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Option Valuation The Black-Scholes-Merton Option Pricing Model

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Derivatives and Risk Management w How Do Stock Prices Evolve Over Time? Random walk: proportional changes in prices over a short period of time are normally distributed Let S = current stock price = expected return on stock = volatility of stock price Over a short time interval t, the mean proportional change in stock price is * t and the standard deviation of the proportional change is * t Formally, S/S ~ N ( * t, * t ) and S /S = * t + * (used for simulation: S = S[ * t + * ]) Where is a random draw from the standard Normal dist. (mean=0, =1)

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Derivatives and Risk Management w The Lognormal Distribution Random Walk implies that stock price at any future date, S T, has a lognormal distribution. This means that the logarithm of the stock price at any future time is normally distributed: Ln(S T ) is normal with Mean = Ln(S 0 ) + ( 2 )T St. Dev. = Thus, E(S T ) = S 0 e T Ln(S T ) - Ln(S 0 ) = Ln(S T /S 0 ), which is the continuously compounded return realized over the period from now till time T, is normal with Mean = ( 2 )T St. Dev. =

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Derivatives and Risk Management w Estimating Volatility (Check P. 238-239 of textbook) Let: S i = Stock price at end of i th time interval t = length of time interval in years Compute return over interval i: u i = ln (S i / S i-1 ) An estimate, s, of the standard deviation of the u i ’s is: s = (u i - u) 2 where u is the mean of the u i ‘s The annual estimate of volatility, , is s / t n i = 1 1 n - 1

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Derivatives and Risk Management w Assumptions Underlying Black-Scholes-Merton Stock prices follow a random walk with constant and No transactions costs or taxes No dividends over the life of the option Trading is continuous Investors can borrow and lend at risk free rate, r, which is constant over the life of the option There are no riskless arbitrage opportunities

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Derivatives and Risk Management w The Black-Scholes-Merton Analysis Similar logic to that for binomial option pricing Create riskless portfolio consisting of position in stock and option The return on the portfolio must be the risk free rate Complication: must adjust hedge portfolio every instant (“portfolio rebalancing”)

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Derivatives and Risk Management w The Black-Scholes-Merton Formula where N(d) = The standard cumulative normal density function (i.e. the probability of standardized normal variable being < d) d 1 = [ ln(S 0 /X)+(r + 1 / 2 2 ) t ] / t ] d 2 = d 1 - t c = S * N(d 1 ) - X* e -r * t * N(d 2 )

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Derivatives and Risk Management w The Black-Scholes-Merton Formula: Puts p = X* e -r*T * N(-d 2 ) – S 0 * N(-d 1 ) Comparing the two models: p+S = c + X*e -r * T Put-call parity

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Derivatives and Risk Management w Measurement of Inputs Current Stock Price S - observable Strike Price X - observable Time to Expiration T - observable Risk-free rate r Annual yield on risk-free, short-term Bill (using continuous compounding) Volatility Standard deviation of historical returns on the stock Implied volatility

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Derivatives and Risk Management w Implied Volatility Set market price today = model price Know S, X, t and r Solve for S=100, X=100, t = 0.5, r = 5%, c = 6.89 = 20% Use 20% to value OTC options with similar maturity

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Derivatives and Risk Management w Option Valuation: General Intuition Call payoff at T = C T =S T – Xif S T > X 0if ST < X E(C T ) = Expected payoff= pr(S T >X)*E(S T – X | S T >X) + 0 = Pr(S T >X)*E(S T | S T >X) – Pr(S T > X)*X = N(d 1 )*E(S T ) – N(d 2 )*X Notice: N(d 1 ) > Pr(S T >X) but E(S T ) X) E(S T ) = S 0 e T Hence, E(C T ) = N(d 1 )* S 0 e T – N(d 2 )*X And C 0 = PV(C T )= [N(d 1 )* S 0 e T ] e - T – N(d 2 )*X* e -rT (Notice we discounted the stock position at the rate of return of the stock while the riskless cash flow X at the risk free rate on interest.)

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Derivatives and Risk Management w General Intuition, Continued Hence, C 0 = N(d 1 )* S 0 – N(d 2 )*X* e -rT Notice: we get this formula regardless of what is. We might as well assume = r, i.e., we are valuing the option in a risk neutral world. This is one more way to see the appropriateness of the risk neutrality assumption: It does not hurt, but it adds a lot of convenience.

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Derivatives and Risk Management w Effect of Dividends (dividend yield) If the underlying asset pays a continuous dividend yield, q: E(S T ) = S 0 e (r-q)T PV of E(S T ) = e -rT S 0 e (r-q)T = S 0 e -qT, then c = S 0 e -qT * N(d 1 ) – X* e -r*T * N(d 2 ) p = X*e -r*T * N(-d 2 ) – S 0 * N(-d 1 ) Where d 1 = [ ln(S 0 /X)+(r –q + 1 / 2 2 ) t ] / t ] d 2 = d 1 - t N(-d) = 1 – N(d)

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Derivatives and Risk Management w Known Dividend Amount Break stock price into two components: S* + PV(div) riskless component, PV(div), used to pay dividends during option life risky component S* Black-Scholes-Merton formula correct if S* = S – PV(div) In previous example, assume firm pays one dividend of $0.5 in 3 months and $0.5 in 6 months S=100, X=100, T = 0.5, r = 5%, = 20% PV(div) = 0.5 * e -.05 *.25 + 0.5 * e -0.5*.5 = 0.98 Revalue call using S* = $99.02 c = $6.32 (6.89 previously)

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Derivatives and Risk Management w American Calls with Dividends It may be optimal to exercise an American call immediately before an ex-dividend date C = Max[c, c 1, c 2,…..,c n ] c is value assuming exercise at maturity T c i is value assuming exercise at i th ex-dividend date In practice, calls are most likely to be exercised just before last dividend Black approximation: C = Max [c, c n ]

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Derivatives and Risk Management w The Black Approximation: An Example t= 3 months, ex-dividend in 1 month, dividend = $1, Stock price = $20, r = 10%, Volatility = 30%, X = 20 S=20, X=20, T=0.0833, r=0.1, = 0.3 c 1 = 0.77 S*= 20 – PV($1) = 19.01, T=0.25 c = 0.92 Call value = Max[0.77, 0.92] = 0.92 nowDiv = $1 1 month2 months T

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