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Put-Call parity and Convexity

Posted: August 1st, 2011, 9:33 pm
by DoubleTrouble
Hello!I've been told that it's possible to show the inequality both using (1) Jensen's inequality (convexity) and using the (2) Put-Call Parity. I would be very grateful to any kind of hint!(1) Regarding the convexity we know that the function is convex so by using Jensen's inequality I can figure out that But that's where I get stuck! Is this even the right approach?(2) Regarding the Put-Call parity. We know that and that I don't know if this last step is necessary. I can't get further than this here either. How do I get the desired inequality?Thank you in advance!

Put-Call parity and Convexity

Posted: August 2nd, 2011, 1:24 am
by bwarren
For (1), use the fact that the price of a call is the risk-neutral expectation of the payoff. Also, you have Jensen's inequality backwards.For (2), you don't need the Black-Scholes formula. Theses inequalities are model-independent. Try using the fact that option prices are nonnegative.

Put-Call parity and Convexity

Posted: August 2nd, 2011, 8:20 am
by GuitarTrader
1. c(t, St, K, T)=exp(-r*(T-t))* E[(ST-K)+] > exp(-r*(T-t))* (E(ST)-K)+ = exp(-r*(T-t))* (St*exp(r*(T-t))-K)+ = (St-K*exp(-r*(T-t))*)+ > (St-K)+2. p=c+k*exp(-rT)-s >0=> c > s-k*exp(-rT) > s-Kand c >0so c >(s-K)+

Put-Call parity and Convexity

Posted: August 2nd, 2011, 11:31 am
by DoubleTrouble
Thank you for your hints bwarren, that was about all the help I needed! I feel really stupid for not figuring out (2)! I just have one question regarding (1). Is it a reasonable restriction to assume that the Stock is modeled by a GBM with constant rate of return and constant volatility like GuitarTrader does? If we assume this, then this is how I reason:The map is obviously convex so we can use Jensen's inequality in the following way:sinceThe only question mark I have is the fact that we assume the stock price to have constant rate of return and volatility. Any comments on this? Is it possible to do in another way without assuming this?Thanks in advance!

Put-Call parity and Convexity

Posted: August 2nd, 2011, 11:37 am
by DoubleTrouble
The easiest way of proving this inequality ought to be the following: The map is increasing. Therefore if t < T we get:

Put-Call parity and Convexity

Posted: August 2nd, 2011, 12:08 pm
by GuitarTrader

Put-Call parity and Convexity

Posted: August 3rd, 2011, 12:05 am
by bwarren
QuoteOriginally posted by: DoubleTroubleThank you for your hints bwarren, that was about all the help I needed! I feel really stupid for not figuring out (2)! I just have one question regarding (1). Is it a reasonable restriction to assume that the Stock is modeled by a GBM with constant rate of return and constant volatility like GuitarTrader does? If we assume this, then this is how I reason:The map is obviously convex so we can use Jensen's inequality in the following way:sinceThe only question mark I have is the fact that we assume the stock price to have constant rate of return and volatility. Any comments on this? Is it possible to do in another way without assuming this?Thanks in advance!I was able to show it using Jensen's inequality just assuming the drift of the stock is >= r, i.e. the market price of risk is nonnegative. I think this is reasonable since a rational person would not invest in a risky asset with a negative excess return.