Option pricing in an investment risk-return setting

ABSTRACT

In this paper, we combine modern portfolio theory and option pricing theory so that a trader taking a position in a European option contract, the underlying assets, and a risk-free bond can construct an optimal portfolio while ensuring that the option is perfectly hedged at maturity. We derive both the optimal holdings in the underlying assets for the trader’s optimal mean-variance portfolio and the amount of unhedged risk prior to maturity. Solutions assuming the price dynamics in the underlying assets follow a discrete binomial model, and continuous diffusions, stochastic volatility, volatility-of-volatility, and Merton’s jump-diffusion model are derived.

KEYWORDS:

• Option pricing
• mean-variance portfolio
• binomial pricing trees
• stochastic continuous diffusions
• stochastic volatility
• volatility-of-volatility
• Merton jump diffusions

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ The method described in the paper can be used with other utility functions.

² If $p_h$ is chosen to be $p_h=\frac{1}{2}+\frac{\mu -\frac{{\sigma }^2}{2}}{2\sigma }\sqrt{h},h\downarrow 0$, the KSRF model is identical to the CRR model for option valuation using the binomial pricing tree. In KSRF, $p_h$ can have any value in $\left(0,1\right)$. Typically, $p_h$ is estimated from historical data on the number of times the asset’s price moves up and the number of times it moves down.

³ See https://us.spdrs.com/en/etf/spdr-sp-500-etf-SPY.

⁴ https://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yieldYear&year=2019.

⁵ The processes ${\mu }_t$, ${\sigma }_t$, and $r_t$ defined on $\left(\mathrm{\Omega },\mathbb{F}=\left({\mathcal{F}}_t,t\in \left[0,T\right]\right),\mathbb{P}\right)$, satisfy the usual regularity conditions, see Duffie (2001, chapter 6).

⁶ As expected, when ${\mu }_t=\mu ,{\sigma }_t=\sigma ,r_t=r$, (20) represents the continuous version of (7).

⁷ As an example for $h(x)$, $x>0$, one can consider $h(x)=x^a$, $a\in \left(0,1\right)$, and $h(x)=log(x)$.

⁸ Examples of market traded $\mathbb{V}$ are CBOE Equity VIX on Apple (VXAPL), on Amazon (VXAZN), on Goldman Sachs (VXGS), and on IBM(VXIBM).

⁹ See Duffie (2001, Section 6.I, p.118) for sufficient conditions implying the existence and uniqueness of an equivalent measure $\mathbb{Q}$.

¹⁰ See Duffie (2001, Section 5.I).

¹¹ See Drimus (2011), Gao and Xue (2017), Huang et al. (2018), Branger, Hulsbusch, and Kraftschik (2018), and Sueppel (2018).

¹² See the CBOE White Paper ‘Double the Fun with CBOE’s VVIX’ available at http://www.cboe.com/products/vix-index-volatility/volatility-on-asset-indexes/the-cboe-vvix-index/vvix-whitepaper.

¹³ As an example for $h(x)$, $g(x)$, $x>0$, one can consider $h(x)=x^a$, $g(x)=x^b$, $(a,b)\in (0,1{)}^2$, and $h(x)=g(x)=log(x)$.

¹⁴ In our vol-of-vol model, we choose the SPDR S&P 500 ETF (SPY) as an example for the market traded security $(\mathbb{S})$. See https://www.morningstar.com/etfs/arcx/spy/quote.html.

¹⁵ We choose the CBOE Equity VIX (the volatility index for SPY) as an example of market-traded volatility $(\mathbb{V})$. See http://www.cboe.com/vix.

¹⁶ We choose as an example of market-traded vol-of-vol $(\mathbb{W})$, the CBOE VIX volatility index (VVIX). See https://finance.yahoo.com/quote/%5Evvix?ltr=1.

¹⁷ See Duffie (2001, Section 5.I).

¹⁸ See also Runggaldier (2003) and Rachev, Stoyanov, and Fabozzi (2017).

¹⁹ See Runggaldier (2003, 202).

²⁰ See Runggaldier (2003, 179 and 196).