volatility smiles in commodity futures - scicomp smiles in commodity futures qimou su scicomp inc....

Volatility Smiles in Commodity Futures

Qimou SuSciComp Inc.

Global Derivatives USA 2012Trading and Risk Management

November 14-15, 2012

Introduction Margins Copula Results

Outline

1 Introduction

2 Marginal Distribution Surface

3 Joint Distribution through Copula

4 Numerical Results


Smiles

We see smiles everywhere: EQ, FX, IR, Commodity...

0

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4

5 −0.5

0

0.5

10.2

0.25

0.3

0.35

0.4

0.45

0.5

log moneyness

Implied volatility vs maturity and log moneyness

maturity

imp

lied

vo

latilit

y

Figure: Smiles in oil futures

How to incorporate the smile info into pricing and risk?


Overview

Much research has been done: EQ, FX, IR ...Local volatility, stochastic volatility and jumps for spot priceCalibration is not always accurate and stable in practice

A market modeling framework:Direct modeling of the forward prices (market observable)Flexibility to model term-structure and correlationDifficulty due to multi-dimension and lack of market data

Objective and requirements:Match the market-implied marginal distributions (smiles)Consistent with the lognormal modelExtend to multi-asset multi-factor frameworkPractical and easy implementation


Log-normal Model

Log-normal Model: Gabillon (1991), Schwartz (1997), Schwartzand Smith (2000), etc.Start from spot price: mean reverting, convenience yield,Samuelson effect, etc.Forward price representation:

dF(t,Ti) = σ(t,Ti)F(t,Ti)dWi(t), t ≤ Ti, (1)

for i = 1, · · · , nModel components:

instantaneous volatility σ(t, T) is a deterministic function of time tcorrelation ρij(t) can be modeled through a multi-factor frameworkinitial forward curve is an exogenous market input

Samuelson effect: captured by the volatility model σ(t,T)


Example: Gabillon model

Spot price dynamics:

dS(t)S(t)

= κ lnL(t)S(t)

dt + σSdWS(t)

dL(t)L(t)

= µLdt + σLdWL(t)

Model components:S(t) is the short-term (spot) priceL(t) is the long-term priceSamuelson effect: σS > σL

Correlation: < dWS, dWL >= ρSL


Forward Price

Gabillon model:Volatility

σ2(t, T) = σ2Se−2κ(T−t) + 2ρSLσSσL

(e−κ(T−t) − e−2κ(T−t)

)+ σ2

L

(1 − e−κ(T−t)

)2

Correlation

dW(t) = σSe−κ(T−t)

σ(t, T)dWS(t) + σL

1 − e−κ(T−t)

σ(t, T)dWL(t)

Schwartz-Smith model (another example):Volatility

σ2(t, T) = σ2Xe−2κ(T−t) + 2ρXYσXσY e−κ(T−t) + σ2

Y

Correlation

dW(t) = σXe−κ(T−t)

σ(t, T)dWX(t) + σY

1σ(t, T)

dWY(t)


Calibrated (ATM) Volatility Term Structure

The model-implied volatility:

σmod(τ,T) =

√1τ

∫ τ

0σ2(t,T)dt, τ ≤ T (2)

Gabillon model: κ = 0.37, ρSL = −0.29, σS = 0.41, σL = 0.29

0 1 2 3 4 5 60.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42Gabillon implied volatility vs maturity

maturity

implie

d v

ola

tility

Figure: Volatility term structure


Humped-Shape Term Structure

Volatility model:

σ(t,T) = [a(T − t) + c]e−b(T−t) + d (3)

Much better fit: a = 1.10, b = 3.38, b = 0.13, d = 0.21

0 1 2 3 4 5 60.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42Calibrated implied volatility vs maturity

maturity

implie

d v

ola

tility

Figure: Volatility term structure


Correlation

Forward model: flexibility to model the correlationA correlation model given by Ronn (2009):

