. . 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. Kijima (TMU) Pricing of VWAP Options AMMF2015 @ 04/09/15 1 / 24
Formulation Let S t and v t be the time- t price and trading volume, respectively, of the underlying asset. VWAP (volume weighted average price) is determined by ∫ T 0 v t S t d t M T = , ∫ T 0 v t d t where M T 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 [( M T − K ) + ] where S = S 0 is the initial price, v = v 0 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 M T directly. Kijima (TMU) Pricing of VWAP Options AMMF2015 @ 04/09/15 2 / 24
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. Kijima (TMU) Pricing of VWAP Options AMMF2015 @ 04/09/15 3 / 24
Literature Review It is common to model the underlying S t by using the geometric Brownian motion (GBM) for simplicity. For the trading volume v t , 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 M T . On the other hand, Novikov and Kordzakhia (2013) derive very tight upper and lower bounds for the price of VWAP options. Kijima (TMU) Pricing of VWAP Options AMMF2015 @ 04/09/15 4 / 24
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 � M T , ∫ T ∫ T � M T ( x ) = v t S t d t − x v t d t, 0 0 when the underling asset price and trading volume processes follow a local volatility model and a mean-reverting model, respectively. Kijima (TMU) Pricing of VWAP Options AMMF2015 @ 04/09/15 5 / 24
The Setup We assume that the price S t and the trading volume v t of the underlying asset are modeled by the following SDE: d S t = r ( t )d t + σ ( S t , t )d W t S t ( θ ( t ) − κ ( t ) v t )d t + γ ( v t )d W v d v t = t under the risk-neutral measure Q , where { W t } and { W v t } are the standard Brownian motions with correlation d W t d W v t = ρ d t . 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 . Kijima (TMU) Pricing of VWAP Options AMMF2015 @ 04/09/15 6 / 24
Key Observation Denote the cumulative distribution function (CDF) of M T by F M ( x ) = Q { M T ≤ x } , x > 0 . The VWAP call price can be written as ∫ ∞ VC( S, v, K, T ) = e − rT (1 − F M ( x ))d x K For each x > 0 , let F � M,x ( y ) be the CDF of the random variable ∫ T ∫ T � M T ( x ) = v t S t d t − x v t d t 0 0 It follows from the definition of VWAP that F M ( 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 � M T ( x ) . Kijima (TMU) Pricing of VWAP Options AMMF2015 @ 04/09/15 7 / 24
Approximation 1 Employing the same idea as in Theorem 3.1 of Funahashi and Kijima (2015a), S t is approximated by the following formula: . Lemma . . . ∫ t 0 r ( u )d u be the forward price of the underlying asset Let F (0 , t ) = S e with delivery date t . Then, ∫ t ∫ t (∫ s [ ) S t ≈ F (0 , t ) 1 + p 1 ( s )d W s + p 2 ( s ) σ 0 ( u )d W u d W s 0 0 0 ∫ t (∫ s (∫ u ) ) + p 3 ( s ) σ 0 ( u ) σ 0 ( r )d W r d W u d W s 0 0 0 ∫ t (∫ s (∫ u ) ) ] + p 4 ( s ) p 5 ( u ) σ 0 ( r )d W r d W u d W s 0 0 0 . . . . . Kijima (TMU) Pricing of VWAP Options AMMF2015 @ 04/09/15 8 / 24
