An Analytical Approximation for Pricing VWAP Options...Motivation VWAP Options are becoming...

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An Analytical Approximation for Pricing

VWAP Options

Hideharu Funahashi and Masaaki Kijima

Graduate School of Social Sciences, Tokyo Metropolitan University

September 4, 2015

Kijima (TMU) Pricing of VWAP Options AMMF2015 @ 04/09/15 1 / 24

This Talk is Based on

...1 Funahashi, H. and Kijima, M. (2015c), “An analytical approximation

for pricing VWAP options,” Working Paper.

...2 Funahashi, H. and Kijima, M. (2015b), “A unified approach for the

pricing of options related to averages,” Working Paper.

...3 Funahashi, H. and Kijima, M. (2014), “An extension of the chaos

expansion approximation for the pricing of exotic basket options,”

Applied Mathematical Finance, 21 (2), 109–139.

...4 Funahashi, H. and Kijima, M. (2015a), “A chaos expansion approach

for the pricing of contingent claims,” Journal of Computational

Finance, 18 (3), 27–58.


Formulation

Let St and vt be the time-t price and trading volume, respectively, of

the underlying asset.

VWAP (volume weighted average price) is determined by

MT =

∫ T0 vtStdt∫ T0 vtdt

,

where MT is called the VWAP of the time interval [0, T ].

The standard definition of a continuous VWAP call option is given by

VC(S, v,K, T ) = e−rTE[(MT − K)+]

where S = S0 is the initial price, v = v0 is the initial trading

volume, K is a strike, T is a maturity, and r is the short rate, E is

the risk-neutral expectation operator.

Existing papers try to approximate the distribution of MT directly.


Motivation

VWAP Options are becoming increasingly popular in options markets,

since they can help corporate firms hedge risks arising from market

disruption when entering large buy or sell orders.

Their prices assign more weight to periods of high trading than to

periods of low trading in its calculation.

Hence, VWAP options differ conceptually from Asian options because

the resulting payoff is not a linear combination of underlying prices.

As a result, the pricing of VWAP options is significantly more difficult

than Asians and few pricing models have been proposed in the

literature, despite their popularity in practice.

See, e.g., Buryak and Guo (2014), Novikov et al. (2013) for details.


Literature Review

It is common to model the underlying St by using the geometric

Brownian motion (GBM) for simplicity.

For the trading volume vt, Stace (2007) proposes a mean-reverting

process; Novikov et al. (2013) use a squared Ornstein–Uhlenbeck

(OU) process; and Buryak and Guo (2014) suggest a simple gamma

process, respectively, for the trading volume process.

Under the GBM assumption, these papers produce approximated

pricing formulas by utilizing the moment-matching technique for MT .

On the other hand, Novikov and Kordzakhia (2013) derive very tight

upper and lower bounds for the price of VWAP options.


In This Talk

Other than the GBM case, these approaches seem difficult to apply

for deriving an approximation formula of VWAP options.

Funahashi and Kijima (2015b) apply the chaos expansion technique

to derive a unified approximation method for pricing any type of

Asian options when the underling process follows a diffusion.

In this talk, not of the VWAP itself, but we try to approximate the

distribution of MT ,

MT (x) =

∫ T

0vtStdt − x

∫ T

0vtdt,

when the underling asset price and trading volume processes follow a

local volatility model and a mean-reverting model, respectively.


The Setup

We assume that the price St and the trading volume vt of the

underlying asset are modeled by the following SDE:dSt

St= r(t)dt + σ(St, t)dWt

dvt = (θ(t) − κ(t)vt)dt + γ(vt)dWvt

under the risk-neutral measure Q, where {Wt} and {W vt } are the

standard Brownian motions with correlation dWtdWvt = ρdt.

The volatility functions σ(S, t) and γ(v) are sufficiently smooth with

respect to (S, t) and v, respectively.

r(t), θ(t) and κ(t) are some deterministic functions of time t.


Key Observation

Denote the cumulative distribution function (CDF) of MT by

FM(x) = Q{MT ≤ x}, x > 0.

The VWAP call price can be written as

VC(S, v,K, T ) = e−rT

∫ ∞

K(1 − FM(x))dx

For each x > 0, let FM,x

(y) be the CDF of the random variable

MT (x) =

∫ T

0vtStdt − x

∫ T

0vtdt

It follows from the definition of VWAP that

FM(x) = FM,x

(0), x > 0

Therefore, it suffices to know the CDF FM,x

(y) at y = 0.

To this end, we apply the chaos expansion approach to approximate

the distribution of the random variable MT (x).


Approximation 1

Employing the same idea as in Theorem 3.1 of Funahashi and Kijima

(2015a), St is approximated by the following formula:.Lemma..

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. ..

