DBJ Discussion Paper Series, No.1212

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

Masayuki Otaki(Institute of Social Science, University of Tokyo)

March2013

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白紙Can bailout improve the economic welfare?
A structural derivation of the option price
Masayuki Otaki
Institute of Social Science, University of Tokyo
Email ohtaki@iss.u‐tokyo.ac.jp

Abstract
I succeed in deriving the Black‐Scholes formula from the payoff functions within asome 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 relevantcase 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.

1Introduction

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 anticipatedbailout 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 thederivative transactions are heavily levied for financing such a policy, and thus, economicwelfare, as a whole, is always worsened by the policy.
The paper consists of three sections. In Section 2, I define the payoff functions of aEuropean 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 4solves the option price that is attached by the money poured into the bailout policy, andconsiders the macroeconomic welfare implication of the policy. In Section 5, we provide
brief concluding remarks.

12 The ModelI consider a European call option the expiry date is T . A European call option is akind 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.
Theorem 1 Let us denote the buyer’s and seller’s payoff functions as Vt B and Vt S ,respectively. They can be written as
Vt B (t : T ) = Et [ max[e  r [T t ] [ ST  X ]  Pt , Pt ]],Vt S (t : T ) = Et [ min[e  r [T t ] [ X  ST ]  Pt , Pt ]],

(1)
(2)

where Pt is the option price at time t , ST is the stock price at the expiry date T , andX is the exercise price. Et denotes the conditional expectation operator on theavailable 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.
t (t < ~t < T ) with some positive
Suppose that an early exercise occurs at time ~probability. Then,~~V B (t : T )  e  r [ t t ] [ S ~t  X ]d ( S ~t | St )  Pt > 0
 r [ t t ](3)holds.  (  ) is the conditional cumulative distribution function of P~t on Pt . The
above inequality exists given the fact that an early exercise occurs because of the chanceof the excess gain.
On the other hand, from (3), the seller’s payoff function satisfies~~V S (t : T )  Pt  e  r [ t t ] [ S ~t  X ]d ( S ~t | St ) < 0.
 r [ t t ](4)

[ S ~  X ]et Pt[ S ~  X ]et Pt

From (4), the seller’s payoff becomes negative whenever an early exercise occurs withsome 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 Pt is expressed as
Pt = Et [ max[e  r [T t ] [ ST  X ],0]].

To prove Theorem 2, the following lemma is of use.
Lemma 22(5)V S (t : T ) = V B (t : T ) = 0(6)holds.
Proof.
By adding up (1) and (2), we obtain
V S (t : T )  V B (t : T ) = 0.
Since both V S and V B are non‐negative, (6) holds.
The proof of Theorem 2
Substituting (6) into (1) and (2), I can obtain (5).

3 The Black‐Scholes Formula
In this section, I derive the Black‐Scholes formula based on the structural
approach explained above.
When the stock price St follows the geometric Brownian motion with drift r
and instantaneous variance  2 , by Ito’s formula, the logarithm of the stock price stfollows the normal distribution,  , the mean and standard deviation of which are[[r 22][T  t ],  T  t ] . In addition, it can be easily shown by some elementary calculusthat S d(S | S ) = S eXTTtt SN ( zt   z * ),St
2][T  t ] [r 22, z*  X,zT  T t T twhere  SN is the standard normal cumulative distribution function.
Since the logarithmic value st of the stock pricesT  s t  [ r 2 r [T t ]][T  t ](7)lnfollows

2][T  t ]2also follows the standard2 T tnormal distribution  SN . Let us define the critical value y * asS2ln t  [r ][T  t ]*2Xy .
 T tSubstituting these results into (5), we obtain
Pt = S t  Pr ( zT   z * )  X  Pr ( yT   y * ).
Finally, by the symmetry of the normal standard distribution,
Pt = St  SN ( z * )  X SN ( y * ).

N ([r 2][T  t ],  T  t ) , yT sT  s t  [ r St3(8)This is the Black‐Scholes formula.

4 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 anticipatedbailout will be consumed to stimulate the zero‐sum game within the option trading. This isbecause 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 thedisparity of income distribution, the bailout policy harms peoples’ well‐being.
To endorse the above discussion, let us define the payoff functions under a bailoutpolicy. The assumed bailout policy is as follows: at expiry date T , money, which is leviedby 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 X  M . This isbecause the bailout money M becomes the lower bound of the seller’s revenue, andthus, the option is exercised only when ST  X  M .
Vt B (t : T , M ) = Et [ max[e  r [T t ][ ST  [ X  M ]]  Pt , e  r [T t ] M  Pt ]]= e  r [T t ] [ X M[ ST  [ X  M ]]d( ST | St )  M( X  M )]  Pt .

(9)

Corresponding to (9), the seller’s payoff function becomes

Vt S (t : T , M ) = Pt  e  r [T t ] X M[ ST  [ X  M ]]d( ST | St ).

Note that when ST  X < M , the option is not exercised. Since there is no loss on thesellerr’s side, the term M corresponding to such a case in the above equation does notappear.
Summing up both sides of (9) and (10), I obtain
V B (t : T , M )  V S (t : T , M ) = e  r [T t ] M ( X  M ) < e  r [T t ] M .
Since the actual total sum of money poured into the bailout is e  r [T t ] M , (11) implies
that the social cost of the bailout policy exceeds the benefits. Consequently, we reach thefollowing 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

4(10)(11)probability 1   ( X  M ) . In such a case the transaction of the option attached thebailout policy becomes a zero‐sum game because the exercise price increases by M , andthere 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 ispassively 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 isan indeterminacy concerning the pricing, because the game is not zero‐sum and thevalues 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 forthem (i.e., V S (t : T , M ) = 0 ). Then, applying the Black‐Scholes formula to (10), I obtain
Pt = S t  SN ( z M* )  [ X  M ] SN ( y M* ),StSt [r  [r ][T  t ]ln][T  t ]22z M*  X  M, y M*  X  M.
 T t T tNote that V B (t : T , M ) = e  r [T t ] > 0 , and hence, the buyers are ready to accept such anoffer price. Thus, the trading is surely settled.
22ln5 Concluding Remarks
This paper presented a game‐theoretic approach concerning option pricing, thevalidity and tractability of such an approach were ascertained by deriving theBlack‐Scholes formula. I also applied this approach to welfare implications of the bailout
policy. I found that such a policy always worsens economic welfare. This is because of themoral hazardous behavior of the buyer owing to the limited liability which has beenpreviously emphasized in studies such as by Arrow [2] and Stiglitz and Weiss [3].

References[1] F. Black and M. Scholes, ``The Pricing of Options and Corporate Liabilities,’’ Journal of
Political Economy Vol 81, 1973, pp.637‐659.
[2] K. J. Arrow, ``Uncertainty and the Welfare Economics of Medical Care,’’ AmericanEconomic Review Vol. 53, 1963, pp. 941‐973.
[2] J. E. Stiglitz and A. Weiss, ``Credit Rationing in Markets with Imperfect Information,’’American Economic Review Vol.71, 1981, pp. 393‐410.

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