Pricing American Options Dilip Madan Department of Finance Robert H. Smith School of Business American Call Options We have seen as a consequence of...

Pricing American Options Dilip Madan Department of Finance Robert H. Smith School of Business American Call Options We have seen as a consequence of call lower bounds that in the absence of dividends there is no early exercise for an American call option. We now consider early exercise in the presence of a …xed dividend of d t dollars at time t < Tthe option maturity. After time tthe stock has no dividends and there will be no early exercise. One may seek to exercise before the dividend date if the dividend is large enough. To determine how large we compare the cash ‡ow on exercise predividend to what one has post dividend. Time Value of Call Option with no dividends In this regard it is useful to de…ne the time value of an option with no dividends. The time value of an option is the excess of the value of the option over giving up the option and commiting to exercise now. The latter is the value of a forward contract.

Hence T V C (t) = C(S (t) ; t ;K; T ) (S (t) K B (t; T )) C (S (t) ; t ;K; T ) = S(t) K B (t; T ) + T V C (t) Figure 1: Time Value of Call Option We present a graph of this time value function As may be observed this time value is always strictly positive as it exceeds the gap between the forward and the payo¤ for large values of the spot. We now recognize that prior to the dividend the stock price is a cum dividend price that we write as80859095100105110115120­20­15­10­50510152025Stock PriceValuesTime Value of CallCall ValueVa lu e  o f F o r w a r dCall Pay offTime Value S c ( t) while after twe have ex-dividend price written as Se (t) with the relation at tof S c ( t) = Se (t) + d t: If we exercise predividend we receive the value Sc ( t) K If we fail to exercise we have the value C(S e (t) ; t ;K; T ) We now note that Sc ( t) K =Se (t) + d t K while C (S e (t) ; t ;K; T ) =Se (t) K B (t; T ) +T V C (t) Early exercise is then called for when dt > K (1 B(t; T )) + T V C (t) The dividend must be large enough to compensate for interest lost on early payment of strike plus loss of time value on option. If interest rates are low and the spot is high the right hand side is small and early exercise gets more likely for any given dividend. American Put Options We consider put options exercise with no dividends and then with dividends. The time value of a put option is the excess of the put value over the value of commiting to exercise now. The latter is the forward sale for the strike.

Hence T V P (t) = P(S (t) ; t ;K; T ) (K B (t; T ) S(t)) P (S (t) ; t ;K; T ) = K B(t; T ) S(t) + T V P (t) We present a graph of the time value for a European put option. For low values of the spot time value vanishes making exercise likely especially if rates are high and the gap between payo¤ and European put value is high. If we exercise we receive the value K S(t) If we hold the option we have P (S (t) ; t ;K; T ) =K B(t; T ) S(t) + T V P (t)5060708090100110120­30­20­1001020304050Stock PriceValuesTime Value of PutPut ValueValue of ForwardPu t Pa y o ffTime Value of Put So we exercise if K(1 B(t; T ))> T V P (t) or the interest earned on the early receipt of the strike dominates the time value of the put option. In the presence of dividends in the interim at say time u; t < u < T in the amountd u the value of the forward changes to the present value of the strike less the value of the forward stock or K B(t; T ) (S (t) d uB (t; u ) ) The exercise condition is revised to K(1 B(t; T ))> T V P (t) + d uB (t; u ) The early exercise interest earnings must also com- pensate for loss of dividends on sale of stock. Pricing American Calls on a Tree We consider a two period one year tree with a 5 dollar dividend at time 1the half year point. The tree is recombining before this time after this time but splits at time 1: We present the tree as follows.

