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1. Simple compounding vs continuous compounding

The inverse discount rate we can express as :
d_i^-1 = 1 + r_si t_si/Y_si = e^r_cct_cc/365
so, we get r_cc=(365/t_cc)Ln(1 + r_si t_si / Y_si) .. where

r_cc is the continuously compounded rate, Y_si is the no of days in one year based on the money market convention
r_si is the simple compounded rate, t_cc is the actual number of days in a period
t_si is the number of days in a period based on the money market convention.

the simple k4/kdb+ function for this can be :

tC:{[rsi;tsi;Ysi;tcc](365f%tcc)*log 1f+rsi*tsi%Ysi} /and we can now define a projection of tC ..
tCact360:{tC . (x;y;360f;y)}

2. The Future curves If a futures market is available for a specific currency, we will get a sequence of implied forward rates
(from the quoted futures prices) f₁,f₂, .. ,f_n-1,f_n and the corresponding forward discount factors p₁,p₂, .. ,p_n-1,p_n

where on the index set I_n holds: p_j=1 + f_j t_{m_j}/Y_si and p₀ is a stub-rate^*.. I_n is equivalent to the set of the first n natural numbers.
t_{m_j} is the number of days per future period j based on the convention of the money market.
The continuously compounded zero coupon rates we can obtain by : z_j+1^cc=(-365/F_j+1)Ln p₀p₁...p_j-1p_j where j is in the index set
I_n including element {0}. In reality represents this product a recursion of pairs: (f_n,t_n), where t_n is the difference
between the 1^st futures IMM dates T_n and T_n+1.

Just a sample....

We assume a given stub-rate of 1.64% on the 17.Sep.2003 (simple compounded), the maturities of the futures we takes as:
(i should mention that we assume here also a significant trading volume of those sample contracts)
The reference spot date of the stub-rate is 04.Aug.2003, and the table-dates below are 1^st IMM dates

	Future Strips of a currency X (act/360)
dates	17.9.2003	17.12.2003	17.3.2004	16.6.2004	22.9.2004	22.12.2004	23.3.2005	15.6.2005	21.9.2005
quotes	98.25	98.05	97.81	97.49	97.11	96.76	96.39	96.05	last 1st IMM

the simple k4 solution for this can be :

T:2003.08.04,2003.09.17,2003.12.17,2004.03.17,2004.06.16,2004.09.22,2004.12.22,2005.03.23,2005.06.15,2005.09.12
R:1.64,100.0-98.25,98.05,97.81,97.49,97.11,96.76,96.39,96.05

k)daysa:+\daysd:1_-':T
r:tCact360 . (0.01*R;daysd)

(r@0) {[x;y]((x*y0-y2)+(y@1)*y2:y@2)%y0:y@0}\ +(1_daysa;1_r;1_daysd)

.. returns the zero points determined by our future prices

^* The stub-rate corresponds to the period from now to the maturity of the first futures contract can be obtained by
linearly interpolating the last two continuously compounded money market rates r_i^cc and r_i+1^cc, so that
F₁ in the closed interval [t_i , t_i+1] where F₁ is the maturity date of the futures contract which falls between the two money market dates
If r_i^cc is not available from money market data, we assume that : r_i^cc = r_i+1^cc = p₀^cc,
where p₀^cc is the continuously compounded stub rate. In the normal case we get it from : p₀^cc=[r_i^cc Dt_i+(F₁ - t_i)(r_i+1^cc - r_i^cc)]/Dt_i
where Dt_i = t_i+1 - t_i.

