Illustration of Option Pricing

A numeraire approach is adopted in pricing here, it appears to make the analysis cleaner. The one innovation here is to introduce a name for a concept : transfer pricing. That is, the exchange price of one unit of asset X for X_B units of asset B.

Consider an economy consisting of two markets such that each trades in a zero-coupon bond D, F denominated in the local currencies \$, \pounds. Also assume the currencies are linked by exchange rate mechanism. The transfer prices are given by D_\$,F_\pounds,\pounds_\$. In the real-world measure \mathbb{P} the dynamics of these prices are given in the natural price basis by

  • dD_\$ = D_\$ r dt
  • dF_\pounds = F_\pounds u dt
  • d\pounds_\$ = \pounds_\$ (\mu dt + \sigma dw(t))

where the constants r is the domestic interest rate, u is the foreign interest rate. Due to the presence of transfer pricing in currency, it is possible to trade the foreign bond in domestic current. Note that F_\$ = F_\pounds \pounds_\$. The dynamics are given by

d F_\$ = F_\$ ( (u + \mu) dt + \sigma dw(t))

Using the domestic bond as numeraire F_B = F_\$ \$_B

d F_B = F_B ( (u + \mu - r) dt + \sigma dw(t))

By Girasinov’s theorem, associated with B is a measure \mathbb{Q}^B that makes any B-transfer price of a (static) asset a martingale such that that the volatility is unchanged. Hence

d F_B = F_B \sigma dv(t)

This SDE has solution F_B(T) = F_B(t) e^{\sigma (v(T)-v(t)) - \frac{1}{2} \sigma^2 \tau }

Example 1

Now consider a T-claim such that V(T) = 1_\pounds - k 1_\$. That is, at expiry, one pound can be exchanged for k dollars. Since p = 1_F - k 1_B is such that p(T) = V(T), it follows by the LOP that V_B(t) = p_B(t). By the FTAP, V_B(t) = \mathbb{Q}^B_t V_B(T). Since in this case V_B(T) = F_B(T) - k, it follows that V_B(t) = F_B(t) - k. But since F_B(t) = F_\pounds (t) \pounds_\$ (t) \$_B(t), and so V_\$(t) = V_B(t) B_\$(t) = F_\pounds(t) \pounds_\$ (t) - k B_\$(t). In this case the LOP and FTAP can be used to obtain the same result.

Example 2

Consider a T-claim such that V(T) = 1_\pounds - k 1_\$ | 0. That is, at expiry the holder has the option of selling cash to buy a pound or doing nothing. Note that the B-transfer price is V_B(T) = (F_B(T) - k)^+ = (F_B(T) - k) \mathbb{I}_{F_B(T) > k} and can be seen as being a long/short position in two T-claims, one an asset binary, the other a bond binary. By the FTAP,

V_B(t) = \mathbb{Q}^B_t F_B(T) \mathbb{I}_{F_B(T) > k} - k \mathbb{Q}^B_t \mathbb{I}_{F_B(T) > k}.

Since F_B(T) > k iff \ln \frac{F_B(t)}{k} - \sigma \tau^{1/2} Z - \frac{1}{2} \sigma^2 \tau> 0 iff Z < \frac{\ln \frac{F_B(t)}{k} - \frac{1}{2} \sigma^2 \tau }{\sigma \tau^{1/2}} = d_0

where Z is a gaussian(0,1) random variable, by the Gaussian shift lemma (Büchen),

\mathbb{Q}^B_t F_B(T) \mathbb{I}_{F_B(T) > k} = F_B(t) N(d_0 + \sigma \tau^{1/2})

\mathbb{Q}^B_t \mathbb{I}_{F_B(T) > k} = N(d_0)

and so the transfer price

V_B(t) = F_B(t) N(d_0 + \sigma \tau^{1/2}) - k N(d_0).

The cash price is the natural price basis and so

V_\$(t) = F_\$(t) N(d_0 + \sigma \tau^{1/2}) - k B_\$(t) N(d_0) \\ = \pounds_\$(t) e^{-u\tau} N(d_0 + \sigma \tau^{1/2}) - k e^{-rt} N(d_0).

furthermore expressing d_0 in natural price basis,

d_0 = \frac{\ln \frac{F_\pounds(t) \pounds_\$(t)}{k} + (r - \frac{1}{2} \sigma^2) \tau }{\sigma \tau^{1/2}}

Asset Flows and Assets

An asset is zero-flow if it generates no asset flows over its lifetime. The FTAP only applies to assets that do not generate asset flows. A company share paying a regular dividend is an example of a positive-flow asset. In this case, the asset is not just the share, but the flows which are obtained from ownership. So the combined asset is a position comprised of the asset and the flows. One way to simplify the analysis is to use the asset as the numeraire and adopt a policy of reinvestment of the flows into the asset. A standard assumption is:

d1_s = \delta dt 1_s

The value of the position p(t) = 1_s has price increment

p_\$(t+dt)-p_\$(t) = (1 + \delta dt) s_\$(t+dt) - s_\$(t) = \delta dt s_\$(t+dt) + ds_\$(t)

assuming continuity of trajectories,

dp_\$(t) = ds_\$(t) + \delta s_\$(t) dt = s_\$(t) ( (\mu + \delta) dt + \sigma dw(t))

Another possibility is d 1_s = \delta dt 1_\$

test for a tradeable process: there is an asset whose price is described by the process. If holding an asset generates multiple flows, that are not incoporated into the price then the process does not represent the true price.

The replicating position 1_V(t) = \phi_t 1_S + \psi_t 1_B. The price V_\$(t) = \phi_t S_\$(t) + \psi_t B_\$(t) and the price increment dV_\$(t) = \phi_t dS_\$(t) + \psi_t dB_\$(t) + \phi_t \delta S_\$(t) dt if self-financing. This can be written as

dV_\$(t) = \phi_t e^{-\delta t} d(e^{\delta t} S_\$(t)) + \psi_t dB_\$(t)

where

d(e^{\delta t} S_\$(t)) = e^{\delta t} S_\$(t) (\mu dt + \sigma dw(t)) + \delta e^{\delta t} S_\$(t) dt = e^{\delta t} S_\$(t) ( (\mu + \delta) dt + \sigma dw(t))

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