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Motion picture derivatives have proved valuable in hedging financial risk of the movie industry. However, the existence of pseudo assets within certain category of movie contingencies makes market trading below capacity due to hyper level pricing arbitrage. This paper analyzes pseudo assets lying within movie contingencies and develops a stochastic pricing strategy under no arbitrage condition. Demonstration of the application of derived formulas is provided as examples and remarks over imposed effects sequel to sub pseudo contingencies. Accordingly, the results derived here are aids of assistance for movie investors relative to financing pseudo asset ventures for the purpose of valuation.

Motion picture is a collection of film making institutions comprising production, screen writing, festivals, distribution and marketing companies. It is the biggest leisure industry in the universe and generally expensive to run. As a result, institutions within the industry connect with other industries for equipments, securities, marketing and so on. The largest movie markets by box office are the US, China, UK, Japan and India. Countries with largest movie productions are India, Nigeria, US, UK, and France and area with largest movie productions is Hong Kong in China (https://en.wikipedia.org/wiki/Film_industry). Motion picture has enjoyed and still is enjoying good market statistics; for instance, Hollywood reporter of 21st March 2019 indicates that worldwide theatrical market has box office worth of 41 billion USD as at 2018. Top 3 regions by box office gross earnings as at 2016 are Asia-Pacific (14 billion USD), North America (11 billion USD) and Europe together with the Middle East and North Africa (11 billion USD).

Hollywood is the name of the US^{1} motion picture. It is the oldest motion picture in history. In 2011, Motion Picture Association of America (MPAA) identifies Hollywood; “a major employer” in the US supporting over 2.2 million jobs worth 137 billion USD in total wages as at 2009. Direct jobs generate 40.5 billion USD in wages with average salary 26% higher than the national average. Jobs in core business of production, marketing, manufacturing and distribution of films stand at 272,000 [

Although, multilingual audiovisual add-ons have been present in movies of any wood as assets and contingencies only recently, mathematicians and other scholars become interested in their forms [

The Koreans love not just English but American English. Korean surjection is passionate to Americanised English [

In his 1964 book; “From Understanding Media” [

One way trading add-ons is positive to motion picture is realistic financing. Movie markets in several spots round the globe trade add-ons (dubbings, subtitles, voice over, translations etc.) with little or no market value. Particular case is the sameness in price of the movie tyrant directed by Gideon Raff with or without add-ons. This price misfit holds for most movies of any wood. There are consequences for this mis-pricing act especially on investment capital. On records, add-on processes are not free lunches since resources are drained in the process. Price insensitivity and mis-fits reflecting as zero add-on prices in movies can result only in market arbitrage [

Moreover, investophobia sequel to risk tendencies capable of crunching movie markets with maximum impacts on movie driven economies is within premise. Clearly, there is need to trade add-ons as contingencies for additional profit. Generally, asset pricing are settled through trading in financial markets where prices of underlying assets sometimes tend to be volatile especially for unawared assets such as pseudo assets and derivatives portfolio. The eventual consequences like market inefficiency present severe negative implications on motion pictures [

Here, arbitrageurs manipulate the market to gain excess returns while real investors lost more than expected leading to deadweight loss [

Add-oning is Herculean in the least so to speak. Darbeltnet’s model presented in [

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[

u i j t = θ j ( χ n ) − λ ( t − r j ) + ξ j t + ζ i t + ( 1 − σ ) ϵ i j t (1)

where θ j is quality of movie j under add-on prices, t − r j is the elapsed time in weeks since release, ξ j t is a week’s random effect. With this extension, the individual error term ζ i t + ( 1 − σ ) ϵ i j t can be transformed by suitable choice of pricing functions enough to include errors from add-on prices so that Einav’s share for movie j in week t logit is

log S j t S 0 t = θ j ( χ n ) + τ t − λ ( t − r j ) + σ log ( S j t 1 − S 0 t ) + ϵ ^ j t (2)

Studios for production, distribution and marketing add-ons may follow [

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Unfortunately, even with these clear cut fitting demonstrations, a pseudo movie contingency can challenge movie pricing sequel to fears of arbitrage and its financial consequences. This is in view of the hidden un-priced sub assets like add-ons within the majority of movie contingencies. This article fills some gaps by defining pseudo assets and contingencies and proves existence of arbitrage therein pseudo assets of movie spaces and finally, provides pricing strategy for such assets and portfolios.

