Option arbitrage question

[Helloall123]

Thanks a lot guys!

It makes much more sense now to understand options arbitrage. I just needed to know which reasoning direction was the correct one to find out arbitrage.

@Kevin Schmit

Indeed, still need to be more comfortable with Ln calculations. Thanks a lot though!

 

[JackRab]

I think they were aiming at comparing a short term ATM straddle with a long term ATM straddle..

Where if the 1yr straddle = 160 (call + put) and the 12 yr straddle is also 160... than the latter is grossly underpriced.

But since the 1yr 1000 call is not the ATM call... the ATM is the 1050 call... that doesn't fly IMO. 1yr ATM 1050 straddle will be about 90... 45 for the call and 45 for the put... which means the 12 year ATM IV sits at about half that of the 1 year IV. Seems fair...

I don't know what the arb is... I don't see a real arb...

@sle ? What's your take?

 

[ajacobson]

OK Hint: European options with a 5% carry. The put gets progressively cheaper - long-term volatility becomes very very unimportant in the equation.

 

[sle]

@sle ? What's your take?

Hmm, I agree. My initial take (without calculating) was that there is a negative forward vol, but the 1y call is ITM by a fair bit (50 points). Back of the envelope on my cell phone calculator:

call(1y) = F * 0.4*vol*sqrt(1) + delta * (F - K) = 0.4 * 1050 * vol(1y) + 0.5 * (1050 - 1000)
vol(1y) = (80*1.05 - 25) / (1050 * 0.4) = 13.8%

put(12y) = F * 0.4 * vol(12) * sqrt(12) = 1822 * 0.4 * vol(12) * 3.45
vol(12)y = (80*1.82)/(1822 * 0.4 * 3.45) = 5.8%


forward volatility is a low but positive number

 

[sle]

OK Hint: European options with a 5% carry. The put gets progressively cheaper - long-term volatility becomes very very unimportant in the equation.

The given strike makes the put smack ATMF, so volatility is important

 

[Kevin Schmit]

Hmm, I agree. My initial take (without calculating) was that there is a negative forward vol, but the 1y call is ITM by a fair bit (50 points). Back of the envelope on my cell phone calculator:

call(1y) = S * 0.4*vol*sqrt(1) + delta * (F - K) = 0.4 * 1000 * vol(1y) + 0.5 * (1050 - 1000)
vol(1y) = 13.5%

put(12y) = S * 0.4 * vol(12) * sqrt(12)
vol(12)y = 5.8%

forward volatility is a low but positive number

On the numbers above (exact is about 13.375 and 5.8 so your cell phone calcs are spot on) the forward vol (yrs 2+) is approximately 2.7. That is verrryyy low for a name with a 13.375 1yr vol.

Also many text book problems are sloppily written. I am half convinced that they meant the 1050 strike not the 1000, which would make the forward vol near zero.

 

[sle]

Also many text book problems are sloppily written. I am half convinced that they meant the 1050 strike not the 1000, which would make the forward vol near zero.

Right, that's what I thought at the first glance. If this is a homework problem indeed and the kid submits "there is not arb", I wonder what would the professor say?

 

[ajacobson]

This is the Buffet long-term put question. ST volatility is easy to calculate. Why the 1822 level? So an index goes from 1000 to 1822 over 12 years - interest rates are a constant 5% and there is continuously computed dividend. Why continuously computed in any other world than academia or textbook. The one year ATM call is $80 - the 12 year ATM put is also $80. Use the S and P as a benchmark and the long-term volatility is HALF the short-term volatility
Why the 1822 level and 5% carry - why 12 years and not say 10. It's IMHO plug numbers. Use the LT S & P dividend. A lot of info that I implied was left out - so my assumptions may be incorrect, but it was pretty close to what the press described at the Buffet put.
The devil is in the details and the details are in the 5% 12-year carry and the LT dividend of 2.5% in the S and P. Obviously change any one number and the whole things changes.
About a 22 ST vol and an 11 LT volatility. Long-Term volatility is always considered to be lower and in the real world, a LT volatility estimate is very tough to estimate so most backsolve from the swap rate and some dividend growth model.

 

[sle]

Use the LT S & P dividend. A lot of info that I implied was left out - so my assumptions may be incorrect, but it was pretty close to what the press described at the Buffet put.

LOL. Having being on the receiving end of the said put, yeah, that's pretty much perfectly true. But this is a homework question, so we are playing by the rules here

 

[JackRab]

@sle and @Kevin Schmit regarding the 1yr IV... That 13.8 or 13.375 is for the 1000 strike correct?

Depending on how much skew there is... If I look at 25% skew to a 1sd strike, that would mean the ATM vol is about 12... bit lower even. So if you compare that to the ATM 12yr vol of about 5.8... that means the fwd vol is about 5.

Which isn't really strange IMO... maybe somewhat on the cheap side.