities for different strikes

Strike Monte carlo Markovian projection difference

–10% 7.50% 7.57% –0.07%

0% 7.03% 7.06% –0.03%

10% 6.84% 6.85% –0.02%

20% 7.27% 7.30% –0.04%

30% 8.07% 8.13% –0.06%

^risk.net 89

dS t ( ) ^{= ∂S t} ( )

∂L _k ( ) t

k=n n+m−1

∑ ^ψ k ( t, L k ( ) t ) dW k A ( ) t

= Σ ( t,L t ( ) ) dW ^A ( ) t , Σ ² ( t,L t ( ) ) ^@ _∂L ^{∂S t ( )}

k ( ) t

k, ′ ∑ k _∂L ^{∂S t ( )}

k ′ ( ) t ^{ψ k} ( ^{t, L k} ( ) ^t ) ^ψ k ′ ( t, L k ′ ( ) t ) ^ρ k ′ k

where dW

is a driftless Brownian motion under P

.

By the Markovian projection method, to value European- style options on S(·) (swaptions), we can use the following dynamics:

dS t ( ) = η ( t,S t ( ) ) dW ^A ( ) t where:

η ( ) t,S = E ( ^A ( Σ ² ( t,L t ( ) ) S t ( ) ⁼ S ) ) ^1/2

≈ E ^A ( Σ ( t,L t ( ) ) ^{S t ( ) = S} ) ⁽¹⁶⁾

To derive closed-form approximations, we proceed with the standard linearisation and Gaussian approximation arguments.

Linearising around the expected value of the Libor rates, we find that:

Σ ( t,L t ( ) ) ^{≈ Σ} ( t,E ^A L t ( ) ) ^{+ ∇Σ } _ ( ^{t,E A L t ( )} ) ^ _ ^T ( ^{L t ( ) − E A L t ( )} )

where ∇ f denotes the gradient of f.

To calculate conditional expected values in (16), we approxi- mate L _

(t), S(t) with Gaussian processes:

d ˆL n ( ) t = ψ _n ( t, L n ( ) 0 ) dW n T

( ) t d ˆS t ( ) ^{= Σ} ( t,L 0 ( ) ) dW ^A ( ) t Then:

E ^A ( L t ( ) ⁻ E ^A L t ( ) S t ( ) )

≈ ˆ L t ( ) , ˆL t ( ) ⁻¹ ˆL t ( ) , ˆS t ( ) ( S t ( ) − S 0 ( ) )

Then the following shifted-lognormal dynamics are obtained:

dS t ( ) = ( a t ( ) + b t ( ) ( S t ( ) − S 0 ( ) ) ) ^{dW A ( ) t}

a t ( ) ^{= Σ} ( t,E ^A L t ( ) )

b t ( ) = ∇Σ ^ _ ( t,E ^A L t ( ) ) ^ _ ^{T ˆL t ( ) , ˆL t ( ) −1 ˆL t ( ) , ˆS t ( )}

The formula is similar to, but not exactly the same as, that obtained in Piterbarg (2005b).

The parameters of this SDE are time-dependent. To obtain a model with time-constant parameters (and, thus, enable closed- form valuation), we apply parameter averaging. It follows that approximately:

dS t ( ) = α + β ( ( S t ( ) − S 0 ( ) ) ) ^{dW A ( ) t}

where:

α ² = 1

T _n ∫ ₀ ^T

a ^{2 ( ) t dt, β = α} ∫ ₀ ^T

^{v t ( )} ( ^{b t ( ) / a t ( )} ) dt v t ( ) dt

0 T

∫

where the skew weights v(·) are given by:

v t ( ) ⁼ a ² ( ) t ∫ ₀ ^t a ^{2 ( ) s ds}

We do not present separate numerical results because the approximation here is closely related to the one for the index in Example 1, and also because extensive testing for similar approxi- mations was performed in Piterbarg (2005b).

Conclusions

We have developed a systematic approach to obtaining approxi- mate closed-form expressions for European-style options in a large class of models, and demonstrated its power on a number of examples relevant to practical applications. We have shown that further advances in this area are intricately linked to the quality of tractable approximations that can be obtained for conditional expectations of a certain type, an area that merits close scrutiny from researchers in the field. n

Vladimir Piterbarg is head of fixed-income quantitative research at barclays capital in london. He would like to thank his colleagues, as well as leif andersen, Martin Forde and alexandre antonov, for valuable comments and discussions. comments from anonymous referees were valuable and are gratefully acknowledged. email: [email protected]

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