Simulating Future Values of a Bond, Stock, and Option Portfolio

Summary

The document asks how to estimate future values for a portfolio containing a stock, a long-term coupon bond, and a European call option. It provides example daily inputs for the stock price, required yield, volatility, and short-term interest rate, and asks for simulated portfolio values over a future horizon. The response recommends specifying a stochastic process for each asset and using Monte Carlo simulation to generate possible portfolio outcomes.

For the stock, it suggests geometric Brownian motion or a broader Lévy process, with the stock process also serving as the option's underlying. For the bond, it proposes modeling interest rates, for example with an Ornstein–Uhlenbeck process, then valuing coupons and principal by present value. The combined simulations produce a distribution of future portfolio values or returns. The answer is a high-level outline rather than a worked procedure: it does not define parameter estimation, dependence between assets, option valuation details, or how the listed data should be used to calibrate the processes. Those choices materially affect the resulting distribution.

Key ideas

• Monte Carlo simulation can generate a distribution of possible future portfolio values.
• The stock and the option's underlying require a specified stock-price process.
• The bond can be valued by discounting its coupon and principal cash flows under a modeled rate process.
• The suggested processes and inputs do not by themselves specify calibration or dependence across assets.

Tags

# Historical Simulation of Bond, Stock and Option Portfolio


# Historical Simulation of Bond, Stock and Option Portfolio












If I have a portfolio consisting of

1-one stock of unit price equal to S,

2-one 9% coupon American bond with 20 years to maturity and a par value of $1000,

3-and one European call option on the stock of unit price C who matures in 3 months and the strike price of the option \$200

How to calculate simulated portfolio value over the next N days, from these inputs:

```
    Date (t) St ($) rt (required yield) σt (volatility) int. (three month rate) 
    Day 1    201    12 %                23 %            0.9 % 
    Day 2    203    12.3 %              20 %            0.6 %
    ...
    Day N    ...
```

## Answer by alexbougias (score 1)

https://quant.stackexchange.com/a/45559

You can assume a stochastic process for each security, and conduct a Monte Carlo simulation, generating different realization of porfolio values. For the stock, a Geometric Browinian Motion or a broader Levy process, would be ok. This should be the process of the underlying stock in case of the option, in order to price the option. For the bond, define a stochastic process for the interest rates (e g Ohrstein-Uhlenbeck), and through the present value of Coupons and Principal, calculate the bond price. That should give you a distribution of future portfolio values ($V_{t+2}$), or equivalently the return distribution of the portfolio.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.