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Let F (0, t) = Se∫ t0 r(u)du be the forward price of the underlying asset

with delivery date t. Then,

St ≈ F (0, t)

[1 +

∫ t

0

p1(s)dWs +

∫ t

0

p2(s)

(∫ s

0

σ0(u)dWu

)dWs

+

∫ t

0

p3(s)

(∫ s

0

σ0(u)

(∫ u

0

σ0(r)dWr

)dWu

)dWs

+

∫ t

0

p4(s)

(∫ s

0

p5(u)

(∫ u

0

σ0(r)dWr

)dWu

)dWs

]


Approximation 1, Continued

where

p1(s) := σ0(s) + F (0, s)σ′0(s)

(∫ s

0σ20(u)du

)+

1

2F 2(0, s)σ′′

0 (s)

(∫ s

0σ20(u)du

)p2(s) := σ0(s) + F (0, s)σ′

0(s)

p3(s) := σ0(s) + 3F (0, s)σ′0(s) + F 2(0, s)σ′′

0 (s)

p4(s) := σ0(s) + F (0, s)σ′0(s)

p5(s) := F (0, s)σ′0(s)

with σ′0(t) := ∂xσ(x, t)|x=F (0,t) and σ′′

0 (t) := ∂xxσ(x, t)|x=F (0,t)


Approximation 2

Employing the successive substitution used in Funahashi (2014), we obtain

the following result. Let E(0, t) = e∫ t0 κ(u)du, E(t) = 1/E(t), and

Vt = E(t)

(v0 +

∫ t

0

E(s)θ(s)ds

).Lemma..

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. ..

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The trading volume vt is approximated as vt ≈

V (0, t) + E(t)∫ t

0p6(s)dW

vs + E(t)

∫ t

0γ′0(s)

(∫ s

0E(u)γ0(u)dW

vu

)dW v

s

+ E(t)∫ t

0E(s)γ′′

0 (s)(∫ s

0E(u)γ0(u)

(∫ u

0E(r)γ0(r)dW

vr

)dW v

u

)dW v

s

+ E(t)∫ t

0γ′0(s)

(∫ s

0γ′0(u)

(∫ u

0E(r)γ0(r)dW

vr

)dW v

u

)dW v

s ,

where γ0(t) := γ(Vt), γ′0(t) := ∂xγ(x)|x=Vt , γ

′′0 (t) := ∂xxγ(x)|x=Vt , and

p6(t) := E(t)γ0(t) +1

2E(t)γ′′

0 (t)

(∫ t

0

E2(s)γ20(s)ds

)


Approximation 3

Using the approximation results for St and vt, we obtain the following..Lemma..

.

. ..

.

.

The traded value vtSt is approximated as

vtSt ≈ I0(t) + I1(t) + I2(t) + I3(t),

where I0(t) = VtF (0, t) + ρF (0, t)E(t)∫ t0 p6(s)p1(s)ds,

I1(t) = V (0, t)F (0, t)

∫ t

0

p1(s)dWs + F (0, t)E(t)

∫ t

0

p10(t, s)dWs

+F (0, t)E(t)

∫ t

0

(p6(s) + p9(t, s)) dWvs ,

and others (omitted).


Approximation 4

By changing the order of integration, MT can be approximated by a

truncated sum of iterated Ito stochastic integrals as follows..Lemma..

.

. ..

.

.

For each x > 0, the random variable MT can be approximated as

MT =

∫ T

0vtStdt − x

∫ T

0vtdt

≈ J0(x, T ) + J1(x, T ) + J2(x, T ) + J3(x, T ),

where

J0(x, T ) =

∫ T

0Vt (F (0, t) − x) dt

+ ρ

∫ T

0F (0, t)E(t)

(∫ t

0p6(s)p1(s)ds

)dt,


Approximation 4, Continued

J1(x, T ) =

∫ T

0

p1(t)

(∫ T

t

V (0, s)F (0, s)ds

)dWt

+

∫ T

0

(∫ T

t

p10(s, t)F (0, s)E(s)ds

)dW v

t

+

∫ T

0

(∫ T

t

(p6(t) + p9(s, t)F (0, s)E(s)

)ds

)dW v

t

−x

∫ T

0

p6(t)

(∫ T

t

E(s)ds

)dW v

t ,

J2(x, T ) =

∫ T

0

r1(t)

(∫ t

0

σ0(s)dWs

)dWt +

∫ T

0

r2(t)

(∫ t

0

p6(s)dWvs

)dWt

+

∫ T

0

r3(t)

(∫ t

0

p1(s)dWs

)dW v

t +

∫ T

0

r4(x, t)

(∫ t

0

E(s)γ0(s)dWvs

)dW v

t ,

and others (omitted).


Option Pricing Formula

Let us define Yt = Mt − J0(x, t), and denote its probability density

function (PDF) by fYT ,x(y).

We can obtain the PDF by applying the following lemma.

.Lemma..

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. ..

.

.

The PDF of YT is approximated as

fYT ,x(y) ≈ n (y; 0, Vx(T )) − ∂

∂y{E[J2(x, T )|J1(x, t) = y]n (y; 0, Vx(T ))}

− ∂

∂y{E[J3(x, t)|J1(x, t) = y]n (y; 0, Vx(T ))}

+1

2

∂2

∂y2

{E[J2(x, t)

2|J1(x, t) = y]n (y; 0, Vx(T ))},

where n(y; a, b) denotes the normal density with mean a and variance b.