ρij = e−b|Ti−Tj| + (1− e−b|Ti−Tj|)e−a/min(Ti,Tj), (a, b > 0) (4)

Nice properties:|ρij| ≤ 1 and ρii = 1ρij is decreasing with |Ti − Tj|ρij increases with min(Ti, Tj) and limTi,Tj→∞ ρij = 1

We assume a well calibrated log-normal modelAim to adjust the model to incorporate the smile info


Outline

1 Introduction



4 Numerical Results


Marginal Distribution

Consider a single forward contract with maturity T

Assume a European call option expires on τ ≤ T

The option price C (without discount) at time 0 is

C(τ,K) = EQ0 [(F(τ,T)− K)+] =

∫ +∞

K(y− K)f (τ, y)dy,

where f (τ, y) is the risk-neutral density of F(τ,T)

The marginal distribution of forward price F at τ :

ψ(τ,K) := P(F(τ,T) < K) = 1 +∂

∂KC(τ,K) (5)

Surface of call price⇒ surface of marginal distribution


Surface of Options Prices

How to construct the surfaces C(t,K; τ,T) for t < τ ≤ T?The main problem is lack of market data:

Option quotes are only available for discrete strikesUsually only one volatility smile is available per forward contract

Traditional method:Fit each slice to some model, SVI/SABR/Heston, with constraintsThen, interpolate the resulted curves with no-arbitrage conditionsPotential issues: accuracy and stability

New method from local volatility model:Andreasen-Huge (2011): local volatility surface for FX spot marketApply the Dupire forward equation to perform the interpolationExtend to commodity futures volatility in multi-dimension setting


Local Volatility Model

Imagine a local volatility model:

dF(t,T) = σloc(t,F; T)F(t,T)dW(t), (t ≤ T) (6)

The Dupire equation:

∂

∂τC(τ,K; T) =

12σ2

loc(τ,K; T)K2 ∂2

∂K2 C(τ,K; T) (7)

In terms of forward log-moneyness x := ln(K/FT):

∂

∂τC(τ, x; T) =

12σ2

loc(τ, x; T)

[∂2

∂x2 −∂

∂x

]C(τ, x; T) (8)

Consider a set of forward maturities: 0 = T0 < T1 < · · · < Tn

Corresponding option expiries: 0 = τ0 < τ1 < · · · < τn (τi ≤ Ti)


Option Prices on Discrete Expiries

Now fix a forward maturity Ti:Given local volatility ϑij(x) := σloc(τj, x; Ti) constant in t

Construct the call prices for all option expiries by solving:[1− 1

2∆τjϑ

2ij(x)

(∂2

∂x2 −∂

∂x

)]C(τj, x; Ti) = C(τj−1, x; Ti) (9)

with C(0, x; Ti) = F(0,Ti)(1− ex)+, (1 ≤ j ≤ i ≤ n)Fully implicit FD scheme for the Dupire PDETri-diagonal solver: A(τj)C(τj; Ti) = C(τj−1; Ti)

A(t) is tri-diagonal, diagonally dominantwith positive diagonal and negative off-diagonalsA−1 ≥ 0, Nabben (1999), Andreasen and Huge (2011)

One-step PDE solver for each option expiry⇒ Fast


Calibrate ϑij(x) to Market Quotes

Objective: bootstrap the local volatility from the market quotesFor a given set of call option prices C(τj, x; Ti) (j ≤ i)

Apply optimization routine to minimize the pricing errors of (9)Quotes are available for C(τi, x; Ti) (one smile per maturity)Use the ATM implied volatility to scale the previous smiles:

λij :=σimp(τj, 0; Ti)

σimp(τj, 0; Tj)=

√∫ τj

0 σ2(t,Ti)dt∫ τj

0 σ2(t,Tj)dt(10)

Define a volatility smile for option expiries τj by scaling:

σimp(τj, x; Ti) = λijσimp(τj, x; Tj) (11)


Calibration: Smile Scaling

Scaled market implied volatilities for forward maturity T = 4.93

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1

2

3

4

5 −0.5

0

0.5

10.2

0.25

0.3

0.35

0.4

0.45

0.5

log moneyness


maturity

imp

lied

vo

latilit

y

(a) Market Volatility

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log moneyness


maturity

imp

lied

vo

latilit

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(b) Scaled Volatility

Remove most of the Samuelson effect


Calibration cont.

