monte carlo methods in structuring and derivatives...

Monte Carlo Methods in Structuring and Derivatives Pricing

Prof.ssa Manuela Pedio

20541– Advanced Quantitative Methods for Asset Pricing and Structuring

Spring 2019

Outline and objectives

2Monte Carlo Methods in Structuring and Derivatives Pricing

▪ The basic Monte Carlo algorithm

▪ Discretization schemes, exact and approximate

▪ Monte Carlo greeks

▪ Monte Carlo in multi-asset and American pricing problems

▪ Speeding up Monte Carlo

▪ Monte Carlo methods for American-style options

This set of slides is based on the book Paul Wilmott introduces Quantitative Finance, by Paul Willmott, Chapter 29

http://www.untag-smd.ac.id/files/Perpustakaan_Digital_1/FINANCE%20Paul%20Wilmott%20Introduces%20Quantitative%20Finance%200470319585.pdf

http://www.untag-smd.ac.id/files/Perpustakaan_Digital_1/FINANCE Paul Wilmott Introduces Quantitative Finance 0470319585.pdf

Curiosity: the origins of Monte Carlo method

3Monte Carlo Methods in Structuring and Derivatives Pricing

▪ Although the use of techniques based on the drawing of random numbers dates back to the beginning of 1900, Monte Carlo algorithm has been formalized in the context of Manhattan Project in the late ’40s

▪ It was suggested by Stanislaw Ulam, while he was investigating radiation shielding and the distance that neutrons would likely travel through various materials

▪ Because the experiments that they were carrying out weresecret, he needed a code name for the algorithm: his colleague Nicholas Metropolis, suggested using the name Monte Carlo, which refers to the Monte Carlo Casino in Monaco

https://en.wikipedia.org/wiki/Monte_Carlo_method

▪ The value of an (basket of) option(s) is the expected present value of its payoff(s) at expiry

o Under a risk neutral random walk for the underlying

▪ How can we “estimate” (approximate) the expectation?

The Monte Carlo algorithm (1/3)

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When a derivative payoff is computable, the no-arbitrage price of anystructure can be computed by averaging over N ⟶ ∞ simulated paths


or

with deterministic discount factor

with stochastic discount factor

④ Calculate the average payoff over all realizations⑤ Take present value of average, this is the value of the structure

▪ How do you recursively update St? In the case of a Geometric Brownian Motion, the obvious choice is an Euler scheme:

o This discretization method has an error of O(δt), i.e., the error disappears at the same speed of the decline in time increments


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▪ By an application of a law of large numbers such that (under some assumptions) sample means asymptotically converge to population expectations, it is enough to follow 5 simple steps:

① Simulate the risk-neutral random walk starting at today's value of the asset S0, over the required time horizon; this gives one realization of the underlying price path

② For this realization, calculate the option payoff

③ Perform many such realizations (N) over the time horizon


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o Let’s see the case of an Asian option


▪ For a log-normal random walk no approximation is needed because there is a rather simple, and exact, time stepping algorithm

▪ Because the risk-neutral stochastic differential equation for (log of) S is

▪ This can be integrated exactly:

▪ Or, over a time step δt,

▪ Advantage is that δt need not be small, since expression is exact

▪ Because it is exact, if we have a payoff that only depends on the final asset value, we can simulate the final price in one giant leap

o If the option is path dependent (e.g. continuous barriers) we will still introduce an error O(δt), decreasing as time-step decrease

Exact vs. approximate discretization (1/2)

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When the assumed stochastic process for the underlying follows a log-normal random walk, then the discrete-time process is exact


When the discretization is approximate, the approximation may introduce errors of O(δt); in addition because, the error due to using a finite number (N) of realizations of the asset price paths is O(N-1/2)

Exact vs. approximate discretization (2/2)

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▪ When it comes to simulating a random walk it doesn't matter very much what distribution we use for the random increments as long as the time step is small and thus we have a large number of steps

o In the limit, as the size of the time step goes to zero the simulations have the same probabilistic properties over a finite time-scale regardless of the distribution over the infinitesimal time-scale

o This is a result of the central limit theorem.

▪ If size of time step is δt then, for complicated products, such as path-dependent ones, the error in the price will be the worst between the error introduced by a finite number of simulations and the error due to discretization:


▪ Monte Carlo methods can also be applied to estimate the greeks

▪ The simplest way to calculate the delta of an option using Monte Carlo simulation is to estimate the option's value twice:

▪ This is an accurate estimate of the first numerical derivative, with an error of O(h2)

▪ However, the error in the measurement of the two option values at S + h and S - h can be larger for the Monte Carlo simulation

o These Monte Carlo errors are then magnified when divided by h, resulting in an error of O(1/(hN1/2))

o To overcome this problem, estimate the value of the option at S + h and S - h using the same values for the random numbers: this way the errors in the Monte Carlo simulation will cancel each other out

Monte Carlo Greeks (1/2)

