Option Pricing With Dividends

Option Pricing With Dividends Betuel Canhanga. Carolyne Ogutu. Analytical Finance I Seminar Report October, 13

Contents 1 Introduction Solution One: Include Any Dividends After Expiration 3.1 Expiry before the dividend......................................... 3.1.1 Examples............................................... 5. Expiry After Dividends........................................... 6..1 Case 1................................................. 6.3 Examples................................................... 6.3.1 Case................................................. 6.3. Examples............................................... 8 3 Treat All Dividends as Proportional 1 3.1 Expiry before dividend............................................ 1 3. Examples................................................... 1 3.3 Expiry After Dividend............................................ 1 1

Chapter 1 Introduction Chash dividends issued by stocks have a big impact on their option prices. This is because the underlying stock price is expected to drop by the said dividend on the ex-dividend date. Options are valued taking into account the projected dividends receivable in the coming weeks and months up to the option expiration date.

Chapter Solution One: Include Any Dividends After Expiration.1 Expiry before the dividend Let S(t) = D e rt + (S D )e (r.5σ )t+σw(t) t < T 1 < t d (.1) τ = T t σ(w (T ) W (t)) = σ T tz At maturity, the stock price is given by S(T ) = D t e rτ + (S t D t )e (r.5σ )τ+σ τz (.) The payoff for a call option is Max(S T K, ). Therefore to calculate the call price we need to solve the following expression e rτ E Q [Max(S T K, ) F t ] (.3) Putting the expression for S T in. into.3 we have (without the discounting factor) E Q [ Max(D t e rτ + (S t D t )e (r.5σ )τ+σ τz K, ) F t ] Note: From here on, we assume the expect ion is in risk-neutral world and is with respect to the filtration at time t Recall that the payoff is always equal to zero when S T K and this further implies that the expectation will also be zero. With this result we have that D t e rτ + (S t D t )e (r.5σ )τ+σ τz < K (S t D t )e (r.5σ )τ+σ τz < K D t e rτ e (r.5σ )τ+σ τz < K Dterτ (S t D t) (.4) 3

Taking ln on both sides, we have (r.5σ )τ + σ τz < ln( K Dterτ (S t D t) ) σ τz < ln( K Dterτ (S t D t) ) (r.5σ )τ Z < ln( K D rτ te (S t D t ) ) (r.5σ )τ σ τ From the last expression, we let Z < and considering expectations for the positive part of the payoff, we have E [Max(S T K, )] = E [ Z I Z> d ] = = Solving the first integral we have that (D t e rτ + (S t D t )e (r.5σ )τ+σ τz K)φ(z)dz (D t e rτ K)φ(z)dz + (S t D t )e (r.5σ )τ+σ τz φ(z)dz (D t e rτ K) φ(z)dz But φ(z) is the standard normal distribution and we know by symmetry of the distribution φ(z)dz = P {Z > } = P {Z < d } = Φ(d ) This implies (D t e rτ K) φ(z)dz = (D t e rτ K)Φ(d ) (.5) Solving the second integral: Let y = z σ τ, therefore σ τ (S t D t )e rτ (S t D t )e (r.5σ )τ+σ τ(y+σ τ) 1 e (y+σ τ) dy σ τ e.5σ τ+σ τ(y+σ τ) 1 e (y+σ τ) dy Solving the exponents inside the integral, many terms will cancel out and thus we have (S t D t )e rτ σ τ 1 e y dy This results in (S t D t )e rτ Φ(d 1) d 1 = d + σ τ (.6) 4