Discretize the local volatility function ϑij(x):For a fixed pair of (i, j), let ϑij(xijk) = θijk,Define the function ϑij(x) through interpolation:

ϑij(x) = h({

xijk, θijk}

k),

where h can be a nearest-neighbor or linear interpolation

Solve the optimization problem:

minΘij

∑k

[C(τj, xijk; Ti, θijk)− C(τj, xijk; Ti)

]2(12)

Calibrated local volatility:

ϑ∗ij(x) = h({

xijk, θ∗ijk

}k)


Calibration results

Calibrated ϑij(x) and pricing errors (maturity T = 4.93)

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logmon

Calibrated theta vs expiry and logmon

expiry

the

ta

(c) Calibrated theta

01

23

45 −0.5

0

0.5

1−1

−0.5

0

0.5

1

x 10−3

logmon

Calibration: pricing error vs expiry and logmon

expiry

pricin

g e

rro

r

(d) Pricing errors

Underlying forward price: $89.15The RMSE of call price: $3.3e-4 (5-year options)


Fill the Gaps between Options Expiries

Use the calibrated local volatility to interpolate the intermediateoption prices at t ∈ (τj−1, τj)

Again, by solving the PDE[1− 1

2(t − τj−1)ϑ2

ij(x)

(∂2

∂x2 −∂

∂x

)]C(t, x; Ti) = C(τj−1, x; Ti),

for 1 ≤ j ≤ i ≤ n

This is the same PDE as in the calibration step with (τj − τj−1)replaced by (t − τj−1)

No optimization involved: simple one-step PDE solver


Marginal Distribution

Interpolated call price surface and marginal distribution surfaceχ(t, x) = ψ(t,F(0,T)ex), (maturity T = 4.93)

(e) Interpolated call option prices (f) Marginal distribution surface

One surface for each forward contract


Local Volatility Surface

Local volatility surface and pricing errors (maturity T = 4.93)

(g) Local volatility surface

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2

3

4

5 −0.5

0

0.5

1−1

−0.5

0

0.5

1

1.5

2

x 10−3

logmon

Pricing error: impvol vs expiry and logmon

expiry

imp

vo

l

(h) Pricing errors

The RMSE of ImpVol = 1.6 bps


Pricing Errors

Re-price the options using the imaginary local volatility surfacesCalculate the implied volatilities and pricing errors (vs market)The resulted RMSE:

Expiry (y) 0.08 0.16 0.25 0.33 0.42 0.50RMSE (bp) 2.0 2.9 2.6 2.2 1.8 1.8Expiry (y) 0.58 0.67 0.75 0.91 1.17 1.42RMSE (bp) 1.7 1.4 1.8 1.5 1.6 2.3Expiry (y) 1.92 2.41 2.92 3.92 4.93RMSE (bp) 2.2 2.0 2.4 2.6 1.6

Table: The RMSE of implied volatility


Outline

1 Introduction



4 Numerical Results


Copula Function

Copula: transforms random variables by their margins to obtainuniform variables that contain the same informationA copula function can be defined

c(x1, · · · , xn) := g(g−11 (x1), · · · , g−1

n (xn))

for a distribution g(x1, · · · , xn) and its margins g1(x1), · · · , gn(xn)

Moreover, a joint distribution can be defined by

g(x1, · · · , xn) := c(g1(x1), · · · , gn(xn)),

for a given copula function c(· · · ) and a set of margins g1, · · · , gn

How to obtain a reasonable copula function?


