# Option Pricing. Chapter 4 Including dividends in the BS model. Stefan Ankirchner. University of Bonn. last update: 6th November 2013

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1 Option Pricing Chapter 4 Including dividends in the BS model Stefan Ankirchner University of Bonn last update: 6th November 2013 Stefan Ankirchner Option Pricing 1

2 Dividend payments So far: we assumed that the underlying assets do not pay any dividends. Question: How do the option pricing formulas change if we allow for dividend payments? Two models for dividend paying stocks: continuously paying dividends lump payments of dividends Stefan Ankirchner Option Pricing 2

3 Continuous dividend payments Consider a stock, or more realistically a basket of stocks, with value S t at time t, providing continuous dividend payments at a constant rate q [0, 1). The latter means that holding Γ shares between t and t + dt entails a dividend yield of ΓqS t dt. Stefan Ankirchner Option Pricing 3

4 Impact of continuous dividend payments Impact of continuously payed dividends: Price dynamics: The dividend payments reduce the stock value. The dynamics are given by ds t = µs t dt + σs t dw t qs t dt. Self-financing condition: Let (, η) be a trading strategy, and V t = (t)s t + η(t)st 0 the portfolio value. The portfolio is self-financing if dv t = (t)ds t + η(t)ds 0 t + (t)qs t dt. Stefan Ankirchner Option Pricing 4

5 Continuous dividend payments Consider a European option with payoff h(x) at expiration T. We follow the steps of our recipe to derive the arbitrage free pricing formula. 1) Assume that the option is replicable and that the time t fair value / replicating portfolio value is equal to f (t, S t ). 2) Applying Ito s formula to f (t, S t ) implies df (t, S t ) = f t (t, S t )dt + f x (t, S t )(µ q)s t dt + f x (t, S t )σs t dw t f xx(t, S t )S 2 t σ 2 dt. (1) The self-financing condition yields df (t, S t ) = (t)µs t dt + (t)σs t dw t + (f (t, S t ) t S t )rdt. (2) Stefan Ankirchner Option Pricing 5

6 Continuous dividend payments 3) By matching Equations (1) and (2), it must holds that (t) = f x (t, S t ), and f t (t, S t ) + f x (t, S t )(µ q)s t f xx(t, S t )St 2 σ 2 = f x (t, S t )µs t + (f (t, S t ) t S t )r. Thus f (t, x) has to satisfy the PDE f t + (r q)xf x σ2 x 2 f xx r f = 0, with terminal condition f (T, x) = h(x). Stefan Ankirchner Option Pricing 6

7 Continuous dividend payments 4) Applying discounted Feynman-Kac we obtain the solution f (t, x) = e r(t t) E t,x [h(x T )] where dx s = (r q)x s ds + σx s dw s. Remark: Note that the drift term of X t depends on q. Stefan Ankirchner Option Pricing 7

8 Example: Call option pricing formula Consider a European call, with strike K and maturity T, on an asset paying continuously dividends at a constant rate q. The time t arbitrage free price is given by BS call(s, K, T t, σ, r, q) = e q(t t) SΦ(d 1 ) Ke r(t t) Φ(d 2 ), (3) where d 1 = log ( ) S K + (r q + σ 2 )(T t) 2 σ, T t d 2 = d 1 σ T t. Proof. Stefan Ankirchner Option Pricing 8

9 Conclusion: Continuous dividend payments We get the same call option value for a stock traded at e q(t t) S t, with no dividend payments, a stock traded at S t, with a continuous dividend yield at a rate of q. Explanation: Stock price is reduced by the dividend payments. Lower growth rate is equivalent to starting from a lower stock price. Dividend payments don t influence the value of a replicating portfolio. Stefan Ankirchner Option Pricing 9

10 Model assumptions: n dividend payments up to expiration T, dividend payment dates are known in advance: 0 < t 1 < < t n < T. a j (0, 1) is the dividend as a percentage of the stock price at time t j, i.e. dividend payment at t j = a j S tj. Stefan Ankirchner Option Pricing 10

