Choosing a Pricing Measure for Mutual Fund Call Options
Summary
The document examines whether to price a call option on a mutual fund using a risk-neutral return assumption or an estimated real-world return. It notes that historical volatility may be used when implied volatility is unavailable, but expected return assumptions can produce materially different valuations. Because the fund cannot be shorted, the usual replication argument for risk-neutral pricing is weaker, though equilibrium arguments may still support it.
The response cautions that a real-world expected payoff requires a discount reflecting the investor’s price of risk; simply inserting a real-world drift into a standard option formula is not sufficient. Under a simple geometric Brownian motion with a market price of risk based on excess return divided by volatility, the calculation reduces to the familiar risk-free drift result. A different risk preference can be represented as an adjusted effective drift. The document also flags fund distributions such as dividends, while offering no general calibration procedure for expected returns or risk preferences.
Key ideas
- Historical volatility can serve as an estimate when market implied data are unavailable.
- Risk-neutral and real-world return assumptions can lead to different option values.
- A real-world expected payoff needs discounting that reflects the price of risk.
- Under a simple geometric Brownian motion assumption, a standard price-of-risk adjustment recovers the risk-free drift result.
- Mutual fund dividends should be included in the valuation.
Tags
Full text
# Call option on a Mutual Fund
# Call option on a Mutual Fund
I am trying to price a call option on a mutual fund.
Given the lack of market implied data, I am going to estimate the fund´s expected volatility using as a reference its historical volatility (calculated over a period comparable to the option maturity).
However, I am not sure how to estimate the fund´s expected return. Some alternatives are:
- Use an appropriate risk free rate and deduct the annual fees of the underlying fund
- Use the historical return as expected return and neglect the annual fees
(1) will be consistent with the traditional risk-neutral framework while (2) can be backed by the lack of market implied information, the potential limitations in hedging the call option, and the fact that historical returns may exhibit some statistical persistence in mutual funds.
Which alternative do you think is more appropriate? Given the fund´s historical returns and the current risk-free rates (1) and (2) produce very different option values.
EDIT: Per comments I understand that there might be no generally accepted answer for this question. Therefore I will use both risk-neutral and real-world distributions.
Regarding the real-world distribution, once I have estimated $\mu$ and $\lambda =\frac{\mu-r_f}{\sigma}$ my idea will be :
- Estimate the real-world terminal distribution using $\mu$ instead of $r_f$. For instance using: $\frac{dS}{S} = \mu dt + \sigma dW_t$
- Calculate the expected payoff under the real-world terminal distribution
- Discount this payoff using an appropriate discount rate based on $\mu$ and $\lambda$.
Do you think this approach is enough or are there other details that I should take into account.
Finally, which will be an appropriate discount rate for the real-world payoffs? I am inclined to use a simple CAPM approach $e^{-r_dt}$ with $r_d = r_f + \beta(E(r_m) - r_f)$, but this do not make any explicit use of $\lambda$, so I am uncertain here.
## Answer by Brian B (score 4, accepted)
https://quant.stackexchange.com/a/10541
In effect, you are wondering whether to price this option on
- risk-free probability distributions (B-S drift $r_f$), or
- real-world ones (B-S drift $\mu$, however calibrated)
One cannot short the mutual fund, so the argument for using risk-free is weakened. But, there are various economic equilibrium arguments why using it may still be OK.
If you use the real-world distribution, it is important to include a discount reflecting your price for risk -- one cannot just use the real-world terminal distribution in the Black-Scholes pricing formula.
If you choose the simplest possible form for the price of risk, $\frac{\mu-r_f}{\sigma}$ in a geometric brownian motion, then you actually end up recovering the Black-Scholes formula with the risk-free rate $r_f$ as drift for fund value $S$. ( A more complicated risk alteration to the Black-Scholes SDE will not necessarily provide so simple a conclusion.)
Therefore, if your own aversion to risk is different from that of other market participants for some reason, characterized by an amount $-\Delta \mu$, you can view this as an adjustment to the effective drift of the fund value $S$, substituting $r_f+\Delta \mu$ as the drift.
Don't forget to correct for dividends, which many mutual funds pay.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.