3. Playing with FRA's

An FRA is simply one pair of cashflows.
Let be t₁ the time from the transaction date to the settlement date and t₂ the time between the transaction date and the maturity date,
and r_c^cc is the continuously compounded FRA contract rate, r₁^cc is the zero-coupon rate at the settlement, r₂^cc is the zero-coupon
rate of the maturity.
The net unrealized profit at a time t = t₀ we simply obtain as the sum of the both cashflows A_s(settlement) and -A_m(maturity).
So, the FRA_pl = A_se^{-r₁^cc(t₁-t₀)}+(-A_m)e^{-r₂^cc(t₂-t₀)} = A_s(e^{-r₁^cc(t₁-t₀)}-e^{r_c^cc(t₂-t₁)-r₂^cc(t₂-t₀)})

We can now use the definition of the delta D _s where the index s refers to the simple compounding, and we let be r(i) the
mapping from the continuous into the simple compounding form.
The D _s = dV_t/di=dV_t/dr_t*dr_t/di where V_t is the present value at the current timepoint t and F₀ the originally invested nominal
in t = 0, so V_t=F₀e^{r₂^cct₂ - r_t^cc(t₂-t)} and where t₂ is the maturity timepoint and the r^cc are as usual the correspondingly observed
cont.compounded zero rates.
As dr_t/di = t_M[Y_M(t₂-t)]^-1e^{- r_t^cc(t₂-t)} .. thus dV_t/di= - V_tt_MY_M^-1e^{-r_t^cc(t₂-t)} .... however in practice we rather use the
socalled basis point value bpv = -0.0001 V_tt_MY_M^-1e^{-r_t^cc(t₂-t)}
In FRA terms we obtain the bpv_FRA as the sum of bpv_S and bpv_M where S refers again to the settlement and M to the maturity.
So, bpv_FRA = 0.0001 A_sY_M^-1[(t₂-t)_Me^{-2r₂^cc(t₂-t)+r_c^cc(t₂-t₁)} - (t₁-t)_Me^{-2r₁^cc(t₁-t)}] where the subscript M relates to the money market day count convetion.

Here a simple kdb script simulating a random portfolio of 300'000 FRA's having 40 currencies. The FRA valuation itself is
actually a one or two-liner, all other lines are just used for comment and for building my sample portfolio, which would
clearly exceed the classical size of an FRA-book of a large bank by many times (10 to 100). The currency variety i took
is higher than in any other "real" portfolio.

FRA simulation Kdb+ script

4. European Options ..just a short introduction...

The payoff(expectation value) of an European call we can express as :

C = e^-r(T-t)J[ max(0;S_T-X)h(X_T)dS_T;0;infinity ] where J[ f ; a; b ] describes the definite integral of f over [a;b] where X is the strike, S is the spot(asset) price, T is the maturity date, t is our timepoint, r is the (riskless) rate

Changes in the spot price are the sum of a random component and a time-dependent deterministic component..
Eg: dS = (r - d )Sdt + s Sdz, where d represents the annual dividend yield. dz represents a sum of independent random
events over the time horizon dt. So, dz must follow a normal distribution, according to the central limit theorem, eg we can
describe dz by its variance (which is dt) and mean (which is 0).

This means the process for x = ln(S) leads us to the equation:

dx = (r - d - 0.5 s ²)dt + s dz
dx is a linear combination of a constant and the normally distributed dz, so must be normally distributed, too.
The mean of dx is (r - d - 0.5 s ²)dt, and its variance is s ²dt.
The natural logarithm of S at maturity is normally distributed with the following parameters:

N [(Ln S) + (r - d - 0.5 s ²)(T - t) ; s (T - t)^0.5
This allows to revisit the integral mentioned above - and to solve it. This way we obtain
the Black-Scholes solution for an

European call option.

