Cash takeover notes
Aloke Mukherjee
June 2006
1. Implementation notes
Numerical integration
Calculating the price of the option subject to takeover requires integrating the
model option value against takeover times up to expiry. In the first iteration
this was done using a fixed step-size. This code is
http://math.nyu.edu/~atm262/files/takeover/attic/takeover.m. It yields
acceptable results in many cases. However, if the takeover intensity is very
high the probability of takeover is concentrated in the next instant.
Integrating this “half-delta” function requires additional resolution in this area.
I first rewrote the function to adaptively adjust the step-size using Richardson
extrapolation: the step-size is halved until the error in the result is less than
the given tolerance. This code is
http://math.nyu.edu/~atm262/files/takeover/attic/take2.m. This yielded
more accurate values at the cost of performance: the overhead of repeated
integrations with smaller and smaller step sizes obviously increases run time. .
The most recent iteration uses MATLAB’s quadv (vectorized quadrature)
function which implements a recursive adaptive Simpson quadrature. This
code is http://math.nyu.edu/~atm262/files/takeover/take2quad.m. It yields
similar results to the adaptive method above but it is faster since it partitions
the integral, increasing resolution only where required. The vectorization
allows results to be calculated for a range of parameter values at once (e.g. for
different volatility levels).
Other changes
MATLAB’s Black-Scholes function blsprice rejects negative risk-free rates.
Inherently, the BS formula has no such restriction. Negative risk-free rates
arise easily in the cash takeover model since takeover intensity induces a
compensating negative drift. The method was altered to remove this
restriction.
As discussed above, a high takeover intensity leads to a rapid decay in the
integrand with time. This can lead to NaNs being returned by the Black-Scholes
function for high lambda and far future times. These values can be treated as
zeros with no impact on accuracy since they are vanishingly small (the normal
CDF is evaluated at very negative values). This change allows options to be
correctly valued at higher takeover intensities.
NaN implied volatilities are also possible in the cash takeover model. The
minimum value of an option as a function of volatility occurs when volatility is
zero. Lower option values are possible in the presence of cash takeover
resulting in undefined BS implied volatilities. These are now treated as zero
when calculating the implied volatility surface.
Jump-to-default
Defaults, like takeovers, are assumed to be a Poisson process with a unique
intensity. We assume the default and takeover processes are independent.
As Merton observed, incorporating default into the option pricing formula
simply requires replacing appearances of the risk-free rate r with r + λ where λ
is the default intensity. This makes intuitive sense since the possibility of
default induces a compensating drift in the diffusion, however this drift affects
only the no-default term (standard Black-Scholes) of the summation over
number of defaults. All other terms contribute nothing since the option value
goes to zero in default. The same applies with takeover, default only
manifests in the drift.
Monte Carlo
Simulating the process where the underlying follows GBM requires first
generating takeover and default times using the standard method for
generating Poisson arrival times from uniform random variables. We make the
simplifying assumption that the processes are independent, which allows the
draws to be made separately. For each sample path one takeover time and one
default time are generated. The following logic is used to determine the
simulation horizon:
If (Default time < takeover time & Default time < expiry)
No need to simulate path (PV of zero is zero)
Else if (Takeover time < default time & Takeover time < expiry)
Simulate to takeover time
Else
Simulate to expiry
We make a draw from the lognormal distribution appropriate for the given
horizon. For paths with no takeover or default, the payoff function is applied
at expiry and discounted to the present. For takeovers, the payoff is applied
at takeover time to the stock price multiplied by the jump factor and
discounted to the present. The option price is the average present value over
all paths.
2. Validation
Reproduce standard Black-Scholes when default and takeover intensity are
zero
Agreement between “closed-form” and MC
MC error convergence
Non-zero default intensity price > zero default intensity
As takeover intensity goes to infinity convergence to Soe^j – K
(doesn’t)
Zero vol
If jump in S upon takeover is 0% call value should be decreasing in arrival
rate of takeover (intuition – lower the takeover rate, greater the range of
possible ending prices)
3. Stylized Facts / Motivation
(show results for combinations of set parameter values, including default)
Term structure of vol (upward sloping for low takeover, downward for high)
When takeover imminent – short term atm options go down, slightly otm
options go up (has to do with change in slope?) / long term options go down,
more so the higher the strike
Peak in option value as a function of takeover intensity
(not for higher lambdas – graph)
Show other graphs – how option value changes with time, strike, vol, etc.