Certificate inQuantitative Finance

Module 6 Assessed Assignment

2012

Luigi Piva

UNCERTAIN VOLATILITY ANDSTATIC 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 strike price , a maturity date, and an underlying. Binaries are sold in exchange for an upfront premium payment, just like 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, Heaviside function are widely used when we are dealing with binary (digital) option payoffs. A Heaviside function is a step function whose value 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) Put Payoff = H(K-S)

We are going to consider a Cash-Or-Nothing Call and 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 strike price and zero otherwise. A standard vanilla option pays out a potentially unlimited variable amount. This difference between a binary and vanilla option has two benefits. First, the option sellers assume a known limited risk. Second, from the 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

Option Pricing: uncertain volatility approach

With regard to the volatility, we will consider the model developed in 1995 by Avellaneda, Levy and Paras . The authors avoid the use of a term structure of volatility as a deterministic function of time and the price of assets. Also, they do not use stochastic volatility models. They have chosen an environment with 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 form extreme values of implied volatilities of a liquid derivatives or from historical volatility spikes.

It 'is therefore useful to find the volatility limits to restrict the spread with the help of static hedging, using the the worst case scenario.

The yield in this worst case scenario is set equal to risk free rate

we will have

.

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 choose the lowest volatility value while when it is negative we will choose the highest value

long position worst case

That is the worst case in a Long position, but the best case scenario in a Short position,while

is the worst case in a Short position.

We are always in the Black-Scholes world, we assume that the price of an asset a process

We use Ito's lemma and obtain a stochastic process

We have a Portfolio:

We are Delta-hedging to eliminate a source of uncertainty from the underlying . Π is an instantaneously risk-free portfolio and as such must have a return, which coincides with the risk-free return.

where r is a risk-free rate of return. Assume that r is a constant, we obtain the Black-Scholes equation , which is

We will use it as our model in the finite differences

The Portfolio to hedge a Binary Call is formed by a Short Call with Strike lower 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 analyticalformula 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.

To applying the method of finite differences, we set up a grid

The grid has typically the same number of asset steps 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 variable and k signed the time variable.

Looking for the value V of the option at any point in time we will march backward starting from expiration, because we 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:

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 asset steps: 3] 3 represents the three vector arrays: underlying price, payoffs, option value and builds an array using nested For Loops ,simulating the movement in the grid , starting from expiration. The function will use finite 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)

'dimension of the distance between nodes

dS = 2 * Strike / NASdt = 0.9 / NAS / NAS / Vol / Vol NTS = Int(Expn / dt) + 1dt = Expn / NTS

'Call / Put condition q = 1If Payoff = "P" Then q = -1

‘for loop for the S array For i = 0 To NAS S(i) = i * dS If S(i) >= Strike Then VOld(i) = Cash ' Payoff Else VOld(i) = 0 End If Dummy(i, 1) = S(i) Dummy(i, 2) = VOld(i) ' PayoffNext 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

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 iEnd IfNext kFor 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. We also assume the 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%. We apply the function on a Binary Call, assuming interest rates at 5% ,with Strike = 100, which pays cash=1 and 1 year .

This is the chart of the Call option value and payoffs computed using finite difference

d.

This is the Binary Put with the same parameters

Volatility effect on Binary 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 instrument approaches 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 the Strike 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 sensitive to the volatility of the underlying, just like 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

Put:15% volatility Put:25%volatility

Uncertain Volatility

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.

Call:10-30% volatility Call:20-25%volatility

Put:10-30% volatility Put:20-25%volatility

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 strike price.

You can't perfectly replicate a Binary option with Vanilla options, even with arbitrary Strike.

The graph of a binary option payouts have infinite slope around the strike price, while all Vanilla options have finite-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 improve the hedging, but that's outside the context of this Project

We will approximate our hedging portfolio for the Binary Call struck at 100 by selling short a Call struck at 90 and buying a Call struck at 110. We are using the uncertain volatility approach. Applying the VBA code we get this Portfolio shape:

The shape of the Portfolio seems to be correct, reflecting the problem mentioned before, the slope of the Binary options around the Strike price.The shape of the single options seems correct too, and probably shows the effectof a relatively high volatility.

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 before the 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 where the 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 difference for 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 assume that the Call 90 goes to Espiration in 0.8 (years) , instead of 1(year):

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.

3. Now we want to 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 have payoff

Λi(S)

We have to take into account the 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 "worst case scenario" and our Call 110 using the "best case scenario" to take into account the bid -ask factor, while it will be the opposite for a position.

This is the Portfolio for Long Position.

This is the Portfolio for Short Position .

We could then back out the value of hedged Binary Option

and, taking a Long Position, we may want to maximize the Binary Option Value