Monte Carlo Simulations Lecture 1 December 7, 2014
Outline
Monte Carlo Methods Monte Carlo methods simulate the random behavior underlying the financial models Remember: When pricing you must simulate the risk-neutral random walk(s) The value of a contract is then the expected present value of all cashflows
When implementing a Monte Carlo method look out for the following 1 Number of dimensions Is the contract an option on a single underlying or many? Is there any strong path dependence in the payoff? For each random factor you will have to simulate a time series The time will only be proportional to the number of factors Monte Carlo methods are ideal for higher dimensions (finite difference crawls)
2 Functional form of coefficients The main difference between an equity option problem and a single-factor interest rate option problem is in the functional form of the drift rate and the volatility These appear in the governing partial differential equations as coefficients Does it matter? As with Finite difference methods, it doesn t matter so much what the drift and volatility functions are in practice
3 - Boundary/final conditions In a numerical scheme the difference between a call and a put is in the final condition You tell the MC scheme how to start at expiration and work towards the present
4 Decision features Early exercise, instalment premiums, chooser features, are all examples of embedded decisions seen in exotic contracts Monte Carlo method becomes cumbersome When using the Monte Carlo method we are only finding the option value at today s stock price and time. But to price an American option correctly, say, we need to know what the option value would be at every point in stock price-time space
We are concerned with simulating or approximating the solutions of stochastic differential equations, in order to compute expected discounted payoffs
Integrable SDEs Example 1: simple Brownian motion ds = µdt + σdw with constant µ and σ This can be integrated to give S(T ) = S(0) + µt + σw (T )
Example 2: Geometric Brownian motion (Black-Scholes model for stock prices) ds = µsdt + σsdw and we can use Ito to convert this to d(log S) = (r 1 2 σ2 )dt + σdw which can be integrated to give log S(T ) = log S(0) + (r 1 2 σ2 )T + σw (T ) S(T ) = S(0) exp ((r 1 2 σ2 )T + σw (T ))
Non Integrable SDEs Example 1: Cox-Ingersoll-Ross model dr = α(β r)dt + σ rdw Example 2: Heston stochastic volatility model (stock prices) ds = rsdt + V SdW 1 dv = λ(σ 2 V )dt + ξ V dw 2 with dw 1 dw 2 = ρdt
Generic non-integrable SDEs Multi-dimensional SDE ds = a(s, t)dt + b(s, t)dw Approximating this using time step h by S n+1 = S n + a(s n, t n )h + b(s n, t n ) W n Such an approximation is also needed for the simple integrable cases when the payoff is path-dependent
Monte Carlo Objective What are we trying to achieve with Monte Carlo simulation? estimate prices which correspond to expectation (under risk neutral measure) of discounted payoff Option Value t = e r(t t) E(Payoff ) estimate price derivatives (Greeks) for hedging V θ where θ may refer to initial asset price (delta), or volatility (vega), or some other quantity
European Option Value Payoff value depends solely on the state of the underlying at maturity Call Option Option Value t = e r(t t) E [max(s(t ) K, 0)] Put Option Option Value t = e r(t t) E [max(k S(T ), 0)]
European Option Value Digital Call Option Option Value t = e r(t t) E [H(S(T ) K )] Digital Put Option Option Value t = e r(t t) E [H(K S(T ))]
Basket Options Value Multi-asset Payoffs Call on weighted basket: [ ( n )] Value t = e r(t t) E max w i S i (T ) K, 0 i=1 Put on weighted basket [ ( Value t = e r(t t) E max K )] n w i S i (T ), 0 i=1
Monte Carlo Algorithm Estimate for the value of an option by following these simple steps: 1 Simulate the risk-neutral random walk as discussed below Starting at today s value of the asset S(0), over the required time horizon This time period starts today and continues until the expiry of the option This gives one realization of the underlying price path 2 For this realization calculate the option payoff
1 Perform many more such realizations over the time horizon 2 Calculate the average payoff over all realizations 3 Take the present value of this average, this is the option value
Monte Carlo Strengths Simple and flexible The mathematics that you need to perform a Monte Carlo simulation can be very basic To get a better accuracy, just run more simulations Easily able to handle high-dimensional problems
Weaknesses of Monte Carlo Not as efficient as finite differences for very low dimensions (1-3?) Not yet efficient for applications with optional exercise (American options, Bermudan options, optimal trading given transaction costs)
Finite Difference or Monte Carlo What is used in industry? FX - finite difference because low-dimensional (1 domestic interest rate, 1 foreign interest rate and 1 exchange rate = 3-dimensional) Fixed income - MC for LIBOR models because of dimensionality Energy options - finite difference because low-dimensional and options with conditional exercise
Credit - MC because high-dimensional (multiple companies) Equities - MC because of high-dimensional baskets