The document discusses static hedging of binary options using a portfolio of vanilla options. Specifically, it examines hedging a binary call option with a strike of 100 using a short call with a strike of 90 and a long call with a strike of 110. The analysis considers uncertain volatility, inhomogeneous maturity between the options, and incorporating bid-ask spreads to maximize the value of the binary option for both long and short positions. Finite difference methods are used to numerically evaluate the option prices under different volatility assumptions and jump conditions.

1. Certificate in
Quantitative Finance
Module 6 Assessed Assignment
2012
Luigi Piva
UNCERTAIN VOLATILITY AND
STATIC HEDGE
1.
We are going to build a portfolio of Binary and Vanilla options.
Just like a standard vanilla American or European style option, Binary options are
defined in terms of a strikeprice , a maturity date, and an underlying. Binaries
are sold in exchange for an upfrontpremium payment, justlike other
options. Both calls and puts are available.
Taking price dynamics as a separate subject, the only difference between a binary
and standard option is its payout profile. A binary option has a Heaviside
payoff. In quantitative finance, Heavisidefunction are widely used when we are
dealing with binary (digital) option payoffs.
A Heaviside function is a step function whosevalue is 0 (zero) when the argument
is negative and 1 (one) when the argument is positive. In the case of Binary
options payoff we have:
Call Payoff= H(S-K)
PutPayoff=H(K-S)

2. We are going to consider a Cash-Or-Nothing Calland Put in our pricing problem.
A "cash or nothing" binary call (put) option pays out a fixed amount, the cash, if
the underlying is grater (lower) then the strikeprice and zero otherwise. A
standard vanilla option pays out a potentially unlimited variableamount. This
difference between a binary and vanilla option has two benefits. First, the
option sellers assumea known limited risk. Second, fromthe perspective of the
buyer, a binary option can offer significantly greater leverage since the premium
is lower.
binary call option payoff diagramm
binary put option payoff diagramm

3. OptionPricing:uncertainvolatility approach
With regard to the volatility, we will consider the model developed in 1995 by
Avellaneda, Levy and Paras . The authors avoid the useof a term structureof
volatility as a deterministic function of time and the price of assets. Also, they do
not use stochastic volatility models. They havechosen an environmentwith
uncertain volatility.
The main assumption is that volatility is in a value limited into a specific set, but it
is unknown.
According to our expectations we will have a band that should represent upper
and lower bounds for volatility. The authors suggested of getting the minimum
and maximum value formextreme values of implied volatilities of a liquid
derivatives or fromhistorical volatility spikes.
It'is therefore usefulto find the volatility limits to restrictthe spread with the
help of static hedging, using the the worstcasescenario.
The yield in this worstcasescenario is set equal to risk free rate
we will have
.

4. the term with the volatility is multiplied by the Gamma of the option, it follows
that the value of the volatility that will give the minimum portfolio value depends
on the sign of Gamma, if it is positive if we choosethe lowestvolatility value
while when it is negative we will choosethe highest value
long position worstcase
That is the worstcasein a Long position, but the best casescenario in a Short
position,while
is the worstcase in a Shortposition.
We are always in the Black-Scholes world, weassumethat the price of an asset a
process
We use Ito's lemma and obtain a stochastic process

5. We have a Portfolio:
We are Delta-hedging to eliminate a sourceof uncertainty fromthe underlying . Π
is an instantaneously risk-freeportfolio and as such musthave a return, which
coincides with the risk-freereturn.
wherer is a risk-freerate of return. Assumethat r is a constant, weobtain the
Black-Scholes equation , which is
We will useit as our model in the finite differences
The Portfolio to hedge a Binary Call is formed by a ShortCall with Strikelower
than the Binary Option and a Long Call with Strike higher than the Binary Option.
In particular, we will examine a Binary Call with a strike equal to 100, a Call short
Struck to 90 and a Call Long Struck to 110.
Numerical Methods:Finite Differences
The differential equations that can be solved by an explicit analytical
formula are few. Consequently, the development of accurate numerical
approximation schemes is essential to extract quantitative information.
With the help of numerical methods we can use models that are valid, but does
not have an explicit solution
The finite difference method can be used to obtain approximations of differential
equations that do not have an analytical solution. These approximations are
obtained by replacing the derivatives in the equation with appropriate numerical
differentiation formulas.