Log-normal Distribution

Fix a time spot t and denote by F(t,T1), · · · ,F(t,Tn) the forwardprices obtained from a log-normal model:Gabillon (1991), Schwartz-Smith (2000), Humped-shape, etc.Define random variable:

Zi(t) :=ln(F(t,Ti)/F(0,Ti))− µ(t,Ti)

ν(t,Ti)

where

µ(t,T) = −12

∫ t

0σ2(s,T)ds and ν(t,T) =

(∫ t

0σ2(s,T)ds

)1/2

The random variables are joint Normal:

(Z1(t), · · · ,Zn(t)) ∼ N(0,Σ)


Gaussian Copula

The normal variable Zi(t) can be transformed to a uniform:

Ui(t) = Φ(Zi(t)), (i = 1, · · · , n)

where Φ(·) is a normal CDFThe joint distribution of the uniforms define a copula function:

c(u1, · · · , un) := FU1(t),··· ,Un(t)(u1, · · · , un)

Use to define a joint distribution with the “skewed” marginsThis can be done by the following transform

Zi(t) = Φ−1i (t,Ui(t)) = Φ−1

i (t,Φ(Zi(t)))

The “skewed” margins are given by

Φi(t, z) = χi(t, µ(t,Ti) + zν(t,Ti))


Simulation Algorithm

Input: the marginal distribution surface χi(t, x)

Output: the “skewed” forward price F(t,Ti)

Generate forward price F(t,Ti) from a log-normal modelTransform the log-normal price F(t,Ti) to a Normal variable Zi(t)

Compute the probability Ui(t) = Φ(Zi(t)) using Normal CDF Φ(z)

Compute the “skewed” random variable Zi(t) = Φ−1i (Ui(t)) using

the “skewed” margin Φi(t, z) = χi(t, µ(t,Ti) + zν(t,Ti))

Recover the “skewed” forward price F(t,Ti) using formula

F(t,Ti) = F(0,Ti) exp{µ(t,Ti) + ν(t,Ti)Zi(t)

}


Outline

1 Introduction



4 Numerical Results


Simulation Results

Re-price the options using MC simulationCalculate implied volatility (total variance) vs market quotes

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1

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3

4

5 −0.5

0

0.5

10.2

0.25

0.3

0.35

0.4

0.45

0.5

log moneyness


maturity

imp

lied

vo

latil

ity

(i) Implied volatility

−0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35Total variance vs log moneyness

log moneyness

To

tal v

aria

nce

(j) Total variance


Simulation Results, cont.

Individual smiles: market vs model (from MC simulation)

−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.40.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48Implied volatility vs log moneyness (maturity = 0.331507, the 4th slice)

log moneyness

imp

lied

vo

latilit

y

−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.34

0.36

0.38

0.4

0.42

0.44


log moneyness

imp

lied

vo

latilit

y

−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.32

0.34

0.36

0.38

0.4

0.42


log moneyness

imp

lied

vo

latilit

y

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.60.32

0.33

0.34

0.35

0.36

0.37

0.38

0.39

0.4


log moneyness

imp

lied

vo

latilit

y

−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.60.32

0.34

0.36

0.38

0.4

0.42


log moneyness

imp

lied

vo

latilit

y

−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.60.3

0.32

0.34

0.36

0.38

0.4


log moneyness

imp

lied

vo

latilit

y


Reference

Qimou Su and Curt Randall, Putting Smiles Back to The Futures,Wilmott, September 2012Qimou Su and Curt Randall, Construction of Volatility Surface forCommodity Futures, Working paper, 2012Qimou Su and Curt Randall, A Market Model for CommodityFutures with Smiles, Working paper, 2012

volatility smiles in commodity futures - scicomp smiles in commodity futures qimou su scicomp inc....

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