Question

A stock price is currently $50. A stock price is currently $50. Over each of the next two three-month periods it is expected to go up by 6% or down by 5%. The risk-free interest rate is 5% per annum with continuous compounding. Use two-period binomial models to value the six-month options on this stock. Remember to show detailed calculations of the option value at each node.

(a) What is the value of a six-month European call option with a strike price of $51?

(b) What is the value of a six-month European put option with a strike price of $51? Verify that the European call and European put prices satisfy put–call parity.

(c) If the put option in part (b) of this question were American, would it ever be optimal to exercise it early at any of the nodes on the tree?

Answer 1

a.

Value of European Call [Method 1: See Note 3] (or from image)	$                                                 1.63
Value of European Call [Method 2: See Note 3]	$                                                 1.63
Value of European Call [Method 3: See Note 3]	$                                                 1.64

b.

Value of European Put [As seen from the image]

$                                                 1.37

Put-Call Parity
Spot + Put = Call + Exercise Price x e^-rt
$50 + $1.37 = $1.63 + $51 x 1 / 1.0253
$51.37 = $1.63 + $49.74 = $51.37. Hence verified.

c. If the put option is American, it is beneficial to exercise it in node C, as at C the pay off would be $3.5 being $51 - $47.5, while the present value at node C in case of European option is $2.866.

Notes:

Note 1: Basic Data
Spot Price	$                                              50.00
Strike Price	$                                              51.00
No. of periods	2
Duration of each period in years	1/4
Total duration in years [No. of periods x Duration of each period]	0.5
Percentage increase per period	6%
Percentage decrease per period	5%
Risk free rate	5%
Maximum price after one period	$                                              53.00	Spot + Increase per period
Minimum price after one period	$                                              47.50	Spot - Decrease per period
Maximum price after end of periods	$                                              56.18
Minimum price after end of periods	$                                              45.13
Profit at the end of the periods [used in Note 3]	$                                                 5.18
Note 2: Calculation of Probability
er t	e^(0.05x0.25)
i = Value of er t = ( 1+ (r x t) )	1.01250	Method 1 [Approximate]
i = Value of er t = ( (1+r) + (r x r)/2 )^t	1.01257	Method 2 [More accurate]
i = Value of er t using Excel formula: =EXP(r x t)	1.01258	Method 3 [Most accurate]
u	1.06	1 + 6%
d	0.95	1 - 5%
Probability [(i - d) / (u - d)]
=      (1.0125-0.95)/(1.06-0.95)	0.5682	i as per Method 1
Probability [(i - d) / (u - d)]
=      (1.0125733715245-0.95)/(1.06-0.95)	0.5688	i as per Method 2
Probability [(i - d) / (u - d)]
=      (1.01257845154063-0.95)/(1.06-0.95)	0.5689	i as per Method 3
Note 3: Call Option Valuation
Value = Profit x Probability for each period / i for each period
=      (5.18000000000001x0.568181818181818)/(1.0125)	$                                                 1.63	i as per Method 1
Value = Profit x Probability for each period / i for each period
=      (5.18000000000001x0.568848832040913)/(1.0125733715245)	$                                                 1.63	i as per Method 2
Value = Profit x Probability for each period / i for each period
=      (5.18000000000001x0.568895014005767)/(1.0125733715245)	$                                                 1.64	i as per Method 3

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