NNon-traded call’s volatility smiles
Marek CapinskiMarch 20, 2019
Abstract
Real life hedging in the Black-Scholes model must be imperfect andif the stock’s drift is higher than the risk free rate, leads to a profit onaverage. Hence the option price is examined as a fair game agreementbetween the parties, based on expected payoffs and a simple measureof risk. The resulting prices result in the volatility smile.
We consider a European call option written on stock satisfying the Black-Scholes equation dS ( t ) = µS ( t ) dt + σS ( t ) dW P ( t ) , (1)where W P ( t ) is a Wiener process in the probability space (Ω , F , P ). Thestrike price is K and exercise time T , and the option payoff is denoted C ( T ) =( S ( T ) − K ) + . The option is sold over the counter for the price C . This option is assumedto be non-tradable, so the arbitrage pricing argument does not apply and theprice will the result of an agreement between the writer (seller) and the holder(buyer). The buyer is an investor who strongly believes that the stock willperform well and is ready to invest in a call, which gives certain leverage. Inparticular, he believes that µ > r, where r is the risk-free rate, and we makethis assumption throughout.The writer will hedge by taking a position in the primary market con-sisting of stock S and the money market account A ( t ) = e rt . The model iscomplete, but this requires continuous rebalancing, which is impossible. Sothe initial position will be taken for a period of time, suppose first this is1 a r X i v : . [ q -f i n . P R ] M a r ept constant till exercise, with x shares. As a result, the expected writer’spayoff, computed with respect to the physical probability, which means thestock price with µ is used, is larger than the expected option payoff (sincethe stock on average grows faster than the risk free asset). The holder mayrequire the initial option price be lower, so that the expected payoffs are thesame. However, this equilibrium price depends on x, so one more conditionis needed. A natural assumption is that writer’s hedging risk is minimised,which requires choosing a risk measure. The resulting x is not binding andonly serves the purpose of negotiating the option price.At any time the writer may change the hedging position and the sameanalysis (based on the assumption that the new position hypothetically staysstatic) will give the current value of the option.It is interesting to see that the implied volatility (that is, the σ whichwould result in the Black-Scholes option price being the same as the equilib-rium price) shows the volatility smile effect.A similar strategy is discussed in [1], where imperfect hedging leads tothe prices found by means of the physical probability but the method isdifferent (it involves utility functions). Various risk measures are used forimperfect hedging in [2] and [3]. A classical example of a non-traded optionare employee’s options, but the key feature is concerned with the exercisetime limitations, so our approach does not cover this. Despite the fact that the price is not based on the no-arbitrage principle, itis a natural idea to use the Black-Scholes formula first.Let r be the risk-free rate and introduce the risk-neutral probability Q, where W Q ( t ) = W ( t ) + µ − rσ t is a Wiener process under Q. The stock pricesfollow the equation dS ( t ) = rS ( t ) dt + σS ( t ) dW Q ( t )and the no-arbitrage call option price is given by the Black-Scholes formula C BS = S (0) N ( d r + ) − e − rT KN ( d r − ) , where d r ± = ln S (0) K + rT ± σ Tσ √ T . C = C BS . At time T the option holder’s profit is P H ( T ) = C ( T ) − C BS e rT . The expectation of P H ( T ) with respect to risk-neutral probability is of coursezero since C BS = e − rT E Q ( C ( T )). However, the risk-neutral world is abstractand in reality the stock follows equation (1) under the physical probability P , so the expectation with respect to this measure is relevant: E P ( P H ( T )) = E P ( C ( T )) − C BS e rT . Proposition 2.1. If µ > r, then E P ( P H ( T )) > . Proof.
It is sufficient to see that E Q ( C ( T )) < E P ( C ( T )) . We have E Q ( C ( T )) = E Q (exp( rT − σ T + σW Q ( T )) , E P ( C ( T )) = E P (exp( µT − σ T + σW P ( T )) , but the distribution of W Q ( T ) under Q is the same as the distribution of W P ( T ) under P, so E Q (exp( σW Q ( T )) = E P (exp( σW ( T ))which gives the claim.Now we look at this from the option writer’s perspective. This amount C BS obtained for the option is used to build a hedging portfolio ( x, y ) with x being the number of shares and y the number of units of the money marketaccount. The risk free position is then y = C BS − xS (0) . Idealy, the hedging position should be continuously rebalanced but this isnot realistic. A practical hedging strategy could be piece-wise constant withrandom rebalancing times. However, since the option price must be agreednow, it is natural to consider first a constant strategy.Assuming that the positions x, y are kept constant over the time interval[0 , T ] , the terminal value of the portfolio is V ( T ) = x [ S ( T ) − S (0) e rT ] + C BS e rT so the profit of the option writer P C W ( T ) = V ( T ) − C ( T ) is P W ( T ) = x [ S ( T ) − S (0) e rT ] + C BS e rT − C ( T ) . x the expected writer’s profit computed with respect to therisk-neutral probability is zero. Indeed, we have E Q ( S ( T )) = S (0) e rT and C BS = E Q ( e − rT ( S ( T ) − K ) + ) , which gives the claim. But we use the physi-cal probability and since E P ( S ( T )) = S (0) e µT , the expected option writer’sprofit is an increasing function of x, provided µ > r, in fact it is a linearfunction with positive slope: E P ( P W ( T )) = xS (0)( e µT − e rT ) + C BS e rT − E P ( C ( T )) . Example 2.2.
