hello,a friend of mine was asked the following question in an interview and we have no idea how to solve it.."what price would you charge for the following:- 100-110 call spread - you can only take a static position in the underlying stock at inception and unwind at maturity?"thank you,j

ok i suppose it's a hard question?!?any thoughts anyone?

Hey Jokeoh,If I needed to hedge out my sale, I would need to buy (once and not modify) a portfolio replicating the call spread payoff at expiry. Consequently, this solely depends on which type of products are available for trading my underlying asset. Indeed, if the tradable assets at my disposal are future/forward contracts and bonds, I simply can not find an efficient portfolio... But, I could get pretty close by buying a strip of Cash-or-Nothing options going from strike 100 to 110.I think this is what they wanted your friend to say...

so what would your PRICE be for this call spread?

clearly the price must be more than zero and less than 10%*discount factor for maturity T, where T is the life of the call spread. So we can write: 0<=P <=10*exp(-r*T), where P= price of call spread, r=effective discounting rate, T=time to maturity.Note, from the above we see that: as T increases P is bounded by a small number. In particular, as T increases to infinity, P becomes zero.

but we know that the P should be a non-decreasing function of T.so P=0.is that a logical argument?-- EDIT: no it's not. P is not necessary a non-decreasing function of T.

Yes, P is bounded by 0 and the price of a bond paying 10 currency unit at maturity but this boundary is clearly not really precise.This is true that P tends to 0 (r > 0) when T tends to infinity. It simply means that seen from today, my future payoff is less and less substantial which makes total sense (Theta call spread < 0).Replicating the call spread 100-110 with a portfolio of digital (Cash-Or-Nothing) calls is easy and efficient. The static portfolio will be all the more efficient as the number of strikes used tends to infinity. Let me explain with an example. The best way to understand this technique is to draw the call spread 100-110 payoff at maturity with those of the replicating portfolios:1. With 1 strike, a rough replication of the call spread would be to use 10 digital call of strike 1052. With 2 strikes, a rough replication of the call spread would be to use 5 digital calls of strike 100 and 5 of strike 110.. .n. With n strikes, a rough replication of the call spread would be to use 10/n digital calls of strike {100 + 10*i/(n-1)} with i -> {0...n-1}As you have guessed, when n tends to infinity, the portfolio price tends to the call spread price.Is it clearer ?

i understand.but you now have reduced the original question to answering the question:what is the PRICE of the digitals you mentioned?

You have to know where the underlying is to be able to answer the question. I'll assume the underlying stock price = 100. In that case, assuming a normal distribution, the probability of ending up at P>100 = 0.5. The maximum payoff of the spread is 10, so the maximum price of the spread is 0.5*10 = 5 (assuming no skewness obviously). This is essentially the price (undiscounted etc etc) of a k=100 digi, so it is going to overestimate the spread price (but at least it gives you an upper bound). How much by depends on the actual shape of the distribution.

you sure? i was told no other assumptions were to be made..we probably need to use the additional constraint about only being able to buy the underlying at inception and unwinding at maturity

any ideas anyone?

The call spread is a portfolio of two calls - long a call with strike K1, and short a call with strike K2. The payoff is bounded by K2-K1, and so the price is bounded by the discounted value of (K2-K1).The payoff is also bounded by (K2-K1)/K2 times the stock price. So the best upper bound for the price is min([1-K1/K2]*Stock, [K2-K1]*Bond).

question looks unsolvable(in terms of finding a number as price) withoutdistributional assumption of risk-neutral probability of different payoffs.

Maybe this is a trading instinct rather than a quantitative question.The question is what do I charge, so I take it to mean they are a buyer.The spread can expire at most at ten, but they might want to buy it for ten, if there is a dividend of some kind (cash, stock, two tiered deal).I'd offer at twenty, If they execise their calls, I think the most I can lose excercising my calls is ten. So Worst case I break even.I'm at twenty, does anyone make a better offer?

On a non-div paying stock, I offer them 10 without any more information. If the current stock price is at 100 or below, I'd offer them anywhere between 5 and 10. But given that Anderman said 20, I'm off that price and I'll only show 19 .