AN INTRODUCTION TO MONTE CARLO METHODS IN FINANCE
AN INTRODUCTION TO MONTE CARLO METHODS IN FINANCE
MATT BRIGIDA
Associate Professor of Finance (SUNY Polytechnic Institute)
<h2>Monte Carlo: Solution by Simulation</h2> <p align="left">The goal of this presentation is to show you when to use Monte Carlo and to provide a couple of interactive examples with visualizations.</p> - The idea is that you should first have a good understanding of <span class="fragment highlight-green">when Monte Carlo is useful</span> before you dive into its math and mechanics.
## Monte Carlo and Analytic Solutions <p align="left"><span class="fragment highlight-current-green">When an analytic solution is available, you should generally use it.</span></p> - If you make the same underlying assumptions, the analytic solution will be the same as the Monte Carlo, though the Monte Carlo solution will have estimation error.
## Monte Carlo is Useful When... <p align="left">An <span class="fragment highlight-current-red">analytic solution is unavailable or difficult to obtain or when the underlying assumptions differ from a continuous-time solution</span>. This is often the case, which makes Monte Carlo an important tool for analysts.</p>
## Example: European Option Pricing <p align="left">Assuming a stock price follows a [geometric Brownian Motion](https://en.wikipedia.org/wiki/Geometric_Brownian_motion), then at time $T$ in the future we have:</p> <span class="fragment highlight-current-green">$S_T = S_0 e^{(\mu - \frac{1}{2}\sigma^2)T + \sigma B_T}$</span> <p align="left">where $S_0$ is the stock price today, $\mu$ and $\sigma$ are the stock's drift and volatility respectively, and $B_T$ is the value of a Brownian Motion at time $T$.</p>
- Since $B_T \sim N(0, T)$, we can <span class="fragment highlight-green">create random draws from $B_T$ by drawing from $N(0, 1)$ and multiplying by $\sqrt{T}$</span>. - For each draw of $B_T$ we get a new value of $S_T$.
## Drift and Volatility <p align="left">We estimate volatility $\sigma$ using historical or forward looking measures. However, in a Black-Scholes world the market is complete, and so we can set the drift $\mu$ equal to the risk free rate $r_f$.</p>
- Therefore each simulated stock price at time $T$ is: <span class="fragment highlight-green">$S_T = S_0 e^{(r_f - \frac{1}{2}\hat{\sigma}^2)T + \hat{\sigma} N(0, 1)\sqrt{T}}$</span> <p align="left">where $N(0, 1)$ is a random draw from a standard Normal distribution.</p>
- We then simulate *i* possible stock prices at time *T*, and calculate the payoff of the call option for each *i*. - The *expected* payoff on the option is the average of these payoffs. - Then the value of the call option today is the expected payoff discounted back to today. <!-- $C_0=e^{-r_fT}\text{mean}(\text{max}(S_{iT} - X, 0))$ -->
<iframe width="560" height="315" src="https://www.youtube.com/embed/HOSf-lk9JOw" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe> <!--- <p align="left">See here for detailed instructions on how to calculate a call option's value by Monte Carlo, including R code and an Excel spreadsheet.</p> -->
## Monte Carlo vs Analytic (Black-Scholes) <p align="left">In both cases we make the same assumptions, namely the stock follows geometric Brownian Motion, $\sigma$ is a constant, and the [market is complete.](https://en.wikipedia.org/wiki/Complete_market).</p> - Therefore <span class="fragment highlight-current-green">the Monte Carlo estimate should be equal to the Black-Scholes analytic solution</span>.
## Black-Scholes Value <span class="fragment highlight-current-green">$C_0 = S_0N(d_1) - Xe^{-rT}N(d_2)$</span> <!-- <div class ="columns-2"> --> <p align="left">where</p> $d_1 = \frac{ln(\frac{S_0}{X}) + (r+\frac{\sigma^2}{2})T}{\sigma\sqrt{T}}$ $d_2 = d_1 - \sigma\sqrt{T}$ <p align="left">$C_0$ are $S_0$ are the values of the call option and underlying stock at time 0. $X$ is the strike price, and $T$ the time until option expiration in years.</p> <!-- </div> -->
<!-- [when binomial option pricing presentation is complete, change link to slide on complete market]. --> - On the next slide we calculate the value of a European call option by both the Black-Scholes formula and Monte Carlo.
## Pricing Path-Dependent Options <p align="left">As mentioned, <span class="fragment highlight-current-green">Monte Carlo makes sense when an analytic solution is unavailable or its solution is intractable</span>. This is often the case for *path-dependent* options, where payoffs are a function of the stock prices over some interval---the path the stock took to get to its present price. Examples of such options are:</p>
- <span class="fragment highlight-blue">American Options</span>: can be exercised prior to expiration. - <span class="fragment highlight-blue">Lookback Options</span>: payoff is a function of the maximum (or minimum) stock price over the interval. - <span class="fragment highlight-blue">Asian Options</span>: payoff is a function of the average stock price over the interval.
## Lookback Option <p align="left">Say an option pays the difference between the *maximum* price of a security over a time period and a set strike price. So the <span class="fragment highlight-green">payoff is $max(S_M - X, 0)$ where $S_M$ is the maximum security price over a set interval</span>. This is a European lookback call option with a fixed strike price (you can also let the strike price float).</p> - If we assume the underlying follows geometric Brownian motion, then this option value [has an analytic solution](https://scholar.google.com/scholar?hl=en&q=goldman+sosin+gatto+1979).
- However what if your boss wants you to value a lookback option on an underlying which is described by the following process? <span class="fragment highlight-green">$dS_t = \kappa(\mu_S - S_t)dt + \sigma S_t dB_t$</span> - An analytic solution might be possible, but it may take a while to find. This is where it makes sense to turn to Monte Carlo.
## Monte Carlo Setup <p align="left">Given the parameters $\mu_S$, $\kappa$, and $\sigma$, and a starting price, we'll <span class="fragment highlight-green">simulate various price paths</span>. We'll calculate the maximum over each of these price paths, which will give us the lookback call option payoff given the path. Taking the average value and discounting at the risk-free rate (assuming a complete market) affords the lookback call option's value.</p> - Remember, here we need to simulate the entire security's path over the year, not just the terminal value like in the earlier valuation of a European call.
## Monte Carlo Setup <p align="left">In the next slide we'll price a lookback call with 1 year to maturity, on a security with a \$35 price today. The strike price is \$37, and we will simulate a price path with 252 points (one for each trading day of the year).</p> - You can set the $\kappa$, $\mu$, and $\sigma$ parameters using the slider inputs. When you change any slider the Monte Carlo valuation is re-run.
## Parting Note <p align="left">Monte Carlo is a good approach in the case where we don't have a closed-form solution give a particular underlying process. However what if we do have a <span class="fragment highlight-green">closed-form solution for a continuous-time process</span>, but the <b class="fragment highlight-red">lookback option samples the price at discrete points?</b></p>
<p align="left">For example, what if the lookback option's payoff was the max closing price of the security minus the strike price. Then the continuous-time solution will likely overvalue the option.</p> - Consider the probability that the maximum value over a period occurs at a closing price.
<p align="left">So in this case, despite having a closed-form solution for a continuous process, it still makes sense to simulate the process in a Monte Carlo solution because this is closer to the actual payoff function.</p> - In sum, a <b class="fragment highlight-green">good understanding of the problem is necessary to choose the most appropriate solution method</b>.