Question

Assume that the price S of a risky asset follows a binomial model with S(0) =
$100, u = 10% and d = -10%. The underlying asset pays a dividend of $5 on the odd times, i.e., 1; 3; 5...,
and only if the price is strictly higher than $95. In this market, the risk-free rate is 0% (zero).

You are called to price a European call with strike price K = 87 and expiry date N = 3 with the additional
restriction that during the life of the call the stock price has not exceeded the value of $110.

Answer 1

solu:

Given

• current price S = 100
• risk free rate (r) = 0%

Assume: 1. a portfolio made up of one European call and h shares of the stock. where investor owns h shares and writes a call option

2. Since risk free rates is zero (ie r = 0), therefore pricing will be same for N =1 or 3

calculating price if goes up by u = Su or goes down by d = Sd

• u = 10% ; d = -10%
• => Su = S + u; Sd = S + d
• => Su = 100 + 10% +10% + 10% = 133.10
• => Sd = 100 + (-10%) + (-10%) + (-10%) = 72.9

calculating payoffs Cu and Cd for Su and Sd respectively given exercise price (k) = 87

• Cu = max (Su - E, 0)
• => max (133.10 - 87,0) = 46.10
• Cd = max (Sd - E, 0)
• => max (72.9 - 87, 0) = 0

Let h be the no. of shares an investor holds or buy. h is chosen so that the portfolio has the same price whether the stock price goes up or goes down. The value of h that achieves this condition is given by

h = (C_u - C_d) / S (u - d)

=> (46.10 - 0) / (133.1 - 72.9)

=>46.1 / 60.2 = 0.766

calculate the probability of an up movement (p) using:

p = r - d / u - d

=> 0 - (-0.1) / 0.1 - (-0.1)

=>0.1 / 0.2 = 0.5

=> therefore downside probability (1-p) = 1-0.5 = 0.5

Let C represents the value of the call

C = (p*Cu + (1-p)*Cd) /(1+r)

=> [(0.5 * 46.1) + (0.5 * 0)] / (1+0)

=> 23.05