Question

In Financial Derivatives

Consider the two-period, Binomial Options Pricing Model. The current stock price S= $105 and the risk-free rate r = 3% per period (simple rate). Each period, the stock price can go either up by 10 percent or down by 10 percent. A European call option (on a non-dividend paying stock) with expiration at the end of two periods (n=2), has a strike price K = $100. The risk-neutral probability of an “up” move is q = (R-D)/(U-D), where R= 1+r.

a). Set out the stock price tree (e.g. in a table), calculate the (no-arbitrage) “fair” price of the call and explain the meaning of “risk neutral valuation” (RNV).

b). Calculate the hedge ratio at t=0 and explain how you can hedge 100 written calls at t=0. Calculate the value of the hedge portfolio at nodes “U” and “D” and hence show that the hedge-portfolio earns the risk-free rate over the first period (i.e. along the path from node t=0, either to node-D or node-U).

c). Calculate the hedge ratio at node-U and explain what this implies for delta hedging (over the next period).

Answer 1

a) The stock price tree is as shown below

			Value of call option
		127.05	27.05
	115.5	103.95	3.95
105	94.5	85.05	0
t=0	t=1	t=2

Risk neutral probability of upmove p = (1.03- 0.9)/(1.1-0.9) = 0.65

So, value of option at t=1 when the stock price is 115.5

= (p*value of call when stock price is 127.05 + (1-p)*value of call when stock price is 103.95)/1.03

=(0.65*27.05+0.35*3.95)/1.03

=18.41262

So, value of option at t=1 when the stock price is 94.5

= (p*value of call when stock price is 103.95 + (1-p)*value of call when stock price is 85.05)/1.03

=(0.65*3.95+0.35*0)/1.03

=2.492718

So, value of option at t=0 when the stock price is 105

= (p*value of call when stock price is 115.5 + (1-p)*value of call when stock price is 94.5)/1.03

=(0.65*18.41262+0.35*2.492718)/1.03

=12.47

the (no-arbitrage) “fair” price of the call option is $12.47

Risk neutral valuation means that the options can be priced as if there was a riskless portfolio and the options payoff are similar to that portfolio. In that case, the risk neutral valuation gives the price of the options as being equal to the current value of the portfolio (=value of the future portfolio discounted at risk free rate)

b) Hedge ratio at t=0

= ( (value of option in upmove at t=1) - value of option in downmove at t=1) / (stock price in upmove at t=1 - stock price in downmove at t=1)

=(18.41262-2.492718)/(115.5-94.5)

=0.758

So, for 100 written calls , one has to purchase 75.8 stocks to perfectly hedge the position

Value of the portfolio today = 75.8*105 - 100*12.47 = $6712

Now, if the stock makes an upmove

Value of the portfolio = 75.8*115.5 - 100*18.41262 = $6914

Return earned on the portfolio = (6914-6712)/6712 = 0.03 or 3%

& if the stock makes a downmove

Value of the portfolio = 75.8*94.5 - 100*2.492718 = $6914

Return earned on the portfolio = (6914-6712)/6712 = 0.03 or 3%

So, it is clearly shown that the hedged portfolio earns a risk free return of 3% over the 1st period no mattter what

c) Hedge ratio at node U (when stock price is $115.5)

= (27.05-3.95)/(127.05-103.95)

=1

So, the hedge ratio has changed from 0.758 to 1

So now to hedge 100 written calls,one has to buy 100 shares to cover the exposure fully

This leads to delta hedging as it is clear from the tree that in the next period the options are going to be exercised at both nodes. So 100 shares must be available to satisfy the written calls. Thereby hedging the exposure