ex-div 5 dollars 117.52 132.23 112.52 100 99.65 88.57 98.21 83.57 74.01 R (h ) = 1 :0304 ; D=:8857 ; U= 1:1752 Call Values The American Call Value Tree for a strike of K= 90 is 42:23 C u 1 9:65 C 0 8:21 C d 1 0 We work back from the end where we know values, but at each node we work out two values, exercise and holding value. The value of the option is the greated of the two and we note the decision related to the value. Time One Down State Consider time one down state. We have a risk neu- tral tree and we may use probabilities of :5 : The holding value is 1 1 :0304 1 2 8:21 = 3 :9839 The value of exercise is (88:57 90) + = 0 So we have Cd 1 = 3 :9839 and we hold the option. The value of mis m = 8 :21 0 98 :21 74 :01 = 0 :3393 and B= 3 :9839 :3393 83 :57 = 24 :3714 Time One Up State The holding value is 1 1 :0304 1 2 42 :23 + 1 2 9:65 = 25 :17 The exercise value is 117:52 90 = 27 :52 Hence Cu 1 = 27 :52 and we exercise. Time 0 The holding value is 1 1 :0304 1 2 27 :52 + 1 2 3:9839 = 15 :2872 The exercise value is 100 90 = 10 So C 0 = 15 :2872 and we hold the option with m =27 :52 3:9839 117 :52 88 :57 = 0 :8130 and B= 15 :2872 :8130 100 = 66 :0128 : American Put on a Tree Consider a put with strike 110on our two period tree. The tree is 138:10 117 :52 100 104 :08 88 :57 78:44 The put value tree is 0 P u 1 P 0 5 :92 P d 1 31:56 Time One Down State The holding value is 1 1 :0304 1 2 5:92 + 1 2 31 :56 = 18 :1871 The exercise value is (110 88 :57) = 21 :43 So Pd 1 = 21 :43 and we exercise. Time One Up State The holding value is 1 1 :0304 1 2 5:92 = 2 :8727 The exercise value is 0 So Pu 1 = 2 :8727 and we hold with m = 5:92 138 :10 104 :08 = :1740 and B= 2 :8727 + :1740 117 :52 = 23 :3212 Time 0 The holding value is 1 1 :0304 1 2 2:8727 + 1 2 21 :43 = 11 :7928 The exercise value is (110 100) = 10 : Hence P 0 = 11 :7928 and we hold with m =2 :8727 21 :43 117 :52 88 :57 = 0:6410 and B= 11 :7928 + :6410 100 = 75 :8928 : American Option Values on a Large Grid We present a typical exercise region for an American Put Option. Value of American Put near exercise region equals value of exercise, hence you do not need to know the exercise region, as you exercise when the value of put is at value of exercise. We graph these values for our one year 90 put. Figure 2: Exercise Region for one year 90 put with r=.06, sg=.2.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 76 78 80 82 84 86 88 90 92 T ime St o ck  Price Exerci se B oundary For A meric an Put r=.06 s g=.2 T =1 K =90 Exerci se Region Holding or Continuati on Region Exerci se oc c urs  on fi rst pas s age into E xerc is e Region Figure 3: American Put Option Values808590951001051101151201250510152025Spot PriceOption PriceValue of Americ an Put as  a func tion of SpotValue at MaturityValue at 9 monthsValue at 6 monthsValue at 3 months The American Put Premium The American Put Option Premium with no inter- mediate dividends P(S (t) ; t ;K; T ) p(S (t) ; t ;K; T ) is equal to the expected interest on the strike for the time spent by the stock price in the exercise region. Suppose we sell the American put for Pand we buy the European put for p:We ask what earnings or cash ‡ows are accessed by this position.

Suppose the stock is outside the exercise region to begin with. If it never enters the exercise region both puts expire unexercised and they both have the same cash ‡ow of zero. Next suppose the stock enters the exercise region at t < T : We exercise the American put, short the stock and receive the strike K:Now if the stock stays in the exercise region till Twe earn the interest on the strike for the time T tand at Tthe European put we sold is exercised, we pay out the strike, receive the stock and cancel the short stock position. Our earnings are the interest on the strike for the period T t:

Next suppose the stock has entered the exercise region t; we have shorted the stock, received the strike and at time u; t < u < T the stock leaves the exercise region.

We have earned the interest on the strike for the time u t:At time uwe spend the strike Kto buy back the stock for S u < K and we buy an American put for K S u:

The short stock position is cancelled, the strike is spent and we are back to holding an American put with the stock outside the exercise region. This is where we started and the argument repeats.

The di¤erence P pis then the expected interest on the strike for the time the stock spends in the exercise region.

If interest rates are low, the stock price is high relative to the strike, then this premium will be small.