Assume S (0) = 100 , µ = 10% , σ = 20% , r = 5% andconsider European call with strike K = 100 , T = 1 . The Black-Scholes priceis C BS = 10 .
45 and using the formula E P ( C ( T )) = e µT N ( d µ + ) − KN ( d µ − )where d µ ± is defines as d r ± with µ replacing r, we find E P ( P H ( T )) = 3 . . With delta hedging x = N ( d r + ) we have E P ( P W ( T )) = − . . However, it iswriter’s decision to choose x and if he tries to maximise the expected profithe may take large x, but this would be an active and risky position typicalfor an investor, not an option writer, whose priority is hedging. As we have seen, the price given by the Black-Scholes formula may not beacceptable for the option writer. The option price of an OTC transactionhas to be determined by an agreement between both parties involved and wepropose some natural criteria. Recall that the expected profits of the partiesinvolved, the holder and the writer respectively, are E P ( P H ( T )) = E P ( C ( T )) − Ce rT , E P ( P W ( T )) = xS (0)( e µT − e rT ) + Ce rT − E P ( C ( T )) . Criterion 3.1. ( Fair play)
The price C should be such that E P ( P W ( T )) = E P ( P H ( T )) . This condition gives us the price as a function of x : C x = e − rT ( E P ( C ( T )) − xS (0)( e µT − e rT )) . (2)We need a mutually acceptable criterion for establishing x, and it is naturalto assume that it will be based on minimising writer’s risk. There are many4hoices of a risk measure, like Value-at-Risk, or Conditional-Value-at-Risk(CVaR) (see [ ? ] for instance), but to use them we have to agree on someconfidence level, which is subjective. So we propose to use a simple andnatural idea of expected loss under the condition that it is positive.The writer’s loss is given by L W ( T ) = − x ( S ( T ) − S (0) e rT ) − Ce rT + C ( T ) (3)and we introduce the conditional expectation of the loss, provided it is posi-tive: γ W ( x ) = E P ( L W ( T ) | L W ( T ) >
0) = E P ( L W ( T ) { L W ( T ) > } ) P ( L W ( T ) > γ W ( x ) is CVaR α ( L W ( T )) at α = P ( L W ( T ) > . Criterion 3.2. ( Risk minimising)
The hedging position x should be suchthat γ W ( x ) considered in the interval [0 ,
1] attains minimum at x . Note that the option writer decides the value of x, and the number re-sulting from this criterion is not binding.In order to investigate the existence of the prices and hedging portfo-lios satisfying the above criteria, we derive a closed form expression for thefunction γ W ( x ) . Proposition 3.3.
Assume x ∈ [0 , . Writer’s risk is given by1. γ W ( x ) = E ( C ( T ) { L W > } ) P ( L W ( T ) > − x E ( S ( T ) { L W > } ) P ( L W ( T ) >
0) + xS (0) e rT − C x e rT , where2. P ( L W ( T ) >
0) = N ( d ) + 1 − N ( d ) , with d = 1 σ √ T (cid:18) ln ( xS (0) − C x ) e rT S (0) x − µT + 12 σ T (cid:19) ,d = 1 σ √ T (cid:18) ln K + ( C x − xS (0)) e rT S (0)(1 − x ) − µT + 12 σ T (cid:19) , E ( C ( T ) { L W ( T ) > } ) = S (0) e µT [1 − N ( d − σ √ T )] − K [1 − N ( d )] , . E ( S ( T ) { L W ( T ) > } ) = S (0) e µT [ N ( d − σ √ T ) + 1 − N ( d − σ √ T )] . The following relation will be used in the proof.
Lemma 3.4.
We have d < d < d , where d = 1 σ √ T (cid:18) ln KS (0) − µT + 12 σ T (cid:19) . The proof is routine and a sketch is given in the appendix. It is based onthe inequality µ > r and employs call-put parity.
Proof of Proposition 3.3.