Therefore, the expectation becomes E [Max(S T K, )] = (S t D t )e rτ + (D t e rτ K)Φ(d ) (.7) But the call price is given by (.3). Therefore replacing (.7) into (.3) we have the call price as C = (S t D t )Φ(d 1) + (D t Ke rτ )Φ(d ) (.8) Where d = ln( St Dt K D te rτ ) + (r.5σ )τ σ τ d 1 = d + σ τ.1.1 Examples 1. Consider a case where the stock price is 1, dividend is, strike price is 1, maturity time is six months, risk-free interest rate is 1%, volatility is 4%. Implementing equation (.8), the European call price is given by [1] 11.37461. Considering the same example but pricing in the binomial model. The European call option price is given by 7.78 53.68 c( depth + 1, depth 1) 4 4 11.38 17.69 5.1 6.66 8.79 1.4 38.7 14.78.78 4. 5.53 37.1 1.99 1 3 4 5 6 c(1, depth) Figure.1: Binomial Lattice for stocks with dividend payment after expiry of the option 5

. Expiry After Dividends..1 Case 1 S(t) = D e rt + (S D )e (r.5σ )t+σw(t) The solution is the same as solution (.8) but the times have changed in the implementation. (.9).3 Examples 1. Consider a case where the stock price is 1, dividend is, strike price is 1, maturity time is six months, risk-free interest rate is 1%, volatility is 4%. Implementing equation (.8), the dividend is paid 3 months before maturity and thus European call price is given by [1] 11.4881. The binomial equivalent to the above example is given by 74.57 c( depth + 1, depth 1) 4 4 1.54 1.7 4.1 34.3 7.98 5.79 15.87 31.56 1 3 4 5 c(1, depth) Figure.: Binomial Lattice for stocks with dividend payment before expiry of the option.3.1 Case (S D )e (r.5σ )t+σw(t) (.1) 6

S(T ) = (S t D t )e (r.5σ )(T t)+σ(w(t ) w(t)) Let τ = T t σ(w (T ) W (t)) = σ T tz At maturity, the stock price is given by S(T ) = (S t D t )e (r.5σ )τ+σ τz (.11) The payoff for a call option is Max(S T K, ). Therefore to calculate the call price we need to solve the following expression e rτ E Q [Max(S T K, ) F t ] (.1) Putting the expression for S T in (.11) into (.1) we have (without the discounting factor) E Q [ Max((S t D t )e (r.5σ )τ+σ τz K, ) F t ] Note: From here on, we assume the expection is in risk-neutral world and is with respect to the filtration at time t Recall that the payoff is always equal to zero when S T K and this further implies that the expectation will also be zero. With this result we have that Taking ln on both sides, we have (S t D t )e (r.5σ )τ+σ τz < K (S t D t )e (r.5σ )τ+σ τz < K e (r.5σ )τ+σ τz < K (S t D t) (r.5σ )τ + σ τz < ln( K (S t D t) ) σ τz < ln( K (S t D t) ) (r.5σ )τ Z < ln( K (S t D t ) ) (r.5σ )τ σ τ From the last expression, we let Z < and considering expectations for the positive part of the payoff, we have E [Max(S T K, )] = E [ Z I Z> d ] = ((S t D t )e (r.5σ )τ+σ τz K)φ(z)dz = (S t D t )e (r.5σ )τ+σ τz φ(z)dz Kφ(z)dz 7

Solving the second integral we have that K φ(z)dz (.13) But φ(z) is the standard normal distribution and we know by symmetry of the distribution φ(z)dz = P {Z > } = P {Z < d } = Φ(d ) This implies K φ(z)dz = KΦ(d ) (.14) Solving the first integral: Let y = z σ τ, therefore σ τ (S t D t )e rτ (S t D t )e (r.5σ )τ+σ τ(y+σ τ) 1 e (y+σ τ) dy (.15) σ τ e.5σ τ+σ τ(y+σ τ) 1 e (y+σ τ) dy Solving the exponents inside the integral, many terms will cancel out and thus we have This results in Therefore, the expectation becomes (S t D t )e rτ σ τ 1 e y dy (S t D t )e rτ Φ(d 1) d 1 = d + σ τ (.16) E [Max(S T K, )] = (S t D t )e rτ + KΦ(d ) (.17) But the call price is given by (.1). Therefore replacing (.17) into (.1) we have the call price as C = (S t D t )Φ(d 1) + Ke rτ Φ(d ) (.18) Where d = ln( St Dt K ) + (r.5σ )τ σ τ d 1 = d + σ τ.3. Examples 1. Consider a case where the stock price is 1, dividend is, strike price is 1, maturity time is six months, risk-free interest rate is 1%, volatility is 4%. Implementing equation (.18), the European call price is given by [1] 4.5885. The equivalent binomial option price is given by 8

