# 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:

1. Use an appropriate risk free rate and deduct the annual fees of the underlying fund
2. 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.

In effect, you are wondering whether to price this option on

1. risk-free probability distributions (B-S drift $r_f$), or
2. 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.

• Thanks Brian, just two comments: First, since the option is a European call, the hedging strategy will require having a non-negative amount of underlying units in the portfolio. Therefore the no shorting constrain may have limited effect on the hedge. In addition, the fund has daily NAV and units can be bought/sell daily. Do you think this is enough to argue that a drift $r_f$ is more appropriate in this case? – sets Mar 11 '14 at 15:11
• Second comment: If I use the real-world distribution, and the price of risk is modeled as $\frac{\mu-r_f}{\sigma}$ in a GBM framework, could you please elaborate how this choice revert to the Black-Scholes formula with a risk-free rate $r_f$ drift. I think I am missing something here. – sets Mar 11 '14 at 15:16
• That's just what you get as the drift correction for changing to risk-neutral probabilities. – Brian B Mar 12 '14 at 1:50
• With respect to the shorting, note that a replication strategy would have positive units, but a hedging strategy would have negative units. – Brian B Mar 12 '14 at 1:51
• Thanks again Brian. You are of course right regarding the hedging vs replication strategy. I have just one last question, but it is a bit long for a comment, I will update the original question. – sets Mar 12 '14 at 8:46