Comparison of an analytical expression of the value of an american put with standard methods
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Comparison of an Analytical Expression of the Value of an American Put with standard Methods. -Vicky Sharma. Background(1). Partial Differential Equation (PDE) A differential equation of a function of 2 or more equations comprising of partial derivatives of the function .

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Comparison of an analytical expression of the value of an american put with standard methods l.jpg

Comparison of an Analytical Expression of theValue of an American Put with standard Methods

-Vicky Sharma


Background 1 l.jpg
Background(1)

  • Partial Differential Equation (PDE)

    • A differential equation of a function of 2 or more equations comprising of partial derivatives of the function.

  • Boundary Value Problem

    • Boundary value problem is a differential equation together with a set of

      additional restraints. A solution to a boundary value problem is a solution tothe differential equation which also satisfies the boundary conditions.


Background 2 l.jpg
Background(2)

  • Black-Scholes PDE for the price of the option

  • The value of the option is known at expiry. This is known as the terminal condition.

    • European Options are only exercised at maturity.

      • Terminal condition becomes the boundary condition. The PDE can be solved for a unique solution – Black-Scholes Formula

    • American Options can be exercised at any time before maturity

      • There is no boundary for each time instant before maturity – there is no analytical solution.


American options l.jpg
American Options

  • Without dividends, it is in-optimal to exercise an american call before maturity.

    • Same as european call, problem can be solved with a combination of binomial tree and black-scholes formula.

  • American Puts are different

    • There exists a critical price Sf(t) at time t below which it is optimal to exercise the put.

      • This price is called the optimal exercise price at time t

    • The optimal exercise price is the boundary condition for the PDE for an american put

      • The optimal exercise boundary is a part of the solution

        • The problem becomes a free boundary problem


Optimal exercise boundary for pricing a put l.jpg
Optimal Exercise Boundary for pricing a put

  • The optimal exercise boundary defines the value of the put option when exercised.

    • If the optimal exercise boundary is known,, monte-carlo simulations are used to compute the present discounted values of the maximum value of the put when it crosses the boundary.

    • An average over multiple simulations (assuming log-normal model) gives the value of the put option.


American put l.jpg
American Put

  • The problem of finding the analytical approximation to the american put becomes the problem of finding optimal exercise boundary.

    • There have been several attempts in the past.

      • Most of the solutions were complex and inaccurate for larger maturities.

  • A recently proposed method has used laplace transformation with pseudo steady state approximation to solve for the optimal boundary and price of american put.

    • We will look at this approach and compare it’s performance to the binomial tree model and the real-world stock prices.


Analytical approximation 1 l.jpg
Analytical Approximation (1)

  • The following transformations transform the black-scholes PDE with a terminal condition to a PDE with an initial condition.

    • X = Strike, τ= T – t.

    • Sf(t) = optimal exercise price.

  • The Next transformation is on V. V is replaced by U, where

    • U = V + S – 1 when S <= 1

    • U = V, when S > 1


Analytical approximation 2 l.jpg
Analytical Approximation (2)

  • The PDE is now changed.

  • There are additional constraints on the continuity and differentiability of U at S = 1.


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Analytical Approximation (4)

  • Pseudo-Steady State Approximation

    • We assume that the optimal exercise boundary changes slower than the diffusion of the put value.

    • Hence, the stock price can be taken as the constant Sf(t) at the boundary.

      • This helps in taking the laplace transform of S around the boundary.

      • L(S) = Sf/p (p = laplace transform variable)


Analytical approximation 3 l.jpg
Analytical Approximation (3)

  • Using Laplace Transformation

    • The author used laplace transformation to convert the PDE to a linear differential equation.

  • A linear transformation of variable t (time from start) to time to expiration translates terminal condition to initial condition.


Analytical approximation 411 l.jpg
Analytical Approximation (4)

The solution for laplace transform

Is calculated easily

The boundary conditions are used to solve for the unknown variables


Analytical approximation 5 l.jpg
Analytical Approximation (5)

  • The laplace transform of the optimal boundary can be computed.

    • The problem is to compute the inverse transform.

      • The laplace transform of Sf has poles at at zero and the negative half plane.

    • Hence, the inverse laplace transform must be computed over a line in the tight-half plane.


Analytical approximation 6 l.jpg
Analytical Approximation (6)

A closed region is used to compute a known

Line integral over the poles.

