Question

Consider a 1-year European call option on 100 shares of SPY with a strike price of $290 per share. The price today of one share of SPY is $285. Assume that the annual riskless rate of interest is 3%, and that the annual dividend yield on SPY is 1%. Both rates are continuously compounded. Finally, SPY annual price volatility is 25%. In answering the questions below use a binomial tree with two steps. a) Compute u, d, as well as p for the binomial model. b) Value the option today using the binomial tree. c) How would you hedge today a long position in this option?

Answer 1

	If the stock is paying continuous dividend yield q,
	a=e^(r-q)*delta t
	So , a=e^(3%-1%)*0.5
	a=e^0.02*0.5
	a=1.01005
	u= e^sigma*Sq rt of delta t
	so u=e^0.25*Sqrt0.5
Ans a	or u=1.1934
	d= e^-sigma*Sq rt of delta t =1/u
Ans a	d=1/1.1934=0.8380
	p= (a-d)/(u-d)
Ans a	p=(1.01005-0.8380)/(1.1934-0.8380)=
	p=0.4842
	Option Price at step A

	f=e^-2rt [ p^2*fuu + 2*p(1-p)*fud + (1-p)^2*fdd ]			Binomial Tree		Step C
	f0=e^-2*0.03*0.5[0.4842^2*115.898+2*0.4842*(1-0.4842)*0+(1-0.4842)^2*0]		delta t=0.5 years each	Suu	405.898
	=1.0305*56.118			Step B		fuu	115.898
	or f0=28				Su	340.119
Ans b	So value of option today =$28 per share			fu	56.966
	Option Price at Sept B =
	fu= e^-rT[p*fuu +(1-p)*fdd]		Step A				Sud	285.0197
	fu=e^-0.03*0.5*[0.4842*115.898+(1-0.4842)*0]	S0	285			fud	0
	fu=56.966	F0	28
Ans c					Sd	238.83
	A long position in the option can be hedged by buying a put option of it.			fd	0
	We can short sale the put option, invest the amount at risk free rate.					Sdd	200.1395
	During exercise , if the share price rises above $290 we can close the position purchase back at $290 per share					fdd	0
	and sale back at price higher than $290.