Underlying assets are three global stock index : Eurostoxx 50, HSI, KOSPI 200
Maturity: 36 months with advanced redemption date in every 6 months if prices of indexes satisfy given conditions at each date.
Payoff : 4.5%(annually)
In the case where prices of each indexes at the maturity date are all less than 60% of the prices at the initial date, you get loss.
I want to use Monte Carlo Simulation using R.
I plan to determine the expected returns and co-variance matrix of three indexes from historical data and generate random numbers which follow multivariate normal distribution.
Using those random numbers, I will generate paths of each stock indexes and determine the total payoff.
However, determining the expected return or variance is hard because, lately, historical returns of some of indexes are not good, not even positive.
How can I determine that? Also, If you give me any idea of improving or implementing this simulation, I'd really appreciate that.
Thank you.
Answer
No offense but it will be much more complicated than what you think... I'm not even sure that you are familiar with risk-neutral pricing in the first place? I'll try to give you some clues.
This security is called a basket option. On top of the multi-asset feature, there are non-trivial mechanisms embedded in the contract you mention:
- an auto-callable feature, meaning early redemption can happen if certain conditions are met at discrete observation dates specified as part of the contract.
- a compo/quanto feature. Because individual indices are not denominated in the same currency, either you define the basket $t$-value by converting each individual index's $t$-value in a fixed reference currency (compo) or you simply view these indices values as plain 'numbers' and express their weighted sum in a fixed reference currency regardless of the original denominations (quanto).
Let's forget about the auto-callable and quanto/compo features (as they would require a post of their own to be correctly addressed) and focus only on the basket part for the moment.
Omitting these features makes the payoff purely European i.e. it only depends on the terminal value of the basket ($T$-value). Let's denote it by $\phi(B_T)$, where in your case $\phi$ looks like the payoff of a put (or possibly a down-and-in put it's not clear from your explanation).
Let's also assume deterministic and constant rates in what follows, along with proportional dividends for the sake of clarity. The model categories I list below only reflect my own views, there is no such distinction in practice.
Pricing 101
Under the risk-neutral measure $\mathbb{Q}$ the price of an option is given by: $$ V_0 = e^{-r T} \mathbb{E}_0^\mathbb{Q} \left[ \phi(B_T) \right] $$ with here $$B_t = \frac{1}{N} \sum_{i=1}^N S_t^{(i)} $$ the $t$-value of the basket, $r$ the risk-free rate (rate at which money can be deposited on the money market, here you can take the EONIA discount curve for instance).
Mathematically, the risk-neutral measure $\mathbb{Q}$ is defined in such way that: $$ F(0,T) = \mathbb{E}_0^\mathbb{Q}[ S_T ] $$ where $F(0,T)$ is the forward price at $T$ of the equity $S$ as seen of $t=0$.
There is a lot of theory hidden behind the last few equations, I would recommend you to investigate that using a good reference book e.g. Shreve. Maybe this post is a good start also. Anyway, the consequence is that your model should eventually read something like:
$$ \frac{dS_t^{(i)}}{S_t^{(i)}} = \frac{\partial \ln F^{(i)}(0,t)}{\partial t} dt + \sigma^{(i)}(...) dW_t^{(i),\mathbb{Q}} $$ with a certain dependence structure between the Brownian motions driving the individual indices' prices.
Basket Pricing 101
From the above you see that: $$ V_0 = e^{-rT} \int_0^\infty \phi(B_T) p(B_T) dB_T $$ in other words, if you know the probability distribution of $B_T$ you are done, since you can either: (1) perform a numerical quadrature and get the option price (2) sample from the terminal distribution and compute the expectation using Monte Carlo simulations. (1) is when $p(B_T)$ is known in closed-form, (2) is more general.