The integrals over C2 and C6 vanish. Integrals

Over C3 and C5 are symmetric and add up to an

Imaginary number.


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Analytical Approximation(7)

The optimal exercise boundary is then calculated by the above expression


Analytical approximation 8 l.jpg
Analytical Approximation(8)

The value of american put is

Approximated by the above

expression


Comparison with binomial model at the money l.jpg
Comparison with Binomial Model – At the money

Interest rate and

Volatility remain

Constant and the

Stock pays

No dividends

Stock Price = $100, risk-free rate = 10%, σ = 0.30 and expiry = 1 year,

The analytical approximation follows the binomial tree model closely.

Black Scholes formula underestimates by a large margin as maturity increases


Comparison with market prices l.jpg
Comparison with Market Prices

  • We used Yahoo Stock and put Prices for our comparison.

    • In order to calibrate the model (deduce volatility), 3 years of daily closing stock prices for yahoo were used.

      • Black Scholes assumption of log-normal stock prices used to estimate volatility.

      • LIBOR Rates were used for risk free rate

        • At the time, it was 4.63%

    • The duration of stock prices from November 26, 2004 through November 26, 2007.

      • The latest stock price was $26.13

  • Price of a put contract (for 100 shares) have been plotted

    • Market price is the arithmetic average of the bid and offer prices


Comparison with binomial model different strikes at same maturity l.jpg
Comparison with Binomial Model – Different Strikes at same maturity

Maturity = 1 Month

In the Money

Out of Money

The analytical approximation compares fairly well with the market and binomial

tree prices.


Comparison with binomial model different strikes at same maturity19 l.jpg
Comparison with Binomial Model – Different Strikes at same maturity

Maturity = 5 Months

At the Money

The approximation compares fairly well with the binomial tree prices. The greatest

error with the market prices is seen around at-the-money mark


Comparison with binomial model different strikes at same maturity20 l.jpg
Comparison with Binomial Model – Different Strikes at same maturity

Maturity = 7 Months

At the Money

The increase in maturity does not affect the comparison with binomial tree model.

The numerical computation results in some under-estimation of the analytical value.

Hence, the analytical approximation is more accurate if more time is provided

computation


Comparison with binomial model different strikes at same maturity21 l.jpg
Comparison with Binomial Model – Different Strikes at same maturity

Maturity = 13 Months

At the Money


Comparison with binomial model different maturities at same strike l.jpg
Comparison with Binomial Model – Different maturities at same strike

Out of Money

Option.

Strike = $20

The analytical approximation underestimates out of money options quite significantly.

In all cases, the approximation performs as good as the binomial model.


Comparison with binomial model different maturities at same strike23 l.jpg
Comparison with Binomial Model – Different maturities at same strike

In the Money

Option.

Strike = $30

The analytical approximation’s accuracy improves as the option gets in the money.

It may also be due to the fact that computation now has larger numbers to process


Comparison with binomial model different maturities at same strike24 l.jpg
Comparison with Binomial Model – Different maturities at same strike

Heavily In the Money Option.

Strike = $40

For seriously in the money options, the difference between market prices and

Analytical approximations is small. This is mainly due to the fact that the value is

Dominated by the intrinsic value of the option.


American put optimal exercise boundary 1 l.jpg
American Put – Optimal Exercise Boundary(1) same strike

  • The computation of optimal exercise boundary results in some interesting results:-

    • The value of the optimal exercise price at expiry is the strike price

  • However, there seems to be a non-zero limit of the optimal exercise price even for large maturities

    • We show the optimal exercise boundaries for a maturity of 1 month (0.0834 years) and 3 different strikes.


American put optimal exercise boundary 2 l.jpg
American Put – Optimal Exercise Boundary(2) same strike

The optimal

Exercise price

Reaches

Strike at

maturity

Optimal boundary seems to approach a

Common minimum value as maturity increases

The optimal exercise boundary rises sharply towards the strike price at maturity.

Hence, the behavior is non-linear. As maturity increases, the optimal exercise

Boundary at t = 0 reaches the same limit for all strikes.


Conclusion s l.jpg

The new approximation formula provides a single expression for the value of american put.

No iterations are needed.

Several concepts, like the perpetual exercise price and value are easily explained.

Compares well with established models

Computation can be slower than the numerical methods.

No significant improvement over binomial tree model.

Seriously under-estimates out-of-money options

Conclusion(s)

The new expression has some advantages and disadvantages

Advantages

Disadvantages


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