1. To find the form of γ W ( x ) all we have to ifto use the expression (3) for L W ( T ) with C = C x , insert it to the definitionof conditional probability, and use the linearity of expectation.2. Write S ( T ) = S (0) e µT − σ T + σ √ T Z where Z ∼ N (0 , . First note that L W ( T ) > x ( S ( T ) − S (0) e rT ) + C x e rT − C ( T ) < C ( T ) . Case 1. S ( T ) < K, that is Z < d, and here C ( T ) = 0 so that L W ( T ) = x ( S ( T ) − S (0) e rT ) + C x e rT . Next, L W ( T ) > S ( T ) < x ( xS (0) − C x ) e rT , which inturn corresponds to Z < d . As a result we get the probability N ( d ) . Case 2. S ( T ) > K that is Z > d, and here C ( T ) = S ( T ) − K. Now,simple algebra shows that L W ( T ) > Z > d with probability1 − N ( d ) , and these two cases give L W ( T ) > Z ≤ d or Z > d , disjoint events, hence the claim.3. For E ( C ( T ) { L W ( T ) > } ) note that C ( T ) = ( S (0) e µT − σ T + σ √ T Z − K ) { Z>d } while { L W ( T ) > } = { Z>d }∪{ Z ≤ d } so we have (using the lemma) E ( C ( T ) { L W ( T ) > } ) = E (( S (0) e µT − σ T + σ √ T Z − K ) { Z>d } ) . Routine integration gives1 √ π (cid:90) ∞ d e σ √ T z − σ T e − z dz = [1 − N ( d − σ √ T )] (4)which readily imples the claim. 6. For stock prices we have E ( S ( T ) { L W ( T ) > } ) = E ( S ( T ) { Z>d }∪{ Z ≤ d } )and all that is left to compute two integrals similar to (4).Option holder may wish to find the risk resulting from the option writer’sdecision and the form of the risk function will be needed. Recall that holder’sloss is L H ( T ) = C x e rT − C ( T ) and the risk is similarly assumed to be theconditional expectation γ H ( x ) = E ( L H ( T ) | L H ( T ) > . Proposition 3.5.
We have γ H ( x ) = C x e rT − S (0) e µT [ N ( d (cid:48) − σ √ T ) − N ( d − σ √ T )] − K [ N ( d (cid:48) ) − N ( d )] N ( d (cid:48) ) , where d (cid:48) = 1 σ √ T (cid:18) ln K + C x e rT S (0) − µT + 12 σ T (cid:19) Proof.
Holder’s loss is positive if S ( T ) < C x e rT + K which corresponds to Z < d (cid:48) (note that d < d (cid:48) ) and gives the probability of the condition. Next E ( L H ( T ) { Z Example 3.6. With the data as in Example 2.2 we find the form of the riskfunction and sketch the graph. The witer’s risk is smallest for x = 0 . C x = 12 . , higher than the Black-Scholesprice.Finally, we repeat this for various strike prices and compute the impliedvolatility using the Black-Scholes formula. Example 3.7. With the same data as before we get the following results(see Figure 2). K 90 95 100 105 110 115 C . 89 15 . 28 12 . 10 9 . 38 7 . 12 5 . σ . 43% 25 . 67% 24 . 38% 23 . 42% 22 . 72% 22 . Remark 3.8. The option writer may wish to rebalance the strategy in thefuture and if the same risk measure is accepted, all that is needed is to usethe expressions derived above for the relevant time period, replacing T by T − t , where t is the time of rebalancing. Proof of Lemma 3.4. We have to see that( xS (0) − C x ) e rT S (0) x < KS (0) < K + ( C x − xS (0)) e rT S (0)(1 − x ) . x = 1 . Using thedefinition of C x we have to show that S (0) e rT − E P ( C ( T )) + 12 S (0)( e µT − e rT ) < K. We use the call-put payoffs parity C ( T ) = P ( T ) + S ( T ) − K, and insert thisabove on the left to get K − E P ( P ( T )) − S (0)( e µT − e rT ) , clearly smallerthan K. The claim follows from the fact that ( xS (0) − C x ) e rT S (0) x is increasing with x which follows after inserting the definition of C x and perfoming some algebra:( xS (0) − C x ) e rT S (0) x = e rT − E ( C ( T )) S (0) x + 12 ( e µT − e rT ) . For the second inequality note that it is obviously true for x = 0 with C = e − rT ( E ( C ( T )) > . It is sufficient to see that the function K +( C x − xS (0)) e rT S (0)(1 − x ) isan incresing function of x. To get this we compute its derivative and somealgebra shows it is positive. References [1] Constantinides, George M., Jens C. Jackwerth, and Stylianos Perrakis.2008. “Option Pricing: Real and Risk-Neutral Distributions”, in Hand-books in OR & MS , Vol. 15, 565–592, J.R. Birge and V. Linetsky (Eds.),Elsevier.[2] Schulerich, Marco, and Siegfried Trautmann. 2003. “Local ExpectedShortfall-Hedging in Discrete Time.” European Finance Review 7: 75–102.[3] Wylie, Jonathan J., Qiang Zhang, and Tak Kuen Siu. 2010. “Can ex-pected shortfall and Value-at-Risk be used to statically hedge options?”