It is straightforward to show that the first 2 moments of $B_T$ under $\mathbb{Q}$ are: $$ B(0,T) = \mathbb{E}_0^\mathbb{Q}[B_T] = \frac{1}{N} \sum_{i=1}^N F^{(i)}(0,T) $$ (i.e. sum of the expectations) $$ \sigma^2_B = \frac{1}{N^2} \left( \sum_{i=1}^N \sigma^{2,(i)} + 2 \sum_{j=1}^i \rho_{ij} \sigma^{(i)} \sigma^{(j)} \right) $$ (i.e. sum of the covariances)
[Model Type 1]
Assume $B_T$ is log-normal with first moments given by the above. This is a mere approximation since we know that the sum of $N$ log-normal variables is not a log-normal. But it allows you to compute the price of a basket using the BS formula: $$ V_0 = e^{-rT} ( B(0,T) N(d_+) - K N(d_-) ) $$ $$ d_\pm = \frac{\ln\left(\frac{B(0,T)}{K}\right) \pm \frac{1}{2}\sigma^2_B T}{\sigma_B \sqrt{T}} $$
[Model Type 2]
Use a shifted log-normal to approximate $\phi(B_T)$, or any other known distribution for that matter. The idea is that you already know the first 2 theoretical moments of $B_T$ and you can easily write the third and even the fourth moments using standard calculus. You then fit the known distribution to the unknown distribution of $B_T$ by matching their moments. This is known as moment-matching. In the case of a shifted-lognormal, this leads to a closed-form formula.
[Model Type 3]
You consider the true marginals of each individual asset $S_t^{(i)}$ i.e. $$ p^{(i)}(x) = \frac{d F^{(i)}(x)}{d x} = \frac{d \mathbb{Q}\left(S_T^{(i)} \leq x\right)}{ dx} $$ where $F^{(i)}$ is the $i^{th}$ asset cumulative distribution function.
You can infer the above risk-neutral probabilities from listed option prices using the Breeden-Litzenberger idendtity, see here.
Now that you have identified the marginal distributions of each asset $S_t^{(i)}$, you need to define their dependence structure so that you can eventually obtain their joint distribution from which you will be able to infer the distribution of $B_t$ since: $$ p(B_t \in A) = \int_{\frac{1}{N} \sum_{i=1}^N x_i \in A} p\left(x_1,...,x_N\right) dx_1 ... dx_N $$ You can use copulas, to form the joint distribution $F^B(x)$ from the knowledge of the marginals $F^{(i)}(x)$.
[Basket model $\infty$] There are of course many variations of the above methods
Depending on the choice of copulas used in model 3 above for instance. Gaussian is the usual go-to choice, but you could pick any other dependence structure. One which accounts for the fact that correlations explode when the market crashes seems like a better option. See the problem of correlation skew.
You could use local volatility + instantaneous correlations as a replacement for coupling individual marginals with a Gaussian copula. Choosing different copulas then amounts to defining different local correlation structures. This is still an open research subject, see the work of Langnau, Guyon etc. in this area.
- Be careful with decorrelation issues and the famous instantaneous vs. terminal correlation debate
[Additional details]
To price in the auto-callable feature it may be interesting to set up a hybrid equity-rates model (i.e. work with stochastic interest rates).
For the same reason, it might also be useful to consider discrete cash dividends instead of discrete proportional dividends.
To price in the compo feature you'll need to know the FX forwards. To price in the quanto feature you'll need to know the FX volatilities (see quanto drift adjustment) and sometimes more depending on whether you regard the equity volatility as a stochastic process or not.
[Your implementation]
You may wonder as to what model your implementation corresponds. Assuming you used the appropriate equity forward curves to build the risk-neutral drifts of the individual indices + used an implied volatility number (i.e. not quantities estimated under the real-world measure $\mathbb{P}$ such as expected returns and historical volatility, but rather quantities inferred from listed option prices provided under $\mathbb{Q}$), what you did corresponds to postulating log-normal marginals combined through a Gaussian copula.
This is in-between model 1 (which is worse than yours because it assumes $B_t$ is log-normally distributed but exhibits correct first 2 moments) and model 3 (which is better than yours because it uses the true implied maringals and not a log-normal assumption for the marginals).
Good luck
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