Question

The current price of a non-dividend-paying stock is $32.59 and you expect the stock price to either go up by a factor of 1.397 or down by a factor of 0.716 over the next 0.7 years.

A European put option on the stock has a strike price of $33 and expires in 0.7 years. The risk-free rate is 3% (annual, continuously compounded).

Part 1. What is the option payoff if the stock price goes down?

Part 2. What is the risk-neutral probability of an up movement?

Part 3. What is the value of the option?

Answer 1

Particulars	Amount
Stock Price	$                32.59
Strike Price	$                33.00
Upside price	$                33.99
Down side Price	$                31.87
Risk Free Rate per period	2.10%
Rater per anum	3.00%
Time period in Years	0.7000
Risk Nuetral Prob to go upside ( P ):
P = [ e^rt - d ] / [ u - d ]
d = Down side price / Stock Price
u = Upside Price / Stock Price
e^rt :
= e^0.03 * 0.7
= e^0.021
= 1.0212
d = $ 31.874 / $ 32.59
= 0.978
u = $ 33.987 / $ 32.59
= 1.0429
P = [ e^rt - d ] / [ u - d ]
= [ e^0.03 * 0.7 - 0.978 ] / [ 1.0429 - 0.978 ]
= [ e^0.021 - 0.978 ] / [ 1.0429 - 0.978 ]
= [ 1.0212 - 0.978 ] / [ 1.0429 - 0.978 ]
= [ 0.0432 ] / [ 0.0648 ]
= 0.6662
Risk Nuetral Prob to go Downside side (1 - P ):
1 - P = 1 - 0.6662
= 0.3338
Value of Put after 0.7 Years :
Future SP	Strike Price	Exercise/ Lapse	Prob	Value of Put	Expected Vp
$                                     33.99	$                33.00	Lapse	0.6662	$                        -	$                 -
$                                     31.87	$                33.00	Exercise	0.3338	$                    1.13	$            0.38
Value of Put after 0.7 Years					$            0.38
Value of put Today:
Value of Put after 0.7 Years * e^-rt
= $ 0.3758588 * e^-0.03 * 0.7
= $ 0.3758588 * e^-0.021
= $ 0.3758588 * 0.9792
= $ 0.368
Value of Put is $0.37