11 Model assumptions cont d Stock price jumps at payment dates: S(t j ) = (1 a j )S(t j ). price dynamics between payment dates: ds t = µs t dt + σs t dw t. Notice that the stock price at time T is given by S T = S 0 { Π n j=1 (1 a j ) } e σw T +(µ σ 2 /2)(T t). At time T the stock price coincides with the price of an asset satisfying the SDE d S t = µ S t dt + σ S t dw t, with initial condition S 0 = S 0 Π n j=1 (1 a j). Stefan Ankirchner Option Pricing 11

12 Self-financing condition Consider a portfolio with a position of stock at time t j. The dividend payments entail: a dividend yield of (t j )a j S(t j ), (4) the value of the stock position marked-to-market declines by (t j )(S(t j ) S(t j )) = (t j )a j S(t j ). Thus the total portfolio value remains unchanged! Stefan Ankirchner Option Pricing 12

13 A single dividend payment Consider a European option with payoff h(x) at expiration T. We follow our recipe to derive the pricing formula. For simplicity: assume that there is only one dividend payment. After the dividend payment: Suppose that the value of a portfolio replicating the option is given by f (t, S t ) for all t [t 1, T ]. As before, we obtain that f satisfies the Black-Scholes PDE on [t 1, T ] f (t, x) = e r(t t) E t,x (h(x T )), for t [t 1, T ], where dx t = rx t dt + σx t dw t. Stefan Ankirchner Option Pricing 13

14 A single dividend payment At the dividend payment date: Denote by g(t, S t ) the value of the replicating portfolio before t 1. Since the portfolio value remains unchanged at t 1, we have g(t 1, S t1 ) = f (t 1, S t1 ) = f (t 1, (1 a 1 )S t1 ). Thus g(t, x) has to satisfy g(t 1, x) = f (t 1, (1 a 1 )x). Stefan Ankirchner Option Pricing 14

15 A single dividend payment Before the dividend payment date: On [0, t 1 ] we need to replicate f (t 1, (1 a 1 )S t1 ). To this end show that g(t, x) satisfies the Black-Scholes PDE with terminal condition g(t 1, x) = f (t 1, (1 a 1 )x), apply Discounted Feynman-Kac to derive for all t [0, t 1 ] g(t, x) = e r(t 1 t) E t,x [f (t 1, (1 a 1 )X t1 )], where dx t = rx t dt + σx t dw t. Stefan Ankirchner Option Pricing 15

16 A single dividend payment Observe that for t t 1 g(t, x) = e r(t 1 t) E t,x [f (t 1, (1 a 1 )X t1 )] = e r(t 1 t) E t,(1 a1 )x[f (t 1, X t1 )] = e r(t 1 t) E t,(1 a1 )x[e r(t t 1) E t1,x t1 (h(x T ))] = e r(t t) E t,(1 a1 )x[h(x T )]. Observation: g(t, x) is also the value of an option on an asset with a price of (1 a 1 )x at time t, no dividend payments up to T. Stefan Ankirchner Option Pricing 16

17 Call option value when a single dividend payment Question: What is the price of a call if one dividend payment up to expiration is expected? The value coincides with the price of a call on a non-dividend paying asset with current price reduced by the factor (1 a 1 ). The price of a call with strike K and exp. T is given by S (1 a 1 )Φ(d 1 ) Ke r(t t) Φ(d 2 ), where d 1 = log ( ) S K + log(1 a1) + (r + σ2 )(T t) 2 σ, T t d 2 = d 1 σ T t. Stefan Ankirchner Option Pricing 17

18 Call option price when finitely many dividends Consider a stock paying dividends of a j S tj at several times t 1 < < t n between t and T. The arbitrage free price of a call option on this stock, with strike K and maturity T, is given by S Π n j=1(1 a j ) Φ(d 1 ) Ke r(t t) Φ(d 2 ), where d 1 = log ( ) S K + n j=1 log(1 a j) + (r + σ2 )(T t) 2 σ, T t d 2 = d 1 σ T t. Stefan Ankirchner Option Pricing 18

19 Conclusion: finitely many dividend payments We get the same call option value for a stock traded at Π n j=1 (1 a j) S, with no dividend payments, a stock traded at S, with dividend payments a 1,..., a n up to expiration T. Rule of thumb for pricing European options with dividends: Reduce the starting price by the dividend payments up to maturity, and then use pricing formulas for non-dividend paying assets. Stefan Ankirchner Option Pricing 19

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