# 1 Pricing options using the Black Scholes formula

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1 Lecture 9 Pricing options using the Black Scholes formula Exercise. Consider month options with exercise prices of K = 45. The variance of the underlying security is σ 2 = The risk free interest rate is r = 6%. The current price of the underlying security is S = 0.. Determine the Black Scholes prices for call and put options. 2. Check that your calculations satisfy put call parity. Solution to Exercise. Call Put prices σ = 0.20 = C BS(S = 0, K = 45, r = 0.06, σ = , (T t) = 0.25) d = ln ( ) ( S K + r + σ2) (T t) 2 σ T t 0 ln( 45 d = ) + ( ) d =.644 N(d ) = 0.05 d 2 = N(d 2) = 0.02 C BS = 0. N( z) = N(z) 2 p = Ke r(t t) N( d 2) SN( d ) = 45e ( 0.02) 0( 0.05) = 4.46 Check this using put call parity r(t t) c p = S Ke p = c S + Ke r(t t) = e = 4.46 Exercise 2. Consider an option contract where the current price S = 00, the exercise price is K = 00, and time to maturity T t is one year. The risk free interest rate is 5%.. Suppose you observe yesterdays call price to be C = What is the volatility implied in this price? Solution to Exercise 2. Given C = 4.97, can find the implied volatility by solving the equation C obs = = C(S = 00, K = 00, σ, (T t) =, r = 0.05) Unfortunately, there is no way to analytically solve for σ in this equation, but it is easy to solve numerically.

2 option price c =.74 σ = 0.5 volatility option price c obs =.74 ĉ < c obs σ < σ σ = 0.5 volatility Look in the programming notes to see a couple of ways to do this. If you do, find σ implied = Financial Structure firm Exercise. Consider a firm that is currently unlevered with 00 shares outstanding, each selling for \$5. The firm will operate for one more period and then be liquidated. The possible cash flows at the end of the period are as follows. Find the total value of the firm. 2 Determine the expected return on equity. Firm Cash Flow \$800 \$800 \$2800 Now suppose the firm has talked with its investment banker and has gotten assurances that it can issue up to \$600 of debt at a 0% interest rate. If the firm takes the advice of its investment banker and issues the \$600 of debt, the proceeds will be used to pay a dividend or repurchase shares. Determine the cash flows to the stakeholders of the firm. 4 Show that this change will not change the total value of the firm. 5 Determine the expected return on equity after the change. Solution to Exercise.. The total value of the firm is thus equal to V U = \$ Determine the expected return on equity. Firm Cash Flow \$800 \$800 \$2800 Return on equity -46.7% 20.0% 86.7% 2

3 ( 0.467) = 0.2 Expected return = 20% The cash flows at the end of the period to the firms security holders will be: Total cash flow to firm \$800 \$800 \$2800 Cash flow to debt Cash flow to equity What effect will this recapitalisation have on the value of the firm and the wealth of the firms shareholders? According to MM, the answer is none. The value of the firm should be unaffected by the recapitalisation and the firms shareholders should be no better off and no worse off that they were before the recapitalisation. To see that this must be so, suppose you bought % of both the debt and equity of the levered firm. This cost you 0.0 V L. Your payoff next period will be: Cash flow to debt Cash flow to equity Total cash flow These cash flows are identical to the cash flows from an investment in % of the equity of the unlevered firm. Consequently, the cost of each investment must be the same. 0.0 V L = 0.0 V U V L = V U = \$500 Since D L = \$600, the value of the equity must be E L = V L D L = \$900 Thus, the returns on the levered equity are: Cash flow to equity Return on equity -84.4% 26.7% 7.8% Expected return = 26.7%. Leverage increases both the risk and expected return on equity. however, is unaffected by leverage. r Exp cashflow to firm = Market value of firm = = 20%. The expected return on the firm s assets, Suppose an investor owned all of the firms debt and equity. The expected return on his portfolio should also equal r : r = 20% = D V rd + E V re = = 20%

4 Exercise 4. Consider two firms U and L. Both firms have the same annual earnings before interest and taxes of \$5,000. Firm U is unlevered firm. Firm L is a levered firm, with \$0,000 of 0% debt. The interest rate for discounting the cash flows r = 20%. Both firms pay taxes at a rate of 4%. Determine the cash flows to bond and equity holders for the two firms 2. Suppose these cash flows are perpetuities. Determine the values of the two firms.. How much would the wealth of equity holders in U increase if the firm issued \$0,000 of debt? 4. What is the cost of capital for firm L? Solution to Exercise 4.. Cash flows Income statement Firm U Firm L EBIT Interest Taxable income Tax (4%) Net income Total income to bondholders and stockholders Income to the government Firm values Annual cash flow CF U + τ c I = = 640 = CF L V L = V U + τ c D = , = 6, 500 +, 400 = The value of the firm s equity is: E = V L D = 9, 900 0, 000 = 9, 900 Thus, if firm U issued \$0,000 of debt, it could increase its market value by \$,400. The increase in firm value flows through directly to the old stockholders. They receive a \$0,000 dividend at the time the debt is issued and are left with equity worth \$9,900. This represent an increase in wealth of \$,400: Wealth of equity holders = (0, , 900) 6, 500 = \$, 400 = τ c D We can estimate the cost of capital for firm L, r, from the cost of capital for firm U, r, and firm L s debt-to-value ratio, D/V. r = r ( τ c ) D ( V ) 0, 000 = , 900 = 6.6%. 4

5 The cost of equity capital for firm L is: r E = r + (r r D)( τ c ) D E 0, 000 = (0.2 0.) ( 0.4) 9, 990 = 26.7%. Assuming that β =.0, and β D = 0, the levered firm s equity beta is Exercise 5. β E = β + (β β D)( τ c ) D E 0, 000 =.0 +.0(.0 0.4) 9, 000 =.67 E[ r m ] = 0% r f = 5%. What is the asset beta β? 2. Find the firms cost of capital Solution to Exercise 5.. What is the asset beta β? β = = Find the firms cost of capital Use the asset beta: r = r f + (E[r m] r f )β = ( )0.95 = 9.75% β Market value Debt Equity Calculate required returns for debt and equity separately: r E = r f + (E[r m] r f )β E = ( ).40 = 2% r D = r f + (E[r m] r f )β D = ( )0.05 = 5.25% r = D V rd + E V re = % % = 9.75% 5

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