C = Se^{-d (T - t)}N(d₁) - Xe^{-r (T - t)}N(d₂)

..and symetrically for the European put-option

P = Xe^{-r (T - t)}N(-d₂) - Se^{-d (T - t)}N(-d₁)

..and where d₁ = s ^-1(T - t)^{- 0.5}[ Ln (S/X) + (r - d + 0.5 s ²)(T - t)] and d₂ = d₁ - s (T - t)^0.5

A solution in Kdb+ can look like...(in case of an european currency option)

optBS:{{[S;X;rf;rd;t;s] /rd=domestic yield(cc), rf=foreign yield(cc), s=implied volatility
d1:((log S%X)+t0*((s*s*0.5)+rd-rf))%a:s*sqrt t0:t%365f;(S*(exp neg rf*t0)*g d1)-X*(exp neg rd*t0)*g d1-a}

..where g is the gauss function ..
pi:acos -1
g:{abs (neg x>0f)+(1f%sqrt 2f*pi)*(exp -0.5*x*x)*t*0.31938153+t*-0.356563782+t*1.781477937+ ..
.. t*-1.821255978+1.330274429*t:1f%1f+0.2316419*abs x}

if we change the Spot S..and let the other dimensions constant..we will obtain (in case of a call) the profile:
x-units are in 0.001 .. (if C=1 we have a call, C=0 we have a put)

call profile

5. American Options

This formula of Black and Scholes is of course widely used and it is quite handy and traceable from the
analytical point of view. Its simplicity has the price that it can be only used for classical European
styled options. American styled options are more "worth" than European styled options, as they can be
exercised at any day until the option maturity. One possible way to price an American option is the
binomial method, which is the common method. The binomial method has been introduced 1978 by Sharpe.
Although the binomial method is not that handy closed a formula, it is a fairly simple and intuitive method.
As the name suggests..there are only two possible states(up and down) in that tree in each time-interval
T/n.(T is the maturity and n the number of chosen intervals). Further we assume recombining nodes on the
tree, so we don't run into a messy mega-tree.
As we stated above already, Ln S_dt is normally distributed, and the mean is Ln S₀ + (r_d - r_f - 0.5 s ²)dt,
the variance is s ²dt
Exactly this finding helps us to find the UP's and DOWN's in the binomial tree. Theoretically we "assume"
that the chosen time-metrics is small enough, that we can find based on the above stated characteristics
of Ln S_dt and the classical relation for independent random variables, VAR(X) = E(X²) - [E(X)]²,
that VAR(S_1,T) = S₀²e^{2T(r_d - r_f)}(e^{s ²T} - 1). We see here also that the standard deviation of S_1,T is
monotonously increasing with s . With this, the future exchange rate movements are defined.
Thus:


	UP	p	S₀u
S₀
	DOWN	1-p	S₀d

Where: u = e^{s (Dt)^0.5}, d = u^-1 and p = (u - d)^-1(e^{(r_d - r_f)Dt} - d). For each last time-point jDt
we calculate the option-tree out of the price-tree: Q_jDt,i= max [0,L*S₀z^2(i-1) - X] ..where i in {1,2,..., j+1}, z in {u,d} and where
L = 1 for a call option, and -1 for a put.
The rest of the option-tree (where m in {j-1,j-2,...,1,0}) we have for each Q_mDt,n the equation that:
Q_mDt,n = max{L*E*(S₀z^{m - 2(n-1)} - X) , e^-r_dDt(pQ_(n+1)Dt,n + (1 - p)Q_(n+1)Dt,n+1) } and where E is the early exercise flag.
eg. E is 1 in case we exercise, and otherwise 0.

The answer using Kdb+ can be..

Xn:neg X;dt:T%n;n1:-1f;l:exp n1*rd*dt
q:1-p:((neg@ d)+exp dt*rd-rf)%(u:1f%d)-d:exp neg@ s*sqrt dt:T%n
On:0f|h*Xn+S*last atree:xexp[u;{x-2*til x+1} each 1+til n];
On {max each flip (n1*Xn+S*y;fx x)}/1 _ reverse 1f,atree

function fx should remain secret, but is only a short expression.

The graph below shows how the binomial model converges against the correct value in case of variable time-intervals
which represent the x-axis(starting from 1 and going up to 300)
The upper path in the graph below is an European put, and the other is an American put.

binomial convergence

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