Approximation 1, Continued where (∫ s ) σ 2 σ 0 ( s ) + F (0 , s ) σ ′ p 1 ( s ) := 0 ( s ) 0 ( u )d u 0 (∫ s ) + 1 2 F 2 (0 , s ) σ ′′ σ 2 0 ( s ) 0 ( u )d u 0 σ 0 ( s ) + F (0 , s ) σ ′ p 2 ( s ) := 0 ( s ) 0 ( s ) + F 2 (0 , s ) σ ′′ σ 0 ( s ) + 3 F (0 , s ) σ ′ p 3 ( s ) := 0 ( s ) σ 0 ( s ) + F (0 , s ) σ ′ p 4 ( s ) := 0 ( s ) p 5 ( s ) := F (0 , s ) σ ′ 0 ( s ) with σ ′ 0 ( t ) := ∂ x σ ( x, t ) | x = F (0 ,t ) and σ ′′ 0 ( t ) := ∂ xx σ ( x, t ) | x = F (0 ,t ) Kijima (TMU) Pricing of VWAP Options AMMF2015 @ 04/09/15 9 / 24
Approximation 2 Employing the successive substitution used in Funahashi (2014), we obtain ∫ t 0 κ ( u )d u , ¯ the following result. Let E (0 , t ) = e E ( t ) = 1 /E ( t ) , and ∫ t ( ) V t = ¯ E ( t ) v 0 + E ( s ) θ ( s )d s 0 . Lemma . . . The trading volume v t is approximated as v t ≈ ∫ t ∫ t (∫ s V (0 , t ) + ¯ s + ¯ 0 p 6 ( s )d W v 0 E ( u ) γ 0 ( u )d W v d W v 0 γ ′ ) E ( t ) E ( t ) 0 ( s ) u s ∫ t (∫ s (∫ u + ¯ 0 ¯ 0 E ( r ) γ 0 ( r )d W v ) d W v ) d W v E ( t ) E ( s ) γ ′′ 0 ( s ) 0 E ( u ) γ 0 ( u ) r u s ∫ t (∫ s (∫ u + ¯ 0 γ ′ 0 γ ′ 0 E ( r ) γ 0 ( r )d W v ) d W v ) d W v E ( t ) 0 ( s ) 0 ( u ) s , r u where γ 0 ( t ) := γ ( V t ) , γ ′ 0 ( t ) := ∂ x γ ( x ) | x = V t , γ ′′ 0 ( t ) := ∂ xx γ ( x ) | x = V t , and (∫ t p 6 ( t ) := E ( t ) γ 0 ( t ) + 1 ) ¯ E ( t ) γ ′′ E 2 ( s ) γ 2 0 ( t ) 0 ( s )d s 2 0 . . . . . Kijima (TMU) Pricing of VWAP Options AMMF2015 @ 04/09/15 10 / 24
Approximation 3 Using the approximation results for S t and v t , we obtain the following. . Lemma . . . The traded value v t S t is approximated as v t S t ≈ I 0 ( t ) + I 1 ( t ) + I 2 ( t ) + I 3 ( t ) , ∫ t where I 0 ( t ) = V t F (0 , t ) + ρF (0 , t ) ¯ E ( t ) 0 p 6 ( s ) p 1 ( s )d s , ∫ t ∫ t p 1 ( s )d W s + F (0 , t ) ¯ I 1 ( t ) = V (0 , t ) F (0 , t ) E ( t ) p 10 ( t, s )d W s 0 0 ∫ t + F (0 , t ) ¯ ( p 6 ( s ) + p 9 ( t, s )) d W v E ( t ) s , 0 and others (omitted). . . . . . Kijima (TMU) Pricing of VWAP Options AMMF2015 @ 04/09/15 11 / 24
Approximation 4 By changing the order of integration, � M T can be approximated by a truncated sum of iterated Itˆ o stochastic integrals as follows. . Lemma . . . For each x > 0 , the random variable � M T can be approximated as ∫ T ∫ T � M T = v t S t d t − x v t d t 0 0 ≈ J 0 ( x, T ) + J 1 ( x, T ) + J 2 ( x, T ) + J 3 ( x, T ) , where ∫ T J 0 ( x, T ) = V t ( F (0 , t ) − x ) d t 0 (∫ t ) ∫ T F (0 , t ) ¯ + ρ E ( t ) p 6 ( s ) p 1 ( s )d s d t, . 0 0 . . . . Kijima (TMU) Pricing of VWAP Options AMMF2015 @ 04/09/15 12 / 24
Approximation 4, Continued ∫ T (∫ T ) J 1 ( x, T ) = p 1 ( t ) V (0 , s ) F (0 , s )d s d W t t 0 ∫ T (∫ T ) p 10 ( s, t ) F (0 , s ) ¯ d W v + E ( s )d s t 0 t ∫ T (∫ T ) p 6 ( t ) + p 9 ( s, t ) F (0 , s ) ¯ ( ) d W v + E ( s ) d s t 0 t ∫ T (∫ T ) ¯ d W v − x p 6 ( t ) E ( s )d s t , 0 t ∫ T (∫ t ∫ T (∫ t ) ) p 6 ( s )d W v J 2 ( x, T ) = r 1 ( t ) σ 0 ( s )d W s d W t + r 2 ( t ) d W t s 0 0 0 0 ∫ T (∫ t ∫ T (∫ t ) ) d W v E ( s ) γ 0 ( s )d W v d W v + r 3 ( t ) p 1 ( s )d W s t + r 4 ( x, t ) t , s 0 0 0 0 and others (omitted). Kijima (TMU) Pricing of VWAP Options AMMF2015 @ 04/09/15 13 / 24
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