4.2 Convex risk based traders
4.2.2 Understanding traders’ demands based on convex risk measures
position x will always be related to the gradient of the corresponding penalty func- tion. I will also show that this relationship does not hold for the demands of a trader in a multi-period market. Later I show that the demands of a first-time trader matches with the potential based definition when using afixed-pricepricing function. 4.2.2.1 Demands of a first-time trader
We now focus on the final position x of a first time trader. If the trader has initial wealthW and makes a purchases, then his final position is equivalent to holding a positionx= (W−Π(s)).1+s. We still adopt thefixed-pricepricing andcost function- based pricing functions we used in Section 4.1, so that we have that the Π(x) = W. Now I show that a first trader makes demands that is linearly related to the negative of the gradient of the penalty function.
Proposition 2. If a convex risk measure based first-time trader with an initial wealth W makes a purchase s in a market where the pricing function is translation invariant (i.e.,
Π(s+α.1) = Π(s) +α for any α ∈ R), then the demand s can be expressed as s =
−∇α(q∗) + (Π(s)−Π(−∇α(q∗))).1for a risk measureρ(x) =supq∈∆
N{hq,−xi −α(q)} and q∗ =arg supq∈∆
N{hq,−xi −α(q)}.
Proof. Let the trader’s equivalent position be x = (W−Π(s)).1+s. This gives
Π(x) = W, using the translation invariance property of the pricing functions (Sec- tion 3.3.3).
Since ρ(x) = supq∈∆N{hq,−xi −α(q)}, we get that −x = ∇α(q∗) +µ.1, for q∗ =
arg supq∈∆N{hq,−xi −α(q)} where µ is a Lagrangian dual variable requiring that
hq,1i = 1. Letting y = −∇α(q∗), we write, x = y−µ.1. This gives that Π(x) =
Π(y)−µ=W. Soµ=Π(y)−W andx= (W−Π(y)).1+y.
§4.2 Convex risk based traders 69
the trader makes demands that is linearly related to the negative of the gradient of the penalty function, proving the proposition.
Recall from Proposition 1 that penalty function is given byα(q) = λ1(f∗(−λq)−u0) in the case of utility based risk based traders. Also from Theorem 3, that the de- mands of MEU traders relates to the negative of the gradient of g∗(−∇Π(x)) =
1 λf
∗(−
λ.∇Π(x)) when λ is treated as a scalar constant. While both the penalty
functionα(q)and the dual function g∗(−∇Π(x))have slightly similar forms, due to
Proposition 2, we have that demands in the risk-measure based model can be natu- rally expressed as related to the negative of the gradient of the penalty function (i.e., without having to depend onλbeing constant as in the MEU trader model).
Also sinceα(q∗) =supx∈RN{hq∗,−xi −ρ(x)}, it gives an interpretation ofxinρ(x)as
arg minx∈RN{ρ(x) +hq∗,xi}for q∗ = arg supq∈∆
N{hq,−xi −α(x)}. So the demands
of a risk-measure based trader can be seen as minimising ρ(x) +hq∗,xi. In Hu and
Storkey [2014],xis interpreted as the minimiser ofρ(x), although it is not clear of the
role or interpretation of q∗ in their setting. In Theorem 4, I will show thatq∗ corre- sponds with the pricesπin fixed-price pricing, soxcan be interpreted as minimising ρ(x), given thatΠπ(x) =W.
4.2.2.2 Demands of a trader in a multi-period market
Following Hu and Storkey [2014], if a trader is allowed to make subsequent trades, we get that the above relationship (given in Proposition 2) does not hold any more. LetWt−1be the traders remaining wealth (cash) andxt−1be the total purchases so far at the time of atth subsequent trade. Then the trader will be holding an equivalent position Xt = Wt−1.1+xt−1−Π(st).1+st, at the end of the tth trade if he makes a purchasestat time t.
We now get that st+xt−1 = y+ (Π(xt−1+st)−Π(y)).1 = xt, so that the trader’s
demand st does not satisfy the linear relationship with the gradient of the penalty function any more (unlessxt−1=k.1for any k∈R).
4.2.3 First-time trader in a fixed-price market makes potential-based de- mands
Usingfixed-pricepricing (Π(s) =Ππ(s):=h
π,si), I now show thatq∗corresponds to
§4.2 Convex risk based traders 70
1. The demands of a first-time trader in fixed-price pricing can be expressed as potential based, thus satisfying the condition for SMD in Theorem 1.
2. x can be interpreted as arg minx∈RN{ρ(x)}, sinceΠ(x) =hπ,xi=W.
Theorem 4. If a convex risk measure based first-time trader with an initial wealth W makes a demand s in a market with a fixed-price pricing function (Π(s) =Ππ(s):=h
π,si), where π ∈int(∆N)(i.e.,∀nπn∈(0, 1)), then
1. q∗ = π, where q∗ = arg supq∈∆N{hq,−xi −α(q)}, for a risk measure ρ(x) =
supq∈∆N{hq,−xi −α(q)},
2. the demand s= −∇α(π) + (Π(s)−Π(−∇α(π))).1, thus potential based, 3. the equivalent position x= (W−Π(s)).1+s is the minimiser ofρ(x).
Proof. First I show thatq∗ = arg supq∈∆N{hq,−xi −α(q)} = π. Let y = −∇α(q∗).
From Proposition 2 we have that x = y−µ.1 for x = (W −Π(s)).1+s, where
Π(x) =Π(y)−µ=W. Soµ=Π(y)−Π(x) =hπ,yi − hπ,xiandx=y−(hπ,yi − hπ,xi).1 = y−(hπ,yi −W).1(since Π(x) =W). Since ρ(x) =hq∗,−xi −α(q∗), we can write ρ(x) =hq∗,−xi −α(q∗) =ρ(y−µ.1) =ρ(y) +µ =ρ(y) +hπ,yi − hπ,xi
By re-arranging we havehπ,xi+ρ(x) =hπ,yi+ρ(y). Now taking the derivative of
both sides with respect toy gives,
∇(x)(hπ,xi+ρ(x)).∇(y)x=∇(y)(hπ,yi+ρ(y))
(π−q∗).∇(y)x= (π−q∗)
(π−q∗).(1−π) = (π−q∗)
for all π. Consider α(q∗) = supx∈RN{hq∗,−xi −ρ(x)} = sup(x+k.1)∈RN{hq∗,−(x+ k.1)i −ρ((x+k.1))}(where we have used the translation invariance property of risk
measures). So we have that that −q∗= ∇ρ(x) = ∇(x+k.1)ρ(x+k.1)for any k ∈ R, which gives the second equality. The third equality is due to x= y−(hπ,yi −W)1.
This gives us thatq∗ =πforπ ∈int(∆N), proving the first part.
For the second part of the theorem, use the result from Proposition 2 that s =