Applications of option pricing analysis CLIFFORD W. SMITH JR. The University of Rochester 1979 Presentation: 應子健 廖尉呈 李承儒 Presentation: 應子健 廖尉呈 李承儒

1. Introduction 歷史背景 : 歷史背景 : 1973 年， Black-Scholes 模型理論剛剛發表， 尚未廣泛應用於市場上。這篇論文的發表 在 1979 年，利用 Black-Scholes 模型定價公 司資產、負債與其他衍生性金融商品。這 篇 paper 在當時算是很創新的概念，不過在 今天看來卻是屬於必備的知識。 1973 年， Black-Scholes 模型理論剛剛發表， 尚未廣泛應用於市場上。這篇論文的發表 在 1979 年，利用 Black-Scholes 模型定價公 司資產、負債與其他衍生性金融商品。這 篇 paper 在當時算是很創新的概念，不過在 今天看來卻是屬於必備的知識。

2. The pricing of European options THE PRICING OF EUROPEAN PUT AND CALL OPTION THE PRICING OF EUROPEAN PUT AND CALL OPTION Assumptions Assumptions 1. There are no penalties for short sales ; 2. Transactions costs and taxes are zero ; 3. The market operates continuously ; 4. The riskless rate is known and constant ; 5. The stock price follows a continuous Ito process ; 6. The stock pays no dividends ; 7. The option can only be exercised at the terminal date of the contract.

European call option pricing European call option pricing The price of call option ： The price of call option ： The solution can be written in general form as The solution can be written in general form as where ; ; ; ;. where ; ; ; ;.

European put option pricing European put option pricing Consider two portfolios ： Portfolio I ： one European call + one share of stock sold short + X pure discount B(T) and face value of one dollar Portfolio I ： one European call + one share of stock sold short + X pure discount B(T) and face value of one dollar Portfolio II ： one European put with same terms as the call Portfolio II ： one European put with same terms as the call

Portfolio Current value Stock price at T=0 I II 0 II 0Relationship between the terminal value of portfolios I and II

This solution can be written in general form as

3. The pricing of corporate liabilities Pricing of the debt and equity of a firm Pricing of the debt and equity of a firm 公司發行債券，實質上相當於股票持有者將所持有的資產賣 給債券持有者，並且再買一個可以在債券到期日時以債券票 面價買回資產的買權。 若票面價為 X ，執行日當天公司價值為 V * ，則在執行日時 股票持有者的收入為 max [ 0, V * - X] ，取決於公司價值 V * 是否大於 X ，與選擇權相當類似。

Assumptions 1. 公司發行還本時付息債券，並且在到期日之前限制任 何股利發放。 2. 公司總價值不會受到資本結構改變而影響。（符合 Modigliani-Miller world ） 3. 公司價值的動態變化有相同的期望值；並且在任何有 限時間之中是一個有著固定變動報酬率的指數常態分 佈。 4. 存在已知的固定無風險利率。 By Black-Scholes call option solution yields

By payoff graph By payoff graph In risk neutral world, They should have equal price, or arbitrage will appear. Thus, the pricing of equity can be compute by the method of call option pricing. E*E* XV*V* P XS

Debt Debt via V = E + D  D = V – E By the result of equity pricing - = - = V*V* XV*V* E*E* XV*V* D*D* XV*V*

Debt can be expressed as Debt can be expressed as and and

※ The option pricing model and the CAPM The pricing of the equity is consistent with the continuous time CAPM The pricing of the equity is consistent with the continuous time CAPM Instantaneous return to the stockholder Instantaneous return to the stockholder

By ito ’ s lemma By ito ’ s lemma Substituting into the definition of the systematic risk of equity That is

Elasticity is greater than 1, thus the absolute value of the systematic risk of the stock is greater than the absolute value of the systematic risk of the firm. Elasticity is greater than 1, thus the absolute value of the systematic risk of the stock is greater than the absolute value of the systematic risk of the firm.

※ Risk structure of interest rates is the expected return rate of the bond is the expected return rate of the bond  The risk premium

※ Coupon Bonds With required interest payments the stockholders ’ equity is like an option on an option on … an option on the assets of the firm. By paying the last coupon, the stockholder buy the option to purchase the firm by paying the face value of the debt. With required interest payments the stockholders ’ equity is like an option on an option on … an option on the assets of the firm. By paying the last coupon, the stockholder buy the option to purchase the firm by paying the face value of the debt.

