Hello everyone!!I hope that my question is simple to answer. I'm reading volume 1 of Steven Shreve's Stochastic Calculus for Finance.On chapter 4, actually on section 4.2 he develops an algorithm for pricing non-path-dependent american derivatives. Let be the instrinsic value of the derivative security. Shreve's call the following formulasas the American Algorithm (for those who are not used to Shreve notation, we are into the binomial model, under risk neutral probabilites, $s$ is the stock price, $u$ is the up factorand $d$ the down factor)then he proceeds using g as , g(s) = K - s , and thus evaluates the fair price of an american put option as an example. Question 1) I wonder if this same algorithm is valid to calculate the fair price of an american call option.At first sight the answer seems to be yes since that section is about non-path-dependent american derivatives and the algoritm is called 'American algorithm'. But later, on section 4.5I learned that american call options (on stocks paying no dividend) have the same initial price as european call options (under the same parameters).And calculating the price of an american CALL option using the scheme of Fig 4.2.1 (page 91) using the American Algorithm and then using the result that it should be the same price as an european price, I found out that those prices are not the same. In fact, using the American algorithmthe price of the american call is $1.76 and using the result of section 4.5 the price of the american call is $1.2. I also note that Shreve's exercice 4.1 ask us to verify that V_S < V_P + V_C , where V_S the price of an straddle , V_P an american put and V_C an american call, (all evaluated at time 0)If you use the american algorithm to calculate V_C, this is not true (in fact, you get an equality!) and using that V_C is the same as the price of an european call, the inequality seems to hold. Shreve's also shows that using the American algorithm you can replicate and hedge a sort position on a american put option. I will do the math later, but I wonder whether is easy to see or not at a glance if that algorithm will work to hedge a position on a american call option. Thank you

an american call on a non-dividend paying stock is the same as a european call. you can use the Shreve algorithm to value both. should make no difference

Hello Dave. Thanks for your reply. I'm sorry, In fact, I did some miscalculations.On the other hand, I'm still confused about the inequality for the straddle option price. Working on Exercice 4.1, and I tested again using the data of example 4.2.1I only get the equality (I can post my calculations if you think that is needed.)I wonder if this is due a possible typo on the book where the strict inequality sign should be replaced by sign.If that is case I would like to if it is easy to think of an example where the strict inequality holds.

i have to say this beats me - i always thought a straddle was a call and put

I wonder if when considering American options, things might change a little bit so that we have a inequality rather than a equality.This question remains open here then.

I edited my earlier response because I think I understand the issue now. According to Shreve S < C + P for american options. this makes sense only in the sense of a straddle being a single contract. In the case where you exercise the put early then the straddle disappears but the call would still be alive in the rhs.

this is still puzzling me!I'll show you one example where the equality holds, what's going wrong then?Let the risk free rate be and the risk neutral probabilities next consider the following binomial schemeand an american put option with strike price 5. the price dynamics of this derivative is given byand thus, as the price of the american call option is the same of the european call optionyou can calculate-it using the risk neutral probabilities, it is $1.76so, summing up 1.76 + 1.36 = 3.120 which is the price of the straddle.To calculate the price of the straddle I used the American algorithm, as followsas you can see, in this case

please see my response earlier.

I saw it. I don't know exactly what you mean by a straddle being a single contract or not (sorry, I'm still too academic). I thought that my last message was a counter example to your message. Why it is not a counter example then?

QuoteOriginally posted by: fniskiI saw it. I don't know exactly what you mean by a straddle being a single contract or not (sorry, I'm still too academic). I thought that my last message was a counter example to your message. Why it is not a counter example then?an american straddle is different to a european straddle becuase the latter can only be exercised on maturity hence can be replicated by a combination of european call and put. an american straddle is different. if for example, you exercise the straddle because the "put" element of the straddle makes it optimal to do so then you lose the call but in a replicating portfolio of a call and put the call will still exist. so lets take an example of a stock at 4 and a straddle struck at 5 replicated with call and put. lets say the stock drops to 2 in which case its optimal to exercise the put and the straddle but lets say the stock then rallied to 16 subsequently. as the straddle no longer exists you will not benefit from this but the call option will be alive and kicking.

I think I understood your point about the difference between an american and an european straddle. But insisting on the numerical example that I showed you, what you're saying implies that I miscalculated the price of the straddle in such a way that it should be lower than $3.120? I just can't figure it out where I've used the American algorithm wrongly....I'm sorry in case I'm still asking the very same question. I'm trying hard but I still couldn't see how exactly your messages are answering it.

i think its because in the tree you have it not possible to go from 2 to 16.

that's right. So do you agree that there is a typo on shreve's book?

i wouldn't or couldn't say because I haven't seen it. is he a lurker here ? perhaps he can tell us.