Question

Consider a two-step binomial tree where the spot price of the underlying is currently $20. In

each of the two time steps, the spot price may go up by 10% or down by 10%. Suppose that

each time step is 3 months long and the risk-free rate is 12% per year.

(a) Value a 6-month European call with a strike price of $21.

(b) How would your analysis change if you were valuing an American call instead?

2

Answer 1

(a) Value of 6-month European call = $1.2822

(b) The basic difference between European option and American option is that American option can be exercised before its maturity. Under Binomial option pricing model, An american call option can be exercised at early stage if at that stage payoff of the option is greater than option value at that stage. In given case, American option would not be exercised at early stage because its payoff is lower than its value at each early nodes. Thus, Both style call option would have same value i.e $1.2822.

Please refer to below spreadsheet for calculation and answer. Cell reference also provided.

D E F G H I $20.00 $21.00 12% Current Price of Stock(SO) Strike Prie (K) volatility (0) risk free rate(r) p.a Expiry period Time each period (t) up factor(u) down factor (d) Risk netural Probability Plu) P(d) Time value Factor (eart) 0.50 0.25 1.1000 0.9000 0.6523 0.3477 1.0305 Stock Price at each Period Period=0 Period = 1 Period = 2 $24.2000 $22.0000 $20.0000 $19.8000 $18.0000 $16.2000 28 (a) (b) Value of European Call option at each Node Period =0 Period = 1 Period = 2 Value of American Call option at each Node Period =0 Period = 1 Period = 2 D $3.2000 | $3.2000 $2.0256 $2.0256 $1.0000 $1.2822 $0.0000 $1.2822 $0.0000 39 $0.0000 $0.0000 41 $ 0.0000 $0.0000

Cell reference -

IEF Current Price of Stock(SO) Strike Prie (K) volatility (0) risk free rate(r) p.a Expiry period Time each period (t) up factor(u) down factor (d) Risk netural Probability Plu) Pld) Time value Factor (eart) 0.12 =6/12 =3/12 =1+10% =1-10% =(EXP(C4*C6)-C8)/(07-08) =1-09 =EXP(C4*C6) 9 10 Stock Price at each Period Period = 1 Period =0 Period = 2 Period = 2 =C19*$C$ 7 =B21*$C$7 -C1 =C23*$C$7 =B21*$C$8 =C23*$C$8 (b) Value of European Call option at each Node Period = 1 Period =0 Period = 1 Value of American Call option at each Node Period =0 Period = 1 Period = 2 Period = 2 Period = 2 =MAX(D17-$C$2,0) =MAX(D17-$C$2,0) =(D33*$C$9+D37*$C$10/$C$11 =(133*$C$9+137*$C$10/$C$11 =MAX(C19-$C$2,0) =(C35*$C$9+C39*$C$10/$C$11 =MAX(D21-$C$2,0) =(H35*$C$9+H39*$C$10)/$C$11 =MAX(D21-$C$2,0) =(D37*$C$9+D41*$C$10/$C$11 =(137*$C$9+141*$C$10/$C$11 =MAX(D25-$C$2,0) =MAX(D25-$C$2,0)

Hope this will help, please do comment if you need any further explanation. Your feedback would be highly appreciated.