We consider a compound3 movie asset A in trade given by

A = ∑ i = 0 n A i (3)

where each A i ∈ A is priced X i ( . ) . Suppose that one investor holds portfolio θ ( A ) when the price X ( A ) = X i ( . ) . Then θ ( A ) is a pseudo portfolio for asset A. Evidently, θ ( A ) has arbitrage tendency in view of its trading price X i corresponding to that of A i ∈ A . If θ ( A ) is admissible then, by [

Intuitively, A in (3) is an arbitrage asset and as earlier motivated, there is need to construct an arbitrage free pricing strategy for such movie assets. To prove that any θ ( A ) is an arbitrage, suppose that the price X ( t , ω , A ) for A is such that

d X ( t , ω , A ) = μ i ( A ) X ( t , ω , A ) d t + σ i ( A ) X ( t , ω , A ) d B ( t ) (4)

where μ ( A ) is the drift coefficient of the movie market and σ ( A ) is the volatility coefficient under the white noise B ( . ) . Applying the arbitrage criteria in [

[ σ 11 ( A ) σ 12 ( A ) σ 21 ( A ) σ 22 ( A ) ] [ U 1 U 2 ] = [ μ 1 ( A ) μ 2 ( A ) ] (5)

So that

σ 11 ( A ) U 1 + σ 12 ( A ) U 2 = μ 1 ( A ) (6)

σ 21 ( A ) U 1 + σ 22 ( A ) U 2 = μ 2 ( A ) (7)

In view of (3), (6) and (7) have the representation that

[ σ 11 ( A 1 ) σ 11 ( A 2 ) ⋮ σ 11 ( A n ) ] [ U 1 0 ⋮ 0 ] + [ σ 12 ( A 1 ) σ 12 ( A 2 ) ⋮ σ 12 ( A n ) ] [ U 2 0 ⋮ 0 ] = [ μ 1 ( A 1 ) μ 1 ( A 2 ) ⋮ μ 1 ( A n ) ] (8)

[ σ 21 ( A 1 ) σ 21 ( A 2 ) ⋮ σ 21 ( A n ) ] [ U 1 0 ⋮ 0 ] + [ σ 22 ( A 1 ) σ 22 ( A 2 ) ⋮ σ 22 ( A n ) ] [ U 2 0 ⋮ 0 ] = [ μ 2 ( A 1 ) μ 2 ( A 2 ) ⋮ μ 2 ( A n ) ] (9)

Each μ i ( A ) depends on A i since each iteration drifts independently of the sub assets. Moreover

σ 1 , j ( A 1 + k ) = { σ 1 , j ; k = 0 0 ; k > 0 (10)

Note that the left hand sides of (8) and (9) are column matrices and multiplication is impossible implying that the U’s do not exist and hence θ ( A ) is an arbitrage. For convenience, we choose the Ornstein-Uhlenbeck price path for X ( t , ω , A ) as a special case of (4) such that

d X ( t , ω , A ) = μ ( A ) X ( t , ω , A ) d t + σ ( A ) d B ( t ) (11)

and provide arbitrage free pricing strategy for A as motivated in this work.

Lemma 4.1 The arbitrage free price p for a pseudo-movie asset A with n independent sub-assets A i ’s i = 1 , 2 , 3 , ⋯ , n is given by

p = X ( 0, ω , A ) e μ ( A ) T + ∫ 0 T σ ( A ) e μ ( A ) ( T − t ) E [ d B ( t ) ] (12)

Proof Suppose p ~ E [ X ( t , A ) ] and define f : A → ℜ where A is the space of assets such that for A ∈ A , we have f ( t , A ) = f t ( A ) with the condition that for any A i , A j ∈ A ; i ≠ j , we have

f ( t , A i + A j ) = f t ( A i + A j ) = f t ( A i ) + f t ( A j ) (13)

Again, for i = j = k = m then

f ( t , A i + A j + A k + ⋯ + A m ) = f t ( m A i ) = m f t ( A i ) (14)

Given (11), the integrating factor is e − μ ( A ) t and multiplying the given equation by this factor and upon basic simplification in the interval [ 0, T ] yields

X ( T , ω , A ) = X ( 0 , ω , A ) e μ ( A ) T + σ ( A ) ∫ 0 T e μ ( A ) ( T − t ) d B ( t ) (15)

with expected price p equals to

p = X ( 0, ω , A ) e μ ( A ) T + K ; K = 0,1,2,3, ⋯ (16)

The lemma follows in view of (13) and (14) directly.

Intuitively, the price p in (16) is arbitrage free since each sub price value X ( A i ) is taken into account in the final pricing of A. We see how this pricing works in the following examples.