6. To applying the method of finite differences, we set up a grid
The grid has typically the samenumber of assetsteps and time steps
The grid is made up of points to the underlying values :
and time
We denote the value of the option at each grid point as:
with i which is the asset variableand k signed the time variable.

7. Looking for the value V of the option at any point in time we will march backward
starting from expiration, becausewe know the payoff which is a function of the
underlying price.
As anticipated, we will use the Black-Scholes partial differential equation. We also
said that, in a nutshell, the method consists in replacing the partial derivatives
with the approximations using the grid points and the Black-Scholes equation.
We are going to aproximate θ with the forward difference;
and we approximate Δ with the central difference:

8. There is obviously an approximation error, as in all numerical methods,
approximating continuous partial equations .
We apply a boundary conditions to be applied at S0, which eliminates drift and
diffusion that
and at the maximum level of S, let's say at infinity:
Evaluating Binary Options
We apply all this to the evaluate a binary option using VBA and Excel.
The function creates a matrix of size[number assetsteps: 3] 3 represents the
three vector arrays: underlying price, payoffs, option valueand builds an array
using nested For Loops ,simulating the movement in the grid , starting from
expiration. The function will usefinite differences and Black-Scholes equation
expressed in Greeks.
Function binary_option(Vol, intrate, Expn, Strike, Cash, Payoff, EarlyEx, NAS)
' variables declaration
ReDim S(0 To NAS)
ReDim VOld(0 To NAS)
ReDim VNew(0 To NAS)
ReDim Dummy(0 To NAS, 1 To 3)
' uncertain volatility assumption
Vol = Application.Max(VolMin, VolMax)

9. 'dimension of the distance between nodes
dS = 2 * Strike / NAS
dt = 0.9 / NAS / NAS / Vol / Vol
NTS = Int(Expn / dt) + 1
dt = Expn / NTS
'Call / Put condition
q = 1
If Payoff = "P" Then q = -1
‘for loop for the S array
For i = 0 To NAS
S(i) = i * dS
If S(i) >= StrikeThen
VOld(i) = Cash ' Payoff
Else
VOld(i) = 0
End If
Dummy(i, 1) = S(i)
Dummy(i, 2) = VOld(i) ' Payoff
Next i
‘for loop for the VOld array
For k = 1 To NTS
For i = 1 To NAS – 1
‘greeks
Delta = (VOld(i+ 1) - VOld(i - 1)) / 2 / dS
Gamma = (VOld(i+ 1) - 2 * VOld(i) + VOld(i - 1)) / dS / dS
‘uncertain volatility condition
Vol = VolMax
If Gamma > 0 Then Vol = VolMin
Theta = -0.5 * Vol * Vol* S(i) * S(i) * Gamma - intrate * S(i) * Delta _
+ intrate * VOld(i) ' BSE
‘using Theta to populate the vnew, array

10. New(i) = VOld(i) - dt * Theta
Next i
VNew(0) = VOld(0) *(1 - intrate * dt) ' PV along S=0
VNew(NAS) = 2 * VNew(NAS - 1) - VNew(NAS - 2) ' Cash alla barriera
For i = 0 To NAS
VOld(i) = VNew(i)
Next i
‘check for early exercise (American options ..)
If EarlyEx= "Y" Then
For i = 0 To NAS
VOld(i) = Application.Max(VOld(i), Dummy(i, 2))
Next i
End If
Next k
For i = 0 To NAS
Dummy(i, 3) = VOld(i)
Next i
optvalue_bin = Dummy
End Function
Excluding volatility, we maintain all the Black-Scholes assumptions. Wealso
assumethe absence of dividends, and we exclude the possibility of early exercise.
Let's say that we estimated the volatility minimum at 15% and the maximum at
25%. Weapply the function on a Binary Call, assuming interest rates at 5% ,with
Strike = 100, which pays cash=1 and 1 year .

