Black's model for future opitons

[miaomiao]

c=exp(-rt) x [F * N(d1) - k x N(d2)]

d1=[ln(F/k) + sigma ^2/2 x t ] / (sigma x t^0.5).

this is different from what you will get if replace s with s x exp(-rt) from B-S model. if you do that, you will get
d1=[ln(S/k) + sigma ^2/2 x t ] / (sigma x t^0.5).

so, for options paying stocks, currencies, we can just replace s with s x exp(-qt), but not for future option, is this true?

Thanks a lot.

[White Teeth]

You could also look at it as replacing s * exp (-qt) for s for options on futures where q = r. That should get you the formula for options on futures.

[miaomiao]

You could also look at it as replacing s * exp (-qt) for s for options on futures where q = r. That should get you the formula for options on futures.

no the d1 is different, isn't it?

[frank_exams]

miao,

1. good luck on 7+8.
2. you're missing a very important term in your 2nd d_1 formula, namely the risk-free rate in the numerator. note that ln(F) = ln(S_0 e^{(r-r_f)T} = ln(S_0) - (r-r_f)T. Once you put that (r - r_f)T term in the numerator, then both formulas are equal.

best,
frank

[miaomiao]

miao,

1. good luck on 7+8.
2. you're missing a very important term in your 2nd d_1 formula, namely the risk-free rate in the numerator. note that ln(F) = ln(S_0 e^{(r-r_f)T} = ln(S_0) - (r-r_f)T. Once you put that (r - r_f)T term in the numerator, then both formulas are equal.

best,
frank

Thnaks, Frank. Well, for part 8, I jsut want to give it a shot, not sure if I can get a chance to pass or not. Got to focus on part 7 starting now.

I understand this as follows:

So=F x e^-rt, so just replace So with F x e^-rt in B-S model, you will get the black mode for future options. Does this make sense?

[frank_exams]

Yes. As long as you put the missing "r" back in the numerator of your second d_1 formula, that's exactly how it works out!

Frank

[carrytheCrøss]

These two things I believe are correct:
1) When we use the Black-Scholes Model, we are to use the volatility of the spot asset price when formulating our d1 and d2 terms.

2) When we use Black's Model, we are to use the volatility of the futures price when formulating our d1 and d2 terms.

Now look at questions 16.20 and 17.10 in Hull (7th edition). Both of these questions give you information about the current futures price, but do NOT give you any information regarding the volatility of the futures price. They just give you the volatility of the spot asset price. Because of this, don't these two questions seem defective, or is my thinking defective? :tfh:

[triplea]

I am scared of seeing a difficult problem on this topic (Hull chapter 16).

So an option on a stock is equivalent to an option on a futures contract on that stock if the maturity of the option and futures contract is the same?

Now what happens when you have an option on a futures contract when the maturity of the futures contract is later? We still value to option using Black's model, correct? And we use the volatility of the futures price, right?

if we are given a problem to calculate the value of an option on a futures contract (with a different maturity), but are only given the stock price, we should calculate the futures price and use Black's model?

:wall:^3

[Actuarius]

These two things I believe are correct:
1) When we use the Black-Scholes Model, we are to use the volatility of the spot asset price when formulating our d1 and d2 terms.

2) When we use Black's Model, we are to use the volatility of the futures price when formulating our d1 and d2 terms.

Now look at questions 16.20 and 17.10 in Hull (7th edition). Both of these questions give you information about the current futures price, but do NOT give you any information regarding the volatility of the futures price. They just give you the volatility of the spot asset price. Because of this, don't these two questions seem defective, or is my thinking defective? :tfh:

If F=Soe^(rT) then aren't the volatilities the same?

[Actuarius]

Now what happens when you have an option on a futures contract when the maturity of the futures contract is later? We still value to option using Black's model, correct? And we use the volatility of the futures price, right?

if we are given a problem to calculate the value of an option on a futures contract (with a different maturity), but are only given the stock price, we should calculate the futures price and use Black's model?

I just think of an option on a future like an option on a stock that pays dividends equal to the risk-free rate. But yes, you do replace S with F everywhere in the regular equations. And remember that T is always the time until the OPTION expires, not the time until the future matures.

[tromba]

If F=Soe^(rT) then aren't the volatilities the same?

I don't think so. If you hold a pencil at one end and wiggle it a little, you'll notice that the far end moves a lot further than the end you're holding. In the same way, a small change in the spot price creates a larger change in the futures prices. So the futures price must have a larger volatility than the spot price.

