[R-SIG-Finance] American option sensitivities
Enrico Schumann
enricoschumann at yahoo.de
Fri Feb 10 10:18:08 CET 2012
Hi all,
(comments below)
Am 10.02.2012 01:02, schrieb J Toll:
> On Thu, Feb 9, 2012 at 5:17 PM, Dirk Eddelbuettel<edd at debian.org> wrote:
>>
>> On 9 February 2012 at 17:06, J Toll wrote:
>> | Hi,
>> |
>> | I'd like to calculate sensitivities on American options. I was hoping
>> | somebody might be able to summarize of the current state of that
>> | functionality within the various R packages. It's my understanding
>> | that the fOptions package can calculate greeks for European options
>> | but not American. RQuantLib appears to have had the ability to
>> | calculate greeks for American options at one point, but it appears
>> | that functionality was removed in Release 0.1.8 sometime around
>> | 2003-11-28.
>>
>> ... because that functionality was removed upstream by QuantLib.
>>
>> |
>> | http://lists.r-forge.r-project.org/pipermail/rquantlib-commits/2010-August/000117.html
>> |
>> | Additionally, from RQuantLib ?AmericanOptions says,
>> |
>> | "Note that under the new pricing framework used in QuantLib, binary
>> | pricers do not provide analytics for 'Greeks'. This is expected to be
>> | addressed in future releases of QuantLib."
>> |
>> | I haven't found any other packages for calculating option
>> | sensitivities. Are there any other packages?
>> |
>> | Regarding RQuantLib, is the issue that that functionality hasn't been
>> | implemented in R yet, or is it QuantLib that's broken?
>>
>> There is a third door behind which you find the price: "numerical shocks".
>>
>> Evaluate your american option, then shift the various parameters (spot, vol,
>> int.rate, time to mat, ...) each by a small amount and calculate the change
>> in option price -- voila for the approximate change in option value for
>> change input. You can also compute twice at 'x - eps' and 'x + eps' etc.
>>
>> Dirk
>
> Dirk,
>
> Thank you for your response. I was hoping you might reply.
>
> I understand the concept of your suggestion, although I don't have any
> practical experience implementing it. I'm guessing this is what's
> generally referred to as finite difference methods. In theory, the
> first order greeks should be simple enough, although my impression is
> the second or third order greeks may be a bit more challenging.
>
> I hate to trouble you for more information, but I'm curious why? Is
> this the "standard" method of calculating greeks for American options?
> Has QuantLib decided not to implement this calculation? Just curious.
>
> Thanks again,
>
A simple forward difference is
[f(x + h) - f(x)] / h
'f' is the option pricing formula; 'x' are the arguments to the formula,
and 'h' is a small offset.
Numerically, 'h' should not be made too small:
(1) Even for smooth functions, we trade off truncation error (which is
large when 'h' is large) against roundoff-error (in the extreme, 'x + h'
may still be 'x' for a very small 'h').
(2) American options are typically valued via finite-difference or tree
methods, and hence 'f' is not smooth and any 'bumps' in the function
will be magnified by dividing by a very small 'h'. So when 'h' is too
small, the results will become nonsensical.
Here is an example. As a first test, I use a European option.
require("RQuantLib")
h <- 1e-4
S <- 100
K <- 100
tau <- 0.5
vol <- 0.3
C0 <- EuropeanOption(type = "call",
underlying = S, strike = K,
dividendYield = 0.0,
riskFreeRate = 0.03, maturity = tau,
volatility = 0.3)
Cplus <- EuropeanOption(type="call",
underlying = S + h, strike = K,
dividendYield = 0.0,
riskFreeRate=0.03, maturity=tau,
volatility=0.3)
Cminus <- EuropeanOption(type="call",
underlying = S - h, strike=K,
dividendYield=0.0,
riskFreeRate=0.03, maturity=tau,
volatility=0.3)
## a first-order difference: delta
(Cplus$value-C0$value)/h
## [1] 0.570159
C0$delta
## [1] 0.5701581
## a second-order difference
(Cplus$delta-C0$delta)/h
## [1] 0.01851474
C0$gamma
## [1] 0.01851475
Now for an American option. Here we don't have the delta, so we first
need to compute it as well.
C0 <- AmericanOption(type="put",
underlying = S, strike=K,
dividendYield=0.0,
riskFreeRate=0.03, maturity=tau,
volatility=vol)
Cplus <- AmericanOption(type="put",
underlying = S + h, strike=K,
dividendYield=0.0,
riskFreeRate=0.03, maturity=tau,
volatility=vol)
Cminus <- AmericanOption(type="put",
underlying = S - h, strike=K,
dividendYield=0.0,
riskFreeRate=0.03, maturity=tau,
volatility=vol)
## a first-order difference: delta
(dplus <- (Cplus$value - C0$value)/h)
(dminus <-(C0$value - Cminus$value)/h)
## a second-order difference
(dplus - dminus)/h
## [1] 0.01905605
I ran a little a experiment with different levels of 'h', where you can
clearly see when the gamma diverges.
| h | gamma |
| 1 | 0.01905385 |
| 0.01 | 0.01905612 |
| 1e-4 | 0.01905605 |
| 1e-5 | 0.01915801 |
| 1e-6 | 0.03463896 |
| 1e-8 | 8.881784 |
In the literatur, you find a number of tricks to smooth the function,
but in my experience, you are fine if you make 'h' small with respect to
'x' -- small, not tiny. So if the stock price is 100, a change of 1 or
0.1 is small. (And think of it: even if we found that a change of
one-thousandth of a cent led to a meaningful numerical difference; if
the stock price never moves by such an amount, such a computation would
not be empirically meaningful.)
Regards,
Enrico
>
> James
>
>
>
>>
>> |
>> | Thanks for any clarification.
>> |
>> | Best,
>> |
>> |
>> | James
>> |
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>>
>> --
>> "Outside of a dog, a book is a man's best friend. Inside of a dog, it is too
>> dark to read." -- Groucho Marx
>
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--
Enrico Schumann
Lucerne, Switzerland
http://nmof.net/
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