Example 4.1 A movie retail shop sells a movie contingency A of three independent sub contingencies: the movie, the voice over and translation that costs E 20, E 3 and E 2 respectively at a price E30. The retail shop operates from noon to 7 p.m. Show whether the price p ( A ) has arbitrage^{4}.

Solution 4.1 Here, the business trading period T = t 24 = 7 24 since there are 7 hours between trading hours. Again, X 0 ( A 1 ) = E 20 , X 0 ( A 2 ) = E 3 and X 0 ( A 3 ) = E 2 . In view of (16), we have

p = X 0 ( A 1 ) exp ( T ) + X 0 ( A 2 ) exp ( T ) + X 0 ( A 3 ) exp ( T ) + K , K = 0 , 1 , 2 , 3 , ⋯ (17)

Upon substituting the values of above in (17), we have

p = E 20 exp ( 7 24 ) + E 3 exp ( 7 24 ) + E 2 exp ( 7 24 ) + K (18)

where K = E [ ∫ 0 T e x p ( T − t ) d B ( t ) ] . So that

p = E 26.774 + E 4.016 + E 2.677 + K = E 33.5 ; K = 0 (19)

In view of (12), p ( A ) has pricing arbitrage since it is traded below the arbitrage free price E 33.5. The extent for this pricing arbitrage for θ ( A ) = 1 is about 12%.

Example 4.2 Estimate the stationary arbitrage free price of a movie pseudo asset A with voice over dubbing and translations costing E 40, E 3, and E2 respectively in a retail market operating from 13:00 Hrs to 15:00 Hrs if p ( A ) follows the geometric Brownian motion.

Solution 4.2 Here, the business trading period T = t 24 = 2 24 since there 2 hours between trading hours noon to 13 Hrs to 15 Hrs. Also, X 0 ( A 1 ) = E 40 , X 0 ( A 2 ) = E 3 and X 0 ( A 3 ) = E 2 . By [

p = X ( 0 , A ) e ( μ − 1 2 σ 2 ) t + σ t (20)

In addition, μ = 1 2 σ 2 is the optimal stationary point of p as t → ∞ . Consequently,

p = X ( 0 , A ) e σ t = X ( 0 , A 1 ) e σ t + X ( 0 , A 2 ) e σ t + X ( 0 , A 2 ) e σ t (21)

Here, μ = X ¯ ( t , A ) = E 15 so that the standard deviation σ = E 18 . Hence,

p = E 40 exp ( 36 24 ) + E 3 exp ( 36 24 ) + E 2 exp ( 36 24 ) (22)

Finally, one obtains that

p = E 180 + E 13.5 + E 9 = E 202.50 (23)

Here, any p ( A ) less than E 202.50 has arbitrage tendency.

From examples (4.1) and (4.2) above, it is clear that pseudo movie markets in the path of the geometric Brownian motion are more gaining to motion pictures owing to the magnitude of the finite variation process. Thus, the higher the volatility, the higher the dispersion of returns coupled with increased investment risk. These examples further demonstrate that pseudo movie markets may be closed or open depending on whether the market interacts with others through the free passage of information.

In this respect, closed pseudo movie markets exhibit more often the geometric pricing properties since there exists closed form pricing procedures in place visible in standard movie shops of mega investments and cinemas. The converse also holds good for open pseudo movie markets in pricing. This is in view of apparent flexibilities of open markets compared to closed markets generally as demonstrated in example (4.1). Essentially, example (4.1) in contrast with example (4.2) explains why open pseudo movie markets posses higher tendency for pricing arbitrage compared to closed pseudo movie markets where arbitrage tendencies might not even exists owing to the size of the price.

In this work, pricing movie industry pseudo assets and portfolio are studied. The work proved that pseudo assets trading in movie markets as add-ons present arbitrage opportunity capable of crunching movie markets. The work designed a stochastic pricing strategies under which such arbitrages can be eliminated. The process of eliminating pseudo asset arbitrage in movie pricing is demonstrated by means of examples for easy understanding. In conclusion, it is important to emphasize that for informed decisions on pseudo assets trading in movie markets, the knowledge of market type is significant. This provides some level of control over the hedging of portfolios to prevent pricing arbitrage and loss of investment resources.

The authors acknowledge all the sources of literature used in this article, the anonymous referees and the editors of JMF involved in this work.

The authors declare no conflicts of interest regarding the publication of this article.

Sani, S., Maseko, S.P., Dlamini, Q. and Abdullahi, F.A. (2020) Pricing Pseudo Contingencies on Motion Picture Assets under No Free Lunch with Vanishing Risk. Journal of Mathematical Finance, 10, 525-535. https://doi.org/10.4236/jmf.2020.104032