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Arthurim
Topic Author
Posts: 13
Joined: August 10th, 2017, 12:22 pm

### How to price a spread option on strategies involving more than two different prices ?

To price a simple spread option, one can use Kirk's approximation combined with Margrabe's formula as a close form formula if you have two different prices in the payoff of your option.
But what if you have more than 2 prices ? Is there any close form formula ? Else how would you price it ?

$$\mathop{\mathbb{E}} \left[\left(\alpha K+\beta F_1(t_1) +\gamma F_1(t_2)+\zeta F_2(t_1) +\lambda F_2(t_2)\right)^+\right]$$

Where $K$ is the strike, $F_1$ and $F_2$ the price of two different assets.

list1
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Re: How to price a spread option on strategies involving more than two different prices ?

Arthurim wrote:
To price a simple spread option, one can use Kirk's approximation combined with Margrabe's formula as a close form formula if you have two different prices in the payoff of your option.
But what if you have more than 2 prices ? Is there any close form formula ? Else how would you price it ?

$$\mathop{\mathbb{E}} \left[\left(\alpha K+\beta F_1(t_1) +\gamma F_1(t_2)+\zeta F_2(t_1) +\lambda F_2(t_2)\right)^+\right]$$

Where $K$ is the strike, $F_1$ and $F_2$ the price of two different assets.

The function does not look like a payoff of an option.It  is difficult to find maturity date and with arbitrary greek constants it is just a linear combination of some prices at two moments of time and K.

outrun
Posts: 3913
Joined: April 29th, 2016, 1:40 pm

### Re: How to price a spread option on strategies involving more than two different prices ?

The + superscript makes it the payoff of a spread option.

Perhaps a moment matching method is simplest? Assume the return distributions are Gaussian, make sure the mean and variance match that of the lognormal. You'll I get a 4d gaussian and the covariance between R(t1) and R(t2) is something like min(t1,t2)/max(t1,t2) (not sure).

But it's all full of biases (Kirk also is), a Monte Carlo with a low discrepancy grid might be better?

Froloos here has been working on conditional moment matching, and Roger Lord has also worked in that for Asian/Basket options. Maybe that's a good start?

frolloos
Posts: 1416
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### Re: How to price a spread option on strategies involving more than two different prices ?

I have not tried moment matching for multi-asset spread options but I suppose that can be done as well.

If sticking to Black scholes assumptions, indeed Monte Carlo is straightforward and accurate for pricing.

There is an interesting paper which claims to give very accurate analytical approximation for multi-asset spread prices and greeks; but I have not verified their result yet:

http://www.tandfonline.com/doi/abs/10.1 ... 014.949289

If you want to incorporate smiles then I'd probably 'just' do local vol.

Alan
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Joined: December 19th, 2001, 4:01 am
Location: California
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### Re: How to price a spread option on strategies involving more than two different prices ?

Might be tedious, but under GBM it is simply (an iterated) numerical integration using the known (bivariate) transition density. First value it at t1, conditional on the prices at that time. Then, take the expected value of that function at time t0. So, at most a 4-fold integral to price it 'exactly'.

Arthurim
Topic Author
Posts: 13
Joined: August 10th, 2017, 12:22 pm

### Re: How to price a spread option on strategies involving more than two different prices ?

outrun: I am not familiar with moment matching methods, but from what I found, I believe I will try the quasi MC.

forolloos: Thank you for the paper, I had the intuition for the generalization of Kirk's approximation but did not know whether this was really working and assumed not as I did not find anything on it. However I am going to try that one as well, and see if it matches the MC results.

Alan: I am afraid I don't really get your point, could you develop a little bit ?
What is the bivariate transition density ?

ppauper
Posts: 64808
Joined: November 15th, 2001, 1:29 pm

### Re: How to price a spread option on strategies involving more than two different prices ?

I'm with alan on this one. It can be priced at the earlier of the two expirations $t_1 < t_2$ using standard methods and then you find the expected present value of that value

Arthurim
Topic Author
Posts: 13
Joined: August 10th, 2017, 12:22 pm

### Re: How to price a spread option on strategies involving more than two different prices ?

ppauper: could you explain a little bit more what you mean?

list1
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Re: How to price a spread option on strategies involving more than two different prices ?

It will be interesting to look at boundary condition(s) for the BSE that are specified by the payoff.

Alan
Posts: 9171
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: How to price a spread option on strategies involving more than two different prices ?

