When performing maturity dimension interpolation in Chapter 6, it will be important to exclude calendar spread arbitrage from a grid of European option prices. As concluded in Condition 1, this is accomplished by ensuring that the inequality ∂C∂T ≥0 is satisfied for all option prices. Further, the underlying asset needs to be aQ-martingale since the model developed in this thesis will be valid only for driftless diffusions. As concluded by Gatheral and Jacquir (2013), it is important to pick an appropriate variable space in order to achieve these results.
Definition 10 Let C(S, t) be the value of a call option at time t with some underlying spot exchange rateSt. Assume that the option expires at timeT and
has strike K. Themoneynessof the option is defined as: KM =
K f(0, T)
and themoneyness-forward as:
Xt=
f(t, T) f(0, T),
wheref(t, T)is the forward exchange rate at time twith the same maturity as the call option.
Since the moneyness-forward has initial value X0 = 1, it can be considered a forward contract which has been normalized at inception. The theorem below describes an important property of the moneyness-forward.
Theorem 10 Assume that the spot exchange rate St is a diffusion process fol-
lowing the dynamics:
dSt=St(rd(t)−rf(t)) +Stσ(St, t)dWtQd, (4.1)
where rd(t),rf(t) andσ(St, t) are deterministic functions bounded from above
and away from zero. Then Xt, as in Definition 10, is a martingale under the
domestic2 risk-neutral probability measure Q
d.
2The phrases “Domestic” and “Foreign” currencies, investors et cetera are very common
in financial literature covering foreign exchange related topics. The domestic currency simply corresponds to the num´eraire currency of the investor, while the foreign currency can be considered some risky asset.
4.2. Variable Space 35
Proof. The martingale property of forward contracts was observed by Black
(1976). Since then, the result has been used in a large quantity of papers. Assume that St follows the dynamics in (4.1). Differentiating Xt using the
standard formula for forward contracts3 yields: dXt= 1 f(0, T)df(t, T) = 1 f(0, t)d(Ste RT t (rd(s)−rf(s))ds) = 1 f(0, T) eRtT(rd(s)−rf(s))dsdS t−(rd(t)−rf(t))Ste RT t (rd(s)−rf(s))dsdt = e RT t (rd(s)−rf(s))ds f(0, T) Stσ(St, t)dW Qd t .
Notice that since the second differential of the forward contract with respect to the spot rate is zero, we get no contribution from quadratic variation. This means that Xt is a driftless diffusion. Since the diffusion function is assumed
to be bounded from above and away from zero, this completes the proof. In Lemma 1, we use Theorem 10 to formulate a very convenient way to use the closed form solution for European call options in the Black & Scholes model, (2.7). A very similar approach is taken by Gatheral and Jacquir (2013) among others.
Lemma 1The expression
E[(ST −K)+|S0=S]
f(0, T)
can be regarded as the price of a call option,C(K, τ), at timet= 0 with under- lying asset Xt, strikeKM and time to maturity τ =T−t. Further, assuming
the volatility to be constant, the undiscounted value of this call option is: C(KM, τ) =N(d1)−KMN(d2),
where
d1/2=
−ln(KM)±12σ2τ
σ√τ .
Proof. First notice that:
E[(ST −K)+|S0=S]
f(0, T) =E[(XT−KM) +
|X0= 1],
where Xt andKM comes from Definition 10. Now, lets assume Xt to have a
constant volatility. Using (2.5) and the fact thatXtis a Qd-martingale it can
be concluded that fort= 0:
C(KM, τ) =X0N(d1)−KMN(d2) =N(d1)−KMN(d2),
whered1 andd2 originates from the no-interest rate version of (2.5).
3With standard formula we refer to the expression:f(t, T) =SteRT
t (rd(s)−rf(s))ds. For a derivation of this formula, see Hull (2008, pp.113).
This choice of variables makes it possible to formulate the proposition below.
Proposition 5Let Xt be a Qd-martingale and τ2 ≥ τ1 >0 two time to ma- turities. Assuming the quotes to be arbitrage-free, the following inequality is true: C(KM, τ2) =E[(Xτ2−KM) +]≥E[(X τ1−KM) +] =C(K M, τ1). (4.2) This result, which is shown by Gatheral and Jacquir (2013) among others, fol- lows from the martingale property ofXtand the fact that the payoff from the
call option is bounded from below but not from above.
In (2.11), interest rates have been disregarded. Hence, model implementations using this version of the Dupire equation should be performed in C(KM, τ)
space, where the underlying asset is assumed to be aQd-martingale. One could
criticize that the prices coming from such implementations do not correspond to the “real” prices. Notice, however, that the input to option pricing models in the FX-market is in implied volatility, and the output should be in implied volatility as well. Therefore, it is important that the conversion between implied volatility and price is performed in a consistent way.
The notation KM, for moneyness, will not be used in the later chapters in
this thesis. Since the implementations will be valid in all models where the underlying asset is aQd-martingale, only the notationK will be used for strike.