Pricing a contingent claim liability with transaction costs using asymptotic analysis for optimal investment

We price a contingent claim liability (claim for short) using a utility indifference argument. We consider an agent with exponential utility, who invests in a stock and a money market account with the goal of maximizing the utility of his investment at the final time T in the presence of a proportional transaction cost ε>0 in two cases: with and without a claim. Using the heuristic computations of Whalley and Wilmott (Math. Finance 7:307–324, 1997), under suitable technical conditions, we provide a rigorous derivation of the asymptotic expansion of the value function in powers of \(\varepsilon^{\frac{1}{3}}\) in both cases with and without a claim. Additionally, using the utility indifference method, we derive the price of the claim at the leading order of \(\varepsilon^{\frac{2}{3}}\). In both cases, we also obtain a “nearly optimal” strategy, whose expected utility asymptotically matches the leading terms of the value function. We also present an example of how this methodology can be used to price more exotic barrier-type contingent claims.

1. We thank an anonymous referee for pointing out this example.

The author acknowledges partial financial support from NSF grant DMS-0739195. The author wants to thank Peter Bank, Paolo Guasoni, Johannes Muhle-Karbe, Steven Shreve, and Stephan Sturm for helpful discussions.

Authors and Affiliations

1. Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA, 01609, USA

   Maxim Bichuch

Corresponding author

Correspondence to Maxim Bichuch.

Appendix

Proof of Lemma 4.1

Another way to express P ₀(S,t) is to use the risk-neutral probability \(\tilde{\mathbb{P}}\) in the complete market with zero transaction costs. Then the Black–Scholes price P ₀(S,t) of the contingent claim is

From Assumption 3.3, g is nonnegative, and it follows that P ₀≥0. Moreover, for i≥2, we calculate

where by g ⁽ⁱ⁾ we mean the ith derivative. We conclude that

By Assumption 3.3, the bound in (A.1) is finite, and in a similar manner we conclude that on \(\mathbb{R}_{++}\times { [0,T]}\) the terms \(|{P_{0}-SP_{0_{S}}}|,~S^{2} |{P_{0_{SS}}}|\), \(|{S^{3}P_{0_{SSS}}}|\), and \(S^{4}|{P_{0_{SSSS}}}|\) are bounded by sup _x>0|g(x)−g′(x)x|, sup _x>0 x ²|g″(x)|, sup _x>0|x ³ g ⁽³⁾(x)|, and sup _x>0 x ⁴|g ⁽⁴⁾(x)|, respectively. We also calculate

and from the definition of \(H_{0}^{{ (j)}}\) in (2.8) conclude, both with and without claim, that \(|{S^{2}y_{S}^{{ (j)^{*}}}}|\) is bounded in \((S,t)\in \mathbb{R}_{++}\times { [0,T]}\), and \(|{S^{2}y_{S}^{{ (j)^{*}}}}|\ge\operatorname{e}^{-rT} \varepsilon_{1}\) there. The latter follows from Assumption 3.4. Similar calculations show that \(S^{3}y_{SS}^{{ (j)^{*}}}\) and \(S^{4}y_{SSS}^{{ (j)^{*}}}\) are also bounded in \((S,t)\in \mathbb{R}_{++}\times { [0,T]}\) for j=1,w. Moreover, from (2.7) we have that

From the boundedness of \(S^{2}P_{0_{SS}}\), \(S^{3}P_{0_{SSS}}\), and \(S^{4}P_{0_{SSSS}}\) on \(\mathbb{R}_{++}\times { [0,T]}\), the boundedness of \(S^{2}P_{0_{SSt}}\) also follows, and we conclude that \(S^{2}y_{St}^{{ (j)^{*}}}=S^{2}P_{0_{SSt}}-\frac{r\delta(\mu-r)}{\gamma \sigma ^{2}}\) is bounded in \((S,t)\in \mathbb{R}_{++}\times { [0,T]}\) in the cases both with and without liability. □

Proof of Lemma 6.5

From (6.7) we see that for \((S,Y,t)\in\overline{\mathbf{NT_{Y}^{(j)}}},~j=1,w\),

