Can bailout improve the economic welfare? A structural derivation of the option price

I succeed in deriving the Black‐Scholes formula from the payoff functions within a some kind of zero‐sum game. Such a structural approach enables us to apply the formula to a more general case than the one presented  in the original paper. The most relevant case  concerns  the bailout policy, which  is weaved  into  investors’  rational expectations. Once  such  a  policy  is  anticipated,  the  price  of  option  becomes  dear  and  investors’ behavior becomes more bullish since the bailout policy eliminates  loss when speculation fails. Since  the  transaction  of  derivatives  is  generically  a  zero‐sum  game,  the  bailout policy  never  improves  economic welfare.  Indeed, when we  consider  that  the  financial source of  the bailout policy  is  levied  from other economic agents,  it becomes apparent that such a policy would surely lower economic welfare. 1    Introduction It  is not self‐evident whether bailout policies that are often adopted  in financial crises  improves  economic welfare.  This  is mainly  because  payoff  functions  of  financial assets are not explicitly defined. This article provides the payoffs function of a European call  option  explicitly  and  considers  the welfare  implication  of  the  bailout  policy.  As  its corollary, I induce the Black‐Scholes [1] formula from the payoff functions. Since the transaction of derivatives is generically a zero‐sum game, an anticipated bailout policy does not rescue its traders. Once such a bailout policy  is rationally weaved into expectations,  it becomes a part of the value of the option. Hence, the bailout policy does  not  affect  investors’  risk‐neutral  expected  utility.  However,  the  outsiders  of  the derivative transactions are heavily  levied for financing such a policy, and thus, economic welfare, as a whole, is always worsened by the policy. The paper consists of three sections. In Section 2, I define the payoff functions of a European  call option  and  analyzes  their properties.  Section 3 derives  the Black‐Scholes formula  to ascertain  the  validity of my  approach. Using  the obtained  results,  Section 4 solves the option price that is attached by the money poured into the bailout policy, and considers the macroeconomic welfare  implication of the policy.  In Section 5, we provide brief concluding remarks.


Introduction
It is not self-evident whether bailout policies that are often adopted in financial crises improves economic welfare. This is mainly because payoff functions of financial assets are not explicitly defined. This article provides the payoffs function of a European call option explicitly and considers the welfare implication of the bailout policy. As its corollary, I induce the Black-Scholes [1] formula from the payoff functions.
Since the transaction of derivatives is generically a zero-sum game, an anticipated bailout policy does not rescue its traders. Once such a bailout policy is rationally weaved into expectations, it becomes a part of the value of the option. Hence, the bailout policy does not affect investors' risk-neutral expected utility. However, the outsiders of the derivative transactions are heavily levied for financing such a policy, and thus, economic welfare, as a whole, is always worsened by the policy.
The paper consists of three sections. In Section 2, I define the payoff functions of a European call option and analyzes their properties. Section 3 derives the Black-Scholes formula to ascertain the validity of my approach. Using the obtained results, Section 4 solves the option price that is attached by the money poured into the bailout policy, and considers the macroeconomic welfare implication of the policy. In Section 5, we provide brief concluding remarks.

The Model
I consider a European call option the expiry date is T . A European call option is a kind of zero-sum game that follows the concept of the Stackerberg equilibrium. The strategy of a buyer (follower) relates to price at which she/he exercises the option. The seller's strategy (leader) is to offer the price of option and its exercise price.
As such, we obtain the following theorem.
where t P is the option price at time t , T S is the stock price at the expiry date T , and X is the exercise price. t E denotes the conditional expectation operator on the available information until time t .
To prove the above theorem, the following lemma is of use.

Lemma 1 Early exercise does not occur with probability one.
Proof. Suppose that an early exercise occurs at time t ( ) with some positive probability. Then, is the conditional cumulative distribution function of t P~ on t P . The above inequality exists given the fact that an early exercise occurs because of the chance of the excess gain.
On the other hand, from (3), the seller's payoff function satisfies From (4), the seller's payoff becomes negative whenever an early exercise occurs with some positive probability, and hence such an asset would never be provided. Accordingly, any early early exercise never occurs with positive probability.
Proof of Theorem 1. From Lemma 1, since the call option is held till the expiry date T with probability one, the values of the call option become the expected discount values at T .
From Theorem 1, the following theorem holds.

Theorem 2
The equilibrium price of the call option t P is expressed as To prove Theorem 2, the following lemma is of use. holds.

The Black-Scholes Formula
In this section, I derive the Black-Scholes formula based on the structural approach explained above.
When the stock price t S follows the geometric Brownian motion with drift r and instantaneous variance 2  , by Ito's formula, the logarithm of the stock price t s follows the normal distribution,  , the mean and standard deviation of which are

The Welfare Economic Implication of the Bailout Policy
Usually, the validity of the bailout policy is judged from the view of income distribution, that is, whether the redistribution of incomes from tax payers to failed speculators is legitimate. Instead, in this paper, I deal with the efficiency of the bailout policy. This is not a self-evident problem because some parts of money of an anticipated bailout will be consumed to stimulate the zero-sum game within the option trading. This is because that the bailout guarantees the minimum return for buyers and causes a kind of moral hazard which is intrinsic to the property of limited liability. Thus, not all the money poured into the bailout cannot be used for the compensation of the capital loss, and thus, the effect of the bailout policy is rather mitigated. Consequently, as I have detailed below , the bailout policy acts conversely and worsens economic welfare. Besides advancing the disparity of income distribution, the bailout policy harms peoples' well-being.
To endorse the above discussion, let us define the payoff functions under a bailout policy. The assumed bailout policy is as follows: at expiry date T , money, which is levied by outsiders and amounts to M , is transferred to the losers (i.e., sellers, when the option is exercised and buyers, when not exercised).
We must note that the effective exercise price rises from X to M X  . This is because the bailout money M becomes the lower bound of the seller's revenue, and thus, the option is exercised only when Corresponding to (9), the seller's payoff function becomes , the option is not exercised. Since there is no loss on the sellerr's side, the term M corresponding to such a case in the above equation does not appear.
Summing up both sides of (9) and (10), I obtain (11) Since the actual total sum of money poured into the bailout is M e , (11) implies that the social cost of the bailout policy exceeds the benefits. Consequently, we reach the following important theorem.

Theorem 3
The bailout policy always harms the economic welfare.
The background of the above theorem is as follows. The option is exercised with . In such a case the transaction of the option attached the bailout policy becomes a zero-sum game because the exercise price increases by M , and there is no substantial effect of the bailout policy. In other words, a kind of moral hazard owing to the limited liability occurs on the buyer's side. Since the value function of seller is passively defined in accordance with the buyer's action, an increase in the effective exercise price enlarges the seller's loss and there is no social gain in such a case.
Finally, I solve the equilibrium option price using the Black-Sholes formula. There is an indeterminacy concerning the pricing, because the game is not zero-sum and the values of the payoff functions differ in the case of the bailout policy. Thence, I consider the case that the competition among sellers is under strain, and there is no surplus for them (i.e., 0 = ) , : ( ). Then, applying the Black-Scholes formula to (10), I obtain