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The standard measurement error incurred when approximating the greeks with numerical derivatives are magnified by MC pricing


Monte Carlo Greeks (2/2)

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▪ The same principal is used to calculate gamma, vega, and theta

o Gamma is a second-order numerical derivative vs. S and theta is the derivative when time to maturity is slightly changed

▪ If we are dealing with a lognormal random walk then things are much simpler than this in practice: simulate many lognormal random paths as usual, starting with asset price at S

o If the asset had started at (1 + ϵ)S, you do not have to resample the paths -- all you need is to multiply the final prices, at expiration, by 1 + ϵ

▪ Monte Carlo simulation is a natural method for the pricing of European-style contracts that depend on many underlying assets

▪ Consider a structure paying off some function of S1, S2 , …, Sd


Multi-asset applications

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▪ We could, in theory, write down a PDE in d + 1 variables, but such a problem would be horrendously time consuming to compute

▪ Using MC methods, all we need to do is to simulate is

▪ The catch is that the ϵ is are correlated

▪ How can we generate correlated random variables? Using Choleski

o Suppose that we can generate d uncorrelated Normally distributed variables ϵ 1, ϵ 2, …, ϵ d

o We can use these variables to get correlated variables with the transformation where Ф and ϵ are d x 1 column vectors and

so that

▪ The Choleski factorization gives one way of choosing this decomposition and results in a matrix M that is lower triangular

▪ Choleski is one of many alternative factorizations, but very commonMonte Carlo Methods in Structuring and Derivatives Pricing

Correlation matrix of returns

Flavors of Monte Carlo to increase its speed

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▪ How can we speed MC up?

▪ When antithetic variables are used, one calculates two estimates for an option value using the one set of random numbers

▪ We do this using Normal random numbers to generate one price and then taking the same set of numbers but changing their signs, simulate a realization, and calculate the option present value

▪ The estimate for the option value is the average of these two values

▪ Under a control variate technique, assumes two similar derivatives:_ the former is the one we want to value by MC and_ the second has a similar (but 'nicer') structure such that we

have an explicit formula for its value

▪ We use the one set of realizations to value both options, call the values estimated by the Monte Carlo simulation V’1 and V’2

▪ If the accurate value of the second option is V2 then a better estimate than V’1 for the value of the first option is


Pros and cons of Monte Carlo methods

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▪ The argument behind control variate is that the error in V1 will be the same as the error in V’2 and the latter is known

▪ The idea of this technique is that V’1 – V’2 will have a small or zero mean but can reduce the variance of the error significantly.

▪ The Monte Carlo technique is clearly very powerful and general

o The concept readily carries over to exotic and path-dependent contracts, just simulate the random walk and the corresponding cash flows, estimate the average payoff and take its present value

▪ The main disadvantages are twofold. First, the method is slow when compared with the finite-difference solution of a PDE

▪ Second, the application to American options is far from straightforward

▪ The reason for the problem with American options is to do with the optimality of early exercise

o One must calculate the price for all values of S and t up to expiry in order to check that at no time is there any arbitrage opportunity


Monte Carlo methods for American-style options

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▪ The Monte Carlo method in its basic form is only used to estimate the option price at one point in S, t-space, now and at today's value

▪ The problem concerns the time direction in which we are solving

o When we use MC to value a European option we only find the option's value at the one point, the current asset level and the current time

o We have no information about the value at any other S price or time

▪ In principle, we could find the option value at each point in asset-time space using Monte Carlo

o For every asset value and time that we require knowledge of the option value we start a new simulation

o When we have early exercise we have to do this at a large number of points in S, t space, keeping track of whether the constraint is violated

o If we find a value for the option that is below the payoff then we mark this point in the S, t space as one where we must exercise the option

▪ Such a procedure is possible, but the time taken grows exponentially with the number of points at which we value the option Monte Carlo Methods in Structuring and Derivatives Pricing

Longstaff and Schwartz’s method (1/2)

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▪ However, one recent algorithms does the magic

▪ Longstaff and Schwartz’s (2001) method combines forward simulation of asset price paths from startup to expiration, with the present valuing of cash flows along paths

▪ At each time step one looks at the benefit of exercising versus holding using a simple regression across asset prices

o Consider an American put with strike price $100, expiring in one year's time; the stock is currently $100, it has a volatility of 20% and the risk-free interest rate is 5%

o Step one, simulate many, N, realizations of the asset path from now to expiration

o Next, calculate the pay-offs at expiration for each of these paths assuming that we haven't exercised the put option at any time before expiration


Longstaff and Schwartz’s method (2/2)


o But we might have exercised earlier than that

o One then uses a regression to calculate the cash flow from holding on to the option, conditional on the stock price being at each possible level

o Now, for each value compare the value of exercising immedia-tely with that from holding on, and based on such forecasts we decide whether exercise is optimal

o We must continue to work backwards through all dates, right back to time zero to get a final cash flow matrix

o The last step in pricing the option is to present value all of these cash flows back to time zero and average

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