hedge portfolio for puts, why two different formulas?

[danv0330]

This really irritated me… Between R49 EOC #3c and EOC#5b they calculate the hedge ratio differently for puts… Im confused about which one to use and when/why??

n=p-(-)p+/S+(-)S-

n=p+(-)p-/S+(-)S-

I think the issue here is the difference between replicating a loan and a risk free hedge… can someone spell this out?

with calls is its always c+ (-) c-?

[tickersu]

@danv0330

This really irritated me… Between R49 EOC #3c and EOC#5b they calculate the hedge ratio differently for puts… Im confused about which one to use and when/why??

n=p-(-)p+/S+(-)S-

n=p+(-)p-/S+(-)S-

I think the issue here is the difference between replicating a loan and a risk free hedge… can someone spell this out? I think it could just be sloppiness on their part. The absolute value is the same for the two formulas you gave. (P-) - (P+) should be the numerator.

with calls is its always c+ (-) c-? C+ > C- and the hedge ratio is positive since the call value changes in the same direction as the stock price move. This lets you know to long one and short the other to have the changes net to zero.

The one for puts should give you a negative hedge ratio since the put value is inversely related to the stock price. This tells you to buy both or sell both to have the effect cancel out.

Edit: Negative hedge ratio for puts in the BSOP framework. It doesn’t truly matter, though. As long as you know how to interpret the calculated value, things will be okay.

[lammy]

@tickersu

The one for puts should give you a negative hedge ratio since the put value is inversely related to the stock price. This tells you to buy both or sell both to have the effect cancel out.

Tickersu, I don’t understand this part of your answer. I was thinking that the numerator is (P-) - (P+) because puts increase in value when stock prices decline, so I need to subtract P+, otherwise I’d have a negative hedge ratio… What do you mean by having a negative hede ratio and buy both or sell both?

BTW, thanks for taking the time to reply to so many of our questions, particulary regarding Quant. You’ve been a tremendous help to all of us.

[tickersu]

@lammy

@tickersu

The one for puts should give you a negative hedge ratio since the put value is inversely related to the stock price. This tells you to buy both or sell both to have the effect cancel out.

Tickersu, I don’t understand this part of your answer. I was thinking that the numerator is (P-) - (P+) because puts increase in value when stock prices decline, so I need to subtract P+, otherwise I’d have a negative hedge ratio…

Edit: Sorry, pretty tired. I see what you mean. The calculation will be positive as you wrote it (and as I intended to describe). But, we know that from BS model the hedge ratio will be between 0 and -1 for puts. I don’t think it matters in this framework, as long as you know how to interpret it! I just think my mind was in Black and Scholes set up for this (although, like I said, I don’t believe it really matters).

I denoted the down state as “-”, and I assumed thats what DanV did. So, (P-) - (P+) would be the same as P(down) - P(up). In this option pricing framework, the hedge ratio we’re calculating is similar to the value of N(d1) for calls and N(d1)-1 for puts in the BSOPM (their hedge ratios). Calls have hedge ratios between 0 and 1 and puts have hedge ratios between -1 and 0.

What do you mean by having a negative hede ratio and buy both or sell both? Sorry, I wasn’t too clear. The negative hedge ratio for the put option tells us (and we know from other assessments) that the put option value increases as the underlying price declines (as you said) and the opposite is also true. Overall, put option value is inversely related to the underlying, ceteris paribus. So, think of a hedge-- we wan’t the transactions to offset one another. If the stock price declines, the put price will increase (assume deep in the money, so it’s dollar for dollar). If I long the put and it increased in value and short the stock, I will have a gain. This is true, because the put value increased, and I shorted the stock before the price dropped. BUT, we want these to cancel out–hedge. So, I should buy both the stock and the put or I should sell both. If I buy the put and the stock, and the price of the stock declines, I will lose this amount, but it will be offset by a gain in the put value.

BTW, thanks for taking the time to reply to so many of our questions, particulary regarding Quant. You’ve been a tremendous help to all of us. No problem-- I’m always glad to help. I have been a little worried that I might be spending too much time on here as the test is drawing near. I know there is some FRA and Swaps material with my name on it…

[gad4]

What I think he is saying is a stock prices go up, calls go up, but puts go down. Therefore inversely related. what confuse me is why this is called an arbitrage opportunity because there is no way we can lock values of the stock in. A new ceo can come in tomorrow and the stock could perform completely different than expected

[danv0330]

if you refer to the actual solution to EOC 5b and EOC 3c you will see what the discrepency is

the formulas are not mathematically equivalent in the case that both p(down) and p(up) have a value different from zero

[tickersu]

@danv0330

if you refer to the actual solution to EOC 5b and EOC 3c you will see what the discrepency is

the formulas are not mathematically equivalent in the case that both p(down) and p(up) have a value different from zero They’re not mathematically equivalent, but they will give the same magnitude (absolute value) answer.

I just looked at the solutions that you mentioned, and I’m still seeing this: the absolute value of the answer will be the same, irrespective of whether p(down) and p(up) are non-zero.

Assume that

S(u) = 20 S(d)=5 , so our denominator is 15

if we say p(down) - p (up) divided by 15, we get the same magnitude answer as p(up) - p (down) divided by 15…

It’s no different than saying 3-8= -5 or 8 - 3 = +5… the magnitude is identical, it doesn’t matter if one or both are non-zero values

[tickersu]

@gad4

What I think he is saying is a stock prices go up, calls go up, but puts go down. Therefore inversely related.

I’m saying that the value of a put is inversely related to movements in the underlying’s price (the stock).

The call value is positively related to movements in the stock’s price.