4.85 c( depth + 1, depth 1) 4 4 4.1 7.45.78 13.3 1.55 3.5 3.9 6.15 1 3 4 5 c(1, depth) Figure.3: Binomial option price for option with dividend before inception 9

Chapter 3 Treat All Dividends as Proportional 3.1 Expiry before dividend Let S e (r.5σ )t+σw(t) S(T ) = S t e (r.5σ )(T t)+σ(w(t ) w(t)) (3.1) τ = T t σ(w (T ) W (t)) = σ T tz At maturity, the stock price is given by S(T ) = S t e (r.5σ )τ+σ τz (3.) The payoff for a call option is Max(S T K, ). Therefore to calculate the call price we need to solve the following expression e rτ E Q [Max(S T K, ) F t ] (3.3) Putting the expression for S T in (3.) into (3.3) we have (without the discounting factor) E Q [ Max(S t e (r.5σ )τ+σ τz K, ) F t ] Note: From here on, we assume the expect ion is in risk-neutral world and is with respect to the filtration at time t Recall that the payoff is always equal to zero when S T K and this further implies that the expectation will also be zero. With this result we have that S t e (r.5σ )τ+σ τz < K S t e (r.5σ )τ+σ τz < K (3.4) e (r.5σ )τ+σ τz < K S t 1

Taking ln on both sides, we have Solving the second integral we have that (r.5σ )τ + σ τz < ln( K S t ) σ τ Z < ln( K S t ) (r.5σ )τ Z < ln( K S t ) (r.5σ )τ σ τ K φ(z)dz (3.5) But φ(z) is the standard normal distribution and we know by symmetry of the distribution φ(z)dz = P {Z > } = P {Z < d } = Φ(d ) This implies K φ(z)dz = KΦ(d ) (3.6) Solving the first integral: Let y = z σ τ, therefore σ τ S t e rτ S t e (r.5σ )τ+σ τ(y+σ τ) 1 e (y+σ τ) dy (3.7) σ τ e.5σ τ+σ τ(y+σ τ) 1 e (y+σ τ) dy Solving the exponents inside the integral, many terms will cancel out and thus we have S t e rτ σ τ 1 e y dy This results in S t e rτ Φ(d 1) d 1 = d + σ τ (3.8) Therefore, the expectation becomes E [Max(S T K, )] = S t e rτ + KΦ(d ) (3.9) But the call price is given by (3.3). Therefore replacing (3.9) into (3.3) we have the call price as C = S t Φ(d 1) + Ke rτ Φ(d ) (3.1) Where d = ln( St K ) + (r.5σ )τ σ τ d 1 = d + σ τ 11

3. Examples 1. Consider a case where the stock price is 1, dividend is, strike price is 1, maturity time is six months, risk-free interest rate is 1%, volatility is 4%. [1] 13.5838. The binomial option for the above example is given by 76.7 c( depth + 1, depth 1) 4 4 1.91 1.68 4.15 35.16 8.6 54.9 16.43 3.69 1 3 4 5 c(1, depth) Figure 3.1: Binomial option price for option with proportional dividend 3.3 Expiry After Dividend (S D )e (r.5σ )t+σw(t) The solution to the expression (3.11) is the same as the one given in equation (.18) (3.11) 1

Bibliography [1] Hull, John C. Options, Futures and Other Derivatives [] Cox, J.C; S.A Ross; and M. Rubenstein Option Pricing: A Simplified Approach The Journal of Financial Economics 7 (1979) [3] Black, F. and M. Scholes The Pricing of Options and Corporate Liabilities The Journal of Political Economy 8 (1973) 13