(Or try a simple numeric example. Assuming S can only be 20 or 30, r=.05, and T=1, I get Var(S) = 25 and Var(F) = 27.6).

That said, I doubt the exam authors have realized this subtle point -- they seem to have trouble copying problems from the textbook correctly. If a problem comes up like this, I'll state "assuming the volatility of the futures equals the volatility of the spot price," and then just solve it like Hull problem 17.10.

In real life I imagine the difference between the two volatilities is well within the margin for error in estimating volatility in the first place.

[CaffeineJunky]

This is the way I think of it, you can always use the B-S equation you just need to replace S sub 0 (everywhere it appears) accordingly.
Stock opt with div, replace with S * e^(-(div rate)*t)
Foreign curr opt, replace with S * e^(-(foreigh rate)*t)
Futures options, replace with S * e^(+risk free*t)

[triplea]

On page 350-351 of Hull, it says:

"When the cost of carry and the convenience yield are functions only of time, it can be shown that the volatility of the futures price is the same as the volatility of the underlying asset."

:shrug:

[triplea]

Futures options, replace with S * e^(+risk free*t)

I thought you replace S with F * e^(-risk free*t) since then the r cancels out in the formula for d1

Plus, doesn't this assume that the option T equals the futures contract T? I know Hull says that Black's model does not require the maturities to be the same.

:wall:

[tromba]

I don't think so. If you hold a pencil at one end and wiggle it a little, you'll notice that the far end moves a lot further than the end you're holding. In the same way, a small change in the spot price creates a larger change in the futures prices. So the futures price must have a larger volatility than the spot price.

(Or try a simple numeric example. Assuming S can only be 20 or 30, r=.05, and T=1, I get Var(S) = 25 and Var(F) = 27.6).

That said, I doubt the exam authors have realized this subtle point -- they seem to have trouble copying problems from the textbook correctly. If a problem comes up like this, I'll state "assuming the volatility of the futures equals the volatility of the spot price," and then just solve it like Hull problem 17.10.

In real life I imagine the difference between the two volatilities is well within the margin for error in estimating volatility in the first place.


Ack! Sorry, never mind. Volatility is based on percentage change in stock price, not actual change. TripleA, you're right when you quote Hull -- the volatilities are the same.

[CaffeineJunky]

I thought you replace S with F * e^(-risk free*t) since then the r cancels out in the formula for d1

Plus, doesn't this assume that the option T equals the futures contract T? I know Hull says that Black's model does not require the maturities to be the same.

:wall:

This might be why I am getting wrong numbers for 2009 #22 :exams:

[Actuarius]

I thought you replace S with F * e^(-risk free*t) since then the r cancels out in the formula for d1

Plus, doesn't this assume that the option T equals the futures contract T? I know Hull says that Black's model does not require the maturities to be the same.

:wall:

Nope, that's what I thought too until I got one wrong on a practice exam so I had to look it up to make sure the solution was right. I didn't dig into the theory, but the r's cancel out when you use F instead of S in the formulas for both c (value of the call) and d1.

d1=[ln(F/K) + 0.5sigma^2*T]/[sigma * sqrt(T)]

c=e^(-rT)[FN(d1)-KN(d2)]

[Jim James]

I thought you replace S with F * e^(-risk free*t) since then the r cancels out in the formula for d1

Plus, doesn't this assume that the option T equals the futures contract T? I know Hull says that Black's model does not require the maturities to be the same.

:wall:


Sticking with the analogy that a futures contract is like a stock with a continuous dividend of the risk-free rate, you only discount from the option expiration date because you only want to remove that much of the dividend yield. Sure, it will provide the dividend stream beyond the option expiration date through the futures expiration date but it doesn't matter to you because your option's already expired.

Discussing the two maturity dates further, it doesn't make a whole lot of practical sense to me to have the two expire at the same time. For example, I want to have the option, over the next 3 weeks, to go long on a futures contract that expires in 3 months. Why would I want the option, over the next 3 weeks, to go long on a futures contract that expires the day my call option expires?

[triplea]

Sticking with the analogy that a futures contract is like a stock with a continuous dividend of the risk-free rate, you only discount from the option expiration date because you only want to remove that much of the dividend yield. Sure, it will provide the dividend stream beyond the option expiration date through the futures expiration date but it doesn't matter to you because your option's already expired.

Thanks, that makes more sense now.

[MathForMarines]

Thanks, that makes more sense now.

:iatp: