Self-Financing and Dynamically Hedged Portfolio – Robert Merton’s Option Pricing. Thought of the Day 153.0

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As an alternative to the riskless hedging approach, Robert Merton derived the option pricing equation via the construction of a self-financing and dynamically hedged portfolio containing the risky asset, option and riskless asset (in the form of money market account). Let QS(t) and QV(t) denote the number of units of asset and option in the portfolio, respectively, and MS(t) and MV(t) denote the currency value of QS(t) units of asset and QV(t) units of option, respectively. The self-financing portfolio is set up with zero initial net investment cost and no additional funds are added or withdrawn afterwards. The additional units acquired for one security in the portfolio is completely financed by the sale of another security in the same portfolio. The portfolio is said to be dynamic since its composition is allowed to change over time. For notational convenience, dropping the subscript t for the asset price process St, the option value process Vt and the standard Brownian process Zt. The portfolio value at time t can be expressed as

Π(t) = MS(t) + MV(t) + M(t) = QS(t)S + QV(t)V + M(t) —– (1)

where M(t) is the currency value of the riskless asset invested in a riskless money market account. Suppose the asset price process is governed by the stochastic differential equation (1) in here, we apply the Ito lemma to obtain the differential of the option value V as:

dV = ∂V/∂t dt + ∂V/∂S dS + σ2/2 S22V/∂S2 dt = (∂V/∂t + μS ∂V/∂S σ2/2 S22V/∂S2)dt + σS ∂V/∂S dZ —– (2)

If we formally write the stochastic dynamics of V as

dV/V = μV dt + σV dZ —– (3)

then μV and σV are given by

μV = (∂V/∂t + ρS ∂V/∂S + σ2/2 S22V/∂S2)/V —– (4)

and

σV = (σS ∂V/∂S)/V —– (5)

The instantaneous currency return dΠ(t) of the above portfolio is attributed to the differential price changes of asset and option and interest accrued, and the differential changes in the amount of asset, option and money market account held. The differential of Π(t) is computed as:

dΠ(t) = [QS(t) dS + QV(t) dV + rM(t) dt] + [S dQS(t) + V dQV(t) + dM(t)] —– (6)

where rM(t)dt gives the interest amount earned from the money market account over dt and dM(t) represents the change in the money market account held due to net currency gained/lost from the sale of the underlying asset and option in the portfolio. And if the portfolio is self-financing, the sum of the last three terms in the above equation is zero. The instantaneous portfolio return dΠ(t) can then be expressed as:

dΠ(t) = QS(t) dS + QV(t) dV + rM(t) dt = MS(t) dS/S + MV(t) dV/V +  rM(t) dt —– (7)

Eliminating M(t) between (1) and (7) and expressing dS/S and dV/V in terms of their stochastic dynamics, we obtain

dΠ(t) = [(μ − r)MS(t) + (μV − r)MV(t)]dt + [σMS(t) + σV MV(t)]dZ —– (8)

How can we make the above self-financing portfolio instantaneously riskless so that its return is non-stochastic? This can be achieved by choosing an appropriate proportion of asset and option according to

σMS(t) + σV MV(t) = σS QS(t) + σS ∂V/∂S QV(t) = 0

that is, the number of units of asset and option in the self-financing portfolio must be in the ratio

QS(t)/QV(t) = -∂V/∂S —– (9)

at all times. The above ratio is time dependent, so continuous readjustment of the portfolio is necessary. We now have a dynamic replicating portfolio that is riskless and requires zero initial net investment, so the non-stochastic portfolio return dΠ(t) must be zero.

(8) becomes

0 = [(μ − r)MS(t) + (μV − r)MV(t)]dt

substituting the ratio factor in the above equation, we get

(μ − r)S ∂V/∂S = (μV − r)V —– (10)

Now substituting μfrom (4) into the above equation, we get the black-Scholes equation for V,

∂V/∂t + σ2/2 S22V/∂S2 + rS ∂V/∂S – rV = 0

Suppose we take QV(t) = −1 in the above dynamically hedged self-financing portfolio, that is, the portfolio always shorts one unit of the option. By the ratio factor, the number of units of risky asset held is always kept at the level of ∂V/∂S units, which is changing continuously over time. To maintain a self-financing hedged portfolio that constantly keeps shorting one unit of the option, we need to have both the underlying asset and the riskfree asset (money market account) in the portfolio. The net cash flow resulting in the buying/selling of the risky asset in the dynamic procedure of maintaining ∂V/∂S units of the risky asset is siphoned to the money market account.