An Analytical Approximation for Pricing VWAP Options . . . . . - - PowerPoint PPT Presentation
. . 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
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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
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1 Funahashi, H. and Kijima, M. (2015c), “An analytical approximation
for pricing VWAP options,” Working Paper. .
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2 Funahashi, H. and Kijima, M. (2015b), “A unified approach for the
pricing of options related to averages,” Working Paper. .
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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. .
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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.
Kijima (TMU) Pricing of VWAP Options AMMF2015 @ 04/09/15 1 / 24
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 = ∫ T
0 vtStdt
∫ T
0 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−rT E[(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.
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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.
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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.
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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 ,
∫ T vtStdt − x ∫ T vtdt, when the underling asset price and trading volume processes follow a local volatility model and a mean-reverting model, respectively.
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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)dW v
t
under the risk-neutral measure Q, where {Wt} and {W v
t } are the
standard Brownian motions with correlation dWtdW v
t = 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.
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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 F
M,x(y) be the CDF of the random variable
∫ T vtStdt − x ∫ T vtdt It follows from the definition of VWAP that FM(x) = F
M,x(0),
x > 0 Therefore, it suffices to know the CDF F
M,x(y) at y = 0.
To this end, we apply the chaos expansion approach to approximate the distribution of the random variable MT (x).
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Employing the same idea as in Theorem 3.1 of Funahashi and Kijima (2015a), St is approximated by the following formula: .
Lemma
. . . . . . . . Let F (0, t) = Se
∫ t
0 r(u)du be the forward price of the underlying asset
with delivery date t. Then,
St ≈ F (0, t) [ 1 + ∫ t p1(s)dWs + ∫ t p2(s) (∫ s σ0(u)dWu ) dWs + ∫ t p3(s) (∫ s σ0(u) (∫ u σ0(r)dWr ) dWu ) dWs + ∫ t p4(s) (∫ s p5(u) (∫ u σ0(r)dWr ) dWu ) dWs ]
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where p1(s) := σ0(s) + F (0, s)σ′
0(s)
(∫ s σ2
0(u)du
) + 1 2F 2(0, s)σ′′
0 (s)
(∫ s σ2
0(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)
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Employing the successive substitution used in Funahashi (2014), we obtain the following result. Let E(0, t) = e
∫ t
0 κ(u)du, ¯
E(t) = 1/E(t), and
Vt = ¯ E(t) ( v0 + ∫ t E(s)θ(s)ds )
.
Lemma
. . . . . . . . The trading volume vt is approximated as vt ≈
V (0, t) + ¯ E(t) ∫ t
0 p6(s)dW v s + ¯
E(t) ∫ t
0 γ′ 0(s)
(∫ s
0 E(u)γ0(u)dW v u
) dW v
s
+ ¯ E(t) ∫ t
0 ¯
E(s)γ′′
0 (s)
(∫ s
0 E(u)γ0(u)
(∫ u
0 E(r)γ0(r)dW v r
) dW v
u
) dW v
s
+ ¯ E(t) ∫ t
0 γ′ 0(s)
(∫ s
0 γ′ 0(u)
(∫ u
0 E(r)γ0(r)dW v r
) 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 2 ¯ E(t)γ′′
0 (t)
(∫ t E2(s)γ2
0(s)ds
)
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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) ∫ t
0 p6(s)p1(s)ds,
I1(t) = V (0, t)F (0, t) ∫ t p1(s)dWs + F (0, t) ¯ E(t) ∫ t p10(t, s)dWs + F (0, t) ¯ E(t) ∫ t (p6(s) + p9(t, s)) dW v
s ,
and others (omitted).
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By changing the order of integration, MT can be approximated by a truncated sum of iterated Itˆ
.
Lemma
. . . . . . . . For each x > 0, the random variable MT can be approximated as
= ∫ T vtStdt − x ∫ T vtdt ≈ J0(x, T ) + J1(x, T ) + J2(x, T ) + J3(x, T ), where J0(x, T ) = ∫ T Vt (F (0, t) − x) dt + ρ ∫ T F (0, t) ¯ E(t) (∫ t p6(s)p1(s)ds ) dt,
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J1(x, T ) = ∫ T p1(t) (∫ T
t
V (0, s)F (0, s)ds ) dWt + ∫ T (∫ T
t
p10(s, t)F (0, s) ¯ E(s)ds ) dW v
t
+ ∫ T (∫ T
t
( p6(t) + p9(s, t)F (0, s) ¯ E(s) ) ds ) dW v
t
− x ∫ T p6(t) (∫ T
t
¯ E(s)ds ) dW v
t ,
J2(x, T ) = ∫ T r1(t) (∫ t σ0(s)dWs ) dWt + ∫ T r2(t) (∫ t p6(s)dW v
s
) dWt + ∫ T r3(t) (∫ t p1(s)dWs ) dW v
t +
∫ T r4(x, t) (∫ t E(s)γ0(s)dW v
s
) dW v
t ,
and others (omitted).
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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
. . . . . . . . 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.
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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 F
M,x(y) = FYT ,x(y − J0(x, T )), we have
FM(x) = F
M,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
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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.
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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.
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5 10 15 20 25 80 90 100 110 120
0.02 0.04 0.06 0.08 0.1
Option Price Diff Strike
MC WIC (2nd) Diff
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5 10 15 20 25 80 90 100 110 120
0.02 0.04 0.06 0.08 0.1
Option Price Diff Strike
MC WIC (2nd) Diff
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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.
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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
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.
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Squared OU Model for Trading Volume as in Novikov et al. (2014). vt = X2
t + γ,
dXt = (θ − κXt)dt + βdW v
t ,
where θ, κ, γ and β are some constants. In this case, we have
X2
t
= V 2
t + 2Vt ¯
E(t) ∫ t βE(s)dW v
s + ¯
E2(t) (∫ t β2E2(s)ds ) + 2 ¯ E2(t) (∫ t βE(s) (∫ s βE(u)dW v
u
) dW v
s
)
Generalized VWAP MT =
∫ T
0 w1 t vtStdt
∫ T
0 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−rT E[(MT − ST )+]
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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
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.
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