We use the estimates in Lemma 4.1 and Remark 6.1 to conclude that \(H_{4}^{{ (j)}}\) is bounded in \(\overline{\mathbf{NT_{Y}^{(j)}}}\), j=1,w. Similarly, using (6.10)–(6.13), Lemma 4.1, and Remark 6.1, we can show that the terms \(|{H^{{ (j)}}_{4_{YS}}}|\), \(|{Sy_{S}^{{ (j)^{*}}}H^{{ (j)}}_{4_{Y}}}|\), \(|{S^{2}y_{S}^{{ (j)^{*}}}H^{{ (j)}}_{4_{YS}}}|\), \(|{H^{{ (j)}}_{4_{t}}}\), \(|SH^{{ (j)}}_{4_{S}}|\), \(|{S^{2}H^{{ (j)}}_{4_{SS}}}|\), \(|{S^{2}y_{SS}^{{ (j)^{*}}}H^{{ (j)}}_{4_{Y}}}|\), and \(|{S^{2} (y_{S}^{{ (j)^{*}}} )^{2}H^{{ (j)}}_{4_{YY}}}|\) are all bounded in their respective domains in both cases j=1,w.

In the case j=1 of no claim, Remark 2.2 implies \((\frac{3\gamma^{2}S^{4}\sigma^{3} (y_{S}^{{(1)^{*}}} ) ^{2}}{2\delta^{2}} )^{\frac{2}{3}} = (\frac{3(\mu-r)^{2}}{2\sigma} )^{\frac{2}{3}}\). It follows that the solution to (3.4) in the case j=1 is given by (6.17) as claimed. Thus, \(H_{2}^{{(1)}}\) is bounded, and \(H_{2_{S}}^{{(1)}}=0\). We conclude that \(H_{3}^{{(1)^{\pm}}}(S,t)= \mp M^{{(1)}}(T-t)\mp M_{1}^{{(1)}}\), as defined in (6.5), will satisfy (6.18) with M ⁽¹⁾ big enough and \(M_{1}^{{(1)}}\) given by (6.6).

We now concentrate on the case of holding a claim (j=w). We rewrite (3.4) as \(H_{2_{t}}^{(j)}+rSH_{2_{S}}^{(j)}+\frac {\sigma^{2}S^{2}}{2} H_{2_{SS}}^{(j)}=-f(S,t)\), where \(f(S,t)=\frac{1}{2} (\frac{3\gamma^{2}S^{4}\sigma^{3} ({y^{(w)^{*}}_{S}} )^{2}}{2\delta^{2}} )^{\frac{2}{3}}\) with the final condition \(H_{2}^{(w)}(S,T)=0\). Lemma 4.1 implies that on \(\mathbb{R}_{++}\times { [0,T]}\), both f(S,t) and Sf _S (S,t) are bounded. Using the standard change of variables \(S=\operatorname{e}^{x}\), \(\tau=\frac {\sigma ^{2}}{2}(T-t)\), and \(k=\frac{2r}{\sigma^{2}}\), let \({\tilde{H}}_{2}(\tau ,x) := \operatorname{e}^{\frac{1}{2}(k-1)x+\frac{1}{4}(k+1)^{2}\tau-k\tau}H_{2}^{(w)}(S,t)\). We see that \({\tilde{H}}_{2}\) satisfies a nonhomogeneous heat equation, namely

with zero initial condition. The solution to this equation for \((\tau,x)\in[0,\frac{\sigma^{2}}{2}T]\times \mathbb{R}\) is

where \(\varPsi(x,t)=\frac{1}{\sqrt{4\pi t}}\exp{ (-\frac{x^{2}}{4t} )}\) is the heat kernel. From the boundedness of f it follows that \({\tilde{H}}_{2}(\tau ,x)=O (\operatorname{e}^{\frac{1}{2}(k-1)x} )\), and we conclude that \(H_{2}^{(w)}\) is bounded on \(\mathbb{R}_{++}\times { [0,T]}\). Moreover, \({\tilde{H}}_{2_{x}}\) also satisfies a nonhomogeneous heat equation, namely

with zero initial condition. Using the fact that \(\frac{\partial x}{\partial S} = \operatorname{e}^{-x}\), we calculate that

