Question

Return to Example 16.1. Use the binomial model to value a one-year European put option with exercise price $110 on the stock in that example. Does your solution for the put price satisfy put-call parity?

Short answer

Answer

The call parity is satisfied (with just minor errors)

Step-by-step solution

Given information

$P_{uu}$= $0 (because the stock price uu$S_0$= $121)

$P_{ud}$=$5.50 (because the stock price ud$S_0$= $104.50 which is less than $110 exercise price)

Calculation of hedge ratio and portfolios worth $121

Hedge ratio (H) = ($P_{uu}-P_{ud}$)/($uu{S}_0-ud{S}_0$)

= (0 – 5.50)/(121 – 104.50)

= -1/3

Thus portfolios worth $121 at option expiration regardless of stock price:

Riskless portfolio	ud$S_0$= $104.50	uu$S_0$= $121.00
Buy 1 share at u$S_0$=$110	$104.50	$121.00
Buy 3 puts at $P_u$	16.50	0.0
Total	$121.00	$121.00

Theportfolio must have current market value equal to present value at $121

110 + 3$P_u$= $121/ 1.05 = $115.238 = $1.746

Calculation of hedge ratio and portfolios worth $110

From this point dS0 = $95

$P_{uu}$= $0 (because the stock price uu$S_0$= $121)

$P_{ud}$=$5.50 (because the stock price du$S_0$= $104.50 or rise to the value of P$P_{ud}$= $19.75 as the stock price ddS0 = $90.25)

Therefore the hedge ratio at this point = -1.0

$H=P_{du}-P_{dd}/du{S}_o-dd{S}_0$

= $5.50 - $19.75 / $104.50 - $90.25

= -1.0

Thus portfolios worth $110 at option expiration regardless of stock price:

Riskless portfolio	dd$S_0$= $90.25	du$S_0$= $104.50
Buy 1 share at d$S_0$=$95	$90.25	$104.50
Buy 1 share at $P_d$	19.75	5.50
Total	$110.00	$110.00

Theportfolio must have current market value equal to present value at $110

95 + $P_0$= $110/ 1.05 = $104.762 = $9.762

Calculation of P using values of Pu and Pd

From this point d$S_0$= $95

$P_d$= $9.762 (rise)

$P_u$=$1.746 (fall)

Therefore the hedge ratio at this point = -1.0

H = $P_u$– $P_d$/ u$S_0$–d$S_0$

= $1.746 - $9.762 / $110 - $95

= -0.5344

Thus portfolios worth $60.53 at option expiration regardless of stock price:

Riskless portfolio	d$S_0$= $95	u$S_0$= $110.00
Buy 0.5344 share at S=$100	$50.768	$58.784
Buy 1 put at P	9.762	1.746
Total	$60.530	$60.530

Theportfolio must have current market value equal to present value at $60.53

$53.44 + P = $60.53/ 1.05 = $57.648 = $4.208

Validation for call parity

C = $4.434

P = C + PV(X) – S

$4.208 = $4.434 + ($110 / $1.{05}^2$) -$100

That implies that the call parity is satisfied (with just minor errors)