A stock is trading at $69.50

and does not pay a dividend.

The implied (expected) volatility is 35%.

Interest rates are 6%.

You buy a call with 60 days to expiration and a strike price of $50.

Management announces a dividend to be paid in 30 days. What happens to the value of the call?

Question

A stock is trading at $69.50

and does not pay a dividend.

The implied (expected) volatility is 35%.

Interest rates are 6%.

You buy a call with 60 days to expiration and a strike price of $50.

Management announces a dividend to be paid in 30 days.  What happens to the value of the call?

BlackTom AI · Accepted Answer

A dividend reduces the stock's expected forward price, so the call’s value falls.

1) Relevant concepts
- Black–Scholes pricing: for a non-dividend-paying stock, C = S0 N(d1) − K e−rT N(d2). For a known discrete dividend D at time td, the spot is adjusted by the present value of the dividend: use S0 − PV(D).
- Intuition: a call gains from a higher underlying; a dividend reduces the underlying at the ex-dividend date, lowering expected future stock price and thus the call value. Volatility and interest rates here are unchanged.

2) Step-by-step reasoning and 3) calculations
Given: S0 = 69.50, K = 50, σ = 0.35, r = 0.06, T = 60/365 ≈ 0.16438, dividend announced at td = 30/365 ≈ 0.08219. The exact dividend amount is not given, but to demonstrate direction and magnitude, assume D = $1 (any positive D gives same sign).

PV(D) = D e−r·td = 1 · e−0.06·0.08219 ≈ e−0.00493 ≈ 0.9951.
Without dividend:
- ln(S/K) = ln(69.5/50) ≈ 0.3290.
- (r + 0.5σ^2)T ≈ 0.12125·0.16438 ≈ 0.01993.
- σ√T = 0.35·√0.16438 ≈ 0.1419.
- d1 = (0.3290 + 0.01993)/0.1419 ≈ 2.459, d2 ≈ 2.317.
- N(d1) ≈ 0.993, N(d2) ≈ 0.990.
- C_no_div ≈ 69.5·0.993 − 50·e−0.06·0.16438·0.990 ≈ 69.033 − 49.002 ≈ $20.03.

With dividend (adjust spot S' = 69.5 − 0.9951 = 68.5049):
- ln(S'/K) ≈ 0.3155; numerator = 0.3155 + 0.01993 = 0.3354.
- d1 ≈ 0.3354/0.1419 ≈ 2.364, d2 ≈ 2.222.
- N(d1) ≈ 0.990, N(d2) ≈ 0.987.
- C_with_div ≈ 68.5049·0.990 − 49.5085·0.987 ≈ 67.819 − 48.860 ≈ $18.96.

Change ≈ 18.96 − 20.03 ≈ −$1.07 (a decrease). Approximate sensitivity equals −PV(D)·N(d1) (negative).

4) Why the correct answer is correct
The dividend reduces the expected future stock price (forward), lowering the intrinsic/expected payoff of the call; hence the call’s fair value decreases.

5) Why other answers would be incorrect
Answers saying the call “Goes up” or “Stays the same” are wrong because a positive dividend reduces the underlying’s expected price. While an American call may have some early-exercise considerations, that only mitigates loss slightly; it does not make the announcement increase the call’s value.

6) Conclusion
The call’s value goes down, which matches the provided answer: Goes down.

类似问题

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