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Option Valuation The Black-Scholes-Merton Option Pricing Model.

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Presentation on theme: "Option Valuation The Black-Scholes-Merton Option Pricing Model."— Presentation transcript:

1 Option Valuation The Black-Scholes-Merton Option Pricing Model

2 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)

3 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. =  

4 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

5 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

6 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”)

7 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 )

8 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

9 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

10 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

11 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.)

12 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.

13 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)

14 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)

15 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 ]

16 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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