11. This is the chart of the Call option value and payoffs computed using finite
difference
d.
This is the Binary Putwith the same parameters

12. Volatility effect onBinary Options
The effect of volatility on the price of a binary option depends on whether the
option is deep out of the money, in the of the money, or far from the money.
When the underlying is far below the exercise price, volatility is only a small
component of the theoretical value of the binary option. However, as the price of
the underlying instrumentapproaches and exceeds the threshold of the strike,
the value of a binary option becomes very sensitive to volatility.
This depends on the fact that a Binary Option has very high Gamma as it
approaches and crosses theStrike level, if the underlying volatility increases, this
will essentially "Boost" the option value. Finally, if the binary option is deep in the
money, it becomes less sensitiveto the volatility of the underlying, justlike a
vanilla option.
Constant Volatility
Let's have a quick look at the individual options with the assumption of constant
and known volatility.
Call:15% volatility Call:25%volatility

13. Put:15% volatility Put:25%volatility
UncertainVolatility
We will now consider the behavior of the portfolio using the uncertain volatility
approach .
We reduced drastically the number of assets steps in order to verify how the
evaluation of the option changes with narrow bands and wider bands.

14. Call:10-30% volatility Call:20-25%volatility
Put:10-30% volatility Put:20-25%volatility

15. 2. Hedging Portfolio
The Problem of hedging or replicating a Binary Call is not so easy, you can only
approximate the hedging with a Vertical Spread.
A vertical spread involves buying a call option (or put) and simultaneously writing
a call option (put) with the same maturity, but a different strikeprice.
You can't perfectly replicate a Binary option with Vanilla options, even with
arbitrary Strike.
The graph of a binary option payouts haveinfinite slope around the strike price,
while all Vanilla options havefinite-slope graphs.
Binary calls can be synthesized using an infinite number of vertical spreads of
standard calls, but this replication strategy is obviously unrealistic.
To find a solution to this issue,we could use a technique called Richardson
Extrapolation and approximate the hedge with a finite number of Vanilla Options
(Avellaneda)
Adding Vanilla Options to our Portfolio ,we will therefore improvethe hedging,
but that's outside the context of this Project
We will approximateour hedging portfolio for the Binary Call struck at 100 by
selling shorta Call struck at 90 and buying a Call struck at 110. Weare using the
uncertain volatility approach. Applying the VBA code we get this Portfolio shape:

16. The shape of the Portfolio seems to be correct, reflecting the problemmentioned
before, the slope of the Binary options around the Strike price.
The shape of the single options seems correcttoo, and probably shows theeffect
of a relatively high volatility.

18. Jump Condition
We have, so far, only considered the hypothesis of Homogeneous Maturity. Now
we introduce the case in which one of the Call of the Vertical Spread, the one
struck at 90, expires beforethe Binary Call, ie the case of inhomogeneous
Maturity.
we define a value
NTS1 = Int(PrftExpn - Van1Expn / dt) + 1
which marks the point on the grid wherethe Call struck at 90 Vanilla went to
espiration
We put a condition IF ... Else inside the loop to check if the Call 90 has expired
If PrftExpn > Van1Expn And k < NTS1 Then
VNew1(i) = VNew1(i - 1) ' Application.Max(q * (S(i) - Strike), 0)
Else
VNew1(i) = VOld1(i) - dt * Theta ' Forward differencefor Theta
End If
Since we're marching backward , as long as we are in the period between Call 90
expiration and other Options expiration, the Call 90 will be worth the initial value,
ie the payoff.
We assumethat the Call 90 goes to Espiration in 0.8 (years) , instead of 1(year):

19. The effect on the Portfolio is, quite obviously, to move it from the previous
equilibrium. The condition of Jump, of course, affects only the Call 90, the first in
the next chart.

21. 3. Now we wantto maximize the Binary Option Value. We may want to find the
most convenient price for which we can sell the Binary Option.
These option will havepayoff
Λi(S)
We have to take into accountthe Bid-Ask difference to set up our static hedged
portfolio. We will have a quantity of each Vanilla. The cost of this portfolio is:
where
We will recalculate our Call 90 using the "worstcasescenario" and our Call 110
using the "bestcase scenario" to take into account the bid -ask factor, while it will
be the opposite for a position.

22. This is the Portfolio for Long Position.

23. This is the Portfolio for ShortPosition .
We could then back out the value of hedged Binary Option
and, taking a Long Position, we may wantto maximize the Binary Option Value