@Arthurim,

Hints.

First learn how to value arbitrary (single-time) payoffs (numerically) under multivariate GBM.

How? Express everything in terms of log F's. Then write down the integral for the expected value of the payoff, using the multivariate normal distribution. Do the integral numerically. That integral is a function of the initial value of the F's, right?

Once you have done all that, ideally by writing some code, then go re-read my and ppauper's posts.

list1
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Re: How to price a spread option on strategies involving more than two different prices ?

Alan wrote:
@Arthurim,

Hints.

First learn how to value arbitrary (single-time) payoffs (numerically) under multivariate GBM.

How? Express everything in terms of log F's. Then write down the integral for the expected value of the payoff, using the multivariate normal distribution. Do the integral numerically. That integral is a function of the initial value of the F's, right?

Once you have done all that, ideally by writing some code, then go re-read my and ppauper's posts.

Arbitrary Multivariate GBM probably implies using corresponding multi strikes and we should also specify its action at maturity whether they should be added or multiply or take their max or something else.

list1
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Re: How to price a spread option on strategies involving more than two different prices ?

list1 wrote:
It will be interesting to look at boundary condition(s) for the BSE that are specified by the payoff.

The BSE and a boundary conditions(s) is the exact solution of the option pricing problem. The only pricing problem with boundary conditions which follow from payoff definition. The solution should be proved to exist and unique. Then if we could not present the solution in a closed form we are trying to apply approximate methods like the finite difference method or another. Other way to make formal assumptions on payoff=boundary conditions which may simplify the problem. This comment relates to the final moment maturity payoff. Path dependent options can affect on basic equation , ie BSE too.
If we decide to change underlying model GBM initially we first need to define option price notion because it might do not exist. In other words we need to prove that there exists a hedged portfolio of an option and other known assets. If classical GBM is generalized by adding jumps or others there is no hedged portfolio in general and using BSE solution formula is informal and it is based on our hope that it is still make somewhat sense. Whether or not this hope rationale or not is a quite open financial problem based primarily on how much one earn by selling it to others vs how much one can lost by applying this idea.

Arthurim
Topic Author
Posts: 13
Joined: August 10th, 2017, 12:22 pm

### Re: How to price a spread option on strategies involving more than two different prices ?

Finally found a generalisation of the Kirk's approximation for multi asset spread options (paper).
Similarly, if we have N prices
$$max(0,F_N-\sum_{k=1}^{N-1}F_k-K)$$
translates into
$$F_N N(d_1)-(\sum_{k=1}^{N-1}F_k+K)N(d_2)$$

But then what if we have $max(0,F_N+F_{N-1}-\sum_{k=1}^{N-2}F_k-K)$?
Is it right to say that it translates into $$(F_N + F_{N-1})N(d_1)-(\sum_{k=1}^{N-2}F_k+K)N(d_2)\ \ \ (1)$$

or into

$$F_N N(d_1)-(\sum_{k=1}^{N-2}F_k+K-F_{N-1})N(d_2)?\ \ \ (2)$$

Looking at the proof of the proposition for the general case, (2) seems to be true, but I am not sure for (1).
What do you think ?

list1
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Re: How to price a spread option on strategies involving more than two different prices ?

Arthurim wrote:
Finally found a generalisation of the Kirk's approximation for multi asset spread options (paper).
Similarly, if we have N prices
$$max(0,F_N-\sum_{k=1}^{N-1}F_k-K)$$
translates into
$$F_N N(d_1)-(\sum_{k=1}^{N-1}F_k+K)N(d_2)$$

But then what if we have $max(0,F_N+F_{N-1}-\sum_{k=1}^{N-2}F_k-K)$?
Is it right to say that it translates into $$(F_N + F_{N-1})N(d_1)-(\sum_{k=1}^{N-2}F_k+K)N(d_2)\ \ \ (1)$$

or into

$$F_N N(d_1)-(\sum_{k=1}^{N-2}F_k+K-F_{N-1})N(d_2)?\ \ \ (2)$$

Looking at the proof of the proposition for the general case, (2) seems to be true, but I am not sure for (1).
What do you think ?

Article looks like a good one. For other types of spread options you should follow the original BS idea constructing hedged portfolio with a certain delta-portions of the underling assets. It brings you a correspondent generalization of the BSE. Next you should write probabilistic representation of the solution. With some mathematical skills you can get the explicit formula.