Lecture 15 Concepts of Black-Scholes options model Agenda: I. Intuition of Black-Scholes Pricing formulas II. III. he impact of stock dilution: an example of stock warrant pricing model he impact of Dividends:
I. Intuition of Black-Scholes Pricing formulas: c = S N(d 1 ) Ke -r N(d ) p= Ke -r N(-d ) S N(-d 1 ) ln( S / K ) ( r / ) d = ln( S / K ) ( r / ) =d 1 - Intuition: N(d ): the probability that the option will be exercised, prob.(s > K) in a riskneutral world. N(d ) K: the cash outflow at maturity if the call is exercised. N(d 1 ): the sensitivity of a call option in response to the underlying asset. S N(d 1 )e r : the expected value of a variable that equals St if S >K and is zero if S K in a risk neutral world. r: zero-coupon rate for a maturity. σ: based on market expectation about the future volatility. As S is very large relatively to K: c= S K e -r = Forward As σ is close to, c =max(s e r K, ) at maturity. (It is almost risk-free.) c = e -r max(s e r K, ) = max(s K e -r, ) 1
~ Black-Scholes model for index, currency, and futures options: Stock Option = f(s, X,, r, q, σ) c = S N(d 1 ) Ke -r N(d ) p= Ke -r N(-d ) S N(-d 1 ) d = ln( S / K ) ( r ln( S / K ) ( r / ) / ) =d 1 - For index options, we need to replace S S e -q, q is the continuously compounded dividend yield. *q is similar to your carrying cost. c = S e -q N(d 1 ) Ke -r N(d ) p= Ke -r N(-d ) S e -q N(-d 1 ) d = ln( S / K ) ( r q / ) ln( S / K ) ( r q / ) =d 1 - For currency options, the foreign currency risk-free rate, r f, is jut like d. *r f is similar to q in index OPM. It s your carrying cost. c = S e -rf N(d 1 ) Ke -r N(d ) p= Ke -r N(-d ) S e -rf N(-d 1 ) d = ln( S / K ) ( r r / ) f ln( S / K ) ( r r / ) f =d 1 - For futures options, F = S e (r-q) Replace S with F e -(r-q) c = e -r [F N(d 1 ) K N(d )] p= e -r [KN(-d ) F N(-d 1 )] d = ln( F / K ) ( / ) ln( F / K ) ( / ) =d 1 -
II. he impact of stock dilution: an example of stock warrant pricing model Normal options do not affect a firm s shares outstanding. herefore, there is dilution impact. However, employees stock options or warrants have dilution effect when they are exercised. hus, the pricing for warrants are slightly different from the valuation of normal options. N: shares outstanding M: number of warrants γ: shares for each warrant. V : equity value K: strike price M γk: cash inflow when warrants are exercised. V + M γk: the equity value of the firm N + M γ: shares outstanding when warrants are exercised: he spot price corresponding to the warrant dilution = (V + M γk) / (N + M γ) = (V NK + NK + M γk) / (N + M γ) =[(V NK)/ (N + M γ) + K] he Payoff for warrant holders : γ { [(V NK)/ (N + M γ) + K] K}: = Nγ / (N + M γ) (V /N-K) he payoff for warrants holders: Nγ / (N + M γ) max(v /N-K, ) herefore, Nγ / (N + M γ) is the number of regular call options V /N is the spot price corresponding to the warrants. V = N S + M W V /N = S + M/N W W: the price of the warrant In Black-Scholes model: 1. Use S + M/N W as the spot price.. Multiply the value by Nγ / (N + M γ) 3
~ he volatility estimation: Volatility could be affected by the random arrival of new information or trading activities. According to some empirical research, we find that volatility is largely affected by trading activities. herefore, in estimating volatility, we should use actual trading days, instead of calendar days. Annualized volatility = volatility per day (5) 1/ ~ Volatility smile in stock options markets: Normal distribution V.S. Fat-tailed distribution ~ Volatility smirk in index options markets: Net buying pressure from institutional hedgers. 4
III. he impact of Dividends In reality, S EX_DIV =S D + tax effect. For European options: ~ Adjustment for dividend: Example: =6 months S =$4 D=$.5 in month. D=$.5 in month 5. R=9% K= $4 σ =.3 European call option: $3.67 S adjusted = 4.9741 = 39.59 ~ Adjustment for volatility: σ adjusted = σ S /(S -D) 5
For American options in a binominal tree: 1 3 n Call options ~ Not exercise the option: C S(t n ) D n Ke -r(-tn) S(t n ) D n Ke -r(-tn) S(t n ) K K (1- e -r(-tn) ) D n K (1- e -r(tn - tn-1) ) D n-1 Kr(t i+1 -t i ) D n-1 Exercise the option: K (1- e -r(-tn) ) < D n Put options ~ K (1- e -r(tn - tn-1) ) < D n-1 Kr(t i+1 -t i ) < D n-1 Example of early exercise American call options: =6 months S =$4 D=$.5 in month. D=$.5 in month 5. R=9% K= $4 σ =.3 K (1- e -r(tn - tn-1) ) = 4 (1- e -9% (5- )/1 ) =.89 >.5 not exercise in month. K (1- e -r( - tn) ) = 4 (1- e -9% (6-5)/1 ) =.3 <.5 exercise in month 5. According to Black s Approximation for American call options: If Option price is exercised in month 5: =5 months S =$4 D=$.5 in month. R=9% K= $4 σ =.3 S adjusted = 4.5 e -9% /1 = 39.574 Call price = 3.5 If Option price is exercised in month 6: Call price = 3.67 hus, the American call price should be $3.67 6