3.2 Convertible bond pricing Assumptions Assumptions The convertible bond and the stock are the only liabilities issued by the company. The convertible bond and the stock are the only liabilities issued by the company. Value of convertible bond at maturity date

By payoff graph By payoff graph B*B* V*V* X/aX

3.3 The pricing of subordinated debt The firm issues two debt, one is senior and the other is junior. The firm issues two debt, one is senior and the other is junior. Assumption: The issues contain restrictions against dividend payments until after both the bond issues are paid off.

Pay off graph Pay off graph E*E* X s +X j V*V* Dj XsXs X s +X j V*V* Ds XsXs X s +X j V*V*

The pricing of equity and senior debt are both unchanged The pricing of equity and senior debt are both unchanged The pricing of junior debt The pricing of junior debt can be expressed as

3.4 Pricing of warrants and rights Assumption Assumption The only liabilities issued by the firm are its common stock and the warrants The only liabilities issued by the firm are its common stock and the warrants

Payoff graph Payoff graph E*E* (1-a)X/aV*V* W*W* V*V*

4. The pricing of other contingent claims 4.1 The pricing of underwriting contract Underwriters submit a bid,, today which specifies that on the offer date, T time periods from now, the underwriter will pay dollars and receive shares of stock representing fraction of the total shares of the firm. He can sell the securities at the offer price and receive, or if the share price is below the offer price at the market price,. If his bid is accepted, he will be notified immediately. Underwriters submit a bid,, today which specifies that on the offer date, T time periods from now, the underwriter will pay dollars and receive shares of stock representing fraction of the total shares of the firm. He can sell the securities at the offer price and receive, or if the share price is below the offer price at the market price,. If his bid is accepted, he will be notified immediately.

V*V* U

4.2 The pricing of collateralized loans Assumption: Assumption: 1. There are homogeneous expectations about the dynamic behavior of the value of the collateral. The distribution at the end of any finite time interval is log normal. The variance rate of return is constant. 2. The collateral provides a continuous flow of service to the borrower. The net value of the flow of the service, S, is a constant fraction, s, of the market value of the assets: s=S/V 3. The dynamic behavior of the value of the assets is independent of the value of the probability of bankruptcy.

4. There are no costs to voluntary liquidation or bankruptcy. Bankruptcy is defined as the state in which the borrower ’ s assets are less than the promised repayment amount of a maturing loan. 5. Capital markets and the market for the collateral are perfect. There are no transactions cost or taxes. All participants have free access to all available information. Participants are price takers.(efficient market) 6. There is a known constant riskless rate, r.

Where

4.3 The pricing of leases suggested: suggested: The value of the borrower ’ s equity in the collateral is equivalent to a call option to purchase the collateral with the exercise price equal to the promised repayment on the loan, plus a lease. Therefore, the value of the lease equal the value of the collateral minus the value of the debt minus the value of the call The value of the borrower ’ s equity in the collateral is equivalent to a call option to purchase the collateral with the exercise price equal to the promised repayment on the loan, plus a lease. Therefore, the value of the lease equal the value of the collateral minus the value of the debt minus the value of the call

This equation has an intuitive interpretation: the value of the lease equals the value of the asset minus a claim on the value of the asset T periods from now. This equation has an intuitive interpretation: the value of the lease equals the value of the asset minus a claim on the value of the asset T periods from now.

4.4 The pricing of insurance The insurance contract calls for the payment of a premium, P, at the current date, t. If at the expiration date of the contract, t*, the market value of the insured asset, V*, is less than its insured value, X, then the insurance contract will pay the holder of the policy the difference, X-V*. If the market value of the insured asset is greater than its insured value, then there is no payment. The insurance contract calls for the payment of a premium, P, at the current date, t. If at the expiration date of the contract, t*, the market value of the insured asset, V*, is less than its insured value, X, then the insurance contract will pay the holder of the policy the difference, X-V*. If the market value of the insured asset is greater than its insured value, then there is no payment.

where P*P* X XV*V* 45 o