Let \(\tilde{\tilde{H}}_{2}=-\frac{1}{2}(k-1){\tilde{H}}_{2}+{\tilde{H}}_{2_{x}}\). Then \(H_{2_{S}}^{(w)}(S,t)=\operatorname{e}^{-\frac{1}{2}(k+1)x-\frac{1}{4}(k+1)^{2}\tau+k\tau }\tilde{\tilde{H}}(\tau,x)\), and \(\tilde {\tilde{H}}_{2}\) satisfies the nonhomogeneous heat equation

with zero initial condition. Hence,

From the boundedness of Sf _S (S,t) it follows that \(\tilde{\tilde{H}}_{2}(\tau ,x)=O(\operatorname{e}^{\frac{1}{2}(k-1)x})\), and we conclude that \(SH_{2_{S}}^{(w)}(S,t)\) is bounded on \(\mathbb{R}_{++}\times { [0,T]}\). Using Remark 6.1, it follows that \(S^{2}Y^{(w)} H_{2_{S}}^{(w)}\) is bounded there, too. Then \(H_{3}{^{(w)^{\pm}}}(S,t)=\mp M^{(w)}(T-t)\mp M_{1}^{(w)}\), as defined in (6.5), will satisfy (6.18) with M ^(w) big enough and \(M_{1}^{(w)}\) as defined in (6.6). In either case, \(SH_{3_{S}}^{{ (j)^{+}}}(S,t)\equiv 0\), j=1,w. □

Proof of Lemma 3.11

From Assumptions 3.8 and 3.9, all the bounds in Lemma 4.1 hold on \(\mathbb{D}\). The reader can now verify that on \(\mathbf{NT}^{\mathbf{{ (b)}}}\) the conclusion of Theorem 5.5 holds, too. Note that this proof is a replication of the original proof as all the necessary boundedness conditions in Lemma 4.1 are satisfied, and the assertion needs to be proved on a deterministic domain \(\mathbf{NT}^{\mathbf{{ (b)}}}\). To complete the proof, we have to show what happens on the boundary {S=b} of the \(\mathbf{NT}^{\mathbf{{ (b)}}}\) region. We show that on this boundary, \(\psi^{{ { (b)}^{+}}} \ge \psi^{{(1)^{+}}}\) and \(\psi^{{ { (b)}^{-}}} \le\psi^{{(1)^{-}}}\), therefore allowing us to complete the proof by continuing the trading in \(\mathbf{NT^{(1)}}\) if necessary. This defines \(Q^{{ { (b)}^{\pm}}}\), and without loss of generality, we may assume that the constant \(M_{1}^{{ (b)}}\) in the definition of \(H_{3}^{{ { (b)}^{\pm}}}\) in (6.5) satisfies

and M ^(b) is big enough. We also define \(\psi^{{ { (b)}^{\pm}}}\) by setting j=b in (5.5). It then follows that if τ<T, using \(P_{b}(b,\tau)=H_{2}^{{ (b)}}(b,\tau)=0\) for ε>0 small enough, we get

We can now prove the equivalent of Theorem 3.6. Hence, on [0,τ] the proof of Theorem 3.6 holds. Namely, we have for \((t,B,y,S)\in { [0,T]}\times \mathbb{R}\times \mathbb{R}\times[0,b )\) that

where we have used that \((\psi^{{ { (b)}^{+}}}(t\wedge\tau,B_{t\wedge\tau}, y_{t\wedge\tau}, S_{t\wedge\tau}))\) is a supermartingale to obtain the first inequality, (A.2) to establish the second, and Theorem 3.6 with j=1 for the last inequality. Maximizing over all admissible strategies, we conclude that

Similarly,

As a corollary, we conclude that Lemma 3.11 holds. □

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