HiWhat's the flaw in the following argument:I want to price a 6 month option. I know the price of a 3 month option. I price the 6 month option as a 3 month option in 3 months time, using current spot + 3 month option price as the spot in 3 months..

well the flaw is that you dont know where the spot will be in 3 months time ... the option price does not tell you what the spot will be ...

You didn't mention whether you were doing this for a call or a put. Should a 6 month put be less valuable because adding the 3-month premium to the current spot made the spot higher?

Let's assume we are doing this for a call option. Will convexity cause the option price determined using this method be higher/ lower than theoritical price? How?

nothing to do with convexity ... you just dont know what the spot price of the asset will be in 3 months time

I agree. However, risk neutraility would mean that if you take spot price to be equal to current spot + 3 month option price. My concern is if I am ignoring the convex profile of option by doing this.

This seems like a strange question to me. The B-S formula gives an explicit solution for the 6 month option, after all. So what could the 3 month option tell us to better price the 6 month? Well, volatility.What is your motivation for this approach? Ultimately, what you want is the forward price of the underlying, say S + dS, and you are suggesting dS = current call.If there is a hedging argument behind your idea, pls. elaborate (I don't see it). But here are a few problems: even if we say that EdS = 3month call, we are ignoring that the expectation is discounted back to today. But suppose interest rates are zero. Then the three month call is actually the expected value of (S-K)+ in 3mo. So S_0 + (S_3-K)+ only approximates S_3 on the upside, and even then only if S_0=K.It does seem okay to me to think of a European call or American on a non-dividend paying underlying as a 3 month call 3 months forward, in principle, using today's spot taken forward 3 months in the usual way. The formula should reduce to standard B-S. (Anyone see anything wrong with that?)The only reason I can see for doing what you suggest with the stock price is to try to use market information from the 3 month rather than assume lognormality of the stock price. If that is the idea, and God told you the volatility, back out the forward price of the option from the 3 month, then try to find the 6 month forward price from that. It is not clear that that approximation is better than just taking the spot 6 months forward assuming lognormality.Also, convexity isn't an issue. The stock price isn't convex in the usual graph, it is y=x. Convexity would enter in if you said c(6) = c(3) + delta(3)*dS(3 to 6) + theta(3)*dT, where the sign of theta is taken so that time value grows rather than decays.

Let me see if I understand. A three month call at $50 on Citigroup sells for $1.44 (interpolating from listed option prices). A six month call at $50 sells for $2.06. A three month call at $50 would sell for $2.30 if the price of Citigroup moved to $49.78 + $1.44 (current spot plus three month option price) assuming the implied volatility stayed the same. Are you saying the $2.30 is a good approximation to the $2.06?That works pretty well at-the-money, but if I use $60 strikes the three month call sells for 0.0178, the three month call at the slightly increased stock price sells for 0.0181 and the six month option sells for 0.0630. At $40 strike the three month sells for $9.99, the augmented three month for $19.97, while the six month sells for only $10.18 (granted the vols this far in the money are not very reliable).Adding the three month option price to the spot multiplies the value of the three month option by approximately one plus delta. Near the money, doubling the time to expiry multiplies the value by about the square root of 2. So near the money this can work pretty well because both multiply the option price by something near 1.5. But as you get in the money, the delta goes up and and expiry multiplier goes down; as you get out of the money the delta goes down and the expiry multiplier goes up. So the corrections go in opposite directions. You would do much better to add the price of a three month put to the spot.

Thanks Aaron. Main motivation to my question is for processes where there is not enough data for observed volatility (or there is no intutive sense for it). Now, if you know prices for various 3M month maturity options and 6M maturity options and want to see relative value (mispricing), what kind of intutive conclusions can you draw?

The obvious approach is to convert the prices to implied volatilities. Options with low implied volatilities look cheap, options with high implied volatilties look expensive. You have to adjust for known patterns, more importantly the skew. But if you can buy an out of the money call at a lower implied volatility than an at-the-money call; or buy a six month at-the-money option for a much lower implied volatility than a three month; you should look hard at the opportunity.

Thanks prettyspecific. I realize that I wasn't really specific in my question.I want to do this for ATM options. If you have an ATM call, it's price should be close to PV of E(dS), right?

It's closer to half the expected move. Figure the underlying equally likely to go up or down. If it goes up, the call option captures the move. If it goes down, the call option is worthless. A better approximation is 40% the volatility (not the expected move) over the period until expiry.