Option Pricing Formula, Continued

The conditional expectations can be evaluated explicitly.

Using the approximated density function of fYT ,x(y), we can

approximate the CDF FYT ,x(y) of YT .

But, from the relation FM,x

(y) = FYT ,x(y − J0(x, T )), we have

FM(x) = FM,x

(0) = FYT ,x(−J0(x, T ))

It follows that the VWAP call option price can be approximated as

VC(S, v,K, T ) ≈ e−rT

∫ ∞

K(1 − FYT ,x(−J0(x, T )))dx


Numerical Examples: CEV Case

We suppose that the volatilities are specified as

σ(S, t) = σSβ−1, γ(v) = ν vλ−1,

where σ, β, ν, and λ are some constants.

The base-case parameters are set to be S = 100, K = 100,

T = 1, r(t) = 3.0%, v = 100, and ρ = 0.3.

Also, we set θ(t) = 10 and κ(t) = 0.1, i.e., the long-run average of

the trading volume is 100.

As to the volatilities, we consider (H) high and (L) low volatility cases

in which we set σSβ−1 = 30% and ν vλ−1 = 30% for case (H)

and σSβ−1 = 15% and ν vλ−1 = 15% for case (L), respectively.


Numerical Examples; Accuracy Check

We consider two cases; (1) log-normal case (β = 1 and λ = 1), and

(2) square-root case (β = 0.5 and λ = 0.5).

Figure 1 shows option prices for (L) with short maturity (T = 0.5)

when (1), whereas Figure 2 depicts for (2).

Through the numerical experiments, it is observed that the effect of

volatility and maturity appears only around ATM (K = 100), and

the volatility effect is stronger than the maturity effect.

As to the accuracy of our approximation, we find that the difference

between our approximation and the Monte Carlo result are very small.

The error becomes slightly larger for long maturity and high volatility

cases; however, for practical uses, the errors are sufficiently small.


Figure 1 (GBM, low vol, T = 0.5)

0

5

10

15

20

25

80 90 100 110 120-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1O

ptio

n P

rice

Dif

f

Strike

MCWIC (2nd)

Diff


Figure 2 (Square-Root, low vol, T = 0.5)

0

5

10

15

20

25

80 90 100 110 120-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1O

ptio

n P

rice

Dif

f

Strike

MCWIC (2nd)

Diff


Other Findings; Effect of Correlation

The effect of correlation gets bigger as κ, the speed of mean

reversion, becomes smaller.

When κ is large, the trading volume vt sticks around the long-run

average so as to behave as if it were uncorrelated to the stock price.

Stace (2007) sets κ = 100 under the assumption ρ = 0.

Our result suggests that, when κ = 100, the impact of correlation on

the VWAP call option prices is negligible.

This result may be an important message for practitioners, because it

is in general very difficult to estimate the correlation accurately.

The effect of correlation gets bigger as the maturity T becomes

longer and the volatility σ of the asset price becomes larger.

These results can be understood by the fact that the effect of

correlation is bigger as more uncertainty is involved.


Other Findings; Effect of Model Choice

Recall that

σ(S, t) = σSβ−1, γ(v) = ν vλ−1

The effect of β gets bigger as κ becomes smaller, the maturity

becomes longer and the asset volatility becomes larger.

These results can be explained by the exactly same reason as above.

The effect of λ gets bigger as κ becomes smaller, and has less impact

on the others.

Compared with the impact of the underlying asset price, the maturity

as well as the volatility of trading volume has less impact on the

VWAP option prices.


Some Extensions

Squared OU Model for Trading Volume as in Novikov et al. (2014).

vt = X2t + γ, dXt = (θ − κXt)dt + βdW v

t ,

where θ, κ, γ and β are some constants. In this case, we have

X2t = V 2

t + 2VtE(t)

∫ t

0

βE(s)dW vs + E2(t)

(∫ t

0

β2E2(s)ds

)+2E2(t)

(∫ t

0

βE(s)

(∫ s

0

βE(u)dW vu

)dW v

s

)

Generalized VWAP MT =∫ T0 w1

t vtStdt∫ T0 w2

t vtdt, where wi

t is a deterministic

function of time t. Consider a floating-strike VWAP option defined by

VC(S, v,K, T ) = e−rTE[(MT − ST )+]


Conclusion

In this talk, we develop a unified approximation method for options

whose payoff depends on a volume weighted average price (VWAP).

Compared to the previous works, our method is applicable to the local

volatility model, not just for the geometric Brownian motion case.

Moreover, our method can be used for any special type of VWAP

option, including ordinary Asian and Australian options, with

fixed-strike, floating-strike, continuously sampled, discretely sampled,

forward starting, and in-progress transactions.

Through numerical examples, we show that the accuracy of the

second-order approximation is high enough for practical use.

Our approximation get slightly worse for long maturity and high

volatility case; in such a case, 3rd-order may be required.


Thank You for Your Attention


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