September 1st, 2011, 5:55 pm
Last try (I swear, I know that you can get this, I believe in you!):Do you understand the following things:1) What arbitrage is?2) If you can construct a portfolio that replicate the payoff of a derivate in *any* future scenario and this portfolio is unique (i.e. no arbitrage is possible) then the value of the derivative is equal to the value of the portfolio.QuoteOriginally posted by: listThe equation used for call option pricing is C ( t , x ) = 0 for scenarios from { S ( T ) < K } and [ S ( T ) / S ( t ) ]* I { S ( T ) > K } = [ S ( T ) - K ] / C ( t , x )Points I think you should address:1) You have two equations for the present price of the option which are not equal.2) Both equations require knowledge of the future. Which means they cannot be used in the present.3) To say that prices can be derived from this equation the return must match under every scenario not just those where the option expires with a value greater than zero.Can you see the inconsistencies here?Finally the example you were given in previous thread:Assume the following:1) Transaction costs are zero2) We have 2 times t (now) and T (some point in the future when the call expires)3) Interest rates are zero (i.e. there exists a zero coupon bond with the following price, B(t)=B(T)=1, its price in the future is known)4) There exists a stock S with current price 100 (S(t)=100), in the future its value is given by a Bernoulli distribution, at time T it can take value S(T)=200 with probability p or S(T)=50 with probability (1-p). Where p is in (0,1) pick any value you like for p, the rest of us don't need one (note that 0 & 1 are excluded since they permit arbitrage between the stock and the bond).5) There is a European call option available that has strike 100, that is at time T it's value is C(T)=max(S(T)-100,0).6) From (3) & (4) C(T)=100 if S(T)=100 and C(T)=0 if S(T)=50We need to find C(t), please show where the fault in the following line of reasoning is:1) If we purchase 66 2/3 of S at time t and sell 33 1/3 of the bond B at time t then we have spent in total 33 1/3 (66 2/3- 33 1/3 = 33 1/3) which is its present value.2) At time T the bond purchase will still be worth 33 1/3 (interest rate is zero), the stock purchase is worth either 133 1/3 or 33.33 (it has either doubled in value S(T)=200 or halved S(T)=50 respectively).3) The value of our portfolio at time T is either 100 if S(T)=200 (133 1/3 - 33 1/3 = 100) or 0 if S(T)=50 (33 1/3 - 33 1/3 = 0). These are the same values are the payoff of the option.4) No other combination of the stock and bond will replicate the payoff of the options under the assumptions above.5) The present value of the call must therefore equal that of the portfolio at time t giving C(t)=33 1/3If you disagree with any point please point it out and explain why it is incorrect (in the thread, don't point to a paper, just copy and paste if you need to use something from one).
Last edited by
ACD on August 31st, 2011, 10:00 pm, edited 1 time in total.